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+Inflation in Weyl Scaling Invariant Gravity with R3 Extensions
+Qing-Yang Wanga, Yong Tanga,b,c,d, and Yue-Liang Wua,b,c,e
+aUniversity of Chinese Academy of Sciences (UCAS), Beijing 100049, China
+bSchool of Fundamental Physics and Mathematical Sciences,
+Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China
+cInternational Center for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China
+dNational Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
+eInstitute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
+(Dated: January 11, 2023)
+Abstract
+The cosmological observations of cosmic microwave background and large-scale structure indicate
+that our universe has a nearly scaling invariant power spectrum of the primordial perturbation.
+However, the exact origin for this primordial spectrum is still unclear.
+Here, we propose the
+Weyl scaling invariant R2 + R3 gravity that gives rise to inflation that is responsible for the
+primordial perturbation in the early universe. We develop both analytic and numerical treatments
+on inflationary observables, and find this model gives a distinctive scalar potential that can support
+two different patterns of inflation. The first one is similar to that occurs in the pure R2 model,
+but with a wide range of tensor-to-scalar ratio r from O(10−4) to O(10−2). The other one is a new
+situation with not only slow-roll inflation but also a short stage of oscillation-induced accelerating
+expansion. Both patterns of inflation have viable parameter spaces that can be probed by future
+experiments on cosmic microwave background and primordial gravitational waves.
+1
+arXiv:2301.03744v1  [astro-ph.CO]  10 Jan 2023
+
+I.
+INTRODUCTION
+Inflation is a hypothetical epoch of exponential expansion introduced in the very early
+universe to solve the cosmological horizon and flatness problems [1, 2]. It is also a reasonable
+scheme to explain the origin of primordial density perturbations, which plays the role of
+the seeds that formed the structure of current universe [3]. In recent years, the precise
+measurement of cosmic microwave background (CMB) presents us with an almost scale
+invariant spectrum of primordial perturbations [4]. This result is usually explained by an
+approximate de Sitter spacetime of the very early universe [5–9]. Moreover, it is theoretically
+explored that there is a more profound and basic principle behind the phenomenon, namely,
+local Weyl scaling invariance of the universe. This symmetry is first proposed by H. Weyl in
+the attempt of understanding gravity and electromagnetism in a unified framework [10, 11],
+and after a century of development, it has been applied extensively to particle physics,
+cosmology [12–30] and gauge theory of gravity [31–34].
+Lately, inflation in the Weyl scaling invariant theory of gravity, especially induced by
+a quadratic curvature term R2, has been of many concern [35–45]. Comparing with the
+conventional R2 model, which is also called Starobinsky model [46–49], the scaling invariant
+version not only allows a viable inflation scenario with good observational agreement, but
+also provides a framework to comprehend another fundamental puzzles, such as hierarchy
+problem [37, 40, 50] and dark matter candidates [41, 45].
+However, inflation with only quadratic scalar curvature might be just a simplistic scenario.
+From the viewpoint of effective field theory, any higher-order curvature effects may exist and
+play a role in the early universe. Hence it is reasonable to evaluate their impacts on inflation.
+Generally, the extensions with high-order tensors, like RµνRµν or RµνρσRµνρσ, can result in
+unacceptable ghost degrees of freedom [51], while the terms of arbitrary functions of the
+Ricci scalar are known to be safe. Therefore, in this paper, we consider a minimal extension
+of Ricci scalar beyond the R2 model with Weyl scaling invariance, namely a cubic term
+coupled with an extra scalar field as denominator R3/ϕ2. We will show that even if this
+term is extremely small, it will have an essential impact on inflation, which even open up a
+completely different inflationary scenario from Weyl R2 and conventional R2 + R3 models.
+The paper is organized as follows. In Sec. II, we develop the analytic formalism of Weyl
+R2 + R3 model and derive the effective scalar potential. We show that in some cases, the
+2
+
+potential has two different kinds of global minima, leading to two distinctive inflationary pat-
+terns. In Sec. III, we investigate the inflation in the pattern of evolving to the side minimum.
+We calculate the spectral index ns and tensor-to-scalar ratio r of the inflationary perturba-
+tions, and give the preferred parameter space allowed by the latest observations. Analytical
+treatments are developed for more transparent, physical understanding of the asymptotic
+behaviors. Then in Sec. IV, we investigate the pattern of evolving to the center minimum.
+A special process called “oscillating inflation” is considered in detail. Finally, conclusions
+are given in Sec. V. We adopt the following conventions: metric ηµν = (−1, +1, +1, +1),
+natural unit ℏ = c = 1 and MP ≡ 1/
+√
+8πG = 2.435 × 1018 GeV = 1.
+II.
+WEYL SCALING INVARIANT R2 + R3 MODEL
+We start with the following Lagrangian for metric field gµν, scalar field ϕ, and Weyl gauge
+field Wµ ≡ gWwµ with local scaling symmetry
+L
+√−g = 1
+2
+�
+ϕ2 ˆR + α ˆR2 + β
+ϕ2 ˆR3
+�
+− ζ
+2DµϕDµϕ −
+1
+4g2
+W
+FµνF µν.
+(1)
+Here g is the determinant of metric, α, β and ζ are constant parameters, Dµ = ∂µ − Wµ
+is the covariant derivative associated with scaling symmetry, gW is the coupling constant,
+Fµν ≡ ∂µWν − ∂νWµ defines the invariant field strength of Wµ, and ˆR is the Ricci scalar
+defined by the local scaling invariant connection
+ˆΓρ
+µν = 1
+2gρσ [(∂µ + 2Wµ)gσν + (∂ν + 2Wν)gµσ − (∂σ + 2Wσ)gµν] .
+(2)
+Explicit calculation shows the relation between ˆR and usual R defined by metric field gµν,
+ˆR = R − 6WµW µ −
+6
+√−g∂µ(√−gW µ).
+(3)
+It is straightforward to verify the invariance of Eq. (1) under the following Weyl scaling
+transformation
+metric : gµν → g′
+µν = f 2(x)gµν,
+scalar : φ → φ′ = f −1(x)φ,
+Ricci scalar :
+ˆR → ˆR′ = f −2(x) ˆR,
+Weyl vector : Wµ → W ′
+µ = Wµ − ∂µ ln f(x),
+(4)
+where f(x) is an arbitrary positive function.
+3
+
+The purpose to explore the Lagrangian in Eq. (1) is two-fold.
+Theoretically, such a
+ˆR3 term constitutes as a simple extension of the ˆR2 theory, motivated from perspective of
+effective field theories and also quantum loop corrections in more fundamental theories [31–
+34]. Phenomenologically, it is worthwhile to explore how such a term would modify the
+cosmological observations related to inflation, and evaluate the likelihood and robustness of
+the predictions in the lowest-order theories.
+A.
+Formalism in Einstein frame
+General f(R) gravity is equivalent to the Einstein gravity with a scalar field [52, 53]. In
+Ref. [41], we have extended the proof in general scaling invariant F( ˆR, ϕ) gravity. We can
+explicitly show that by introducing an auxiliary scalar field χ and rewrite the high-order
+curvature terms as
+F( ˆR, ϕ) ≡ ϕ2 ˆR + α ˆR2 + β
+ϕ2 ˆR3 = F ˆR( ˆR → χ2, ϕ)( ˆR − χ2) + F( ˆR → χ2, ϕ).
+(5)
+Here F ˆR denotes the derivative over ˆR, F ˆR = ∂F( ˆR, ϕ)/∂ ˆR. We can verify that the equiv-
+alence relation χ2 = ˆR can be obtained from the Euler-Lagrange equation, δL
+δχ = 0. Substi-
+tuting Eq. (5) into Eq. (1), we find
+L
+√−g = 1
+2
+�
+ϕ2 + 2αχ2 + 3β
+ϕ2 χ4
+�
+ˆR − 1
+2
+�
+αχ4 + 2β
+ϕ2 χ6
+�
+− ζ
+2DµϕDµϕ −
+1
+4g2
+W
+FµνF µν.
+(6)
+Now we have demonstrated that linearization of ˆR has led to the non-minimal coupling of
+the scalar field, χ.
+We can transform the above Lagrangian into the Einstein frame by making a Weyl or
+conformal transformation of the metric field. However, we note that scaling invariance is
+still preserved in our model with χ → χ′ = f −1(x)χ. Therefore, we can directly normalize
+the coefficient before the Ricci scalar as
+ϕ2 + 2αχ2 + 3βχ4/ϕ2 = 1,
+(7)
+due to the scaling invariance of Eq. (6). This is equivalent to making a Weyl transformation
+with f(x) =
+�
+ϕ2 + 2αχ2 + 3βχ4/ϕ2 in Eq. (4). Further dropping the total derivative term
+4
+
+in Eq. (3) due to its null surface integral, we can write the Lagrangian as
+L
+√−g =1
+2R − ζ
+2DµϕDµϕ − V (ϕ) −
+1
+4g2
+W
+FµνF µν − 3W µWµ
+=R
+2 −
+∂µϕ∂µϕ
+2/ζ + ϕ2/3 − V (ϕ) −
+1
+4g2
+W
+FµνF µν − 6 + ζϕ2
+2
+�
+Wµ − ∂µ ln |6 + ζϕ2|
+2
+�2
+,
+(8)
+with the scalar potential
+V (ϕ) = α
+2 χ4 + β
+ϕ2χ6 = α
+6β
+�
+ϕ4 − ϕ2�
++ α3ϕ4
+27β2
+��
+1 − 3β
+α2
+�
+1 − ϕ−2��3/2
+− 1
+�
+,
+(9)
+where we have solved χ from Eq. (7)
+χ2 = αϕ2
+3β
+��
+1 − 3β
+α2 (1 − ϕ−2) − 1
+�
+.
+(10)
+It is now clear that we have a minimally-coupled scalar ϕ with a non-canonical kinetic
+term. To further simplifying the theoretical formalism, we introduce the following redefini-
+tions for the scalar and the Weyl gauge field
+ϕ2 ≡
+�
+�
+�
+�
+�
+6
+|ζ| sinh2 �
+±Φ
+√
+6
+�
+for ζ > 0,
+6
+|ζ| cosh2 �
+±Φ
+√
+6
+�
+for ζ < 0,
+(11)
+˜Wµ ≡ Wµ − 1
+2∂µ ln |6 + ζϕ2| ≡ gW ˜wµ.
+(12)
+Then the final Lagrangian turns into a more compact form
+L
+√−g = 1
+2R − 1
+2∂µΦ∂µΦ − V (Φ) −
+1
+4g2
+W
+˜Fµν ˜F µν − 1
+2m2(Φ) ˜W µ ˜Wµ,
+(13)
+with the mass term of Weyl gauge field
+m2(Φ) =
+�
+�
+�
+�
+�
++6 cosh2 �
+Φ
+√
+6
+�
+for ζ > 0,
+−6 sinh2 �
+Φ
+√
+6
+�
+for ζ < 0.
+(14)
+We should note that m2(Φ) is negative when ζ < 0. Therefore, to avoid Weyl gauge boson
+becoming tachyonic in this case, it requires some other mechanisms to obtain a real mass,
+for example, introducing other scalar field, which we do not explore in this paper. For viable
+inflation, both positive and negative are possible, as we shall show later.
+In the above discussion, we have demonstrated that Weyl scaling invariant ˆR2+ ˆR3 model
+can be written equivalently as the Einstein gravity coupled with a self-interacting scalar Φ
+5
+
+and a massive vector ˜Wµ with a field-dependent mass. This conclusion is also true for any
+Weyl scaling invariant model of gravity with high-order curvature ˆRn as the above formalism
+applies straightforwardly. It is also worth to point out that Weyl vector boson can serve
+as a dark matter candidate [27, 28, 41], with details of the relic abundance being discussed
+in [45]. In this paper, we shall concentrate on the scalar potential Eq. (9) and discuss the
+viable inflation scenarios with the presence of ˆR3.
+B.
+Effective scalar potentials
+There are two necessary requirements for the potential Eq. (9). The first one is ϕ2 > 0
+since ϕ is a real scalar field. The other is 1 − 3β
+α2
+�
+1 −
+1
+ϕ2
+�
+≥ 0, otherwise an imaginary
+potential will emerge. Consequently, there are some constraints on the parameters and the
+viable value of Φ. We can rewrite the second requirement as
+sinh
+�±Φ
+√
+6
+�
+≥ or ≤
+�
+|ζ|
+6 − 2α2/β , for ζ > 0,
+cosh
+�±Φ
+√
+6
+�
+≥ or ≤
+�
+|ζ|
+6 − 2α2/β , for ζ < 0,
+(15)
+where “ ≥ ” for β < α2
+3 and “ ≤ ” for β ≥ α2
+3 . For convenience, we define λ ≡
+�
+|ζ|
+6−2α2/β
+and γ ≡
+3β
+α2, then discuss the possible ranges of the potential corresponding to different
+parameters. The results are listed in the Table. I. To ensure the theoretical stability, we
+require that Φ can only evolve within these ranges where the potential is real. Fig. 1 shows
+some instances of the scalar potential for several values of ζ and γ.
+We first discuss the case of positive ζ. When γ = 0, it is a hill-top-like potential with two
+minima at Φ = ±
+√
+6 sinh−1 �
+ζ
+6. However, as long as there is a tiny cubic curvature, whether
+positive or negative, the shape of potential will be affected significantly. When γ > 0, the
+potential turns to decrease near Φ = 0, and a third vacuum can form there. This behavior
+is transparent, because when ζ > 0, Φ = 0 corresponds to ϕ2 = 0 according to Eq. (11),
+then substituting it in Eq. (9) will obtain V |Φ=0 = 0. When γ < 0, the potential turns to
+rise near Φ = 0 and become imaginary and unphysical in −
+√
+6 sinh−1 λ < Φ <
+√
+6 sinh−1 λ,
+which has been listed in Table. I.
+Next, we switch to the case of negative ζ. It is evident in Fig. 1 that when ζ < 0 and |ζ|
+or |γ| is relatively small, the modification of ˆR3 term on the Weyl R2 potential is moderate,
+6
+
+TABLE I. Effective potential range of the Weyl R2 + R3 model.
+ζ
+γ or β
+real V (ϕ)
+ζ > 0
+γ ≥ 1
+|Φ| ≤
+√
+6 sinh−1 λ
+0 ≤ γ < 1
+fully real
+γ < 0
+|Φ| ≥
+√
+6 sinh−1 λ
+−6 < ζ < 0
+γ >
+1
+1+ζ/6
+fully imaginary
+1 < γ ≤
+1
+1+ζ/6
+|Φ| ≤
+√
+6| cosh−1 λ|
+γ ≤ 1
+fully real
+ζ ≤ −6
+γ ≥ 1
+|Φ| ≤
+√
+6| cosh−1 λ|
+1
+1+ζ/6 < γ < 1
+fully real
+γ ≤
+1
+1+ζ/6
+|Φ| ≥
+√
+6| cosh−1 λ|
+unlike the dramatic change near Φ = 0 in the case of positive ζ. This is because the mapping
+of Φ ⇒ ϕ2 does not cover the interval of ϕ2 < 1 for ζ < 0 according to Eq. (11). In other
+words, for negative ζ with modest |γ|, Φ → 0 does not lead to ϕ2 → 0, which brings the
+violent behavior of the potential around here in the case of ζ > 0. However, when ζ is
+excessively negative or |γ| is large enough, the violent variation will reappear to a certain
+extent. For γ > 0, the potential will return to a downward trend near Φ = 0, albeit there
+is no true vacuum formed (but a false vacuum is formed). And for excessively negative γ,
+the imaginary potential will reappear in the range of −
+√
+6| cosh−1 λ| < Φ <
+√
+6| cosh−1 λ|,
+which we have listed this situation in Table. I (see ζ ≤ −6 with γ ≤
+1
+1+ζ/6 case).
+Generally, inflation takes place when the potential is flat and Φ evolves to the vacuum
+(Φ|V =0). The cosmological observations would restrict the potential and the initial value Φi
+when inflation starts, here the Φi is defined as the value when the comoving horizon of the
+inflationary universe shrinks to the same size as today.
+For ζ > 0 and γ > 0, the scalar potential contains three separate vacua, one lying at the
+center and the other two at both sides. Therefore, there are two different viable inflationary
+patterns. One pattern refers to the evolution into the central minimum, and the other into
+the side minima. We can calculate the value of Φ which corresponds to the hill-top of the
+7
+
+-10
+-5
+0
+5
+10
+0
+0.5
+1
+1.5
+2
+10-10
+-10
+-5
+0
+5
+10
+0
+0.5
+1
+1.5
+2
+10-10
+-10
+-5
+0
+5
+10
+0
+0.5
+1
+1.5
+2
+10-10
+ 
+-10
+-5
+0
+5
+10
+0
+0.5
+1
+1.5
+2
+10-10
+FIG. 1. Effective potentials of Weyl R2 + R3 model with α = 109 and various γ and ζ. Here we
+only depict the real ranges of potentials.
+potential in this case
+Φh = ±
+√
+6 sinh−1
+�
+ζ
+12
+√3γ − 2γ
+3 − 4γ
+,
+(16)
+which is the critical point of two inflationary patterns. Neglecting the velocity, if the initial
+value of inflation field satisfies |Φi| > |Φh|, it will evolve towards the side vacua. If |Φi| < |Φh|
+at the beginning, the inflation field will evolve towards the central vacuum.
+For other cases of ζ and γ, there are only the global side minima. Hence the only feasible
+inflationary pattern is that Φ evolves to either one of the side minimum. The initial value
+Φi has to correspond to a real potential, and when there is a false vacuum in ζ < 0 case, it
+requires a large enough |Φi| outside two local maxima of the potential to ensure the gradient
+of V (Φi) towards the true vacuum. Next, we are going to discuss the inflation in these two
+patterns respectively.
+8
+
+III.
+INFLATION TO THE SIDE
+In this inflation pattern, ϕ2 (defined as Eq. (11)) is usually not very close to 0, and as
+we shall show later, observations generally would require an extremely small cubic curva-
+ture, namely |γ| ≪ 1. Therefore in many cases, |γ(1 − ϕ−2)| ≪ 1 is satisfied. Under this
+condition, we are able to have analytical treatment and expand the potential Eq. (9) as
+V (ϕ) =ϕ4 − ϕ2
+2αγ
++
+ϕ4
+3αγ2
+�
+−3γ
+2
+�
+1 − 1
+ϕ2
+�
++ 3γ2
+8
+�
+1 − 1
+ϕ2
+�2
++ γ3
+16
+�
+1 − 1
+ϕ2
+�3
++ O
+�γ4
+ϕ8
+��
+= 1
+8α
+�
+1 − ϕ2�2
+�
+1 + γ
+6
+�
+1 − 1
+ϕ2
+�
++ O
+�γ2
+ϕ4
+��
+.
+(17)
+Then with Eq. (11), we derive
+V (Φ) =
+�
+�
+�
+�
+�
+1
+8α
+�
+1 − 6
+|ζ| sinh2 �
+Φ
+√
+6
+��2 �
+1 + γ
+6
+�
+1 − |ζ|
+6 csch2 �
+Φ
+√
+6
+��
++ O(γ2)
+�
+for ζ > 0,
+1
+8α
+�
+1 − 6
+|ζ| cosh2 �
+Φ
+√
+6
+��2 �
+1 + γ
+6
+�
+1 − |ζ|
+6 sech2 �
+Φ
+√
+6
+��
++ O(γ2)
+�
+for ζ < 0.
+(18)
+The first term is exactly the effective potential of Weyl ˆR2, which has been shown in [41, 45],
+and the rest originates from the cubic curvature term ˆR3, to the leading order of γ. Next
+we shall calculate the inflationary physical quantities, the spectral index ns and tensor-to-
+scalar ratio r, and contrast them with the latest observations. We first give an analytical
+calculation for two limiting cases, then show the full numerical results for general cases.
+A.
+Analytical approach of γ → 0 case
+We first discuss the γ → 0 case and show how ζ affects ns and r. The slow-roll parameters
+in this case can be derived as
+ϵ ≡ 1
+2
+�V ′(Φ)
+V
+�2
+=
+12 sinh2 �
+2Φ
+√
+6
+�
+�
+|ζ + 3| − 3 − 6 sinh2 �
+Φ
+√
+6
+��2,
+(19)
+η ≡ V ′′(Φ)
+V
+=
+12 cosh
+�
+4Φ
+√
+6
+�
+− 4|ζ + 3| cosh
+�
+2Φ
+√
+6
+�
+�
+|ζ + 3| − 3 − 6 sinh2 �
+Φ
+√
+6
+��2
+.
+(20)
+Generally, the slow-roll inflation occurs when ϵ and |η| is small enough, and it will end when
+any of them evolves to ∼ 1. For the situation we are concerned with, ϵ breaks the slow-roll
+9
+
+limit before the other. Thus we derive the value of Φ when inflation ends according to ϵ = 1
+Φe =
+�
+3
+2 ln
+�
+2
+�
+|ζ + 3|2 + 3
+√
+3
+− |ζ + 3| +
+�
+7
+3|ζ + 3|2 − 4|ζ + 3|
+√
+3
+�
+|ζ + 3|2 + 3 + 3
+�
+. (21)
+When |ζ| > O(102), which is a preferred range by the observational constraints as we will
+show shortly, the above equation can be approximated as
+Φe ≃
+�
+3
+2 ln
+� 1
+√
+3
+�
+2 +
+�
+7 − 4
+√
+3 −
+√
+3
+�
+|ζ + 3|
+�
+≃
+�
+3
+2 ln (0.3094|ζ + 3|) .
+(22)
+It is now clear that when |ζ| is large enough, Φe will be almost independent of the sign of ζ.
+Next, we calculate initial value Φi, which is defined when the size of comoving horizon
+during inflation shrinks to the present size. We first focus on the e-folding number of the
+slow-roll inflation
+N ≡ ln ae
+ai
+≃
+� Φe
+Φi
+dΦ
+√
+2ϵ,
+(23)
+where ai/e ≡ a(Φi/e) is the cosmic scale factor when inflation starts/ends.
+Substituting
+Eq. (19) into it, we find
+N =
+(|ζ + 3| − 3) ln
+�
+tanh
+�
+Φ
+√
+6
+��
+− 6 ln
+�
+cosh
+�
+Φ
+√
+6
+��
+4
+�����
+Φe
+Φi
+= |ζ + 3| − 3
+4
+ln
+�
+�tanh
+� 1
+2 ln(0.3094|ζ + 3|)
+�
+tanh
+�
+Φi
+√
+6
+�
+�
+� − 3
+2 ln
+�
+�cosh
+� 1
+2 ln(0.3094|ζ + 3|)
+�
+cosh
+�
+Φi
+√
+6
+�
+�
+� .
+(24)
+For the circumstances we are concerned with, namely N ∼ (50, 60) and |ζ| > O(102), the
+second term of Eq. (24) is much smaller than the first term, and it can be estimated as
+∼ −2.3. Thus we derive
+Φi ≃
+√
+6 tanh−1
+��
+1 −
+2
+0.3094|ζ + 3| + 1
+�
+e
+−4(N+2.3)
+|ζ+3|−3
+�
+≡
+√
+6 tanh−1 Ω(ζ, N).
+(25)
+Here we have defined Ω(ζ, N) for later convenience.
+When |ζ| ≫ 4N, it can be further approximated as Φi ≃
+�
+3
+2 ln
+|ζ|
+2N+7.8. Substituting
+Eq. (25) into Eq. (19) and (20), we find
+ϵi =
+48Ω2
+[(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2,
+(26)
+ηi =4 [(Ω4 − 1)|ζ + 3| + 3(Ω4 + 6Ω2 + 1)]
+[(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2
+.
+(27)
+10
+
+As a result, the tensor-to-scalar ratio r and spectral index ns of inflationary perturbations
+in the γ → 0 limit are finally calculated as
+r = 16ϵi =
+768Ω2
+[(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2,
+(28)
+ns = 1 − 6ϵi + 2ηi = 1 + 8(Ω4 − 1)|ζ + 3| + 24(Ω4 − 6Ω2 + 1)
+[(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2
+.
+(29)
+For N ∼ (50, 60) and |ζ| > O(102), We can approximate the expressions as
+r ≃ r∗ − 54
+ζ2 ,
+(30)
+ns ≃ n∗
+s − 11N
+ζ2 ,
+(31)
+where
+r∗ ≃
+12
+(N + 3.55)2, n∗
+s ≃ 1 −
+2
+N + 3.55 −
+3
+(N + 3.55)2
+(32)
+are the predictions of Starobinsky model (see Appendix A for an analytical derivation.).
+Thus it is evident that the predictions of inflationary perturbations in our model will converge
+to that of Starobinsky model when γ → 0 and ζ → ∞. As |ζ| decreases, the value of r and
+ns will also decrease. We show this trend as the pink area in Fig. 2. According to the latest
+observation [54], the lower limit of ns has been constrained to ∼ 0.959, hence it requires
+|ζ| > 270 in this γ → 0 case.
+B.
+Analytical approach of ζ → ∞ case
+Now we discuss the ζ → ∞ case and show how γ affects r and ns. When ζ is large enough,
+the potential is greatly widened. The side vacua are far away from 0 and so are Φi and Φe
+(e.g., Φi ∼ 5.4MP, Φe ∼ 9.8MP for ζ = 104). Therefore Eq. (11) can be approximated as
+ϕ2 = 6
+|ζ|
+�
+eΦ/
+√
+6 ± e−Φ/
+√
+6
+2
+�2
+≃ e
+√
+2/3
+�
+Φ−√
+3/2 ln(2|ζ|/3)
+�
+≡ e
+√
+2/3(Φ−Φ0).
+(33)
+Here and after, without losing generality, we may choose to evolve in the positive Φ region,
+and denote Φ0 as the minimum in this region. Substituting it into Eq. (17), we have the
+scalar potential for Φ ≫ 0
+V (Φ) = 1
+8α
+�
+1 − e
+√
+2/3(Φ−Φ0)�2 �
+1 + γ
+6
+�
+1 − e−√
+2/3(Φ−Φ0)�
++ O(γ2)
+�
+.
+(34)
+11
+
+FIG. 2. The predictions of spectral index ns combined with tensor-to-scalar ratio r in the Weyl
+R2 + R3 model with e-folding number N ∼ (50, 60). The pink area shows the results in the γ → 0
+case with various ζ. The yellow and green areas respectively show the ζ → ∞ and ζ = −650 cases
+with various γ. The red line is the result with both γ → 0 and ζ → ∞, which is equivalent to the
+Starobinsky model. The blue area is the latest observation constraint given by the BICEP/Keck
+collaboration [54].
+Ignoring the O(γ2) terms, we give an approximate expression for the slow-roll parameters
+ϵ ≡ 1
+2
+�V ′(Φ)
+V
+�2
+≃
+�
+γe
+√
+2/3(Φ−Φ0) − 2(γ + 6)e
+√
+8/3(Φ−Φ0) + γ
+�2
+3
+�
+e
+√
+2/3(Φ−Φ0) − 1
+�2 �
+γ − (γ + 6)e
+√
+2/3(Φ−Φ0)�2,
+(35)
+η ≡ V ′′(Φ)
+V
+≃ 6(γ + 4)e
+√
+8/3(Φ−Φ0) − 8(γ + 6)e
+√
+6(Φ−Φ0) + 2γ
+3
+�
+e
+√
+2/3(Φ−Φ0) − 1
+�2 �
+γ − (γ + 6)e
+√
+2/3(Φ−Φ0)�.
+(36)
+In this case, the slow-roll inflation also ends at ϵ ∼ 1. To find the expression of Φe, we
+further approximate Eq. (35) as
+ϵ ≃
+e−√
+8/3(Φ−Φ0) �
+γ − 12e
+√
+8/3(Φ−Φ0)�2
+108
+�
+e
+√
+2/3(Φ−Φ0) − 1
+�2
+.
+(37)
+12
+
+.......
+→ 60= -650
+95% CL
+0.03
+68% CL
+5×1
+0.01
+500
+X
+ = 250
+0.003
+3x 10
+3 × 10
+0.001
+0.955
+0.96
+0.965
+0.97
+0.975
+ns0.980.1
+0←
+N = 50
+8个VThen Φe can be derived as
+Φe = Φ0 −
+�
+3
+2 ln
+�√
+3
+γ
+��
+2(2 +
+√
+3)γ + 9 − 3
+��
+.
+(38)
+If γ is extremely small, we will find Φe ≃ Φ0 − 0.94MP.
+Next, we derive the analytic formula for Φi in this case. The e-folding number of the
+slow-roll inflation can be calculated with Eq. (37) as
+N = −
+�27
+4γ tanh−1
+�� γ
+12e−√ 2
+3 (Φ−Φ0)
+�
+− 3
+8 ln
+�
+12 − γe−√ 8
+3 (Φ−Φ0)�
+−
+√
+6
+4 (Φ − Φ0)
+����
+Φe
+Φi
+. (39)
+Considering N ∼ (50, 60) and γ < O(10−3), the first term of the integral is dominant, while
+the rest are the marginal terms which can be approximately treated as a constant, ∼ −2.7.
+Hence we have
+N ≃
+�27
+4γ
+�
+tanh−1
+�� γ
+12e−√
+2/3(Φi−Φ0)
+�
+− tanh−1
+�� γ
+12e−√
+2/3(Φe−Φ0)
+��
+− 2.7,
+(40)
+and derive
+Φi = Φ0 −
+�
+3
+2 ln
+�����
+�12
+γ tanh
+�
+tanh−1
+�� γ
+12e−√
+2/3(Φe−Φ0)
+�
++
+�
+4γ
+27(N + 2.7)
+������
+≃ Φ0 −
+�
+3
+2 ln
+�����
+�12
+γ tanh
+�
+tanh−1 (0.622√γ) +
+�
+4γ
+27(N + 2.7)
+������
+≡ Φ0 −
+�
+3
+2 ln Θ(γ, N),
+(41)
+where we have defined Θ(γ, N) for later convenience. Then substituting it into Eq. (35) and
+(36), we find
+ϵi = [γΘ(1 + Θ) − 2(γ + 6)]2
+3 [1 − Θ]2 [γΘ − (γ + 6)]2,
+(42)
+ηi = 2γΘ3 + 6(γ + 4)Θ − 8(γ + 6)
+3 [1 − ��]2 [γΘ − (γ + 6)]
+.
+(43)
+Finally, we derive r and ns of the inflationary perturbations in the ζ → ∞ limit
+r = 16ϵi = 16 [γΘ(1 + Θ) − 2(γ + 6)]2
+3 [1 − Θ]2 [γΘ − (γ + 6)]2 ,
+(44)
+ns = 1 − 6ϵi + 2ηi = 1 − 2 [γΘ(1 + Θ) − 2(γ + 6)]2
+[1 − Θ]2 [γΘ − (γ + 6)]2 + 4γΘ3 + 3(γ + 4)Θ − 4(γ + 6)
+3 [1 − Θ]2 [γΘ − (γ + 6)]
+. (45)
+13
+
+If γ is extremely small, smaller than O(10−4), the above expressions can be linearly approx-
+imated as
+r ≃ r∗ − 2.4γ,
+(46)
+ns ≃ n∗
+s − 0.42γN,
+(47)
+where r∗ and n∗
+s have been defined in the last paragraph of Sec. III.A. We can see that
+compared with the predictions of Starobinsky model, a positive γ will reduce both r and
+ns, while a negative γ will increase them. We show this trend as the yellow area in Fig. 2.
+It is manifest that the observations have constrained |γ| ≲ 5 × 10−4 in this ζ → ∞ case.
+Actually, this result agrees with other numerical investigations of the R3-extended Starobin-
+sky model [55–61], since the potential Eq. (34) is the same as the R3-extended Starobinsky
+model with a vacuum shift. Moreover, compared with Eq. (30) and (31), we note that the
+predictions of r and ns in the γ → 0 case is similar to that of the ζ → ∞ and γ > 0 case
+with a simple replacement of γ ↔ 24
+ζ2. This can be seen more clearly from Fig. 2, where the
+pink area overlaps with the yellow area with γ > 0.
+C.
+General cases
+Now we discuss the general cases with various ζ and γ by numerical treatment. The
+results are shown in Fig. 3. Here the parameter ranges satisfying observational constraints
+(see blue area in Fig. 2) are marked with colored areas, where the color gradient from blue to
+red corresponds to ascending value of r. The gray areas represent that the potential defined
+by these parameters cannot support an adequate inflation. In other words, their maximal
+e-folding number is unable to reach N = 50 or 60. The white areas are the parameter ranges
+that can give rise to ample inflation, but their prediction of ns or r has been excluded by
+the observation constraints. Here we mark two dotted lines to distinguish the boundaries of
+constraints. Beyond the pink one indicates a large ns that exceeds the observational upper
+limit, while beyond the green one signifies a too small prediction.
+Let us focus on the colored parameter ranges that are allowed by observations. In the
+|ζ| ≫ 1000 case, the result is roughly equivalent to the analytical calculation shown in the
+last subsection. The prediction of r is limited to 0.002 < r < 0.006. However, distinctive
+situations appear when |ζ| is small. First, when −1000 < ζ < −200, the restrictions on
+γ is relaxed, which can stand |γ| ∼ 6 × 10−3 at most. Besides, the upper limit of r is
+14
+
+FIG. 3. Possible parameter space for Weyl R2 + R3 model when Φ evolves to the side vacuum.
+The colored areas are the parameter ranges allowed by the latest observations of BICEP/Keck
+collaboration [54], where the color gradient from blue to red corresponds to r increases from 0.001
+to the observational upper limit 0.036. The dotted lines are the boundaries that ns exceeds the
+observational upper (pink line) or lower (green line) limit. The gray areas represent the parameter
+ranges with inadequate inflation, namely, the maximal e-folding number of inflation cannot reach
+N = 50 or 60.
+greatly expanded. There is even a small parameter range that gives r > 0.01. We show an
+example as the green area in Fig. 2. It clearly shows a distinguishable feature from the Weyl
+R2 model and the R3-extended Starobinsky model. If the next generation experiment of
+CMB B-mode polarization detects the primordial gravitational waves with r > 0.01, it may
+support Weyl R2 + R3 model. Another notable feature emerges at 0 < ζ < 200, where the
+15
+
+r0
+-2
+ns > 0.974
+-4
+inadequate e-folds
+-6
+-3000
+-2000
+-1000
+0
+1000
+2000
+× 10-3
+4
+N = 60
+2
+ns < 0.959
+0
+-2
+ns > 0.974
+-4
+inadequate e-folds
+9-
+-3000
+-2000
+-1000
+0
+1000
+20000.000
+0.015
+3000
+0.005
+0.001
+3000×10-3
+4
+N = 50
+2
+ns < 0.959negative γ, even if very small, can greatly affect the predictions of primordial perturbations.
+Actually, there are some cases with small positive ζ and small negative γ can give proper
+r and ns that match the observation constraints, and generally, r is extremely small. For
+instance, when ζ = 80, γ = −4 × 10−8, and N = 60, we have ns = 0.963 and r = 3 × 10−4.
+IV.
+INFLATION TO THE CENTER
+As we mentioned earlier, the third vacuum appears at Φ = 0 in the case of ζ > 0 and
+γ > 0, and if the initial value satisfies |Φi| < |Φh| (Φh is defined in Eq. (16)), inflation can
+happen in the evolution of Φ to 0. Actually, the situation is more complicated. A process
+called “oscillating inflation” [62–74] will continue immediately after the end of slow-roll
+inflation because the scalar potential in this case is a non-convex function in the region close
+to the vacuum, which means there is d2V
+dΦ2 < 0 when Φ nears 0. In other words, for such a
+non-convex potential, despite the slow-roll conditions (ϵ ≪ 1 and |η| ≪ 1) has been violated
+during the bottom oscillation of the inflaton potential, the universe can keep accelerating
+expansion until the average amplitude of the inflaton’s oscillation becomes lower than the
+borderline of d2V
+dΦ2 from negative to positive (if there is a rounded transition in a small enough
+∆Φ at the bottom to connect the left and right sides of the potential, see [62]), or until the
+contribution of the radiation produced in reheating process becomes non-negligible.
+It is helpful to understand the behavior of oscillating inflation from the perspective of the
+effective equation of state. For an oscillating scalar field Φ, its effective equation of state in
+one oscillating period is defined as
+⟨w⟩ ≡ ⟨p⟩
+⟨ρ⟩ = ⟨ ˙Φ2 − ρ⟩
+⟨ρ⟩
+= ⟨ ˙Φ2⟩
+Vm
+− 1 = ⟨Φ dV
+dΦ⟩
+Vm
+− 1 = 1 − 2⟨V ⟩
+Vm
+,
+(48)
+where ⟨⟩ means the average value in one oscillation period, and Vm represents the maximal
+potential of this oscillation period.
+The accelerating expansion of the universe requires
+⟨w⟩ < − 1
+3, which is equivalent to the following relation
+U ≡ ⟨V − ΦdV
+dΦ⟩ > 0.
+(49)
+In fact, U amounts to the intercept of the tangent to the potential at a certain Φ, shown
+as the upper part of Fig. 4. As long as the intercept is positive and the contribution of
+radiation is insignificant, the accelerating expansion will proceed successfully. This is the
+reason why a non-convex potential can bring about oscillating inflation.
+16
+
+For the process with oscillating inflation, the definition of e-folding number should be
+replaced to
+˜N ≡ ln afHf
+aiHi
+≡ ln aeHe
+aiHi
++ ln aoHo
+aiHi
+≃ N + No,
+(50)
+where the subscripts i and e have been defined in the last section, af and Hf represent
+the cosmic scale and Hubble parameter when the full inflationary period ends, ao and Ho
+represent their multiple of increase or decrease during the oscillating inflation. It indicates
+that the new definition is equivalent to adding a correction No based on the e-folding number
+of slow-rolling period if we take He ≈ Hi. Generally, No is related to the shape of potential
+near its vacuum, reheating efficiency, and the scale of the aforementioned rounded bottom.
+Given that our model does not possess an explicit rounded bottom, No depends only on
+the first two aspects.
+For the shape of potential, actually, our model has the following
+approximate form near the center vacuum
+V (Φ) ≃ ξ(Φ4 − Φ2)
+2α
++ ξ2Φ4
+3α
+��
+1 +
+1
+ξΦ2
+�3/2
+− 1
+�
+,
+(51)
+where ξ ≡
+α2
+3βζ. Since α determines the height of the potential, which has been fixed for
+each set of ζ and β according to the observation result of ∆2
+s ∼
+V
+24π2ϵ ∼ 2.1 × 10−9 [75], the
+shape of the potential is essentially determined by ξ in the oscillatory region. For reheating
+efficiency, we consider a constant transfer rate Γ and the transferred energy all turns to
+radiation ρr
+¨Φ + (3H + Γ) ˙Φ + dV
+dΦ = 0,
+(52)
+˙ρr + 4Hρr − Γ ˙Φ2 = 0.
+(53)
+Then No is substantially related to the parameters ξ and Γ.
+We numerically solve the above equations, and visualize in the lower part of Fig. 4. It
+is transparent that if ξ ≫ 0.1, oscillating inflation will bring appreciable correction to the
+e-folding number.
+Because an inefficient reheating process will postpone the end of the
+oscillating inflation, we can see a smaller Γ corresponds to a larger No for a certain ξ.
+However, No will tend to a fixed value as Γ decreases. This property can be understood as
+follows. We can prove that the potential has a quasi-linear form when Φ → 0
+V |Φ→0 ≃
+√ξ
+3α |Φ|,
+(54)
+17
+
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+10-2
+100
+102
+104
+0
+0.5
+1
+1.5
+FIG. 4. Oscillating inflation in the center-evolving pattern of Weyl R2 + R3 model. The upper
+part is a diagram for visualizing the condition of oscillating inflation, where the effective equation
+of state ⟨w⟩ < − 1
+3 is equated with that the intercept U of the tangent to a certain point on the
+potential corresponding to the average amplitude is positive. The lower part shows the increased
+e-folding number during the oscillating inflation for various ξ and reheating efficiency Γ.
+which implies that U|Φ→0 → 0 according to its definition as the intercept of the tangent to
+the potential. Hence ⟨w⟩ will quickly converge to − 1
+3 as the oscillation proceeds, and No will
+soon grow to a nearly constant maximum if Γ is too small to make the universe promptly
+produce enough radiation to stop the oscillating inflation. This is the reason why No has an
+extreme for each ξ.
+Now we consider the reheating is inefficient, that is to adopt No with Γ → 0, to derive
+the slow-roll e-folding number N corresponding to ˜N ∼ (50, 60), and then to calculate ns
+18
+
+103
+104
+105
+106
+107
+108
+10-7
+10-5
+10-3
+103
+104
+105
+106
+107
+108
+10-7
+10-5
+10-3
+<10-4
+10-3
+10-2
+FIG. 5. Possible parameter space for Weyl R2 + R3 model when Φ evolves to the center vacuum.
+Here the total e-folding number ˜N ≡ N + No is considered with Γ → 0. The meaning of markers
+is the same as that in Fig. 3, except for the color correspondence of r.
+and r for various parameters ζ and γ. The viable parameter space is depicted in Fig. 5,
+where the meaning of markers is the same as that in Fig. 3, except for the scale of color
+bar. It is evident that the observation constraint on ns limits the parameters to ζ > 103 and
+γ < 5 × 10−4. r has an upper limit ∼ 0.006, but no lower limit in this case.
+V.
+CONCLUSIONS
+Cosmological observations have suggested that our universe has a nearly scaling invariant
+power spectrum of the primordial density perturbation, which motivates the scaling sym-
+19
+
+metry as the possible feature of the underlying fundamental theories that lead to inflation.
+We present the theoretical formalism of the Weyl scaling invariant gravity, ˆR2 + ˆR3. We
+show this model in Eq. (1) can be rewritten equivalently to the Einstein gravity coupled
+with a massive gauge boson, and a scalar field as the inflaton. We further discuss the viable
+ranges of the scalar potential according to the requirement for reality and demonstrate how
+the R3 term would affect the shape of potentials. Compared with the Weyl R2 inflationary
+potential [41, 45] with two side minima, the R3 extension brings an additional minimum at
+center. Hence, there are two viable scenarios for the inflation in this model. The first is
+to roll towards the side minima, while the other is a new situation of rolling towards the
+center minimum. Both scenarios allows viable parameter spaces that be probed by future
+experiments on cosmic microwave background and primordial gravitational wave.
+For the first scenario, we calculate the spectral index ns and tensor-to-scalar ratio r
+of primordial perturbations both analytically and numerically, and contrast the parameter
+spaces with the latest observational constraints. The results manifest that the level of cubic
+curvature is limited to |γ| < 6×10−3, and the prediction of r in this pattern has a wide range
+from O(10−4) to the upper limit of the observations, O(10−2). These results are significantly
+different from the R3-extended Starobinsky model.
+For the second scenario, a special process called oscillating inflation emerges after the
+familiar slow-roll inflation because the potential near the center minimum is a non-convex
+function that can lead to a sufficiently negative value of average equation of state.
+We
+calculate the correction of e-folding number in the oscillating inflation stage, and then derive
+the viable parameter spaces. The results indicate that the parameters are limited to γ <
+5 × 10−4 and ζ > 103. Moreover, r has an upper limit ∼ 0.006, but no lower limit.
+ACKNOWLEDGMENTS
+QYW and YT thank Shi Pi for helpful discussions. YT is supported by National Key Re-
+search and Development Program of China (Grant No.2021YFC2201901), and Natural Sci-
+ence Foundation of China (NSFC) under Grants No. 11851302. YLW is supported by the Na-
+tional Key Research and Development Program of China under Grant No.2020YFC2201501,
+and NSFC under Grants No. 11690022, No. 11747601, No. 12147103, and the Strategic Prior-
+ity Research Program of the Chinese Academy of Sciences under Grant No. XDB23030100.
+20
+
+Appendix A: Analytical treatment of Starobinsky inflation
+We give an analytical calculation of the tensor-to-scalar ratio r and spectral index ns in
+the Starobinsky inflationary model, namely, the Einstein gravity modified by a R2 term.
+The effective scalar potential can be written as
+V (φ) = 1
+8α
+�
+1 − e−√
+2/3φ�2
+,
+(A1)
+where α is the coefficient of R2. The relevant two slow-roll parameters are computed as
+ϵ = 4
+3
+1
+�
+e
+√
+2/3φ − 1
+�2,
+η = −4
+3
+e
+√
+2/3φ − 2
+�
+e
+√
+2/3φ − 1
+�2.
+(A2)
+Since inflation ends when ϵ ∼ 1 is reached first (η ≃ −0.15), we have
+φe =
+�
+3
+2 ln
+�
+1 + 2
+√
+3
+�
+≃ 0.94MP.
+(A3)
+Then according to Eq. (23), the e-folding number is
+N =
+�
+3
+4
+�
+e
+√
+2/3φ −
+�
+2
+3φ
+��φe
+φi
+= 3
+4
+�
+e
+√
+2/3φi − e
+√
+2/3φe −
+�
+2
+3(φi − φe)
+�
+.
+(A4)
+For N ∼ (50, 60), we find that approximately
+φi ≃
+�
+3
+2 ln
+�4
+3(N + 4.3)
+�
+.
+(A5)
+Substituting it into Eq. (A2), we finally derive
+r = 16ϵ =
+12
+(N + 3.55)2,
+(A6)
+ns = 1 − 6ϵ + 2η = 1 −
+2
+N + 3.55 −
+3
+(N + 3.55)2.
+(A7)
+These results are shown as the red line in Fig. 2.
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+Draft version January 10, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+Evolution of elemental abundances in hot active region cores from Chandrayaan-2 XSM observations
+Biswajit Mondal,1, 2 Santosh V. Vadawale,1 Giulio Del Zanna,3 N. P. S. Mithun,1 Aveek Sarkar,1
+Helen E. Mason,3 P. Janardhan,1 and Anil Bhardwaj1
+1Physical Research Laboratory, Navrangpura, Ahmedabad, Gujarat-380 009, India
+2Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, Gujarat-382 355, India
+3DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
+ABSTRACT
+The First Ionization Potential (FIP) bias, whereby elemental abundances for low FIP elements in
+different coronal structures vary from their photospheric values and may also vary with time, has been
+widely studied. In order to study the temporal variation, and to understand the physical mechanisms
+giving rise to the FIP bias, we have investigated the hot cores of three ARs using disk-integrated
+soft X-ray spectroscopic observation with the Solar X-ray Monitor (XSM) onboard Chandrayaan-2.
+Observations for periods when only one AR was present on the solar disk were used so as to ensure that
+the AR was the principal contributor to the total X-ray intensity. The average values of temperature
+and EM were ∼3 MK and 3×1046 cm−3 respectively. Regardless of the age and activity of the AR,
+the elemental abundances of the low FIP elements, Al, Mg, and Si were consistently higher than their
+photospheric values. The average FIP bias for Mg and Si was ∼3, whereas the FIP bias for the mid-FIP
+element, S, was ∼1.5. However, the FIP bias for the lowest FIP element, Al, was observed to be higher
+than 3, which, if real, suggests a dependence of the FIP bias of low FIP elements on their FIP value.
+Another major result from our analysis is that the FIP bias of these elements is established in within
+∼10 hours of emergence of the AR and then remains almost constant throughout its lifetime.
+Keywords: Solar X-ray corona, Solar abundances, FIP bias, FIP effect, Active Region
+1. INTRODUCTION
+The earlier study of the Sun as a star by Pottasch
+(1963) revealed that solar coronal abundances are dif-
+ferent from those of the photosphere. The differences
+are correlated to the First Ionization Potential (FIP) of
+the element, in the sense that the abundance ratio of
+a low-FIP (less than 10 eV) element versus that of a
+high-FIP element is higher in the corona. A measure
+of the difference is the so called FIP bias, i.e. the ratio
+between the coronal and the photospheric abundance of
+an element.
+In most of the available literature, the FIP bias has
+been (and still is) estimated by measuring the relative
+abundances between elements, and not relative to hy-
+drogen. This is due to the fact that abundance mea-
+surements with respect to Hydrogen in the low corona,
+and on-disk is non-trivial, due to the lack of H-emission
+Corresponding author: Biswajit Mondal
+[email protected], [email protected]
+lines at a few million Kelvin. Hence, whether it is the
+low-FIP elements that have an increased abundance or
+the high-FIP elements that have a reduced one (com-
+pared to their photospheric values) has been a subject
+of continued debate.
+Further, it has become clear that different solar struc-
+tures have different FIP biases. There are also indica-
+tions that the FIP bias depends on the temperature of
+the plasma. For a long time, it has been widely accepted
+that coronal abundances in active regions increase with
+time. We refer the reader to the recent reviews by Lam-
+ing (2015); Del Zanna and Mason (2018) for more de-
+tails. We also provide in the following section a brief
+summary of available measurements related to active re-
+gions.
+Knowledge of the elemental abundances in different
+atmospheric layers of the Sun is a topic of great inter-
+est to the solar physics community mainly due to the
+following two reasons. The first is that they provide, in
+principle, a way to link the solar source regions to the
+various components of the solar wind. In fact, elemental
+abundance variations are also clearly observed in-situ.
+arXiv:2301.03519v1  [astro-ph.SR]  9 Jan 2023
+
+2
+The slow-speed solar wind has a high FIP bias simi-
+lar to that measured in AR core loops, 3MK, whereas
+the high-speed wind has a near unit FIP bias, similar
+to that of coronal holes (see, e.g., Brooks et al. 2015;
+Gloeckler and Geiss 1989; Feldman et al. 1998; Bochsler
+2007; Brooks and Warren 2011).
+The second reason is that studying abundance vari-
+ations might contribute to a better understanding of
+the physical processes at play in the solar corona. In
+fact, we know that the FIP bias is closely related to
+the magnetic field activity of the Sun (see, e.g. Feld-
+man and Widing 2002; Brooks et al. 2017; Baker et al.
+2018). The Ponderomotive force model (Laming 2004,
+2009, 2012, 2017) is now widely accepted, as it is able to
+reproduce the main characteristics of the FIP effect, as
+measured in-situ and remotely. According to this model,
+the separation of ions from neutral atoms within closed
+loops in an upward direction is caused by the reflec-
+tion of downward propagating Alfv’en waves at chromo-
+spheric heights, causing an enhancement of the low-FIP
+elements in the corona. Since coronal waves can be pro-
+duced by mechanisms that heat the solar corona, it is
+thought that the mechanism underlying the FIP effect
+is inextricably linked to processes that heat the solar
+corona.
+Hence, measuring the FIP bias is an impor-
+tant diagnostic for coronal plasma characteristics (Lam-
+ing 2015; Dahlburg et al. 2016).
+In this paper, we focus on the elemental abundances
+of hot, quiescent AR core emission at 3 MK, by provid-
+ing line-to-continuum measurements of the Sun in the
+soft X-ray energy band using data from the Solar X-ray
+Monitor (XSM: Vadawale et al. 2014; Shanmugam et al.
+2020). It may be noted here that the XSM is the only
+spectrometer to have observed the Sun in the 1-15 keV
+range during the minimum of solar cycle 24 with an en-
+ergy resolution better than 180 eV at 5.9 keV. This reso-
+lution is sufficient to measure the abundances of several
+elements. The soft X-ray continuum is dominated by
+free-free radiation (with some free-bound emission, see
+e.g. Figure 12b of Mondal et al. 2021), which primarily
+originates from H. Hence, measuring the abundances of
+an emission line with respect to the continuum provides
+the absolute abundance of that element. It should be
+noted that the measurement of free-free emission can
+also be carried out in the EUV energy band, but it is
+limited to large flares (e.g., Feldman et al. 2003).
+The XSM energy band is sensitive to temperatures
+above 2 MK. When the Sun was at minimum activity
+levels, without any ARs, the XSM observed a steady sig-
+nal originating from X-ray Bright Points (XBPs), with
+a peak emission around 2 MK (Vadawale et al. 2021b).
+When a single non-flaring AR is present, the signal is
+dominated by the AR’s near-isothermal ∼ 3 MK emis-
+sion (see, e.g. Del Zanna 2013). This provides an ex-
+cellent opportunity to measure the FIP bias of the hot
+quiescent core for individual active regions during their
+evolution.
+In the literature, very few abundance measurements
+are know to be associated specifically with the 3 MK
+emission from quiescent AR cores.
+These are sum-
+marised in Del Zanna and Mason (2018). X-ray spectra
+in the 10–20 ˚A range have provided the relative abun-
+dances of the low-FIP Fe, Mg vs. O, Ne. Most stud-
+ies provided results on single active regions. Saba and
+Strong (1993) reported a significant variability of the
+FIP bias using SMM/FCS observations of several active
+regions. On the other hand, a re-analysis of several qui-
+escent AR cores with improved atomic data and using
+a multi-thermal DEM technique by Del Zanna and Ma-
+son (2014) indicated the same FIP bias, around 3, for
+all active regions, irrespective of their age and size.
+Since 2006, EUV spectra from the Hinode EIS instru-
+ment have provided an opportunity to measure the rel-
+ative FIP bias between low-FIP elements (e.g. Fe, Si)
+and the high-FIP Ar, as well as the mid-FIP S, which
+actually shows the same abundance variations as the
+high-FIP elements. An example case was discussed by
+Del Zanna (2013), showing that the FIP bias in the EUV
+of 3 MK plasma was the same as in the X-rays. Con-
+sidering the size of the emitting plasma and its emission
+measure, Del Zanna (2013) concluded that it should be
+the low-FIP elements that are over-abundant by about
+a factor of 3.
+Del Zanna et al. (2022) carried out a multi-wavelength
+study of an AR as it crossed the solar disk which was ob-
+served by XSM as well as by SDO/AIA, Hinode/EIS and
+Hinode/XRT. The relative FIP bias obtained from Hin-
+ode/EIS observations confirmed the Del Zanna (2013)
+results, and showed no variation with the disk passage.
+The analysis of simultaneous XSM spectra on two days
+also indicated no significant variability, and provided an
+absolute FIP bias for Si of 2.4, i.e. close to the value
+suggested by Del Zanna (2013), and also very close to
+the prediction of Laming’s model.
+In the present study, we extend the previous XSM
+analysis to all the quiescent periods of the same active
+region, and also investigate two other active regions dur-
+ing their disk crossings. One AR in particular is of in-
+terest as it emerged on-disk, and hence offers the op-
+portunity to study the elemental abundances during the
+early phase of the evolution of an AR.
+The rest of the paper is organized as follows: Sec-
+tion 2 provides a short overview of previous abundance
+measurements in active regions. Section 3 describes the
+
+3
+observations and data analysis.
+Section 4 provides a
+detailed spectral analysis. After obtaining the results,
+these are discussed in Section 5. Section 6 provides a
+brief summary of the article.
+2. HISTORICAL OVERVIEW
+Spatially resolved measurements of the relative FIP
+bias have been carried out by several authors (see,e.g.
+Widing and Feldman 1993; Sheeley 1995, 1996; Widing
+1997; Widing and Feldman 2001) using Skylab spectro-
+heliograms with Mg, Ne transition region lines. These
+are formed well below 1 MK, in the legs of active re-
+gion ‘cool’ (1 MK) loops. They found photospheric com-
+position (FIP bias=1) for newly emerged closed loops,
+but increasing FIP bias reaching a value of 3-4 within a
+timescale of 1-2 days (Widing and Feldman 2001), and
+much higher values, up to about 10, within a few more
+days. Differing FIP biases were also obtained by Young
+and Mason (1997) and Dwivedi et al. (1999) using Mg
+and Ne line ratios observed by the CDS and SUMER
+spectrometers onboard the Solar and Heliospheric Ob-
+servatory (SOHO).
+The large values are hard to reconcile with in-situ
+measurements, where the FIP bias is at most 3, and
+also with theory. However, Del Zanna (2003) pointed
+out that as the cool AR loops are almost isothermal in
+their cross-section, the assumption that a smooth emis-
+sion measure distribution was present in the plasma,
+used to interpret the Skylab data, was not justified.
+Del Zanna (2003) took the intensities measured by Wid-
+ing and Feldman (1993), and using an emission measure
+loci approach, showed that a FIP bias of 3.7 was con-
+sistent with the data, much lower than the value of 14
+reported by Widing and Feldman.
+Del Zanna (2003)
+also analysed the legs of several cool loops observed
+by SoHO/CDS and found photospheric abundances, al-
+though a similar analysis for other loops by Del Zanna
+and Mason (2003) found a FIP bias of 4.
+In summary, the legs of cool AR loops do show a range
+of FIP bias values, between 1 and 4, and perhaps occa-
+sionally larger. However, the very high FIP biases found
+from Skylab data were largely overestimated.
+As shown by Del Zanna and Mason (2003), active re-
+gion cores are composed not only of cool 1 MK loops
+and unresolved, almost isothermal 3 MK loops, but also
+unresolved emission in the 1–3 MK range. The plasma
+at different temperatures is generally not cospatial.
+There is evidence from Hinode EIS observations of e.g.
+Si X, S X lines that this ≃2 MK emission has a lower
+relative FIP bias, around 2 (see,e.g. Del Zanna 2012).
+Further studies using the same lines (e.g., Baker et al.
+2013, 2015; Doschek and Warren 2019; Mihailescu et al.
+2022; Ko et al. 2016; Testa et al. 2022) have shown some
+variation (around the value of 2) of the relative FIP bias
+within each active region, but little variability in time,
+except during the decay phase, when an AR effectively
+disappears and the relative abundances become photo-
+spheric.
+In summary, active region structures formed at tem-
+peratures below 2 MK show a range of relative FIP bi-
+ases, and some temporal variability. The few observa-
+tions of the hotter, 3 MK, AR cores have in contrast
+shown a remarkable consistency, with relative FIP bi-
+ases around 3.
+Finally, to interpret observations of the Sun as a star,
+one needs to take into account the above (and other)
+issues. As shown by Del Zanna (2019), when the Sun’s
+actvity is at a minimum with no active region present
+on the solar disk, the corona around 1 MK shows near
+photospheric abundances, whereas in presence of active
+regions, the FIP bias for the 1 MK emission stays the
+same, but the hotter emission shows a higher relative
+FIP bias.
+When active regions flare, the high tem-
+perature plasma shows nearly photospheric composition
+around the peak X-ray emission (see e.g., Mondal et al.
+2021).
+3. OBSERVATIONS AND DATA ANALYSIS
+Observations of the Sun were carried out with the
+XSM during the minimum of solar cycle 24, when no
+active regions were present, covering the years 2019-
+2020. Results are given in Vadawale et al. (2021b). They
+reported intermediate abundances of low-FIP elements
+(Mg, Al, and Si) of 2 MK plasma, primarily originating
+from X-ray Bright Points, XBPs (Mondal et al. 2022).
+Frequent micro-flaring activity was observed and found
+to be occurring everywhere on the solar disk, even when
+no ARs were present (Vadawale et al. 2021a). During
+the minimum of solar cycle 24, XSM observed the disk
+passage of a few individual, isolated ARs in the absence
+of any other major activity. When ARs were present
+on-disk, XSM recorded hundreds of small flares of dif-
+ferent classes. Elemental abundance variations during
+these small flares were found, for the first time, to ini-
+tially drop to photospheric values, then rapidly return
+to coronal values, as described by Mondal et al. (2021),
+Mithun et al. (2022), and Lakshitha et al. (2022). In
+this paper, we analyze the temporal evolution of active
+regions outside of flaring activity and for this we have
+chosen to study three isolated active regions: AR12749,
+AR12758, and AR12759.
+XSM data contain spectra at 1 s cadence in a raw
+(level-1) daily file. Since the visibility of the Sun varies
+within the XSM field-of-view (FOV), with the Sun be-
+
+4
+ing sometimes outside the FOV or being occulted by the
+Moon, the data include both solar and non-solar spectra.
+The XSM Data Analysis Software (XSMDAS: Mithun
+et al. (2021)) has been used to generate the level-2 sci-
+ence data product using the appropriate Good Time
+Intervals (GTIs) and the other necessary instrumental
+parameters. The available default level-2 data contains
+the effective area corrected light curves for every second
+and spectra for every minute. XSMDAS also provides
+the functionality to generate the light curves and spec-
+tra for a given cadence and energy range, which we have
+used in the present analysis.
+Using the XSMDAS, we have generated 2 min av-
+eraged XSM light curves in the energy range of 1-15
+keV during the disk passage of the AR12749, AR12758,
+and AR12759, as shown in the three panels of Figure 1.
+During the evolution of these three ARs, representative
+full disk X-ray images taken by the XRT Be-thin fil-
+ter are shown in the top row of each panel. AR12749
+(Figure 1a) appeared from the east limb on Sept 29,
+2019. Whilst crossing the solar disk, it became fainter
+towards the west limb and went behind the limb on 14
+Oct. AR12758 (Figure 1b) appears to form on disk on
+06 Mar 2020 and fully emerged after 08 Mar. It decays
+whilst crossing the solar disk and finally goes behind the
+west limb on 18 Mar. AR12759 appeared from the east
+limb on 29 Mar 2020 and transited the solar disk until
+14 Apr 2020, before disappearing behind the west limb.
+The full disk XRT images show that during the pas-
+sage of these three ARs, no other major activity was
+present on the solar disk. Thus, we conclude that these
+three ARs were primarily responsible, during their disk
+passage, for the enhanced X-ray emission observed by
+the XSM. These ARs produced many small B/A-class
+flares, seen as multiple spikes in the XSM light curves.
+Detailed studies of these small flares were reported by
+Mondal et al. (2021) and Lakshitha et al. (2022).
+In the present study, we have selected only the quies-
+cent periods from the observed light curves by exclud-
+ing the periods when the small flares occurred using a
+semi-automated graphical algorithm. For example, Fig-
+ure 2 shows the representative selection (orange shaded
+regions) for the AR quiescent durations on 2020-04-06.
+These identified time intervals were used as user-defined
+GTIs to generate the spectra for quiescent ARs on a
+daily basis in order to carry out the detailed spectral
+analysis as discussed in Section 4.
+4. SPECTRAL ANALYSIS
+Broad-band soft X-ray spectra of the solar corona con-
+sist of a continuum as well as the emission lines of the
+different elements. Modeling the soft X-ray spectrum
+provides the measurements of the temperature, emission
+measure, and elemental abundances (with respect to hy-
+drogen) of the emitting plasma (Del Zanna and Mason
+2018). We use the chisoth model (Mondal et al. 2021)
+for the spectral fitting.
+The chisoth is a local model
+of the X-ray spectral fitting package (XSPEC: Arnaud
+et al. (1999)), and it estimates the theoretical spectrum
+using the CHIANTI atomic database. It takes temper-
+ature, emission measure (EM: which is related to the
+density of the plasma), and the elemental abundances
+of the elements from Z=2 to Z=30 as free variables for
+the spectral fitting.
+After generating the spectra for the quiescent peri-
+ods, we fitted them with an isothermal emission model.
+For the spectral fitting, we ignored the spectra below 1.3
+keV where the XSM response is not well-known (Mithun
+et al. 2020), and above the energy where the solar spec-
+trum is dominated by the non-solar background spec-
+trum. During the spectral fitting, the temperature, EM,
+along with the abundances of Mg, Al, and Si (whose
+emission lines are prominent in the XSM spectrum) were
+kept as variable parameters. The 1σ uncertainty of each
+free parameter was also estimated using the standard
+procedure in XSPEC.
+Although the S line complex is visible in the spectra,
+including it in the spectral fits as a free parameter causes
+a large uncertainty in the measurement of the S abun-
+dance because of its poor statistics.
+Hence, we fixed
+the S abundances along with the abundances of other
+elements (whose emission lines are not visible in the ob-
+served spectra) with the coronal abundances of Feldman
+(1992). However, we found that the measurement of the
+S abundance is possible for the summed spectrum of the
+entire AR period.
+Figure 3 shows the representative XSM spectra, for
+the three ARs fitted, in different colours, with an isother-
+mal model.
+The points with error bars represent the
+observed spectra, whereas the solid curves represent the
+best-fit modeled spectra. The grey error bars represent
+the non-solar background spectrum, which is subtracted
+from the observed spectra during the spectral analysis.
+The lower panel shows the residual between the observed
+and model spectra. We have fitted all the spectra in a
+similar way and found that all of them are well described
+by isothermal model.
+The X-rays observed by XSM originated from both
+the AR and the background quiet Sun regions (outside
+the AR). To determine how much emission is due to the
+background quiet Sun regions, we estimate the average
+quiet Sun spectrum using an average quiet-Sun temper-
+ature, EM, and abundances, as reported by Vadawale
+et al. (2021b). The average quiet Sun spectrum is shown
+
+5
+Sep-29
+Oct-01
+Oct-03
+Oct-05
+Oct-07
+Oct-09
+Oct-11
+Date (2019)
+101
+102
+103
+XSM Counts (s
+1)
+AR12749
+a
+Mar-06
+Mar-08
+Mar-10
+Mar-12
+Mar-14
+Mar-16
+Mar-18
+Date (2020)
+101
+102
+103
+XSM Counts (s
+1)
+AR12758
+b
+Mar-26
+Mar-28
+Mar-30
+Apr-01
+Apr-03
+Apr-05
+Apr-07
+Apr-09
+Apr-11
+Apr-13
+Date (2020)
+101
+102
+103
+XSM Counts (s
+1)
+AR12759
+c
+Figure 1. XSM 1-15 keV light curves during the disk passage of AR12749 (panel a), AR12758 (panel b) and AR12759 (panel
+c). The top row of each panel shows representative full disk X-ray images (negative intensities) taken with the XRT Be-thin
+filter during the evolution of the ARs. The vertical dashed lines represent the timing of the XRT images.
+
+6
+05:33
+11:06
+16:40
+22:13
+hh:mm on 2020-04-05
+100
+101
+102
+103
+Rate(c/s)
+c
+05:33
+11:06
+16:40
+22:13
+hh:mm on 2020-03-11
+100
+101
+102
+103
+Rate(c/s)
+b
+05:33
+11:06
+16:40
+22:13
+hh:mm on 2019-10-01
+100
+101
+102
+103
+Rate(c/s)
+a
+Figure 2. Selection of the quiescent AR periods (orange-
+shaded regions) from the XSM light-curves for one represen-
+tative day of AR12749 (panel a), AR12758 (panel b), and
+AR12759 (panel c).
+by the black dashed curve in Figure 3. The quiet Sun
+spectrum is found to be almost an order of magnitude
+lower than the spectra of the active period when the
+ARs were very bright on the solar disk. We thus con-
+clude that the X-ray emission of the active periods is
+primarily dominated by the AR emission.
+Separating the AR emission from the background
+quiet Sun emission would be possible by subtracting the
+quiet-sun spectra from the AR spectra. But, as the ef-
+fective area of the XSM varies with time, this is not
+recommended. It is possible to model the AR spectra
+using a two-temperature (2T) component model rather
+than subtracting the quiet Sun spectra. This is what we
+have chosen to do. One temperature corresponds to the
+background solar emission originating from the regions
+outside the AR and the second temperature corresponds
+to the AR plasma. We have modeled a few AR spec-
+tra with a two-temperature (2T) model. During the 2T
+spectral fitting, the parameters of the background solar
+emission were kept fixed to the average quiet-Sun values
+reported by Vadawale et al. (2021b). For the AR compo-
+nent, the temperature, EM, along with the abundances
+of Mg, Al, and Si, were kept as variable parameters. We
+found that the 2T model can describe the XSM spectra
+for the active periods with similar best-fitted parameters
+as those obtained by the isothermal model. This verifies
+that the AR emission dominates the spectra of the AR
+periods. Thus, in this study, we show the results of the
+isothermal analysis in Figure 5 and 6. This is discussed
+in Section 5.
+It is interesting to study how the plasma parameters
+vary during the emerging phase of the AR12758, i.e.,
+from 07-Mar-2020 to 09-Mar-2020. Figure 4 shows the
+evolution of the photospheric magnetograms (top row)
+and the X-ray emission (bottom row) as observed by
+SDO/HMI and the Be-thin filter of Hinode/XRT re-
+spectively.
+These images were created by de-rotating
+the synoptic data of HMI1 and XRT2 to a common date
+(08-Mar-2020) using the standard procedure of Solar-
+SoftWare (SSW; Freeland and Handy 1998). We also
+determined the total unsigned photospheric magnetic
+flux for the regions ±10 G within the field-of-view shown
+in Figure 4. During this emerging flux period, we car-
+ried out a time-resolved spectroscopic study using the
+XSM observations with finer time bins of less than a
+day. However, during this period, as the emission from
+the AR was not very bright, the emission from the AR
+and the rest of the Sun could be mixed together. Thus
+to derive the evolution of the plasma parameters during
+this period, we modeled the observed XSM spectra with
+a 2T model, where one component represents the emis-
+sion from the AR, and the other represents the emission
+from the rest of the Sun, as discussed in the previous
+paragraph. The results are shown in Figure 7 and dis-
+cussed in Section 5.
+5. RESULTS AND DISCUSSION
+In this study, we have performed the X-ray spectral
+analysis for the evolution of three ARs as observed by
+the XSM. The AR spectra (Figure 3) show a clear sig-
+nature of the thermal X-ray emission from the line com-
+1 http://jsoc.stanford.edu/data/hmi/synoptic/
+2 http://solar.physics.montana.edu/HINODE/XRT/SCIA/
+
+7
+10
+2
+10
+1
+100
+101
+Counts (s
+1keV
+1)
+Mg
+Mg / Al
+Si
+Si
+S
+Quiet Sun
+AR12749 (Oct-01)
+AR12758 (Mar-11)
+AR12759 (Apr-05)
+1.50
+1.75
+2.00
+2.25
+2.50
+2.75
+3.00
+3.25
+Energy (keV)
+5
+0
+5
+Figure 3. Soft X-ray spectra measured by the XSM for three
+representative days of the AR period are shown. Solid lines
+represent the best-fit isothermal model, and the residuals are
+shown in the bottom panel. Gray points correspond to the
+non-solar background spectrum.
+plexes of Mg, Al, Si, and S, along with the continuum
+emission up to ∼3.0 keV. The red points in Figure 5
+show the evolution of the temperature and EM through-
+out the evolution of the three ARs. Figure 6 shows the
+evolution of abundances of Mg (panel a), Al (panel b),
+and Si (panel c). The error bars associated with all the
+parameters along the y-axis represent the 1σ uncertain-
+ties.
+We also derived the average S abundance along
+with the other elements from the summed spectrum for
+the duration when the ARs were very bright on the solar
+disk (bounded by the vertical dashed lines in Figures 5
+and 6).
+This provides the average parameters associ-
+ated with each AR, as shown by magenta bars and also
+given in Table 1. The primary findings of the paper are
+discussed below.
+5.1. Temperature and emission measure
+Temperatures (T) and emission measures (EM) are
+close to the quiet Sun levels (black dashed lines in Fig-
+ure 5) when the ARs were absent from the solar disc
+or only partially present, e.g., 30 September 2019 and
+6 March 2020. Once the ARs appear, the temperature
+rises to more than ∼3 MK from the ∼2 MK of the quiet
+Sun. As the ∼3 MK emission is predominantly derived
+from a smaller volume of AR plasma, the presence of
+the AR reduces the EM from the quiet Sun values. The
+average temperatures for all the ARs are determined to
+be ∼3 MK (blue error bars in Figure 5a), which is close
+to the “basal” temperature of the AR core reported in
+earlier research (e.g., Del Zanna and Mason 2018; Del
+Zanna 2012; Winebarger et al. 2012). The temperature
+and EM do, however, vary slightly over the course of the
+AR’s evolution, which is consistent with the observed X-
+ray light curve. Following the arrival of AR12749 and
+AR12758, their activity decayed while rotating on the
+solar disk (Figure 1), which is why the temperature and
+EM decreased during their evolution, as indicated by the
+dashed vertical lines in Figure 5. After October 6, 2019,
+the EM for AR12749 begins to rise as the AR weakens
+and the quiet Sun emission takes precedence over the
+AR emission. Thus, after the AR has almost died and is
+very faint, the EM and temperature reach values close
+to the quiet Sun temperature and EM. The temperature
+and EM for the AR12759 remain almost constant with
+time, as this AR crossed the solar disk without much
+decay in activity (Figure 1c).
+5.2. Abundance evolution
+In contrast to the temperature and EM, the abun-
+dances of Mg, Al, and Si do not follow the X-ray light
+curve of any of the three ARs throughout their evolu-
+tion (Figure 6). The abundances obtained for low-FIP
+elements Al, Mg, and Si are consistently greater than
+the photospheric values, demonstrating a persistent FIP
+bias during the course of the AR. After the emergence
+of AR12758, the FIP bias is found to be almost constant
+throughout its decay phase. Similarly, during the decay
+of the AR12749, the FIP bias remains nearly constant,
+in contrast to certain earlier studies, such as Ko et al.
+(2016).
+They suggested decreasing FIP bias in high-
+temperature plasma of more than two million degrees
+during the decay phase of an AR. The more established
+AR, AR12759, which evolved without decaying much
+during its transit across the solar disk, also shows an
+almost constant FIP bias, similar to the other two ARs.
+We do not find any relationship between the age of
+the AR and the FIP bias, as suggested in some previous
+papers, e.g.,Del Zanna and Mason 2014; Doschek and
+Warren 2019.
+The measured abundances for Mg, Si,
+and S are comparable to those given by (Feldman 1992)
+and Fludra and Schmelz (1999) (orange shaded regions
+in Figure 6). However, the Al abundance is ∼30%-60%
+higher than the coronal abundances reported in the lit-
+erature. We note that the Al lines in the XSM spectra
+are blended with Mg lines. From Markov Chain Monte
+Carlo (MCMC) analysis (discussed in Appendix A), we
+find that there is no anti-correlation between Mg and
+Al abundances. This suggests that the observed spectra
+does indeed require higher abundances of Al and cannot
+be explained by an enhancement of Mg abundances.
+5.3. FIP bias at the onset of AR core
+Though we do not find any relationship between the
+age of the AR cores and their FIP biases (Section 5.2),
+
+8
+ 5-Mar 17:58
+ 6-Mar 05:58
+ 7-Mar 02:58
+ 7-Mar 12:58
+ 7-Mar 06:58
+ 8-Mar 01:58
+Figure 4. Evolution of the AR12758 during its emergence phase on the solar disk. Top row shows the evolution of photospheric
+magnetograms as observed by HMI and bottom row shows the evolution of X-ray emission as observed by XRT Be-thin filter.
+Figure 5.
+Evolution of the temperature (red points in panel a) and EM (red points in panel b) during the evolution of
+AR12749, AR12758, and AR12759. When the ARs were very bright, as bounded by the vertical dashed lines, the magenta bars
+represent the average values of the temperature and EM. The black horizontal dashed lines represent the average temperature
+and emission measure for the quiet Sun in the absence of any AR reported by Vadawale et al. (2021b). The XSM lightcurves
+of the ARs are shown in grey color, and the lightcurves for the quiescent regions are shown in blue colors.
+which remain constant, it is interesting to study the
+timescale on which the FIP bias developed during the
+emergence of the AR core. Such a study has been made
+possible using the finer (< one day) time-resolved spec-
+troscopy during the emerging phase (07-Mar-2020 to 09-
+Mar-2020) of AR12758.
+During this period, we esti-
+mated the total unsigned photospheric magnetic flux as
+measured by HMI/SDO and shown in Figure 7a (black
+color).
+The peak in the magnetic flux represents the
+time when the AR completely emerged into the solar
+disk.
+After the emergence, the unsigned magnetic flux is
+found to (temporarily) decrease. Figures 7b and 7c show
+the evolution of the AR core temperature and emission-
+measure. With the emergence of the AR. The temper-
+ature becomes close to the AR core temperature of ∼3
+
+AR12749
+AR12758
+AR12759
+4.0
+a
+3.5
+(MK)
+3.0
+2.5
+2.0
+b
+12
+10
+8
+６
+4
+Sep-30 Oct-03
+Oct-06 0ct-09 Mar-06
+Mar-10
+Mar-14 Mar-28 Apr-01 
+Apr-05 Apr-09
+Date (2019)
+Date (2020)
+Date (2020)9
+Figure 6. Panels a-c (red error bars) show the evolution of abundance in the logarithmic scale with A(H)=12 for Mg, Al, and Si
+during the evolution of AR12749, AR12758 and AR12759. The magenta bars represented the average abundances when the ARs
+were very bright, as bounded by the vertical dashed lines. The y-error bars represent 1σ uncertainty for each parameter, and the
+x-error bars represent the duration over which a given spectrum is integrated. The black horizontal dashed lines represent the
+average abundances for the quiet Sun in the absence of any AR reported by Vadawale et al. (2021b). XSM light curves for each
+AR are shown in gray in the background, and the blue color on the XSM light curves represents the time duration excluding the
+flaring activities. The range of coronal and photospheric abundances from various authors compiled in the CHIANTI database
+are shown as orange and green bands. The right y-axis shows the FIP bias values for the respective elements with respect to
+average photospheric abundances.
+MK, and the EM increases as the emitting plasma vol-
+ume increase until it has emerged completely. We also
+derived the evolution of the FIP bias during this period,
+shown in Figure 7d for Si. During this period, as the
+emission from the Mg and Al line complex was weak
+compared with the background solar emission, the de-
+rived FIP bias for Mg and Al has a large uncertainty
+and is not shown here. Within ∼10 hours of the AR
+emergence, the FIP bias was already close to 3, and re-
+mained almost constant throughout the evolution. So
+the emerging hot core loops do not show any variation,
+in agreement with previous suggestions. Recall that the
+variations in FIP bias reported earlier (e.g., Widing and
+Feldman 2001) were observed in the cool loops, not the
+core loops.
+5.4. Enhanced bias for Al
+Figure 8 shows the average values of the FIP bias
+(relative to the photospheric abundance Asplund et al.
+(2009)) for all the elements as a function of their FIP
+values.
+The lower FIP element, Al (FIP = 5.99), is
+found to have the highest FIP bias of 6-7, whereas the
+low-FIP elements, Mg (FIP = 7.65) and Si (FIP = 8.15),
+are found to have a lower FIP bias of ∼3. The mid/high
+FIP element, S, is found to have a much lower FIP bias
+of a factor of ∼ 1.5. A higher FIP bias for Al is note-
+worthy and may point to an intriguing physical process.
+However, this may also be a modeling artifact.
+One of the possibilities could be due to missing flux
+caused by the presence of multi-thermal plasma provid-
+ing strong signals from emission lines of Al or Mg formed
+
+AR12749
+AR12758
+AR12759
+8.2
+a
+3
+8.0
+2
+7.8
+7.6
+b
+7.2
+6
+4
+bias
+6.9
+3
+N
+FIP 
+2
+6.6
+1
+6.3
+c
+4
+8.0
+3
++
+S
+7.8
+7.6
+7.4
+Sep-30 (
+Oct-03
+Oct-06 Oct-09 Mar-06 Mar-10
+Mar-14 Mar-18Mar-28 Apr-01 Apr-05 Apr-09
+Date (2019)
+Date (2020)
+Date (2020)10
+Figure 7. Results showing the emerging phase of AR12758.
+The black curve in panel a shows the evolution of the total
+unsigned photospheric magnetic flux. Panel b and c show the
+evolution of temperature and EM. Panel d shows the evolu-
+tion of FIP bias for Si. The dashed lines in panels b-d repre-
+sent the corresponding parameter for the background solar
+emission from the rest of the solar-disk except AR. The back-
+ground grey curves in each panel represent the X-ray light
+curve observed by XSM. Whereas the blue curves represent
+the selected times excluding the flaring period, representing
+the quiescent AR.
+at different temperatures. To verify this we have simu-
+lated the emission lines in the energy range of the Mg/Al
+line complex by considering the isothermal model and a
+multi-thermal model using the AR DEM of AR12759,
+reported by Del Zanna et al. (2022) (see Figure B.1 in
+Appendix B). Similar line intensities from various ion-
+ization stages of Al and Mg can be seen in both the
+isothermal and multi-thermal models, confirming that
+the absence of the flux is not the result of multi-thermal
+plasma.
+Another possibility is that missing flux is caused by
+missing lines of Al or Mg (mostly satellite lines) that
+are not yet present in CHIANTI version 10. We have
+analysed the high-resolution spectroscopic observations
+described by Walker et al. (1974) and found several ob-
+served lines that are missing in the database. However,
+the total missing flux, compared to the predicted flux
+by CHIANTI is not enough to explain the anomalous
+Al abundance. However, the Walker et al. (1974) obser-
+vations were taken during a high level of solar activity,
+so it is possible that the missing lines have a stronger
+contribution at 3 MK. The Al abundance is currently
+clearly overestimated by some degree.
+Although this analysis is not conclusive enough to rule
+out Al’s high FIP bias as an artifact, it is also not suf-
+ficient to conclude that it is not real. A higher Al FIP
+bias could be real. This might be explained by examin-
+ing a few particular scenarios from the Ponderomotive
+force model (Laming 2015) proposed by Laming (pri-
+vate communication), which could be investigated in a
+subsequent study.
+We have also compared the AR core FIP bias obtained
+with that of the different solar activity levels measured
+by the XSM in previous research. These are overplot-
+ted in Figure 8.
+The blue points show the FIP bias
+during the quiet Sun period, which is dominated by X-
+ray Bright Points (XBP), as reported by Vadawale et al.
+(2021b). While the green points depict the FIP bias dur-
+ing the peak of the solar flares as reported by Mondal
+et al. (2021). The FIP bias of the AR core (red points)
+shows a consistently higher value for the elements Al,
+Mg, and Si compared with the FIP bias of XBPs (green
+points). Since ARs have substantially higher magnetic
+activity than the XBPs, the increased FIP bias of the
+ARs relative to the XBPs is expected from the Pondero-
+motive force model. On the other hand, chromospheric
+evaporation during the flaring mechanism results in a
+near unit FIP bias during the peak of the flares (Mon-
+dal et al. 2021).
+6. SUMMARY
+We present the evolution of plasma characteristics for
+three ARs using disk-integrated soft X-ray spectroscopic
+observations from the XSM to make simultaneous line
+and continuum measurements. Carrying out a compre-
+hensive study of an AR using the Sun-as-a-star mode
+observation is challenging because of the presence of
+multiple activities throughout the solar cycle. Unique
+
+a
+Mx)
+20
+(×1021
+10
+B-flux
+b
+4
+(MK)
+3 
+C
+10
+5
+EM
+0
+d
+6
+4
+bias
+FIP
+2
+-20
+0
+20
+40
+Hours from 07-Mar-202011
+Table 1. Best fitted parameters for the average spectrum of each AR.
+AR
+T
+EM
+Mg
+Al
+Si
+S
+(MK)
+(1046 cm−3)
+12749
+3.14+0.04
+−0.05
+2.46+0.24
+−0.19
+8.00+0.02
+−0.03
+7.28+0.05
+−0.06
+8.00+0.02
+−0.02
+7.23+0.06
+−0.05
+12759
+3.22+0.04
+−0.02
+4.30+0.21
+−0.28
+7.95+0.02
+−0.02
+7.26+0.04
+−0.04
+8.04+0.01
+−0.01
+7.23+0.02
+−0.03
+12758
+2.99+0.05
+−0.03
+3.48+0.25
+−0.31
+7.95+0.03
+−0.02
+7.23+0.06
+−0.05
+8.02+0.02
+−0.02
+7.32+0.05
+−0.06
+6
+7
+8
+9
+10
+11
+FIP (eV)
+2
+0
+2
+4
+6
+8
+FIP bias
+Al
+Mg
+Si
+S
+AR
+XBP
+Flare
+Figure 8. Variation of the FIP bias with the FIP of the ele-
+ments. The red points are the averaged FIP bias for the ARs
+reported in the present study. The blue points are the FIP
+bias for the XBPs as reported by Vadawale et al. (2021b).
+The green points are the measured FIP bias during the peak
+of solar flares as reported by Mondal et al. (2021).
+XSM observations made during the minimum of Solar
+Cycle 24 allowed the study of the evolution of temper-
+ature, EM, and the abundances of Mg, Al, and Si for
+the individual ARs in the absence of any other notewor-
+thy activity on the solar disk. Since the ARs were the
+principal contributors of disk-integrated X-rays during
+their evolution, the temperature and EM followed their
+X-ray light curve. The average temperature of all the
+AR is ∼3 MK, close to the well-known temperature of
+the AR core. Irrespective of the activity and age of the
+ARs, the abundances or the FIP biases of Al, Mg, and Si
+were found to be consistently greater than their photo-
+spheric values without much variation. The abundance
+values develop within ∼10 hours of the appearance of
+the AR during its emerging phase. Throughout the AR
+evolution, the low FIP elements, Mg and Si, have a FIP
+bias close to 3, whereas the mid-FIP element, S, has an
+average FIP bias of ∼1.5. The lowest FIP element, Al,
+has a greater FIP bias of ∼6-7. After discussing vari-
+ous modeling artifacts, the Al abundance appears to be
+overestimated, although the exact factor is unknown.
+Increased Al abundance could be real, implying that
+low-FIP elements degree of FIP bias is linked to their
+FIP values. Future spectroscopic studies to measure the
+FIP bias for more low-FIP elements (for example, Ca,
+whose FIP bias is between Al and Mg) would help us
+to better understand this phenomenon. In this regard,
+recent and upcoming X-ray spectrometers (for example,
+DAXSS: (Schwab et al. 2020) onboard INSPIRESat-1,
+SoLEXS (Sankarasubramanian et al. 2011) onboard up-
+coming Aditya-L1 observatory, and rocket-borne spec-
+trometer MaGIXS (Champey et al. 2022)) will be use-
+ful.
+ACKNOWLEDGMENTS
+We acknowledge the use of data from the Solar X-
+ray Monitor (XSM) on board the Chandrayaan-2 mis-
+sion of the Indian Space Research Organisation (ISRO),
+archived at the Indian Space Science Data Centre
+(ISSDC). The XSM was developed by the engineer-
+ing team of Physical Research Laboratory (PRL) lead
+by Dr.
+M. Shanmugam, with support from various
+ISRO centers.
+We thank various facilities and the
+technical teams from all contributing institutes and
+Chandrayaan-2 project, mission operations, and ground
+segment teams for their support.
+Research at PRL
+is supported by the Department of Space, Govt.
+of
+India.
+We acknowledge the support from Royal So-
+ciety through the international exchanges grant No.
+IES\R2\170199. GDZ and HEM acknowledge support
+from STFC (UK) via the consolidated grant to the
+atomic astrophysics group at DAMTP, University of
+Cambridge (ST\T000481\1). AB was the J C Bose Na-
+tional Fellow during the period of this work. We thank
+Dr. Martin Laming for the useful discussion on anoma-
+lous Al abundance.
+APPENDIX
+
+12
+A. RESULTS OF MCMC ANALYSIS
+We carried out Markov Chain Monte Carlo (MCMC) analysis of the spectra to obtain the regions of parameter space
+that best fits the observed spectra. This was done using the ‘chain’ method available within XSPEC. Figure A1 shows
+the corner plot of the results for the spectrum on 01-Oct-2019. The results show that all parameters are well constrained
+by the spectra. Particularly, we note that there is no anti-correlation observed between Al and Mg abundances showing
+that the enhances Al abundances obtained cannot be adjusted by enhancements in Mg abundances. Similar trends
+are observed for spectra of other days as well.
+7.80
+7.85
+7.90
+7.95
+Mg
+7.0
+7.1
+7.2
+7.3
+Al
+7.92
+7.95
+7.98
+8.01
+Si
+3.30
+3.36
+3.42
+3.48
+3.54
+T
+3.5
+4.0
+4.5
+5.0
+5.5
+EM
+7.80
+7.85
+7.90
+7.95
+Mg
+7.0
+7.1
+7.2
+7.3
+Al
+7.92
+7.95
+7.98
+8.01
+Si
+3.5
+4.0
+4.5
+5.0
+5.5
+EM
+Figure A.1. Corner plot depicting the results of MCMC analysis for the fitted spectrum on 01-Oct-2019. The histograms
+depict the marginalized distribution associated with each parameter. The scatter-plots are overlaid with contours representing
+1σ, 2σ, and 3σ levels to show correlations between all parameters. The best-fit parameters are represented by green lines.
+B. SIMULATED SPECTRUM
+To check the effect of temperatures on the Mg/Al line fluxes in the XSM energy range of 1.55 to 1.70 keV, we
+have compared the simulated spectra in the same energy range by considering the isothermal and multi-thermal DEM
+models. Figure B.1 shows the simulated 3 MK spectrum (blue) overplotted with the multithermal spectrum (red).
+The isothermal spectrum is generated for an emission measure of 1027 cm−5. The multithermal spectrum is derived by
+
+13
+1.56
+1.58
+1.60
+1.62
+1.64
+1.66
+1.68
+1.70
+Energy (keV)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized intensity
+Mg XI, Al XI-XII
+Al XII
+Mg XI
+Mg XI
+DEM
+Isothermal
+Figure B.1. Simulated spectra from CHIANTI v 10 in the energy range of Mg/Al line complex of XSM observed spectrum.
+Solid blue curve show the multi-thermal spectrum and dashed orange curve shows the isothermal spectrum.
+using the reported quiescent AR DEM by Del Zanna et al. (2022), which was obtained from the Hinode EIS observation
+of AR12759. For the comparison of both spectra, we have normalized them with the corresponding line flux of Mg XI,
+and Al XI-XII. Similar line intensities predicted by both isothermal and multithermal models indicates that spectra
+are insensitive to temperature in this case.
+
+14
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+Temperature Measurements. ApJL, 746(2):L17.
+Young, P. R. and Mason, H. E. (1997). The Mg/Ne
+abundance ratio in a recently emerged flux region
+observed by CDS. SoPh, 175(2):523–539.
+

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+arXiv:2301.03520v1  [math.FA]  9 Jan 2023
+CLASSIFYING WEAK PHASE RETRIEVAL
+P. G. CASAZZA AND F. AKRAMI
+Abstract. We will give several surprising equivalences and consequences of
+weak phase retrieval. These results give a complete understanding of the dif-
+ference between weak phase retrieval and phase retrieval. We also answer two
+longstanding open problems on weak phase retrieval: (1) We show that the
+families of weak phase retrievable frames {xi}m
+i=1 in Rn are not dense in the
+family of m-element sets of vectors in Rn for all m ≥ 2n − 2; (2) We show
+that any frame {xi}2n−2
+i=1
+containing one or more canonical basis vectors in Rn
+cannot do weak phase retrieval. We provide numerous examples to show that
+the obtained results are best possible.
+1. Introduction
+The concept of frames in a separable Hilbert space was originally introduced by
+Duffin and Schaeffer in the context of non-harmonic Fourier series [14]. Frames
+are a more flexible tool than bases because of the redundancy property that make
+them more applicable than bases. Phase retrieval is an old problem of recovering
+a signal from the absolute value of linear measurement coefficients called intensity
+measurements. Phase retrieval and norm retrieval have become very active areas of
+research in applied mathematics, computer science, engineering, and more today.
+Phase retrieval has been defined for both vectors and subspaces (projections) in all
+separable Hilbert spaces, (e.g., [3], [4], [5], [6], [9], [10] and [11]).
+The concept of weak phase retrieval weakened the notion of phase retrieval and it
+has been first defined for vectors in ([8] and [7]). The rest of the paper is organized
+as follows: In Section 2, we give the basic definitions and certain preliminary results
+to be used in the paper. Weak phase retrieval by vectors is introduced in section
+3. In section 4 we show that any family of vectors {xi}2n−2
+i=1
+doing weak phase
+retrieval cannot contain a unit vector. In section 5, we show that the weak phase
+retrievable frames are not dense in all frames. And in section 6 we give several
+surprising equivalences and consequences of weak phase retrieval. These results
+give a complete understanding of the difference between weak phase retrieval and
+phase retrieval.
+2. preliminaries
+First we give the background material needed for the paper. Let H be a finite
+or infinite dimensional real Hilbert space and B(H) the class of all bounded linear
+operators defined on H. The natural numbers and real numbers are denoted by
+“N” and “R”, respectively. We use [m] instead of the set {1, 2, 3, . . ., m} and use
+[{xi}i∈I] instead of span{xi}i∈I, where I is a finite or countable subset of N. We
+2010 Mathematics Subject Classification. 42C15, 42C40.
+Key words and phrases. Real Hilbert frames, Full spark, Phase retrieval, Weak phase retrieval.
+The first author was supported by NSF DMS 1609760.
+1
+
+2
+P. G. CASAZZA AND F. AKRAMI
+denote by Rn a n dimensional real Hilbert space. We start with the definition of a
+real Hilbert space frame.
+Definition 1. A family of vectors {xi}i∈I in a finite or infinite dimensional separable
+real Hilbert space H is a frame if there are constants 0 < A ≤ B < ∞ so that
+A∥x∥2 ≤
+�
+i∈I
+|⟨x, xi⟩|2 ≤ B∥x∥2,
+for all
+f ∈ H.
+The constants A and B are called the lower and upper frame bounds for {xi}i∈I,
+respectively. If an upper frame bound exists, then {xi}i∈I is called a B-Bessel
+seqiemce or simply Bessel when the constant is implicit. If A = B, it is called an
+A-tight frame and in case A = B = 1, it is called a Parseval frame. The values
+{⟨x, xi⟩}∞
+i=1 are called the frame coefficients of the vector x ∈ H.
+It is immediate that a frame must span the space. We will need to work with
+Riesz sequences.
+Definition 2. A family X = {xi}i∈I in a finite or infinite dimensional real Hilbert
+space H is a Riesz sequence if there are constants 0 < A ≤ B < ∞ satisfying
+A
+�
+i∈I
+|ci|2 ≤ ∥
+�
+i∈I
+cixi∥2 ≤ B
+�
+i∈I
+|ci|2
+for all sequences of scalars {ci}i∈I. If it is complete in H, we call X a Riesz basis.
+For an introduction to frame theory we recommend [12, 13].
+Throughout the paper the orthogonal projection or simply projection will be a self-
+adjoint positive projection and {ei}∞
+i=1 will be used to denote the canonical basis
+for the real space Rn, i.e., a basis for which
+⟨ei, ej⟩ = δi,j =
+�
+1
+if i = j,
+0
+if i ̸= j.
+Definition 3. A family of vectors {xi}i∈I in a real Hilbert space H does phase
+(norm) retrieval if whenever x, y ∈ H, satisfy
+|⟨x, xi⟩| = |⟨y, xi⟩|
+for all i ∈ I,
+then x = ±y
+(∥x∥ = ∥y∥).
+Phase retrieval was introduced in reference [4]. See reference [1] for an introduc-
+tion to norm retrieval.
+Note that if {xi}i∈I does phase (norm) retrieval, then so does {aixi}i∈I for any
+0 < ai < ∞ for all i ∈ I. But in the case where |I| = ∞, we have to be careful to
+maintain frame bounds. This always works if 0 < infi∈I ai ≤ supi∈Iai < ∞. But
+this is not necessary in general [1]. The complement property is an essential issue
+here.
+Definition 4. A family of vectors {xi}i∈I in a finite or infinite dimensional real
+Hilbert space H has the complement property if for any subset J ⊂ I,
+either span{xi}i∈J = H
+or
+span{xi}i∈Jc = H.
+Fundamental to this area is the following for which the finite dimensional case
+appeared in [10].
+
+WEAK PHASE RETRIEVAL
+3
+Theorem 1. A family of vectors {xi}i∈I does phase retrieval in Rn if and only if it
+has the complement property.
+We recall:
+Definition 5. A family of vectors {xi}m
+i=1 in Rn is full spark if for every I ⊂
+[m] with |I| = n , {xi}i∈I is linearly independent.
+Corollary 1. If {xi}m
+i=1 does phase retrieval in Rn, then m ≥ 2n− 1. If m = 2n− 1,
+{xi}m
+i=1 does phase retrieval if and only if it is full spark.
+We rely heavily on a significant result from [2]:
+Theorem 2. If {xi}2n−2
+i=1
+does weak phase retrieval in Rn then for every I ⊂ [2n−2],
+if x ⊥ span{xi}i∈I and y ⊥ {xi}i∈Ic then
+x
+∥x∥ +
+y
+∥y∥ and
+x
+∥x∥ −
+y
+∥y∥ are disjointly
+supported. In particular, if ∥x∥ = ∥y∥ = 1, then x + y and x − y are disjointly
+supported. Hence, if x = (a1, a2, . . . , an) then y = (ǫ1a1, ǫ2a2, . . . , ǫnan), where
+ǫi = ±1 for i = 1, 2, . . ., n.
+Remark 2.1. The above theorem may fail if ∥x∥ ̸= ∥y∥. For example, consider the
+weak phase retrievable frame in R3:
+
+
+1
+1
+1
+−1
+1
+1
+1
+−1
+1
+1
+1
+−1
+
+
+Also, x = (0, 1, −1) is perpendicular to rows 1 and 2 and y = (0, 1
+2, 1
+2) is orthogonal
+to rows 2 and 3. But x + y = (0, 3
+2, 1
+2) and x − y = (0, −1
+2 , −3
+2 ) and these are not
+disjointly supported. But if we let them have the same norm we get x = (0, 1, −1)
+and y = (0, 1, 1) so x + y = (0, 1, 0) and x − y = (0, 0, 1) and these are disjointly
+supported.
+3. Weak phase retrieval
+The notion of “Weak phase retrieval by vectors” in Rn was introduced in [8] and
+was developed further in [7]. One limitation of current methods used for retrieving
+the phase of a signal is computing power. Recall that a generic family of (2n − 1)-
+vectors in Rn satisfies phaseless reconstruction, however no set of (2n − 2)-vectors
+can (See [7] for details). By generic we are referring to an open dense set in the set
+of (2n − 1)-element frames in Rn.
+Definition 6. Two vectors x = (a1, a2, . . . , an) and y = (b1, b2, . . . , bn) in Rn weakly
+have the same phase if there is a |θ| = 1 so that phase(ai) = θphase(bi) for all
+i ∈ [n], for which ai ̸= 0 ̸= bi.
+If θ = 1, we say x and y weakly have the same signs and if θ = −1, they weakly
+have the opposite signs.
+Therefore with above definition the zero vector in Rn weakly has the same phase
+with all vectors in Rn. For x ∈ R, sgn(x) = 1 if x > 0 and sgn(x) = −1 if x < 0.
+Definition 7. A family of vectors {xi}m
+i=1 does weak phase retrieval in Rn if for
+any x = (a1, a2, . . . , an) and y = (b1, b2, . . . , bn) in Rn with |⟨x, xi⟩| = |⟨y, xi⟩| for
+all i ∈ [m], then x and y weakly have the same phase.
+A fundamental result here is
+
+4
+P. G. CASAZZA AND F. AKRAMI
+Proposition 1. [8] Let x = (a1, a2, . . . , an) and y = (b1, b2, . . . , bn) in Rn.
+The
+following are equivalent:
+(1) We have sgn(aiaj) = sgn(bibj), for all 1 ≤ i ̸= j ≤ n
+(2) Either x, y have weakly the same sign or they have the opposite signs.
+It is clear that if {xi}m
+i=1 does weak phase retrieval in Rn, then {cixi}m
+i=1 does
+weak phase retrieval as long as ci > 0 for all i = 1, 2, . . ., m.
+The following appears in [7].
+Theorem 3. If X = {xi}m
+i=1 does weak phase retrieval in Rn, then m ≥ 2n − 2.
+Finally, we have:
+Theorem 4. [7] If a frame X = {xi}2n−2
+i=1
+does weak phase retrieval in Rn, then X
+is a full spark frame.
+Clearly the converse of above theorem is not hold, for example {(1, 0), (0, 1)} is
+full spark frame that fails weak phase retrieval in R2.
+If {xi}i∈I does phase retrieval and R is an invertible operator on the space
+then {Rxi}i∈I does phase retrieval. This follows easily since |⟨x, Rxi⟩| = |⟨y, Rxi⟩|
+implies |⟨R∗x, xi⟩| = |⟨R∗y, xi⟩|, and so R∗x = θR∗y for |θ| = 1.
+Since R is
+invertible, x = θy. This result fails badly for weak phase retrieval. For example,
+let e1 = (1, 0), e2 = (0, 1), x1 = ( 1
+√
+2,
+1
+√
+2, x2 = ( 1
+√
+2, −1
+√
+2) in R2. Then {e1, e2} fails
+weak phase retrieval, {x1, x2} does weak phase retrieval and Uei = xi is a unitary
+operator.
+4. Frames Containing Unit Vectors
+Theorem 5. Any frame {xi}2n−2
+i=1
+whith one or more canonical basis vectors in Rn
+cannot do weak phase retrieval.
+Proof. We proceed by way of contradiction. Recall that {xi}2n−2
+i=1
+must be full spark.
+Let {ei}n
+i=1 be the canonical orthonormal basis of Rn. Assume I ⊂ {1, 2, . . ., 2n−2}
+with |I| = n − 1 and assume x = (a1, a2, . . . , an), y = (b1, b2, . . . , bn) with ∥x∥ =
+∥y∥ = 1 and x ⊥ X = span{xi}i∈I and y ⊥ span{xi}2n−2
+i=n . After reindexing {ei}n
+i=1
+and {xi}2n−2
+i=1 }, we assume x1 = e1, I = {1, 2, . . ., n−1 and Ic = {n, n+1, . . . , 2n−
+2}. Since ⟨x, x1⟩ = a1 = 0, by Theorem 2, b1 = 0. Let P be the projection on
+span{ei}n
+i=2. So {Pxi}2n−2
+i=n
+is (n − 1)-vectors in an (n − 1)-dimensional space and
+y is orthogonal to all these vectors. So there exist {ci}2n−2
+i=n
+not all zero so that
+2n−2
+�
+i=n
+ciPxi = 0 and so
+2n−1
+�
+i=n
+cixi(1)x1 −
+2n−2
+�
+i=n
+cixi = 0.
+That is, our vectors are not full spark, a contradiction.
+□
+Remark 4.1. The fact that there are (2n− 2) vectors in the theorem is critical. For
+example, e1, e2, e1 + e2 is full spark in R2, so it does phase retrieval - and hence
+weak phase retrieval - despite the fact that it contains both basis vectors.
+The converse of Theorem 5 is not true in general.
+Example 1. Consider the full spark frame X = {(1, 2, 3), (0, 1, 0), (0, −2, 3), (1, −2, −3)}
+in R3. Every set of its two same coordinates,
+{(1, 2), (0, 1), (0, −2), (1, −2)}, {(1, 3), (0, 0), (0, 3), (1, −3)}, and
+
+WEAK PHASE RETRIEVAL
+5
+{(2, 3), (1, 0), (−2, 3), (−2, −3)}
+do weak phase retrieval in R2, but by Theorem 5, X cannot do weak phase retrieval
+in R3.
+5. Weak Phase Retrievable Frames are not Dense in all Frames
+If m ≥ 2n − 1 and {xi}m
+i=1 is full spark then it has complement property and
+hence does phase retrieval. Since the full spark frames are dense in all frames, it
+follows that the frames doing phase retrieval are dense in all frames with ≥ 2n − 1
+vectors. We will now show that this result fails for weak phase retrievable frames.
+The easiest way to get very general frames failing weak phase retrieval is:
+Choose x, y ∈ Rn so that x + y, x − y do not have the same or opposite signs.
+Let X1 = x⊥ and Y1 = y⊥. Then span{X1, X2} = Rn. Choose {xi}n−1
+i=1 vectors
+spanning X1 and {xi}2n−2
+i=n
+be vectors spanning X2. Then {xi}2n−2
+i=1
+is a frame for
+Rn with x ⊥ xi, for i = 1, 2, . . ., n − 1 and y ⊥ xi, for all i = n, n + 1, , . . . , 2n − 2.
+It follows that
+|⟨x + y, xi⟩| = |⟨x − y, xi⟩|, for all i = 1, 2, . . . , n,
+but x, y do not have the same or opposite signs and so {xi}2n−2
+i=1
+fails weak phase
+retrieval.
+Definition 8. If X is a subspace of Rn, we define the sphere of X as
+SX = {x ∈ X : ∥x∥ = 1}.
+Definition 9. If X, Y are subspaces of Rn, we define the distance between X
+and Y as
+d(X, Y ) = supx∈SXinfy∈SY ∥x − y∥.
+It follows that if d(X, Y ) < ǫ then for any x ∈ X there is a z ∈ SY so that
+∥ x
+∥x∥ − z∥ < ǫ. Letting y = ∥x∥z we have that ∥y∥ = ∥x∥ and ∥x − y∥ < ǫ∥x∥.
+Proposition 2. Let X, Y be hyperplanes in Rn and unit vectors x ⊥ X, y ⊥ Y . If
+d(X, Y ) < ǫ then min{∥x − y∥, ∥x + y∥} < 6ǫ.
+Proof. Since span{y, Y } = Rn, x = ay + z for some z ∈ Y . By replacing y by −y
+if necessary, we may assume 0 < a. By assumption, there is some w ∈ X with
+∥w∥ = ∥z∥ so that ∥w − z∥ < ǫ. Now
+a = a∥y∥ = ∥ay∥ = ∥x − z∥ ≥ ∥x − w∥ − ∥w − z∥ ≥ ∥x∥ − ǫ = 1 − ǫ.
+So, 1 − a < ǫ. Also, 1 = ∥x∥2 = a2 + ∥w∥2 implies a < 1. I.e. 0 < 1 − a < ǫ.
+1 = ∥x∥2 = ∥ay + z∥2 = a2∥y∥2 + ∥z∥2 = a2 + ∥z∥2 ≥ (1 − ǫ)2 + ∥z∥2.
+So
+∥z∥2 ≤ 1 − (1 − ǫ)2 = 2ǫ − ǫ2 ≤ 2ǫ.
+Finally,
+∥x − y∥2 = ∥(ay + z) − y∥2
+≤ (∥(1 − a)y∥ + ∥z∥)2
+≤ (1 − a)2∥y∥2 + ∥z∥2 + 2(1 − a)∥y∥∥z∥
+< ǫ2 + 2ǫ + 2
+√
+2ǫ2
+< 6ǫ.
+
+6
+P. G. CASAZZA AND F. AKRAMI
+□
+Lemma 1. Let X, Y be hyperplanes in Rn, {xi}n−1
+i=1 be a unit norm basis for X and
+{yi}n−1
+i=1 be a unit norm basis for Y with basis bounds B. If �n−1
+i=1 ∥xi − yi∥ < ǫ
+then d(X, Y ) < 2ǫB.
+Proof. Let 0 < A ≤ B < ∞ be upper and lower basis bounds for the two bases.
+Given a unit vector x = �n−1
+i=1 aixi ∈ X, let y = �n−1
+i=1 aiyi ∈ Y . We have that
+sup1≤i≤n−1|ai| ≤ B. We compute:
+∥x − y∥ = ∥
+n−1
+�
+i=1
+ai(xi − yi)∥
+≤
+n−1
+�
+i=1
+|ai|∥xi − yi∥
+≤ (sup1≤i≤n−1|ai|)
+n−1
+�
+i=1
+∥xi − yi∥ ≤ Bǫ.
+So
+∥y∥ ≥ ∥x∥ − ∥x − y∥ ≥ 1 − Bǫ.
+����x −
+y
+∥y∥
+���� ≤ ∥x − y∥ +
+����y −
+y
+∥y∥
+����
+≤ Bǫ +
+1
+∥y∥∥(1 − ∥y∥)y∥
+= Bǫ + (1 − ∥y∥)
+≤ 2Bǫ.
+It follows that d(X, Y ) < 2Bǫ.
+□
+Lemma 2. Let {xi}n
+i=1 be a basis for Rn with unconditional basis constant B and
+assume yi ∈ Rn satisfies �n
+i=1 ∥xi − yi∥ < ǫ. Then {yi}n
+i=1 is a basis for Rn which
+is 1 + ǫB-equivalent to {xi}n
+i=1 and has unconditional basis constant B(1 + ǫB)2.
+Proof. Fix {ai}n
+i=1 and compute
+∥
+n
+�
+i=1
+aiyi∥ ≤ ∥
+n
+�
+i=1
+aixi∥ + ∥
+n
+�
+i=1
+|ai|(xi − yi)∥
+≤ ∥
+n
+�
+i=1
+aixi∥ + (sup1≤i≤n|ai|)
+n
+�
+i=1
+∥xi − yi∥
+≤ ∥
+n
+�
+i=1
+aixi∥ + (sup1≤i���n|ai|)ǫ
+≤ ∥
+n
+�
+i=1
+aixi∥ + ǫB∥
+n
+�
+i=1
+aixi∥
+= (1 + ǫB)∥
+n
+�
+i=1
+aixi∥.
+
+WEAK PHASE RETRIEVAL
+7
+Similarly,
+∥
+n
+�
+i=1
+|ai|yi∥ ≥ (1 − ǫB)∥
+n
+�
+i=1
+aixi∥.
+So {xi}n
+i=1 is (1 + ǫB)-equivalent to {yi}n
+i=1.
+For ǫi = ±1,
+∥
+n
+�
+i=1
+ǫiaiyi∥ ≤ (1 + ǫB)∥
+n
+�
+i=1
+ǫiaixi∥
+≤ B(1 + ǫB)∥
+n
+�
+i=1
+aixi∥
+≤ B(1 + ǫB)2∥
+n
+�
+i=1
+aiyi∥.
+and so {yi}n
+i=1 is a B(1 + ǫB) unconditional basis.
+□
+Theorem 6. The family of m-element weak phase retrieval frames are not dense in
+the set of m-element frames in Rn for all m ≥ 2n − 2.
+Proof. We may assume m = 2n−2 since for larger m we just repeat the (2n-2) vec-
+tors over and over until we get m vectors. Let {ei}n
+i=1 be the canonical orthonormal
+basis for Rn and let xi = ei for i = 1, 2, . . . , n. By [10], there is an orthonormal
+sequence {xi}2n−2
+i=n+1 so that {xi}2n−2
+i=1
+is full spark. Let I = {1, 2, . . ., n − 1}. Let
+X = span{xi}n−1
+i=1 and Y = span{xi}2n−2
+i=n .
+Then x = en ⊥ X and there is a
+∥y∥ = 1 with y ⊥ Y .
+Note that ⟨x − y, en⟩ ̸= 0 ̸= ⟨x + y, en⟩, for otherwise,
+x = ±y ⊥ span{xi}i̸=n, contradicting the fact that the vectors are full spark. So
+there is a j = n and a δ > 0 so that |(x + y)(j)|, |(x − y)(j)| ≥ δ. We will show
+that there exists an 0 < ǫ so that whenever {yi}2n−2
+i=1
+are vectors in Rn satisfying
+�n
+i=1 ∥xi − yi∥ < ǫ, then {yi}n
+i=1 fails weak phase retrieval.
+Fix 0 < ǫ. Assume {yi}2n−2
+i=1
+are vectors so that �2n−2
+i=1
+∥xi−yi∥ < ǫ. Choose unit
+vectors x′ ⊥ span{yi}i∈I, y′ ⊥ span{yi}i∈Ic. By Proposition 2 and Lemma 1, we
+may choose ǫ small enough (and change signs if necessary) so that ∥x−x′∥, ∥y−y′∥ <
+δ
+4B . Hence, since the unconditional basis constant is B,
+|[(x + y) − (x′ + y′)](j)|
+≤ |(x − x′)j| + |(y − y′)(j)|
+< B∥x − x′∥ + B∥y − y′∥
+≤ 2B δ
+4B = δ
+2.
+It follows that
+|(x′ + y′)(j)| ≥ |(x + y)(j)| − |[(x + y) − (x′ + y′)](j)| ≥ δ − 1
+2δ = δ
+2.
+Similarly, |(x′ − y′)(j)| > δ
+2. So x′ + y′, x′ − y′ are not disjointly supported and so
+{yi}2n−2
+i=1
+fails weak phase retrieval by Theorem 2.
+□
+6. Classifying Weak Phase Retrieval
+In this section we will give several surprising equivalences and consequences of
+weak phase retrieval. These results give a complete understanding of the difference
+between weak phase retrieval and phase retrieval.
+Now we give a surprising and very strong classification of weak phase retrieval.
+
+8
+P. G. CASAZZA AND F. AKRAMI
+Theorem 7. Let {xi}2n−2
+i=1
+be non-zero vectors in Rn. The following are equivalent:
+(1) The family {xi}2n−2
+i=1
+does weak phase retrieval in Rn.
+(2) If x, y ∈ Rn and
+(6.1)
+|⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, . . . , 2n − 2,
+then one of the following holds:
+(a) x = ±y.
+(b) x and y are disjointly supported.
+Proof. (1) ⇒ (2): Given the assumption in the theorem, assume (a) fails and we will
+show that (b) holds. Let x = (a1, a2, . . . , an), y = (b1, b2, . . . , bn). Since {xi}2n−2
+i=1
+does weak phase retrieval, replacing y by −y if necessary, Equation 6.1 implies
+aj = bj whenever aj ̸= 0 ̸= bj.
+Let
+I = {1 ≤ i ≤ 2n − 2 : ⟨x, xi⟩ = ⟨y, yi⟩.
+Then
+x + y ⊥ xi for all i ∈ Ic and x − y ⊥ xi for all i ∈ I.
+By Theorem 2,
+x + y
+∥x + y +
+x − y
+∥x − y∥ and
+x + y
+∥x + y∥ −
+x − y
+∥x − y∥ are disjointly supported.
+Assume there is a 1 ≤ j ≤ n with aj = bj ̸= 0. Then
+(x + y)(j)
+∥x + y∥
++ (x − y)(j)
+∥x − y∥
+=
+2aj
+∥x + y∥ and (x + y)(j)
+∥x + y∥
+− (x − y)(j)
+∥x − y∥
+=
+2aj
+∥x + y∥,
+Contradicting Theorem 2.
+(2) ⇒ (1): This is immediate since (a) and (b) give the conditions for weak phase
+retrieval.
+□
+Phase retrieval is when (a) in the theorem holds for every x, y ∈ Rn. So this the-
+orem shows clearly the difference between weak phase retrieval and phase retrieval:
+namely when (b) holds at least once.
+Corollary 2. If {xi}2n−2
+i=1
+does weak phase retrieval in Rn, then there are disjointly
+supported non-zero vectors x, y ∈ Rn satisfying:
+|⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, . . . , 2n − 2.
+Proof. Since {xi}2n−2
+i=1
+must fail phase retrieval, (b) of Theorem 7 must hold at least
+once.
+□
+Definition 10. Let {ei}n
+i=1 be the canonical orthonormal basis of Rn. If J ⊂ [n],
+we define PJ as the projection onto span{ei}i∈J.
+Theorem 8. Let {xi}m
+i=1 be unit vectors in Rn. The following are equivalent:
+(1) Whenever I ⊂ [2n − 2] and 0 ̸= x ⊥ xi for i ∈ I and 0 ̸= y ⊥ xi for i ∈ Ic,
+there is no j ∈ [n] so that ⟨x, ej⟩ = 0 = ⟨y, ej⟩.
+(2) For every J ⊂ [n] with |J| = n − 1, {Pjxi}2n−2
+i=1
+does phase retrieval.
+(3) For every J ⊂ [n] with |J| < n, {PJxi}2n−2
+i=1
+does phase retrieval.
+
+WEAK PHASE RETRIEVAL
+9
+Proof. (1) ⇒ (2): We prove the contrapositive. So assume (2) fails. Then choose
+J ⊂ [n] with |J| = n − 1, J = [n] \ {j}, and {PJxi}2n−2
+i=1
+fails phase retrieval. In
+particular, it fails complement property. That is, there exists I ⊂ [2n− 2] and span
+{PJxi}i∈I ̸= PJRn and span {Pjxi}i∈Ic ̸= PJRn. So there exists norm one vectors
+x, y in PJRn with PJx = x ⊥ PJxi for all i ∈ I and PJy = y ⊥ PJxi for all i ∈ Ic.
+Extend x, y to all of Rn by setting x(j) = y(j) = 0. Hence, x ⊥ xi for i ∈ I and
+y ⊥ xi for i ∈ Ic, proving (1) fails.
+(2) ⇒ (3): This follows from the fact that every projection of a set of vectors
+doing phase retrieval onto a subset of the basis also does phase retrieval.
+(3) ⇒ (2): This is obvious.
+(3) ⇒ (1): We prove the contrapositive. So assume (1) fails. Then there is a
+I ⊂ [2n− 2] and 0 ̸= x ⊥ xi for i ∈ I and 0 ̸= y ⊥ xi for i ∈ Ic and a j ∈ [n] so that
+⟨x, ej⟩ = ⟨y, ej⟩ = 0. It follows that x = PJx, y = PJy are non zero and x ⊥ Pjxi
+for all i ∈ I and y ⊥ Pjxi for i ∈ Ic, so {PJxi}2n−2
+i=1
+fails phase retrieval.
+□
+Remark 6.1. The assumptions in the theorem are necessary. That is, in general,
+{xi}m
+i=1 can do weak phase retrieval and {PJxi}m
+i=1 may fail phase retrieval. For
+example, in R3 consider the row vectors {xi}4
+i=1 of:
+
+
+1
+1
+1
+−1
+1
+1
+1
+−1
+1
+1
+1
+−1
+
+
+This set does weak phase retrieval, but if J = {2, 3} then x = (0, 1, −1) ⊥ PJxi for
+i = 1, 2 and y = (0, 1, 1) ⊥ xi for i = 3, 4 and {PJxi}4
+i=1 fails phase retrieval.
+Corollary 3. Assume {xi}2n−2
+i=1
+does weak phase retrieval in Rn and for every J ⊂ [n]
+{PJxi}2n−2
+i=1
+does phase retrieval. Then if x, y ∈ Rn and
+|⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, . . . , 2n − 2,
+then there is a J ⊂ [n] so that
+x(j) =
+�
+aj ̸= 0 for j ∈ J
+0 for j ∈ Jc
+y(j) =
+�
+0 for j ∈ J
+bj ̸= 0 for j ∈ Jc
+Proposition 3. Let {ei}n
+i=1 be the unit vector basis of Rn and for I ⊂ [n], let PI be
+the projection onto XI = span{ei}i∈I. For every m ≥ 1, there are vectors {xi}m
+i=1
+so that for every I ⊂ [1, n], {PIxi}m
+i=1 is full spark in XI.
+Proof. We do this by induction on m. For m=1, let x1 = (1, 1, 1, . . ., 1). This
+satisfies the theorem. So assume the theorem holds for {xi}m
+i=1. Choose I ⊂ [1, n]
+with |I| = k. Choose J ⊂ I with |J| = k − 1 and let XJ = span{xi}i∈J ∪ {xi}i∈Ic.
+Then XJ is a hyperplane in Rn for every J. Since there only exist finitely many
+such J′s there is a vector xm+1 /∈ XJ for every J. We will show that {xi}m+1
+i=1
+satisfies the theorem.
+Let I ⊂ [1, n] and J ⊂ I with |J| = |I|. If PIxm+1 /∈ XJ, then {PIxi}i∈J is
+linearly independent by the induction hypothesis. On the other hand, if m + 1 ∈ J
+then xm+1 /∈ XJ. But, if PIxm+1 ∈ span{PIxi}i∈J\m+1, since (I − PI)xm+1 ∈
+span{ei}i∈Ic, it follows that xm+1 ∈ XJ, which is a contradiction.
+□
+
+10
+P. G. CASAZZA AND F. AKRAMI
+Remark 6.2. In the above proposition, none of the xi can have a zero coordinate.
+Since if it does, projecting the vectors onto that coordinate produces a zero vector
+and so is not full spark.
+References
+[1] F. Akrami, P. G. Casazza, M. A. Hasankhani Fard, A. Rahimi, A note on norm retrievable
+real Hilbert space frames, J. Math. Anal. Appl. 2021. (517)2, (2023) 126620.
+[2] P. G. Casazza, F. Akrami, A. Rahimi, fundamental results on weak phase retrieval, Ann.
+Funct. Anal, arXiv: 2110.06868, 2021.
+[3] S. Bahmanpour, J. Cahill, P.G. Casazza, J. Jasper, and L. M. Woodland, Phase retrieval and
+norm retrieval, arXiv:1409.8266, (2014).
+[4] R. Balan, P. G. Casazza, D. Edidin, On signal reconstruction without phase, Appl. Comput.
+Harmonic Anal. 20, 3, (2006), 345-356.
+[5] S. Botelho-Andrade, Peter G. Casazza, D. Cheng, J. Haas, and Tin T. Tran, Phase retrieval
+in ℓ2(R), arXiv:1804.01139v1, (2018).
+[6] S. Botelho-Andrade, Peter G. Casazza, D. Cheng, J. Haas, and Tin T. Tran, J. C. Tremain,
+and Z. Xu, Phase retrieval by hyperplanes, Am. Math. Soc, comtemp. math. 706, (2018),
+21-31.
+[7] S. Botelho-Andrade, P. G. Casazza, D. Ghoreishi, S. Jose, J. C. Tremain, Weak phase retrieval
+and phaseless reconstruction, arXiv:1612.08018, (2016).
+[8] S. Botelho-Andrade, P. G. Casazza, H. V. Nguyen, And J. C. Tremain, Phase retrieval versus
+phaseless reconstruction, J. Math. Anal. Appl, 436, 1, (2016), 131-137.
+[9] J. Cahill, P.G. Casazza, and I. Daubechies, Phase retrieval in infinite dimensional Hilbert
+spaces, Transactions of the AMS, Series B, 3, (2016), 63-76.
+[10] J. Cahill, P.G. Casazza, J. Peterson and L. Woodland, Phase retrivial by projections, Houston
+Journal of Mathematics 42. 2, (2016), 537-558.
+[11] P. G. Casazza, D. Ghoreishi, S. Jose, J. C. Tremain, Norm retrieval and phase Retrieval by
+projections, Axioms, 6, (2017), 1-15.
+[12] P. G. Casazza and G. Kutyniok, Finite Frames, Theory and applications, Birkhauser, (2013).
+[13] O. Christensen, An introduction to frames and Riesz bases, Birkhauser, Boston (2003).
+[14] R. J. Duffin, A. C. Schaeffer. A class of nonharmonic Fourier series, Trans. Am. Math. Soc,
+72, (1952), 341-366.
+Department of Mathematics, University of Missouri, Columbia, USA.
+Email address: [email protected]
+Department of Mathematics, University of Maragheh, Maragheh, Iran.
+Email address: [email protected]
+

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+Audio-Visual Efficient Conformer for Robust Speech Recognition
+Maxime Burchi, Radu Timofte
+Computer Vision Lab, CAIDAS, IFI, University of W¨urzburg, Germany
+{maxime.burchi,radu.timofte}@uni-wuerzburg.de
+Abstract
+End-to-end Automatic Speech Recognition (ASR) sys-
+tems based on neural networks have seen large improve-
+ments in recent years. The availability of large scale hand-
+labeled datasets and sufficient computing resources made it
+possible to train powerful deep neural networks, reaching
+very low Word Error Rate (WER) on academic benchmarks.
+However, despite impressive performance on clean audio
+samples, a drop of performance is often observed on noisy
+speech. In this work, we propose to improve the noise ro-
+bustness of the recently proposed Efficient Conformer Con-
+nectionist Temporal Classification (CTC)-based architec-
+ture by processing both audio and visual modalities. We im-
+prove previous lip reading methods using an Efficient Con-
+former back-end on top of a ResNet-18 visual front-end and
+by adding intermediate CTC losses between blocks. We con-
+dition intermediate block features on early predictions us-
+ing Inter CTC residual modules to relax the conditional in-
+dependence assumption of CTC-based models. We also re-
+place the Efficient Conformer grouped attention by a more
+efficient and simpler attention mechanism that we call patch
+attention. We experiment with publicly available Lip Read-
+ing Sentences 2 (LRS2) and Lip Reading Sentences 3 (LRS3)
+datasets. Our experiments show that using audio and visual
+modalities allows to better recognize speech in the presence
+of environmental noise and significantly accelerate training,
+reaching lower WER with 4 times less training steps. Our
+Audio-Visual Efficient Conformer (AVEC) model achieves
+state-of-the-art performance, reaching WER of 2.3% and
+1.8% on LRS2 and LRS3 test sets. Code and pretrained
+models are available at https://github.com/burchim/AVEC.
+1. Introduction
+End-to-end Automatic Speech Recognition based on
+deep neural networks has become the standard of state-of-
+the-art approaches in recent years [25, 47, 18, 16, 17, 31, 7].
+The availability of large scale hand-labeled datasets and suf-
+ficient computing resources made it possible to train power-
+40 ms rate
+Visual Conformer
+Stage 2
+20 ms rate
+Visual Conformer
+Stage 1
+Visual Front-end 
+Conv3d + ResNet-18 
+Audio Front-end 
+STFT + Conv2d 
+Audio Conformer
+Stage 1
+Audio Conformer
+Stage 2
+Audio Conformer
+Stage 3
+Audio-Visual 
+Fusion Module
+Audio-Visual
+Conformer Stage
+Visual 
+Back-end
+Audio 
+Back-end
+80 ms rate
+CTC loss
+40 ms rate
+80 ms rate
+80 ms rate
+Figure 1: Audio-Visual Efficient Conformer architec-
+ture. The model is trained end-to-end using CTC loss and
+takes raw audio waveforms and lip movements from the
+speaker as inputs.
+ful deep neural networks for ASR, reaching very low WER
+on academic benchmarks like LibriSpeech [34]. Neural ar-
+chitectures like Recurrent Neural Networks (RNN) [15, 19],
+Convolution Neural Networks (CNN) [10, 28] and Trans-
+formers [12, 23] have successfully been trained from raw
+audio waveforms and mel-spectrograms audio features to
+transcribe speech to text.
+Recently, Gulati et al. [16]
+proposed a convolution-augmented transformer architec-
+ture (Conformer) to model both local and global dependen-
+cies using convolution and attention to reach better speech
+recognition performance. Concurrently, Nozaki et al. [33]
+arXiv:2301.01456v1  [cs.CV]  4 Jan 2023
+
+++improved CTC-based speech recognition by conditioning
+intermediate encoder block features on early predictions us-
+ing intermediate CTC losses [14]. Burchi et al. [7] also pro-
+posed an Efficient Conformer architecture using grouped
+attention for speech recognition, lowering the amount of
+computation while achieving better performance. Inspired
+from computer vision backbones, the Efficient Conformer
+encoder is composed of multiple stages where each stage
+comprises a number of Conformer blocks to progressively
+downsample and project the audio sequence to wider fea-
+ture dimensions.
+Yet, even if these audio-only approaches are breaking
+the state-of-the-art, one major pitfall for using them in the
+real-world is the rapid deterioration of performance in the
+presence of ambient noise. In parallel to that, Audio Visual
+Speech Recognition (AVSR) has recently attracted a lot of
+research attention due to its ability to use image process-
+ing techniques to aid speech recognition systems. Preced-
+ing works have shown that including the visual modality of
+lip movements could improve the robustness of ASR sys-
+tems with respect to noise while reaching better recognition
+performance [41, 42, 36, 1, 45, 29]. Xu et al. [45] pro-
+posed a two-stage approach to first separate the target voice
+from background noise using the speakers lip movements
+and then transcribe the filtered audio signal with the help of
+lip movements. Petridis et al. [36] uses a hybrid architec-
+ture, training an LSTM-based sequence-to-sequence (S2S)
+model with an auxiliary CTC loss using an early fusion
+strategy to reach better performance. Ma et al. [29] uses
+Conformer back-end networks with ResNet-18 [20] front-
+end networks to improve recognition performance.
+Other works focus on Visual Speech Recognition (VSR),
+only using lip movements to transcribe spoken language
+into text [4, 9, 48, 3, 49, 37, 30]. An important line of
+research is the use of cross-modal distillation. Afouras et
+al. [3] and Zhao et al. [49] proposed to improve the lip read-
+ing performance by distilling from an ASR model trained
+on a large-scale audio-only corpus while Ma et al. [30]
+uses prediction-based auxiliary tasks. Prajwal et al. [37]
+also proposed to use sub-words units instead of characters
+to transcribe sequences, greatly reducing running time and
+memory requirements. Also providing a language prior, re-
+ducing the language modelling burden of the model.
+In this work we focus on the design of a noise robust
+speech recognition architecture processing both audio and
+visual modalities.
+We use the recently proposed CTC-
+based Efficient Conformer architecture [7] and show that
+including the visual modality of lip movements can suc-
+cessfully improve noise robustness while significantly ac-
+celerating training. Our Audio-Visual Efficient Conformer
+(AVEC) reaches lower WER using 4 times less training
+steps than its audio-only counterpart.
+Moreover, we are
+the first work to apply intermediate CTC losses between
+blocks [27, 33] to improve visual speech recognition perfor-
+mance. We show that conditioning intermediate features on
+early predictions using Inter CTC residual modules allows
+to close the gap in WER between autoregressive and non-
+autoregressive AVSR systems based on S2S. This also helps
+to counter a common failure case which is that audio-visual
+models tend to ignore the visual modality. In this way, we
+force pre-fusion layers to learn spatiotemporal features. Fi-
+nally, we replace the Efficient Conformer grouped attention
+by a more efficient and simpler attention mechanism that
+we call patch attention. Patch attention reaches similar per-
+formance to grouped attention while having a lower com-
+plexity. The contributions of this work are as follows:
+• We improve the noise robustness of the recently pro-
+posed Efficient Conformer architecture by processing
+both audio and visual modalities.
+• We condition intermediate Conformer block features
+on early predictions using Inter CTC residual modules
+to relax the conditional independence assumption of
+CTC models. This allows us to close the gap in WER
+between autoregressive and non-autoregressive meth-
+ods based on S2S.
+• We propose to replace the Efficient Conformer
+grouped attention by a more efficient and simpler at-
+tention mechanism that we call patch attention. Patch
+attention reaches similar performance to grouped at-
+tention with a lower complexity.
+• We experiment on publicly available LRS2 and LRS3
+datasets and reach state-of-the-art results using audio
+and visual modalities.
+2. Method
+In this section, we describe our proposed Audio-Visual
+Efficient Conformer network. The model is composed of
+4 main components: An audio encoder, a visual encoder,
+an audio-visual fusion module and an audio-visual encoder.
+The audio and visual encoders are separated into modality
+specific front-end networks to transform each input modal-
+ity into temporal sequences and Efficient Conformer back-
+end networks to model local and global temporal relation-
+ships. The full model is trained end-to-end using intermedi-
+ate CTC losses between Conformer blocks in addition to the
+output CTC layer. The complete architecture of the model
+is shown in Figure 1.
+2.1. Model Architecture
+Audio front-end.
+The audio front-end network first
+transforms raw audio wave-forms into mel-spectrograms
+using a short-time Fourier transform computed over win-
+dows of 20ms with a step size of 10ms. 80-dimensional
+
+mel-scale log filter banks are applied to the resulting fre-
+quency features. The mel-spectrograms are processed by
+a 2D convolution stem to extract local temporal-frequency
+features, resulting in a 20ms frame rate signal. The audio
+front-end architecture is shown in Table 1.
+Table 1: Audio Front-end architecture, 1.2 Millions param-
+eters. Ta denotes the input audio sample length.
+Stage
+Layers
+Output Shape
+Fourier
+Transf
+STFT: 400 window length
+160 hop length, 512 ffts
+(257, Ta//160 + 1)
+Mel
+Scale
+Mel Scale: 80 mels
+(80, Ta//160 + 1)
+Stem
+Conv2d: 32, 180 filters, 22 stride
+(180, 40, Ta//320 + 1)
+Proj
+Linear, 180 units
+(Ta//320 + 1, 180)
+Visual front-end.
+The visual front-end network [29]
+transforms input video frames into temporal sequences. A
+3D convolution stem with kernel size 5 × 7 × 7 is first ap-
+plied to the video. Each video frame is then processed inde-
+pendently using a 2D ResNet-18 [20] with an output spatial
+average pooling. Temporal features are then projected to
+the back-end network input dimension using a linear layer.
+The visual front-end architecture is shown in Table 2.
+Table 2: Visual Front-end architecture, 11.3 Millions pa-
+rameters. Tv denotes the number of input video frames.
+Stage
+Layers
+Output Shape
+Stem
+Conv3d: 5 × 72, 64 filters, 1 × 22 stride
+MaxPoo3d: 1 × 32, 1 × 22 stride
+(64, Tv, 22, 22)
+Res 1
+2 ×
+�
+Conv2d: 32, 64 filters
+Conv2d: 32, 64 filters
+�
+(Tv, 64, 22, 22)
+Res 2
+2 ×
+�
+Conv2d: 32, 128 filters
+Conv2d: 32, 128 filters
+�
+(Tv, 128, 11, 11)
+Res 3
+2 ×
+�
+Conv2d: 32, 256 filters
+Conv2d: 32, 256 filters
+�
+(Tv, 256, 6, 6)
+Res 4
+2 ×
+�
+Conv2d: 32, 512 filters
+Conv2d: 32, 512 filters
+�
+(Tv, 512, 3, 3)
+Pool
+Global Average Pooling
+(Tv, 512)
+Proj
+Linear, 256 units
+(Tv, 256)
+Back-end networks. The back-end networks use an Ef-
+ficient Conformer architecture.
+The Efficient Conformer
+encoder was proposed in [7], it is composed of several
+stages where each stage comprises a number of Conformer
+blocks [16] using grouped attention with relative positional
+encodings. The temporal sequence is progressively down-
+sampled using strided convolutions and projected to wider
+feature dimensions, lowering the amount of computation
+while achieving better performance. We use 3 stages in the
+audio back-end network to downsample the audio signal to
+a 80 milliseconds frame rate. Only 2 stages are necessary
+to downsample the visual signal to the same frame rate. Ta-
+ble 6 shows the hyper-parameter of each back-end network.
+Table 3: Back-end networks hyper-parameters. InterCTC
+blocks indicates Conformer blocks having a post Inter CTC
+residual module.
+Network
+Visual
+Back-end
+Audio
+Back-end
+Audio-Visual
+Encoder
+Num Params (M)
+13.6
+17.9
+15.9
+Num Stages
+2
+3
+1
+Blocks per Stage
+6, 1
+5, 6, 1
+5
+Total Num Blocks
+7
+12
+5
+Stage Feature Dim
+256, 360
+180, 256, 360
+360
+Conv Kernel Size
+15
+15
+15
+Stage Patch Size
+1, 1
+3, 1, 1
+1
+InterCTC Blocks
+3, 6
+8, 11
+2
+Audio-visual fusion module. Similar to [36, 29], we
+use an early fusion strategy to learn audio-visual features
+and reduce model complexity. The acoustic and visual fea-
+tures from the back-end networks are concatenated and fed
+into a joint feed-forward network. The concatenated fea-
+tures of size 2 × dmodel are first expanded using a linear
+layer with output size dff = 4 × dmodel, passed through
+a Swish activation function [38] and projected back to the
+original feature dimension dmodel.
+Audio-visual encoder. The audio-visual encoder is a
+single stage back-end network composed of 5 Conformer
+blocks without downsampling.
+The encoder outputs are
+then projected to a CTC layer to maximize the sum of prob-
+abilities of correct target alignments.
+2.2. Patch Attention.
+The Efficient Conformer [7] proposed to replace Multi-
+Head Self-Attention (MHSA) [44] in earlier encoder lay-
+ers with grouped attention. Grouped MHSA reduce atten-
+tion complexity by grouping neighbouring temporal ele-
+ments along the feature dimension before applying scaled
+dot-product attention. Attention having a quadratic com-
+putational complexity with respect to the sequence length,
+this caused the network to have an asymmetric complexity
+with earlier attention layers requiring more flops than latter
+layers with shorter sequence length. In this work, we pro-
+pose to replace grouped attention with a simpler and more
+efficient attention mechanism that we call patch attention
+(Figure 2). Similar to the pooling attention proposed by the
+Multiscale Vision Transformer (MViT) [13] for video and
+image recognition, the patch attention proceed to an average
+Table 4: Attention variants complexities including query,
+key, value and output linear projections. n and d are the
+sequence length and feature dimension respectively.
+Attention
+Variant
+Hyper
+Parameter
+Full Attention
+Complexity
+Regular
+-
+O(n · d2 + n2 · d)
+Grouped
+Group Size (g)
+O(n · d2 + (n/g)2 · d · g)
+Patch
+Patch Size (k)
+O(n/k · d2 + (n/k)2 · d)
+
+AvgPool 
+1
+2
+3
+4
+5
+6
+7
+8
+9
+a
+a
+a
+b
+b
+b
+c
+c
+c
+Upsample 
+a
+c
+Attention
+b
+a
+c
+b
+Figure 2: Patch Multi-Head Self-Attention. The input sequence is downsampled using an average pooling before applying
+multi-head self-attention. The output sequence is then upsampled via nearest neighbor upsampling, reducing attention com-
+plexity from O(n2 · d) to O((n/k)2 · d) where k defines the pooling / upsampling kernel size. Patch attention is equivalent
+to regular attention when k = 1.
+pooling on the input sequence before projection the query,
+key and values.
+X = AvgPooling1d(Xin)
+(1)
+with Q, K, V = XW Q, XW K, XW V
+(2)
+Where W Q, W K, W V ∈ Rd×d are query, key and value
+linear projections parameter matrices. MHSA with relative
+sinusoidal positional encoding is then performed at lower
+resolution as:
+MHSA(X) = Concat (O1, ..., OH) W O
+(3)
+with Oh = softmax
+�QhKT
+h + Srel
+h
+√dh
+�
+Vh
+(4)
+Where Srel ∈ Rn×n is a relative position score matrix that
+satisfy Srel[i, j] = QiET
+j−i. E is the linear projection of
+a standard sinusoidal positional encoding matrix with posi-
+tions ranging from −(nmax − 1) to (nmax − 1). The atten-
+tion output sequence is then projected and up-sampled back
+to the initial resolution using nearest neighbor up-sampling.
+Xout = UpsampleNearest1d(MHSA(X))
+(5)
+In consequence, each temporal element of the same patch
+produce the same attention output. Local temporal relation-
+ships are only modeled in the convolution modules while
+global relationships are modeled by patch attention. We
+use 1-dimensional patches in this work but patch attention
+Audio back-end Conformer Stage
+Module Giga FLOPs
+0
+0.1
+0.2
+0.3
+Stage 1 (d=180, n=500)
+Stage 2 (d=256, n=250)
+Stage 3 (d=360, n=125)
+attention
+grouped attention (g=3)
+patch attention (k=3)
+feed-forward
+Figure 3: Audio-only back-end modules FLOPs (Billion).
+could also be generalized to image and video data using
+2D and 3D patches. We leave this to future works. The
+computational complexity of each attention variant is shown
+in Table 4. Path attention further reduce complexity com-
+pared to grouped attention by decreasing the amount of
+computation needed by Query, Key, Value and Output fully
+connected layers while keeping the feature dimension un-
+changed. Similar to previous work [7], we only use patch
+attention in the first audio back-end stage to reduce com-
+plexity while maintaining model recognition performance.
+Figure 3 shows the amount of FLOPs for each attention
+module variant with respect to encoded sequence length n
+and model feature dimension d. Using patch or grouped at-
+tention variants instead of regular MHSA greatly reduce the
+amount of FLOPs in the first audio back-end stage.
+2.3. Intermediate CTC Predictions.
+Inspired by [27] and [33] who proposed to add interme-
+diate CTC losses between encoder blocks to improve CTC-
+based speech recognition performance, we add Inter CTC
+residual modules (Figure 4) in encoder networks. We con-
+dition intermediate block features of both audio, visual and
+audio-visual encoders on early predictions to relax the con-
+ditional independence assumption of CTC models. During
+both training and inference, each intermediate prediction is
+summed to the input of the next layer to help recognition.
+We use the same method proposed in [33] except that we do
+not share layer parameters between losses. The lth block
+output Xout
+l
+is passed through a feed-forward network with
+residual connection and a softmax activation function:
+Zl = Softmax(Linear(Xout
+l
+))
+(6)
+Xin
+l+1 = Xout
+l
++ Linear(Zl)
+(7)
+Where Zl ∈ RT ×V is a probability distribution over the
+output vocabulary. The intermediate CTC loss is then com-
+puted using the target sequence y as:
+Linter
+l
+= −log(P(y|Zl))
+(8)
+with P(y|Zl) =
+�
+π∈B−1
+CT C(y)
+T
+�
+t=1
+Zt,πt
+(9)
+
+Conformer Block
+Inter CTC 
+Residual Module 
+Conformer Block
+Linear
+Softmax
+Linear
+CTC loss
++  
+Figure 4: Inter CTC residual module. Intermediate pre-
+dictions are summed to the input of the next Conformer
+block to condition the prediction of the final block on it.
+Intermediate CTC losses are added to the output CTC loss
+for the computation of the final loss.
+Where π ∈ V T are paths of tokens and BCT C is a many-to-
+one map that simply removes all blanks and repeated labels
+from the paths. The total training objective is defined as
+follows:
+L = (1 − λ)LCT C + λLinter
+(10)
+with Linter = 1
+K
+�
+k∈interblocks
+Linter
+k
+(11)
+Where interblocks is the set of blocks having a post Inter
+CTC residual module (Figure 4). Similar to [33], we use
+Inter CTC residual modules every 3 Conformer blocks with
+λ set to 0.5 in every experiments.
+3. Experiments
+3.1. Datasets
+We use 3 publicly available AVSR datasets in this
+work. The Lip Reading in the Wild (LRW) [8] dataset is
+used for visual pre-training and the Lip Reading Sentences
+2 (LRS2) [1] and Lip Reading Sentences 3 (LRS3) [2]
+datasets are used for training and evaluation.
+LRW dataset. LRW is an audio-visual word recogni-
+tion dataset consisting of short video segments containing a
+single word out of a vocabulary of 500. The dataset com-
+prise 488,766 training samples with at least 800 utterances
+per class and a validation and test sets of 25,000 samples
+containing 50 utterances per class.
+LRS2 & LRS3 datasets. The LRS2 dataset is composed
+of 224.1 hours with 144,482 videos clips from the BBC tele-
+vision whereas the LRS3 dataset consists of 438.9 hours
+with 151,819 video clips extracted from TED and TEDx
+talks. Both datasets include corresponding subtitles with
+word alignment boundaries and are composed of a pre-train
+split, train-val split and test split. LRS2 has 96,318 utter-
+ances for pre-training (195 hours), 45,839 for training (28
+hours), 1,082 for validation (0.6 hours), and 1,243 for test-
+ing (0.5 hours). Whereas LRS3 has 118,516 utterances in
+the pre-training set (408 hours), 31,982 utterances in the
+training-validation set (30 hours) and 1,321 utterances in
+the test set (0.9 hours). All videos contain a single speaker,
+have a 224 × 224 pixels resolution and are sampled at 25
+fps with 16kHz audio.
+3.2. Implementation Details
+Pre-processing Similar to [29], we remove differences
+related to rotation and scale by cropping the lip regions us-
+ing bounding boxes of 96 × 96 pixels to facilitate recog-
+nition. The RetinaFace [11] face detector and Face Align-
+ment Network (FAN) [6] are used to detect 68 facial land-
+marks. The cropped images are then converted to gray-scale
+and normalised between −1 and 1. Facial landmarks of the
+LRW, LRS2 and LRS3 datasets are obtained from previous
+work [30] and reused for pre-processing to get a clean com-
+parison of the methods. A byte-pair encoding tokenizer is
+built from LRS2&3 pre-train and trainval splits transcripts
+using sentencepiece [26]. We use a vocabulary size of 256
+including the CTC blank token following preceding works
+on CTC-based speech recognition [31, 7].
+Data augmentation Spec-Augment [35] is applied on
+the audio mel-spectrograms during training to prevent over-
+fitting with two frequency masks with mask size parameter
+F = 27 and five time masks with adaptive size pS = 0.05.
+Similarly to [30], we mask videos on the time axis using one
+mask per second with the maximum mask duration set to 0.4
+seconds. Random cropping with size 88×88 and horizontal
+flipping are also performed for each video during training.
+We also follow Prajwal et al. [37] using central crop with
+horizontal flipping at test time for visual-only experiments.
+Training Setup We first pre-train the visual encoder on
+the LRW dataset [8] using cross-entropy loss to recognize
+words being spoken. The visual encoder is pre-trained for
+30 epochs and front-end weights are then used as initializa-
+tion for training. Audio and visual encoders are trained on
+the LRS2&3 datasets using a Noam schedule [44] with 10k
+warmup steps and a peak learning rate of 1e-3. We use the
+Adam optimizer [24] with β1 = 0.9, β2 = 0.98. L2 regular-
+ization with a 1e-6 weight is also added to all the trainable
+weights of the model. We train all models with a global
+batch size of 256 on 4 GPUs, using a batch size of 16 per
+GPU with 4 accumulated steps. Nvidia A100 40GB GPUs
+are used for visual-only and audio-visual experiments while
+RTX 2080 Ti are used for audio-only experiments. The
+audio-only models are trained for 200 epochs while visual-
+only and audio-visual models are trained for 100 and 70
+epochs respectively. Note that we only keep videos shorter
+than 400 frames (16 seconds) during training. Finally, we
+average models weights over the last 10 epoch checkpoints
+using Stochastic Weight Averaging [22] before evaluation.
+
+Table 5: Comparison of WER (%) on LRS2 / LRS3 test sets with recently published methods using publicly and non-publicly
+available datasets for Audio-Only (AO), Visual-Only (VO) and Audio-Visual (AV) models.
+Method
+Model
+Criterion
+Training
+Datasets
+Total
+Hours
+test WER
+AO
+VO
+AV
+(↓) Using Publicly Available Datasets (↓)
+Petridis et al. [36]
+CTC+S2S
+LRW, LRS2
+381
+8.3 / -
+63.5 / -
+7.0 / -
+Zhang et al. [48]
+S2S
+LRW, LRS2&3
+788 / 790
+-
+51.7 / 60.1
+-
+Afouras et al. [3]
+CTC
+VoxCeleb2clean, LRS2&3
+1,032 / 808
+-
+51.3 / 59.8
+-
+Xu et al. [45]
+S2S
+LRW, LRS3
+595
+- / 7.2
+- / 57.8
+- / 6.8
+Yu et al.[46]
+LF-MMI
+LRS2
+224
+6.7 / -
+48.9 / -
+5.9 / -
+Ma et al. [29]
+CTC+S2S
+LRW, LRS2&3
+381 / 595
+3.9 / 2.3
+37.9 / 43.3
+3.7 / 2.3
+Prajwal et al. [37]
+S2S
+LRS2&3
+698
+-
+28.9 / 40.6
+-
+Ma et al. [30]
+CTC+S2S
+LRW, LRS2&3
+818
+-
+27.3 / 34.7
+-
+Ours
+CTC
+LRW, LRS2&3
+818
+2.8 / 2.1
+32.6 / 39.2
+2.5 / 1.9
++ Neural LM
+CTC
+LRW, LRS2&3
+818
+2.4 / 2.0
+29.8 / 37.5
+2.3 / 1.8
+(↓) Using Non-Publicly Available Datasets (↓)
+Afouras et al. [1]
+S2S
+MVLRS, LRS2&3
+1,395
+9.7 / 8.3
+48.3 / 58.9
+8.5 / 7.2
+Zhao et al. [49]
+S2S
+MVLRS, LRS2
+954
+-
+65.3 / -
+-
+Shillingford et al. [40]
+CTC
+LRVSR
+3,886
+-
+- / 55.1
+-
+Makino et al. [32]
+Transducer
+YouTube-31k
+31,000
+- / 4.8
+- / 33.6
+- / 4.5
+Serdyuk et al. [39]
+Transducer
+YouTube-90k
+91,000
+-
+- / 25.9
+- / 2.3
+Prajwal et al. [37]
+S2S
+MVLRS, TEDxext, LRS2&3
+2,676
+-
+22.6 / 30.7
+-
+Ma et al. [30]
+CTC+S2S
+LRW, AVSpeech, LRS2&3
+1,459
+-
+25.5 / 31.5
+-
+Language Models. Similarly to [28], we experiment
+with a N-gram [21] statistical language model (LM) and a
+Transformer neural language model. A 6-gram LM is used
+to generate a list of hypotheses using beam search and an
+external Transformer LM is used to rescore the final list.
+The 6-gram LM is trained on the LRS2&3 pre-train and
+train-val transcriptions. Concerning the neural LM, we pre-
+train a 12 layer GPT-3 Small [5] on the LibriSpeech LM
+corpus for 0.5M steps using a batch size of 0.1M tokens
+and finetune it for 10 epochs on the LRS2&3 transcriptions.
+3.3. Results
+Table 5 compares WERs of our Audio-Visual Effi-
+cient Conformer with state-of-the-art methods on the LRS2
+and LRS3 test sets.
+Our Audio-Visual Efficient Con-
+former achieves state-of-the-art performances with WER of
+2.3%/1.8%. On the visual-only track, our CTC model com-
+petes with most recent autoregressive methods using S2S
+criterion. We were able to recover similar results but still
+lack behind Ma et al. [30] which uses auxiliary losses with
+pre-trained audio-only and visual-only networks. We found
+our audio-visual network to converge faster than audio-only
+experiments, reaching better performance using 4 times less
+training steps. The intermediate CTC losses of the visual
+encoder could reach lower levels than in visual-only experi-
+ments showing that optimizing audio-visual layers can help
+pre-fusion layers to learn better representations.
+3.4. Ablation Studies
+We propose a detailed ablation study to better understand
+the improvements in terms of complexity and WER brought
+by each architectural modification. We report the number
+of operations measured in FLOPs (number of multiply-and-
+add operations) for the network to process a ten second au-
+dio/video clip. Inverse Real Time Factor (Inv RTF) is also
+measured on the LRS3 test set by decoding with a batch
+size 1 on a single Intel Core i7-12700 CPU thread. All abla-
+tions were performed by training audio-only models for 200
+epochs and visual-only / audio-visual models for 50 epochs.
+Efficient Conformer Visual Back-end. We improve the
+recently proposed visual Conformer encoder [29] using an
+Efficient Conformer back-end network. The use of byte pair
+encodings for tokenization instead of characters allows us to
+further downsample temporal sequences without impacting
+the computation of CTC loss. Table 6 shows that using an
+Efficient Conformer back-end network for our visual-only
+model leads to better performances while reducing model
+complexity and training time. The number of model param-
+eters is also slightly decreased.
+Table 6: Ablation study on visual back-end network.
+Visual
+Back-end
+#Params
+(Million)
+LRS2
+test
+LRS3
+test
+#FLOPs
+(Billion)
+Inv
+RTF
+Conformer
+43.0
+39.53
+47.14
+87.94
+5.17
+Eff Conf
+40.4
+37.39
+44.96
+84.52
+5.26
+
+Reference
+the authors looked at papers written over a 10 year period and hundreds had to be thrown out
+Outputs
+Block   3: the otho looing pa people we over s any your per and conndries that aboutent threghow
+Block   6: the autthherss looking paperss we overai year paiod and hundreds that about thrououtow
+Block   9: the authors looked at papers witen over ainght year period and hundreds that to been throw out
+Block 12: the authors looked at papers written over 10 year period and hundreds had to be thrown out
+Figure 5: Output example of our Visual-only model using greedy search decoding on the LRS3 test set with intermediate
+CTC prediction every 3 blocks. The sentence is almost correctly transcribed except for the missing ’a’ before ’10 year’.
+Inter CTC residual modules. Similar to [33], we exper-
+iment adding Inter CTC residual modules between blocks
+to relax the conditional independence assumption of CTC.
+Table 7 shows that using intermediate CTC losses every 3
+Conformer blocks greatly helps to reduce WER, except for
+the audio-only setting where this does not improve perfor-
+mance. Figure 5 gives an example of intermediate block
+predictions decoded using greedy search without an exter-
+nal language model on the test set of LRS3. We can see
+that the output is being refined in the encoder layers by con-
+ditioning on the intermediate predictions of previous lay-
+ers. Since our model refines the output over the frame-level
+predictions, it can correct insertion and deletion errors in
+addition to substitution errors. We further study the im-
+pact of Inter CTC on multi-modal learning by measuring
+the performance of our audio-visual model when one of
+the two modalities is masked. As pointed out by preced-
+ing works [8, 1, 32], networks with multi-modal inputs can
+often be dominated by one of the modes. In our case speech
+recognition is a significantly easier problem than lip reading
+which can cause the model to ignore visual information. Ta-
+ble 8 shows that Inter CTC can help to counter this problem
+by forcing pre-fusion layers to transcribe the input signal.
+Table 7: Ablation study on Inter CTC residual modules.
+Model
+Back-end
+#Params
+(Million)
+LRS2
+test
+LRS3
+test
+#FLOPs
+(Billion)
+Inv
+RTF
+Audio-only (↓)
+Eff Conf
+31.5
+2.83
+2.13
+7.54
+51.98
++ Inter CTC
+32.1
+2.84
+2.11
+7.67
+50.30
+Visual-only (↓)
+Eff Conf
+40.4
+37.39
+44.96
+84.52
+5.26
++ Inter CTC
+40.9
+33.82
+40.63
+84.60
+5.26
+Audio-visual (↓)
+Eff Conf
+60.9
+2.87
+2.54
+90.53
+4.84
++ Inter CTC
+61.7
+2.58
+1.99
+90.66
+4.82
+Table 8: Impact of Inter CTC on audio-visual model WER
+(%) for LRS2 / LRS3 test sets in a masked modality setting.
+Inter CTC
+Audio-Visual Eval Mode
+masked video
+masked audio
+no mask
+No
+4.48 / 3.22
+52.77 / 59.10
+2.87 / 2.54
+Yes
+3.39 / 2.38
+37.62 / 46.55
+2.58 / 1.99
+Patch multi-head self-attention.
+We experiment re-
+placing grouped attention by patch attention in the first
+audio encoder stage. Our objective being to increase the
+model efficiency and simplicity without harming perfor-
+mance. Grouped attention was proposed in [7] to reduce
+attention complexity for long sequences in the first encoder
+stage. Table 9 shows the impact of each attention variant
+on our audio-only model performance and complexity. We
+start with an Efficient Conformer (M) [7] and replace the
+attention mechanism. We find that grouped attention can be
+replaced by patch attention without a loss of performance
+using a patch size of 3 in the first back-end stage.
+Table 9: Ablation study on audio back-end attention.
+Attention
+Type
+Group /
+Patch Size
+LRS2
+test
+LRS3
+test
+#FLOPs
+(Billion)
+Inv
+RTF
+Regular
+-
+2.85
+2.12
+8.66
+49.86
+Grouped
+3, 1, 1
+2.82
+2.13
+8.06
+50.27
+Patch
+3, 1, 1
+2.83
+2.13
+7.54
+51.98
+3.5. Noise Robustness
+We measure model noise robustness using various types
+of noise and compare our Audio-Visual Efficient Conformer
+with recently published methods. Figure 6 shows the WER
+evolution of audio-only (AO), visual-only (VO) and audio-
+visual (AV) models with respect to multiple Signal to Noise
+Ratio (SNR) using white noise and babble noise from the
+NoiseX corpus [43]. We find that processing both audio and
+visual modalities can help to significantly improve speech
+recognition robustness with respect to babble noise. More-
+over, we also experiment adding babble noise during train-
+ing as done in previous works [36, 29] and find that it can
+further improve noise robustness at test time.
+Robustness to various types of noise. We gather var-
+ious types of recorded audio noise including sounds and
+music. In Table 10, we observe that the Audio-Visual Ef-
+ficient Conformer consistently achieves better performance
+than its audio-only counterpart in the presence of various
+noise types. This confirm our hypothesis that the audio-
+visual model is able to use the visual modality to aid speech
+recognition when audio noise is present in the input.
+
+SNR (dB)
+Word Error Rate (%)
+0
+10
+20
+30
+40
+50
+-5
+0
+5
+10
+15
+20
+VO LRS2
+AO LRS2
+AV LRS2
+AV* LRS2
+VO LRS3
+AO LRS3
+AV LRS3
+AV* LRS3
+(a) Babble noise
+SNR (dB)
+Word Error Rate (%)
+0
+10
+20
+30
+40
+50
+-5
+0
+5
+10
+15
+20
+VO LRS2
+AO LRS2
+AV LRS2
+AV* LRS2
+VO LRS3
+AO LRS3
+AV LRS3
+AV* LRS3
+(b) White noise
+Figure 6: LRS2 and LRS3 test WER (%) as a function
+of SNR (dB). * indicates experiments being trained with
+babble noise. We measure noise robustness by evaluating
+our models in the presence of babble and white noise.
+Table 10: LRS3 test WER (%) as a function of SNR (dB).
+Noise
+Mode
+SNR (dB)
+-5
+0
+5
+10
+15
+20
+babble
+AO
+75.9
+32.4
+9.3
+4.1
+2.7
+2.3
+AV
+33.5
+14.8
+5.4
+3.0
+2.3
+2.0
+AV*
+11.2
+4.9
+3.1
+2.5
+2.2
+2.0
+white
+AO
+77.6
+34.0
+15.5
+7.3
+4.1
+2.8
+AV
+28.9
+14.7
+5.5
+3.0
+2.3
+2.0
+AV*
+17.4
+8.9
+3.6
+2.8
+2.3
+2.0
+birds
+AO
+51.8
+23.9
+10.9
+5.9
+3.7
+2.8
+AV
+21.6
+11.5
+6.2
+4.1
+2.9
+2.4
+AV*
+15.9
+8.3
+4.9
+3.4
+2.7
+2.4
+chainsaw
+AO
+82.9
+41.2
+14.8
+5.5
+3.7
+2.7
+AV
+37.8
+17.3
+7.6
+3.9
+2.6
+2.3
+AV*
+25.8
+10.8
+5.0
+3.2
+2.4
+2.3
+jazz
+AO
+25.3
+9.7
+4.1
+3.1
+2.6
+2.3
+AV
+13.9
+6.0
+3.2
+2.4
+2.3
+2.0
+AV*
+10.6
+4.2
+2.8
+2.4
+2.2
+2.0
+street
+raining
+AO
+58.4
+23.8
+8.9
+4.6
+3.0
+2.5
+AV
+27.12
+10.8
+5.7
+3.1
+2.7
+2.3
+AV*
+15.9
+6.9
+3.8
+2.7
+2.3
+2.2
+washing
+dishes
+AO
+47.8
+24.5
+11.5
+6.0
+3.7
+2.8
+AV
+21.3
+11.5
+6.1
+3.6
+2.8
+2.3
+AV*
+14.2
+7.3
+4.3
+2.2
+2.6
+2.3
+train
+AO
+51.3
+18.6
+7.0
+4.0
+2.9
+2.5
+AV
+23.1
+10.1
+4.7
+3.0
+2.4
+2.2
+AV*
+14.5
+6.2
+3.5
+2.6
+2.3
+2.2
+Comparison with other methods.
+We compare our
+method with results provided by Ma et al. [29] and
+Petridis et al. [36] on the LRS2 test set. Table 11 shows that
+our audio-visual model achieves lower WER in the pres-
+ence of babble noise, reaching WER of 9.7% at -5 dB SNR
+against 16.3% for Ma et al. [29].
+Table 11: Comparison with Ma et al. [29]. LRS2 test WER
+(%) as a function of SNR (dB) using babble noise.
+Method
+Mode
+SNR (dB)
+-5
+0
+5
+10
+15
+20
+Ma et al. [29]
+VO
+37.9
+37.9
+37.9
+37.9
+37.9
+37.9
+AO*
+28.8
+9.8
+7
+5.2
+4.5
+4.2
+AV*
+16.3
+7.5
+6.1
+4.7
+4.4
+4.2
+Ours
+VO
+32.6
+32.6
+32.6
+32.6
+32.6
+32.6
+AO
+70.5
+27
+8.6
+4.7
+3.4
+3.1
+AV
+25
+11.2
+5.1
+3.2
+2.8
+2.6
+AV*
+9.7
+5
+3.4
+2.9
+2.8
+2.6
+Table 12: Comparison with Petridis et al. [36]. LRS2 test
+WER (%) as a function of SNR (dB) using white noise.
+Method
+Mode
+SNR (dB)
+-5
+0
+5
+10
+15
+20
+Petridis et al. [36]
+VO
+63.5
+63.5
+63.5
+63.5
+63.5
+63.5
+AO*
+85.0
+45.4
+19.6
+11.7
+9.4
+8.4
+AV*
+55.0
+26.1
+13.2
+9.4
+8.0
+7.3
+Ours
+VO
+32.6
+32.6
+32.6
+32.6
+32.6
+32.6
+AO
+73.1
+32.3
+14.3
+7.2
+4.4
+3.5
+AV
+22.5
+11.5
+6.2
+4.1
+3.2
+2.9
+AV*
+14.4
+8.0
+5.1
+3.9
+3.1
+2.9
+4. Conclusion
+In this paper, we proposed to improve the noise robust-
+ness of the recently proposed Efficient Conformer CTC-
+based architecture by processing both audio and visual
+modalities. We showed that incorporating multi-scale CTC
+losses between blocks could help to improve recognition
+performance, reaching comparable results to most recent
+autoregressive lip reading methods. We also proposed patch
+attention, a simpler and more efficient attention mechanism
+to replace grouped attention in the first audio encoder stage.
+Our Audio-Visual Efficient Conformer achieves state-of-
+the-art performance of 2.3% and 1.8% on the LRS2 and
+LRS3 test sets.
+In the future, we would like to explore
+other techniques to further improve the noise robustness
+of our model and close the gap between recent lip reading
+methods. This includes adding various audio noises during
+training and using cross-modal distillation with pre-trained
+models. We also wish to reduce the visual front-end net-
+work complexity without arming recognition performance
+and experiment with the RNN-Transducer learning objec-
+tive for streaming applications.
+Acknowledgments
+This work was partly supported by The Alexander von
+Humboldt Foundation (AvH).
+
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+

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+  
+Abstract— Objective: The objective of this study is to develop a 
+deep-learning based detection and diagnosis technique for carotid 
+atherosclerosis using a portable freehand 3D ultrasound (US) 
+imaging system. Methods: A total of 127 3D carotid artery datasets 
+were acquired using a portable 3D US imaging system. A U-Net 
+segmentation network was firstly applied to extract the carotid 
+artery on 2D transverse frame, then a novel 3D reconstruction 
+algorithm using fast dot projection (FDP) method with position 
+regularization was proposed to reconstruct the carotid artery 
+volume.  Furthermore, a convolutional neural network was used 
+to classify the healthy case and diseased case qualitatively. 3D 
+volume analysis including longitudinal reprojection algorithm and 
+stenosis grade measurement algorithm was developed to obtain the 
+clinical metrics quantitatively. Results: The proposed system 
+achieved sensitivity of 0.714, specificity of 0.851 and accuracy of 
+0.803 respectively in diagnosis of carotid atherosclerosis. The 
+automatically 
+measured 
+stenosis 
+grade 
+illustrated 
+good 
+correlation (r=0.762) with the experienced expert measurement.   
+Conclusion:   the developed technique based on 3D US imaging can 
+be applied to the automatic diagnosis of carotid atherosclerosis. 
+Significance: The proposed deep-learning based technique was 
+specially designed for a portable 3D freehand US system, which 
+can 
+provide 
+carotid 
+atherosclerosis 
+examination 
+more 
+conveniently and decrease the dependence on clinician’s 
+experience.  
+Index Terms—3D ultrasound imaging, automatic carotid 
+atherosclerosis 
+diagnosis, 
+carotid 
+artery 
+segmentation, 
+reconstruction with regularization. 
+I. INTRODUCTION 
+AROTID atherosclerosis is one of the major causes of 
+stroke which is the world’s second leading cause of death 
+[1]. The prevalence rate of carotid atherosclerosis is 36.2% in 
+Chinese people over 40 years old [2]. The pathological features 
+of carotid atherosclerosis are increase of intima-media 
+thickness and appearance of atherosclerosis plaque. Magnetic 
+resonance imaging (MRI), computed tomography angiography 
+ 
+This work was sponsored by Natural Science Foundation of China (NSFC) 
+under Grant No.12074258. (Jiawen Li and Yunqian Huang are co-first authors.) 
+(Corresponding authors: Rui Zheng, Man Chen.) 
+Jiawen Li, Sheng Song, Duo Xu and Haibin Zhang are with School of 
+Information Science and Technology, ShanghaiTech University, Shanghai, 
+China. 
+Hongbo Chen is with School of Information Science and Technology, 
+ShanghaiTech University, Shanghai 201210, China, also with Shanghai 
+Advanced Research Institute, Chinese Academy of Sciences, Shanghai 200050, 
+(CTA) and digital subtraction angiography (DSA) are several 
+commonly used methods for visualizing and characterizing 
+carotid artery features [3]–[5]. However, these methods still 
+have some limitations during application due to invasiveness, 
+ionizing radiation, heavy equipment etc.; and the approaches 
+are very time-consuming and expensive which can’t satisfy the 
+need of large scale of examinations in different environments 
+especially for community and countryside areas. 2D Ultrasound 
+(US), as a non-invasive and low-cost method, is widely used in 
+the examination of carotid plaque. However, there are several 
+disadvantages of traditional 2D US in the current ultrasound 
+examination of carotid atherosclerosis. (1) It is mainly carried 
+out by experienced sonographers in hospital, and becomes a 
+huge burden for health care system. (2) Routine health check is 
+difficult for carotid atherosclerosis patients especially in rural 
+or undeveloped area. (3) Routine ultrasound examination is a 
+tedious, 
+laborious, 
+experience-dependent 
+work 
+for 
+sonographers. (4) Clinically, some metrics such as intima-
+media thickness (IMT), plaque thickness, plaque area, usually 
+assess the severity of the carotid atherosclerosis in 2D US 
+images, which is prone to variability and lack of 3D 
+morphology of carotid plaque [6], [7]. 3D US carotid artery 
+imaging approaches mainly include mechanical scanning and 
+tracked freehand scanning using various sensors e.g., magnetic 
+tracked senor, optical tracked sensor, etc., [8] which can 
+provide plaque volume estimation, 3D morphology of plaque 
+and other 3D metrics for carotid atherosclerosis diagnosis.  The 
+3D techniques are found to be more accurate to evaluate the 
+progress of carotid atherosclerosis [9]–[12]. Therefore, it is of 
+great importance to develop a portable, reliable and cost-
+effective automatic ultrasound diagnostic technique for carotid 
+atherosclerosis screening.  
+ The automatic diagnosis of carotid atherosclerosis focuses on 
+finding the biomarkers on the ultrasound images, for example 
+China, and also with University of Chinese Academy of Sciences, Beijing 
+100049, China. 
+Yunqian Huang and Junni Shi are with Tongren Hospital, Shanghai Jiao 
+Tong University School of Medicine, Shanghai, China. 
+Man Chen is with Tongren Hospital, Shanghai Jiao Tong University School 
+of Medicine, Shanghai, China (e-mail: [email protected]) 
+Dr. Rui Zheng is with School of Information Science and Technology, 
+Shanghai Engineering Research Center of Energy Efficient and Custom AI IC, 
+ShanghaiTech University, Shanghai, China (phone: 86 21-2068 4452, e-mail: 
+[email protected]) 
+Automatic Diagnosis of Carotid Atherosclerosis 
+Using a Portable Freehand 3D Ultrasound 
+Imaging System 
+Jiawen Li, Yunqian Huang, Sheng Song, Hongbo Chen, Junni Shi, Duo Xu, Haibin Zhang, Man 
+Chen*, Rui Zheng* 
+C 
+
+ 
+2 
+vessel wall area, vessel wall volume or total plaque volume 
+[13]–[15]. These biomarkers are all bounded by the two 
+boundaries of vessels, the media-adventitia boundary (MAB) 
+and the lumen-intima boundary (LIB), thus identifying these 
+two boundaries is an important issue during the carotid 
+atherosclerosis diagnosis. In recent years, deep learning 
+methods has achieved excellent performance in medical image 
+processing. Jiang et al. [16]–[18]designed a novel adaptive 
+triple loss for carotid artery segmentation. To utilize 3D 
+information in 3D volume of carotid artery, Jiang et al. [19] 
+introduced a fusion module to the U-Net segmentation network 
+and yielded promising performance on carotid segmentation 
+task.  Zhou et al.[20] proposed a deep learning-based MAB and 
+LIB segmentation method, and a dynamic convolutional neural 
+network (CNN) were applied to image patches in every slice of 
+the 3D US images. LIB segmentation was performed by U-Net 
+based on the masks of the MAB since the LIB is inside the 
+MAB. The method achieved high accuracy but initial anchor 
+points were still manually placed. Ruijter et al. [21] created a 
+generalized method to segment LIB using CNN. Several U-
+Nets were compared and the experiments showed that the 
+combination of various vessels such as radial, ulnar artery, or 
+cephalic vein improved the segmentation performance of 
+carotid artery. After segmentation, a 3D-geometry can be 
+obtained for further therapy. Van Knippenberg et al [22] 
+proposed an unsupervised learning  method to solve the lack of 
+data in carotid segmentation task. Azzopardi et al. [23] 
+designed a novel geometrically constrained loss functions and 
+received improved segmentation results. Zhou et al.[24] 
+proposed a voxel based 3D segmentation neural network to 
+segment the MAB and LIB in 3D volume directly. Although the 
+proposed algorithm achieved high accuracy with fast process, 
+user’s interaction is yet required to identify ROI in the first and 
+last slice of the volume.  
+After region of interest (ROI) i.e., carotid artery is identified, 
+further analysis needs to be performed to get significant clinical 
+information for carotid atherosclerosis diagnosis such as the 
+existence of plaque, carotid stenosis grade, type of the plaque, 
+etc. Zhou et al.[25],[26] applied 8 different backbone and 
+UNet++ segmentation algorithm trained on 2D longitudinal US 
+images to segment the plaque region and calculate the total 
+plaque area. Xia et al. [27] employed a CNN to categorize 
+segmented carotid images into normal cases, thickening vessel 
+wall cases and plaque cases. Ma et al.[28] proposed a multilevel 
+strip pooling-based convolutional neural network to investigate 
+the echogenicity of plaque which was found to be closely 
+correlated with the risk of stroke. Shen et al. [29] proposed a 
+multi task learning method, the authors combined ultrasound 
+reports and plaque type label to train a CNN to classify four 
+different plaque type. Zhao et al. [30] utilized a novel vessel 
+wall thickness mapping algorithm to evaluate the therapeutical 
+performance on carotid atherosclerosis. Zhou et al. [31] utilized 
+the unsupervised pretrained parameters of U-Net to train a 
+plaque segmentation network with a small 3D carotid artery 
+ultrasound dataset. Saba et al. [32] used a deep learning based 
+method to measure the carotid stenosis, three deep learning 
+based systems were evaluated on 407 US  dataset,  and achieved 
+AUC of 0.90, 0.94 and 0.86 on the longitudinal US images 
+respectively. Biswas et al. [33] proposed a two-stage artificial 
+intelligence model for jointly measurement of atherosclerotic 
+wall thickness and plaque burden in longitudinal US images. 
+The results showed that the proposed method achieved the 
+lowest error compared to previous method.  
+The current 3D carotid imaging device was mainly based 
+on mechanical system and hard to transport which was almost 
+impossible to apply in community or rural area, therefore the 
+portable freehand 3D ultrasound imaging system was required 
+which can be easily applied for various scenarios. However, for 
+the freehand 3D ultrasound reconstruction, the requested small 
+voxel size and various noise would lead to reconstruction 
+artifacts[34], [35]. On the other hand, the clinicians in different 
+scenarios were usually inexperienced so that the diagnosis 
+results might be inaccurate and hard to reproduce compared 
+with sonographers in clinical ultrasound department. In this 
+paper, we developed a new detection and classification 
+technique based on deep-learning algorithms for carotid 
+atherosclerosis diagnosis which can be employed to a portable 
+freehand 3D US imaging system for fast screening. Compared 
+to other 3D ultrasound carotid artery imaging methods mainly 
+focusing on carotid vessel wall segmentation  [18], [20], [21], 
+[24], the proposed method aimed at exploring an automatic and 
+experience-independent technique and framework for fast 
+carotid arteriosclerosis diagnosis.  
+The main contributions are outlined as follows. Firstly, a 
+portable freehand 3D US carotid imaging and diagnosis 
+framework including deep-learning based segmentation, 3D 
+reconstruction and automatic volume analysis was developed 
+for fast carotid atherosclerosis diagnosis. Secondly, a novel 
+position regularization algorithm was designed to reduce the 
+reconstruction error caused by freehand scan. Lastly, post 
+analysis including automatic reprojection and stenosis 
+measurement from 3D volume data provided visible qualitative 
+results and quantitative results for atherosclerosis diagnosis.  
+II. METHODS  
+Fig. 1 showed the overview of data processing procedure 
+including transverse image segmentation, 3D volume 
+reconstruction, detection of carotid atherosclerosis and 3D 
+carotid volume analysis. 
+A. MAB and LIB Segmentation 
+Three consecutive frames were concatenated in channel 
+dimension which is proved to be useful to improve the 
+segmentation accuracy [36].  
+Since the adjacent frames contained lots of redundant 
+information, the pseudo labels were generated using pseudo-
+labeling method to reduce the work load [37]. One of every 5 
+neighbor frames were selected to be manually labeled by 
+experienced sonographers and the other four frames were 
+inferred by the network which was trained using the labeled 
+frames. All generated pseudo labels were checked visually, the 
+labels would be corrected if the segmentation is incorrect.  
+The intensity of the image was normalized to [0,1] as follows: 
+𝐼 = 
+𝐼 − 𝐼𝑚𝑖𝑛
+𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛
+ 
+(1) 
+
+ 
+3 
+where I represented the intensity of the image. Imax and Imin 
+represent the max and minimum value of the intensity in the US 
+image. All images and corresponding labels were resized to 
+224*224 for segmentation network training. 
+U-Net  was employed to segment the MAB and LIB in the 
+transverse US image sequence [38]. The architecture of the 
+network was illustrated in Fig. 2. The segmentation module 
+consisted of two symmetrical sub-module which were encoder 
+and decoder. The number of channels for each convolutional 
+layer were set to (64, 128, 256, 512, 512). Each convolutional 
+layer was followed by a batch normalization module and a 
+rectification linear unit (ReLU) module. The two modules were 
+connected using skip connection to exploit all resolution 
+features. The loss function of the segmentation module was the 
+combination of DSC loss and cross-entropy loss: 
+𝐿𝑜𝑠𝑠 = 𝐿𝑜𝑠𝑠𝑑𝑖𝑐𝑒 + 𝐿𝑜𝑠𝑠𝑐𝑒
+(2) 
+B. 3D Reconstruction with Regularization 
+After the MAB and LIB were identified in every slice of US 
+image sequence, the 3D carotid artery volume was 
+reconstructed using the Fast Dot Projection (FDP) method [39]. 
+However, some disturbances caused by the low precision of the 
+magnetic sensor, inevitable hand shaking and breathing 
+movement during carotid swept, would lead to the 
+reconstruction errors and artifacts. The major problem was the 
+repeated acquisition at the same or very close positions, and it 
+caused large uncertainty at volume voxels and discontinuity in 
+the reconstructed volume [40]. To improve the image quality 
+and decrease the uncertainty of 3D reconstructed volume, a 
+total variation regularization [41] method was integrated with 
+FDP reconstruction algorithm.  
+(1) For all the position information obtained from 3DUS 
+device, it could be formulated as a set of rotation matrix 𝑅 and 
+a translation 𝑡. The tuple (𝑹, 𝒕) consisting of all 𝑅 and 𝑡 formed 
+the special Euclidean group 𝑆𝐸(3) which was the semi-direct 
+product of the rotation group 𝑆𝑂(3) and the translation group. 
+Therefore, the 𝑆𝐸(3) can be formulated as: 
+𝑆𝐸(3) = {(𝑅    𝑡
+0    1) : 𝑅 ∈ 𝑆𝑂(3), 𝑡 ∈ ℝ3}
+(3) 
+ 
+Fig. 2. The architecture of the segmentation module. 
+ 
+Fig. 1. The pipeline of the proposed system and corresponding algorithm. The top row demonstrated the process of the data acquisition, extraction of ROI and 
+3D reconstruction. The bottom row represented the process of 3D carotid volume analysis. The original image sequence and corresponding position 
+information were firstly obtained by the 3D US device. U-Net segmentation algorithm and regularized Fast-Dot Projection algorithm was applied to extract 
+the ROI and 3D carotid volume. Then 3D carotid volume analysis included automatic stenosis grade measurement, longitudinal image reprojection and 
+healthy/diseased cases classification was conducted based on the reconstructed volume. 
+ 
+
+:Conv+Batchnorm+ReLU × 2
+:Maxpooling
+: Upsampling
+:Concatenate
+:1×1 ConvSegmented masks
+Data collection
+Pre-processed images
+of LIB and MAB
+Image
+Segmentation
+Fast-Dot Projection
+3D
+Sequence
+Reconstruction
+Reconstruction
+Algorithm
+Total
+Position
+Variation
+Information
+Fast-Dot Projection
+Regularization
+3D
+Reconstruction
+Algorithm
+Reconstruction
+Stenosis Grade
+Stenosis rate
+-30°
+Measurement
+Projection
+Reprojection
+0°
+Module
+Projection
+Exist Plaque
+Diagnosis
+or Not
++30°
+Projection 
+4 
+ (2) The position signal obtained by the 3DUS system could 
+be considered as a set of entries which forms a vector 𝑷 =
+(𝒑𝟏, 𝒑𝟐, … , 𝒑𝒌) ∈ 𝑴𝒌, where 𝑘 was the number of entries and 
+𝑘 ∈  𝑁,  𝑀𝑘 was a manifold and 𝑀 = 𝑆𝐸(3). Another signal 
+set X were considered to be found when the following formula 
+is minimal. 
+E(𝐱) = D(𝐱, 𝐩) + αR(𝐱), α > 0
+(4) 
+Where 𝐷(𝑥, 𝑝) was the penalizing term to reduce the variation 
+between original signal P and resulted signal X. 𝑅(𝒙) was a 
+regularized term to penalize the position saltation in the signal 
+X.  
+ (3) The deviation penalized term D(x,p) could be defined as: 
+𝐷(𝐱, 𝐟) = ∑  
+𝑘
+𝑖=1
+(ℎ ∘ 𝑑)(𝐱𝑖, 𝐩𝑖)
+(5) 
+Where d(xi,pi) was the length of the geodesic which was 
+defined as a shortest path on 𝑀 between two pose p and q [41]. 
+ℎ was defined as following: 
+ℎ(𝑠) = {𝑠2,     
+𝑠 < 1/√2
+√2𝑠 − 1/2,      otherwise 
+(6) 
+Which was the Huber-Norm. 
+ (4) For the regularized term, it could be defined as the 
+following: 
+𝑅(𝐱) = ∑  
+𝑘−1
+𝑖=1
+(ℎ ∘ 𝑑)(𝐱𝑖, 𝐱𝑖+1)
+(7) 
+Where d(xi,xi+1) could be considered as the first-order forward 
+difference. The optimize problem in (4) could be solved using 
+a cyclic proximal point algorithm. 
+However, the original regularized algorithm couldn’t handle 
+the scanning positions with large backward movements. In this 
+case, the position array was not sequential according to the 
+coordinates, therefore pose re-rank algorithm was proposed. 
+Concretely, considering the centroid point of every frame from 
+the 2D segmented image sequence as 𝑪𝒌 = (𝒄𝟏, 𝒄𝟐, … , 𝒄𝒌) , the 
+PCA (principal components analysis) algorithm was conducted 
+in 𝐶𝑘 and a new matrix 𝐷𝑘 was obtained. The first column of 
+the matrix was the principal vector 𝑣𝑘, then a set of vectors 𝑐𝑑 
+could be acquired by projecting every centroid vector 𝑐𝑘 to 𝑣𝑘. 
+𝒄𝒅 = 𝒄𝑘 − 𝒄𝒌 ⋅ 𝑣𝑘
+𝑣𝑘 ⋅ 𝑣𝑘
+𝑣𝑘
+(8) 
+The new position sequence was finally obtained by sorting the 
+l2-norm of the 𝒄𝒅 set. 
+C.  Carotid Atherosclerosis Diagnosis  
+The US scans including the segmented and reconstructed 
+volume were classified into healthy case and carotid 
+atherosclerosis case using a diagnosis network. As illustrated in 
+Fig. 3, there were two inputs for the diagnosis module. It had 
+been proved that the morphological information was helpful for 
+the network to classify the normal or abnormal (diseased) image 
+[42], therefore the mask of LIB and MAB extracted from each 
+slice of the reconstructed volume was used as one input. The 
+other input was the cropped ROI which was determined by the 
+max bounding rectangular of the mask, and in the cropped 
+image, the intensity in the region between LIB and MAB was 
+set to the original value, while the region inside lumen and 
+outside vessel wall were set to 0. For each input stream, it 
+consisted of three repeated blocks, each block consisted of two 
+consequent basic convolutional sub-block and a max pooling 
+layer. The basic convolutional sub-block was composed of a 
+convolutional layer, a batch normalization module and a linear 
+rectification unit. The number of channels for each repeated 
+block were set to (24, 48, 96). The fusion block concatenated 
+the high-level features of two streams and combined 
+information by introducing a basic convolutional sub-block. 
+After fusion block, the remaining layers were global average 
+pooling (GAP) layers and a fully connected layer to output the 
+diagnosis result. We used focal loss in the diagnosis module. 
+The scan would be diagnosed as a carotid atherosclerosis 
+case if the consecutive 5 transverse slices from the 
+reconstructed volume were classified as existing plaque.  
+D. 3D Carotid Volume Analysis 
+ The clinical diagnostic parameters such as plaque thickness, 
+plaque length, stenosis grade, plaque area, plaque type, etc. can 
+be directly calculated from the reconstructed carotid artery 
+volume. To validate accuracy of the proposed method, the 
+longitudinal US images of carotid artery were obtained by 
+projecting the volume in different angles, and the stenosis grade 
+was calculated.   
+Stenosis rate was usually used to evaluate the stenosis grade. 
+For the slices which were diagnosed as atherosclerosis, stenosis 
+degree can be evaluated using the LIB and MAB masks. The 
+diameter stenosis rate was usually calculated to evaluate the 
+stenosis grade in clinic. We denote it as  
+𝑆𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 =
+𝐿𝑤𝑎𝑙𝑙
+𝐿𝑤𝑎𝑙𝑙 + 𝐿𝑙𝑢𝑚𝑒𝑛
+(9) 
+where 𝐿 represented the length of respective area. The metric 
+was ranged from 0 to 1, the greater number indicated the more 
+severe stenosis. The length of vessel wall 𝐿𝑤𝑎𝑙𝑙 and length of 
+ 
+Fig. 3. The architecture of the diagnosis module. 
+ 
+Fig. 4 The illustration of the approach to calculate the diameter stenosis. 
+
+24X128X128
+128X128
+48X64X64
+96X32X32
+96X16X16
+96X16X16
+Exist
+plaque
+24X128X128
+48X64X64
+or not
+96X32X32
+:Conv+Batchnorm+ReLU
+96X16X16
+:Maxpooling
+128X128
+:Global average pooling
+:Concatenate
++ :Fully connectRadially Sampling
+:lumen
+:Vessel Wall 
+5 
+lumen 𝐿𝑙𝑢𝑚𝑒𝑛 were illustrated as Fig. 4. The diameter stenosis 
+rate was the max 𝑆𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟  which was calculated using all 
+points in MAB boundary. 
+The longitudinal carotid US images were usually used to 
+calculate plaque size and evaluate the morphology of plaque.  
+Since the carotid artery is curved volume, the direct projection 
+along a fixed axis may lead to missing of some structure. 
+Therefore, centroid points of carotid artery in transverse slices 
+were selected to determine the projection plane. Specifically, as 
+illustrated in Fig. 5, denoting the centroid point of i-th slice in 
+the volume as 𝐶𝑖,  the line which was 𝜃 degree angled with y-
+axis through the centroid point 𝐶𝑖 , was sampled as the i-th 
+column of projected longitude image. In our experiment, the 
+longitudinal US images were obtained by reprojecting the 3D 
+carotid volume at the angles of 0°, ±15° and ±30°. 
+III. EXPERIMENTAL SETUP 
+A. Data Acquisition and 3D Ultrasound Scan 
+A portable freehand 3D ultrasound imaging system was used 
+to obtain three-dimensional images of carotid artery as shown 
+in Fig. 6. The system consisted of a 2D linear probe (Clarius, 
+L738-K, Canada), an electromagnetic tracking system 
+(Polhemus, G4 unit, U.S.A) and a host laptop computer (Intel 
+i7-8700k CPU @ 3.70GHz, 32GB RAM) [43]. The 2D 
+transverse US images were acquired by the probe while the 
+corresponding position and orientation information were 
+captured by the magnetic sensor. The images and orientation 
+were acquired with a frame rate of 24 Hz. 
+During the acquisition, the subjects took the supine position 
+and was scanned as shown in Fig. 6 (d), and the probe swept 
+consistently along the long axis of common carotid artery from 
+the proximal end to the distal end at the speed of approximate 
+10-15 seconds per scan. To reduce the reconstruction artifacts, 
+fallback along the swept direction and large movement normal 
+to the swept direction should be avoided. The inclusion criteria 
+were based on visible plaques which was identified by expert. 
+The stenosis grade larger than 70% was excluded to the dataset. 
+A total of 127 3D carotid artery scans from 83 subjects with 
+stenosis range from 0% to 70% were obtained from local 
+hospital, and all subjects consented to participate in this 
+experiment, which was approved by the local ethics committee. 
+The age of the subjects was ranged from 51 to 86 years old 
+(Male: 38, Female: 45).  
+Each scan contained 122-250 2D transverse US images with 
+resolution of 640*480. 7596 2D images from 40 scans were 
+manually delineated for MAB and LIB and labeled for healthy 
+or diseased (with plaque) by experienced sonographers for 
+further training of segmentation and classification network. All 
+Fig. 5. The illustration of the reprojection process. The centroid point was calculated by the segmented MAB mask for each slice in the volume. Then the line 
+segment crosses the centroid point was set to conduct the reprojection. The red and green line segment represent the different resample angle respectively. 
+    
+ 
+(a)                                          (b) 
+    
+ 
+(c)                                            (d) 
+Fig. 6. Ultrasound scan using the freehand imaging system. (a) a handheld 
+US scanner (left), a host laptop computer (middle) and an iPhone SE2 
+(right). (b) Tracking system including a hub (left) and a RF/USB module 
+(right). (c) The sensor (left) and the magnetic source (right). (d) Ultrasound 
+scan using the freehand imaging system. 
+ 
+
+Get Centroid
+0
+Theindexofthecolumn
+n
+GetCentroid
+i-1
+Reprojection longitude image, reprojection angle=0o
+i
+i+1
+Get Centroid
+Theindexofthecolumn
+n
+The index of the slice 
+6 
+127 scans were labeled for healthy or diseased (with plaque) by 
+the same raters examining 2D images. In addition, stenosis 
+grade and plaque size of randomly selected 20 scans were 
+manually measured by expert using clinical 2D US device for 
+verification of the proposed system and algorithm. 
+B. Training Methods 
+25 scans (4694 2D images) were randomly chosen for CNN 
+training and 15 scans (2362 2D images) for validation in order 
+to build and verify the segmentation module. The original 
+images were resized to 224*224. All training process were 
+performed using Pytorch 1.5.1 and Python 3.7 on a NVIDIA 
+RTX 4000 GPU. The two networks were trained separately. For 
+the segmentation module, the applied data augmentation 
+strategies including gamma transformation, rotation, zoom, 
+horizontal and vertical flip, and Adam optimizer were used. The 
+network was trained for 100 epochs with learning rate and batch 
+size set to 0.005 and 8 respectively. For the diagnosis module, 
+the cropped and resized 2D US image segmented with the mask 
+and the corresponding vessel wall mask were used for network 
+training. Gamma transformation and horizontal & vertical flip 
+were applied for data augmentation. The diagnosis network was 
+trained for 50 epochs using Adam optimizer with learning rate 
+and batch size set to 0.005 and 64 respectively. 
+C. Diagnosis parameter measurement  
+To verify the regularized reconstruction and longitudinal 
+images reprojection algorithm, the longitudinal images from 20 
+clinical patients with and without regularization were compared 
+with clinical images acquired by experienced sonographers 
+visually, and the projection angles were set as 𝜃 =
+−30°, −15°, 0°, 15°, 30°. 
+The plaque length and thickness were manually measured on 
+the 3D pseudo volume, the reconstructed 3D volume and the 
+clinical images acquired by experienced sonographers 
+respectively, where 3D pseudo volume was the volume which 
+were stacked directly by the 2D US images sequence. The 
+manual measurement of plaque length and thickness was 
+conducted on the reprojected longitudinal images, among 
+which the reprojection angle was chosen based on the carotid 
+structural integrity and maximum stenosis grade. For plaque 
+size measurement in reprojected image of reconstructed 3D 
+volume, the pixel size was 0.2 × 0.2𝑚𝑚2. For the pseudo 3D 
+volume, the velocity of the swept was assumed constant, 
+therefore the pixel size of reprojected image was determined by 
+the distance of the swept which could be calculated by the 
+magnetic sensor. 
+The whole system in clinical metric measurement was also 
+verified by comparing stenosis rate automatically measured by 
+the 
+system 
+and 
+manually 
+measured 
+by 
+experienced 
+sonographers using clinical US device on 20 random clinical 
+atherosclerosis patients according to formula (9). 
+D. Evaluation Metrics and Statistic Analysis 
+The dice similarity coefficient (DSC) and 95% hausdorff 
+distance (HD95) were used to evaluate the performance of the 
+carotid sequence segmentation. DSC indicated the quantitative 
+metric of the overlap region between the ground truth and 
+prediction mask which was defined as follows: 
+𝐷𝑆𝐶 = 2(𝑃 ∩ 𝐿)
+𝑃 ∪ 𝐿
+(10) 
+where P, L were the prediction mask and ground truth. The 
+hausdorff distance was defined as Eq (11), which indicated the 
+largest point-wise matching discrepancy: 
+𝐻𝐷(𝐴, 𝐵) = 𝑚𝑎𝑥(ℎ𝑑(𝐴, 𝐵), ℎ𝑑(𝐵, 𝐴)) 
+(11) 
+where 
+ℎ𝑑(𝐴, 𝐵) = 𝑚𝑎𝑥𝑎∈𝐴(𝑚𝑖𝑛𝑏∈𝐵||𝑎 − 𝑏||) 
+(12) 
+ℎ𝑑(𝐵, 𝐴) = 𝑚𝑎𝑥𝑏∈𝐵(𝑚𝑖𝑛𝑎∈𝐴||𝑏 − 𝑎||) 
+(13) 
+ For the evaluation of the diagnosis module. The specificity, 
+sensitivity and accuracy were calculated for both 2D US image 
+and scans.   
+ The mean absolute difference (MAD) and standard deviation 
+(SD) between results from the pseudo/reconstructed 3D 
+volumes and results from experienced sonographers were 
+investigated. The metrics in verification of the system, i.e., the 
+stenosis grade, were compared between manual or automatic 
+approach using the proposed 
+technique and manual 
+measurement using the clinical US device with the Pearson 
+correlation analysis.  
+IV. RESULTS 
+A. Segmentation and Diagnosis Accuracy 
+The comparison between nine typical segmented images 
+from U-Net and experienced sonographers was illustrated as 
+Fig. 7, and the images were selected from different scans at 
+some specific locations. Table I showed the average DSC and 
+HD95 between the ground truth and prediction results.   
+TABLE I.  
+THE RESULTS OF VESSEL SEGMENTATION 
+Metrics 
+category 
+MAB 
+Lumen 
+DSC 
+95.00% 
+93.30% 
+HD95(pixel) 
+4.34 
+4.65 
+Table II showed the contingency table of the validation set of 
+2362 2D transverse images, and the sensitivity, specificity and 
+accuracy were 0.73, 0.97 and 0.91 respectively. Table III 
+showed the diagnostic results of carotid atherosclerosis for all 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 7. Comparison of the auto segmentation from U-net (red) and manual 
+segmentation from ground truth (green). 
+
+ 
+7 
+scans, and the sensitivity, specificity and accuracy of carotid 
+atherosclerosis detection was 0.71, 0.85 and 0.80 respectively. 
+TABLE II.  
+THE RESULTS OF DETECTION TEST FOR 2D IMAGES  
+Labels 
+Predictions 
+Positive (plaque) 
+Negative 
+Positive 
+(plaque) 
+454 
+171 
+Negative 
+50 
+1687 
+TABLE III.  
+THE DETECTION RESULTS OF CA FOR SCANS 
+Labels 
+Predictions 
+Positive (plaque) 
+Negative 
+Positive 
+(plaque) 
+25 
+10 
+Negative 
+10 
+57 
+B. Reconstruction and Reprojection Accuracy 
+Fig. 8 illustrated three representative examples of the 
+longitudinal images without regularization, with regularization 
+clinical US images acquired by experienced sonographers, and 
+the corresponding orientation information. The results revealed 
+that the regularized reconstructed volume was smoother with 
+less image artifacts. Fig. 9 demonstrated an example with large 
+fallback of trajectory, the results showed there were still 
+artifacts if directly apply the regularized algorithm and the 
+proposed re-rank algorithm could remove the reconstruction 
+artifacts. Fig. 10 illustrated the 3D volumes reconstructed from 
+the auto-segmentation and ground truth respectively. The 
+volumes were rendered by 3D-slicer (www.slicer.org). The 
+results showed that the segmentation module achieved good 
+agreement with human label. Furthermore, the sunken of the 
+lumen area indicated the existence of the plaque. Fig. 11 
+        
+ 
+Fig. 8. Illustration of the US longitudinal images and the corresponding orientation information from three carotid atherosclerosis patients (by rows). The 
+images in the first column were reconstructed without regularization algorithm while the images in the third column were reconstructed with regularization 
+algorithm. The second column demonstrated the smoother results of the proposed algorithm.  The fourth column represents the images acquired by 
+sonographers using clinical US devices. The images in fifth column illustrate the corresponding original position information and the images in sixth column 
+show the regularized position information.  
+ 
+Fig. 9. Illustration of the proposed re-rank algorithm, the first row 
+demonstrated the longitudinal image and corresponding position 
+information without regularized algorithm. The second row represented 
+the images which applied regularized algorithm and the third row showed 
+the images which used re-rank and regularized algorithm. 
+
+ 
+8 
+demonstrated comparison among 5 projected images in 
+different angles ( 𝜃 = −30°, −15°, 0°, 15°, 30° ), the image 
+directly projected to sagittal plane and the manually acquired 
+image by expert from the same atherosclerosis patient. The 
+results showed that the projected images in different angles 
+could reveal more structures of the carotid than the images only 
+projected to sagittal plane. On the other hand, in Fig. 11, the 
+image in 15° projection angle was most consistent with the 
+clinical image obtained by expert using clinical US device, 
+which indicated that the reprojection of 3D volume could 
+simulate the different scan angles operated by expert to locate 
+the best observation view.  
+The plaque size (length and thickness) measured from the 
+pseudo volume, reconstructed volume and images acquired by 
+expert were compared in Table IV. The results showed good 
+agreement between the automatic measurement from the 
+reconstructed volume and the manual method, while the plaque 
+size measured by pseudo volume showed large difference with 
+the expert measurement. The results indicated that the 3D 
+reconstruction could reveal the true geometry and clinical 
+metric of the carotid artery. 
+TABLE IV.  
+MAD MEASUREMENTS (N=20) BETWEEN CLINICAL US DEVICE 
+AND THE PROPOSED TECHNIQUES 
+ 
+          
+           
+ 
+     
+                
+                
+ 
+ 
+Fig. 10. The 3D volumes from the auto-segmentation (the first row) and ground truth (the second row). The translucent outer wall represents the vessel wall 
+area, the inside red 3D volume represents the lumen area. The sunken of the lumen area indicated the existence of the plaque.  The resolution of reconstruction 
+is set to 0.2×0.2×0.2 𝑚𝑚3. 
+  
+  
+  
+ 
+ 
+ 
+  
+  
+   
+Fig. 11. Illustration of the projected 
+images in different angles, from left 
+top to right bottom were 5 projected 
+images (𝜃 = −30°, −15°, 0°, 15°,
+30° ), direct sagittal image and 
+clinical image respectively. It could 
+be observed the sagittal image 
+missed the part of vessel wall (in the 
+red box) and the reprojected image 
+with  𝜃 = 15 °  showed the most 
+consistent structure of plague with 
+clinical image (in the green boxes 
+shows).  
+
+ 
+9 
+ 
+Plaque 
+length(mm) 
+Plaque 
+thickness(mm) 
+Plaque 
+length 
+(relative 
+error) 
+Plaque 
+thickness 
+(relative 
+error 
+3D 
+Reconstructed 
+volume 
+2.65±2.36 
+0.842±0.617 
+15.4%±
+13.6% 
+26.0%±
+13.2% 
+Direct stacked 
+Pesudo volume  
+6.54±7.23 
+0.976±0.648 
+40.0%±
+48.0% 
+29.4%±
+14.0% 
+C. Stenosis Measurement Accuracy 
+Fig. 12 demonstrated the linear correlation (r=0.762) of the 
+stenosis grade measured by the system and experienced 
+sonographers using the clinical US device on 20 carotid 
+atherosclerosis patients, which indicated the proposed 
+technique had the strong consistency with expert manual 
+approach in carotid atherosclerosis diagnosis. 
+V. DISCUSSION 
+In this study, we proposed a portable freehand 3D US 
+imaging technique for carotid artery diagnosis which could 
+achieve real 3D geometry of carotid artery, and the method 
+showed good agreements with manual measurement of stenosis 
+rate and classification of diseased and healthy case. The system 
+was transportable and less dependent on operator’s experience, 
+which make it possible for routine health check in different 
+environments such as community or rural area. In addition, the 
+3D reconstructed geometry could provide visualized carotid 
+artery structure for further atherosclerosis evaluation.  
+Since the large position variation or fallback movement 
+during scan would cause reconstruction artifacts, we designed 
+a standard scan protocol for 3D carotid US data acquisition and 
+analysis. The whole processing steps included automatic 3D US 
+data 
+acquisition, 
+MAB 
+and 
+LIB 
+segmentation, 
+3D 
+reconstruction, automatic classification and measurement. In 
+practice, the intermediate results of each step could be reviewed 
+and manually corrected by operator if necessary to ensure the 
+accurate final results. The diagnosis result was based on two 
+key points: one was the accurate segmentation of vessel area, 
+and the other was the correct reconstruction volume. The 
+segmentation determined the region of interest (ROI) used for 
+following analysis including automatic stenosis evaluation, 
+plaque size measurement and 3D geometry visualization. The 
+wrong mask might crop regions out of the carotid artery, 
+mislead the diagnosis network and cause confusing diagnosis 
+results. However, if the 3D volume was directly reconstructed 
+from 
+original 
+2D 
+frames 
+before 
+segmentation, 
+the 
+reconstruction artifacts around MAB and LIB such as 
+misplacement or severe blurring could lead to segmentation 
+error of vessels, especially for some cases with large position 
+variation as Fig. 13. showed. Therefore, we conducted 
+segmentation on the original 2D US image sequence before 3D 
+reconstruction to extract the vessel area firstly to reduce the 
+influence of reconstruction artifacts.  
+For the reconstruction process, the failure reconstruction 
+caused by large position variation could result in severe image 
+artifacts which totally deviated the structure of the carotid artery 
+as shown in Fig. 14 For the freehand US scan, theoretically, the 
+position information recorded by magnetic sensor would be 
+consistent with US probe motion i.e., the position of every US 
+ 
+Fig. 12. correlation of stenosis grade between the manual measurement by 
+expert using the clinical US device and the automatic measurement from 
+the proposed technique on 20 carotid atherosclerosis patients. 
+     
+ 
+ 
+    
+ 
+Fig. 14. Severe reconstruction artifacts caused by the large position 
+variation. The image in first row represented the reconstructed volume and 
+the orientation information with regularized algorithm while the images in 
+the second row represented the results without regularized algorithm. The 
+left image shows the transverse image of the locations in the reconstructed 
+volume marked in red boxes in the right image, the large distortion could 
+be observed in the image while the distortion was alleviated using the 
+regularized algorithm. 
+  
+ 
+Fig. 13. Segmentation results on a transverse image collected from the 3D 
+volume reconstructed by the original image sequence (left) and on an 
+original transverse frame data (right). It could be observed that the severe 
+artifacts on the left image led to wrong segmentation result. 
+
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Stenosis grade measured by expert using clinicla US device 
+10 
+image. However, the low precision of the sensor and inevitable 
+hand jitter would lead to the noticeable measurement 
+uncertainty of the position information along the scan direction 
+and influence the reconstruction accuracy. Therefore, we 
+adopted a novel total variation regularization algorithm to 
+smooth the track of the position information and decrease 
+distortion and disconnection of the image volume. The position 
+of the freehand scan can be regarded as continuous and 
+sequential array; therefore, the proposed regularization 
+algorithm could reduce the uncertainty by constructing and 
+minimize a regularized formulate in the manifold of Euclidean 
+transformations. Meanwhile, a re-rank strategy was designed to 
+solve the unordered image sequence caused by fallback 
+movement during scan. In the future, the reconstruction 
+accuracy could be further improved using the neural network. 
+After segmentation and reconstruction, the carotid artery 
+volume could be obtained for further analysis such as healthy 
+or diseased case diagnosis, plaque thickness, length area 
+measurement, 
+plaque 
+type 
+identification 
+and 
+stenosis 
+measurement etc. In the diagnosis module, the cropped and 
+resized images instead of the whole US images were used as the 
+input. Since the plaque was only located inside vessel wall area, 
+removing useless information outside the vessel wall could 
+accelerate network training and improve the detection accuracy. 
+On the other hand, there may be low intensity area in the vessel 
+region which could mislead the network and result in wrong 
+classification since negative sample (no plaque) usually had 
+low intensity in lumen area. Therefore, the MAB and LIB mask 
+were introduced to combine the morphological information 
+with original image information to improve the detection 
+accuracy. However, the proposed approach just utilized the 
+consecutive 2D reconstructed transverse US images to detect 
+plaque cases, thus some cases with small plaque size or severe 
+artifacts were wrongly classified as no plaque. In the future, we 
+will take the z axis information into account and use the whole 
+3D volume as input instead of detecting plaque by limited 
+consecutive transverse slices to improve the accuracy of the 
+diagnosis module.  
+We utilized a reprojection algorithm to project the carotid 
+artery volume to longitudinal planes, so that the clinical metric 
+such plaque length, thickness could be directly measured from 
+the 3D volume with no need of new acquisition in sagittal 
+direction. The traditional clinical carotid artery US examination 
+required appropriate positioning and angle between transducer 
+and neck, which greatly relies on the operator’s experience to 
+localize the plaque and obtain a high-quality US image, the 
+proposed reprojection approach in our method was not only 
+relatively convenient but could reveal the complete structure of 
+the carotid artery with only one scan, and the images obtained 
+by our automatic method achieved great agreement with the 
+images obtained by expert using clinical US device. 
+In segmentation module, we used U-Net to segment the 
+MAB and LIB in 2D US image sequence. Every image in the 
+sequence was treated as a single image for the segmentation 
+network. However, this approach didn’t exploit the context 
+information in the adjacent frames. In addition, some cases with 
+severe noise or shadowing would result in wrong segmentation 
+as Fig. 15 showed. In the future, 3D convolution will be 
+considered to correct the segmentation mistake by utilizing the 
+context information of the adjacent frames, and sample size will 
+be enlarged to improve the accuracy and robustness of the 
+segmentation algorithm. More 3D metrics such total plaque 
+volume, vessel wall volume, etc. would be evaluated for more 
+accurate validation. On the other hand, the learning-based 3D 
+reconstruction algorithm would be taken into account to 
+improve the performance of reconstruction. 
+VI. CONCLUSION 
+We have proposed an automatic 3D carotid artery imaging 
+and diagnosis technique specially designed for the portable 
+freehand ultrasound device. The technique applied a novel 3D 
+reconstructed algorithm and a robust segmentation algorithm 
+for automatic carotid atherosclerosis analysis. The results 
+demonstrated that the technique achieved good agreement with 
+manual expert examination on plaque diagnosis and stenosis 
+grade measurement, which showed the potential application on 
+fast carotid atherosclerosis examination and the follow-ups, 
+especially for those scenarios where professional medical 
+device and experienced clinicians are hard to acquire such as 
+rural area or community with large population. 
+ACKNOWLEDGEMENT 
+ This work was sponsored by Natural Science Foundation of 
+China (NSFC) under Grant No.12074258. 
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+

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+Noise Resistant Phase Imaging with Intensity Correlation
+Jerzy Szuniewicz1, Stanisław Kurdziałek1, Sanjukta Kundu1, Wojciech Zwolinski1,
+Radosław Chrapkiewicz2, Mayukh Lahiri3, Radek Lapkiewicz1∗
+1Institute of Experimental Physics, Faculty of Physics, University of Warsaw,
+ul. Pasteura 5, 02-093 Warszawa, Poland,
+2CNC Program, Stanford University, Palo Alto, CA 94304, United States
+3Oklahoma State University, Stillwater, OK 74078-3072, United States
+∗[email protected]
+Interferometric methods, renowned for their reliability and precision, play a vital role
+in phase imaging. Interferometry typically requires high coherence and stability be-
+tween the measured and the reference beam. The presence of rapid phase fluctua-
+tions averages out the interferogram, erasing the spatial phase information. This diffi-
+culty can be circumvented by shortening the measurement time. However, shortening
+the measurement time results in smaller photon counting rates precluding its applica-
+bility to low-intensity phase imaging. We introduce and experimentally demonstrate
+a phase imaging technique that is immune to position-independent, time-dependent
+phase fluctuation. We accomplish this by measuring intensity correlation instead of
+intensity. Our method enables using long measurement times and is therefore advan-
+tageous when the photon flux is very low. We use a Fisher information-based approach
+to show that the precision of phase reconstruction achieved using our method is in fact
+the best achievable precision in the scenario when two photons are detected per phase
+stability time.
+Introduction
+Phase imaging is important for applications spanning many diverse fields, including biological imaging
+(1), and phase microscopy (2,3). Measurements of the phase shifts within samples can yield information
+about the refractive index, thickness, and structure of an object. Interferometry (4) is a very powerful tool
+that is often used in phase imaging of an object (5). Interferometric measurements allow the detection
+of small variations in optical paths. There are numerous interferometric techniques such as the ones
+regularly used in optical coherence tomography (6,7) or quantitative phase microscopy (8). Some of the
+techniques, especially those related to biology, require very low photon fluxes. For an interferometric
+measurement a wave field that has interacted with an object is superposed with a reference field and the
+resulting interference pattern is detected by a camera. If the object field (probe field) and the reference
+1
+arXiv:2301.11969v1  [physics.optics]  27 Jan 2023
+
+field are mutually coherent, the time-averaged intensity on camera is given by (9,10):
+I(x, y) = Ir + Io + 2
+�
+IrIo cos[φin + φ(x, y)],
+(1)
+where Ir and Io are the averaged intensity of the reference and the object fields, respectively, φin is the
+interferometric phase that can be changed by introducing spatial or temporal delays between the two
+fields, and φ(x, y) is the phase map of the object. Standard interferometric phase imaging techniques are
+based on the signature of φ(x, y) left in the detected intensity pattern. However, for any such method
+to be applicable, the object field and the reference field need to be mutually coherent. Time-dependent,
+uncontrollable phase fluctuations introduce incoherence between object and reference fields. The method
+is therefore vulnerable to time-dependent, uncontrollable phase fluctuations that introduce incoherence
+between object and reference fields.
+When the phase fluctuates much faster compared to the detection time, the coherence between the
+object and image fields is practically lost and, no interference will be observed, i.e.,
+I(x, y) = Ir + Io.
+(2)
+Since there is no information of φ(x, y) in this intensity pattern, the standard phase imaging scheme
+becomes inapplicable to this case. One way to avoid the effect of this time-dependent phase fluctuation
+is to shorten the duration of measurement (11). A short measurement time, however, reduces the amount
+of detected light and is therefore impractical for imaging photo-sensitive biological specimens, which
+require low-intensity light. Furthermore, for interferometric fluorescence super-resolution microscopy
+(12), often very low-intensity light (13) needs to be superposed. In such cases, any time-dependent
+phase fluctuations must be avoided due to the relatively long detection time requirement.
+Here, we introduce a method of phase imaging that is fully resistant to time-dependent phase fluctu-
+ations as long as it is possible to measure at least two photons per phase stability time. Our method is
+fundamentally different from the standard phase imaging techniques (14), as we do not need interfero-
+metric phase stability due to the fact that we measure intensity correlation instead of intensity.
+The scheme of our experiment is illustrated in Fig. 1. A wave field that has interacted with an object
+(object field) is superposed with a reference field and the resulting interference pattern is detected by
+a camera. A time-dependent phase fluctuation Θ(t) is introduced in the reference field. Under these
+circumstances, no information on φ(x, y) can be retrieved from the intensity pattern given by Eq. (2),
+and therefore the standard phase imaging techniques become inapplicable. In the present article, we
+introduce a method of phase imaging that is resistant to time-dependent phase fluctuations, provided that
+phase change is uniform throughout the entire sample (15). Our method relies on measuring intensity
+correlations of light and is inspired by the intensity interferometry technique introduced by Hanbury
+Brown and Twiss (HBT) (16). The HBT method and its generalizations were applied to a variety of light
+sources (17–25) and similarly our technique might be applied in various scenarios including laser and
+thermal light as important examples.
+We determine the correlation function between the intensities measured at a pair of points (x, y) and
+(x′, y′)
+�˜I(x, y; t)˜I(x′, y′; t)
+�
+∝ 1 ± 1
+2 cos [φ(x, y) − φ(x′, y′)] ,
+(3)
+where ˜I(x, y; t) is the instantaneous intensity measured at a point (x, y) at time t. On the right hand side
+2
+
+Figure 1: (a) Simplified schematic of the experiment: we divide input light into two paths, an object
+path(φ(x)), and a reference path. In the object path, we introduce a spatially varying phase that we want
+to image. A time-fluctuating interferometric phase can be introduced to the system (Θ(t)) with no loss
+in the quality of the phase retrieval. For slowly fluctuating phase Θ(t), we can measure high visibility
+interference fringes (b), but no interferogram can be recorded due to insufficient photon statistics and
+rapid fluctuations of (Θ(t)) - depicted in the image (c) - where fringes average to the intensity profile of
+the beam having no phase information. Images (b) and (c) depict normalized one photon interference
+fringes for slowly and highly fluctuating cases respectively. We also depict second-order correlation
+interferograms (d) for the same photons constituting the interferograms in image (c). Even for this
+highly fluctuating case, where we record only a few photons within the stability time of the phase Θ(t),
+we can retrieve high visibility second-order correlation interferograms preserving full phase information
+about the measured phase φ(x).
+of Eq. (3), the plus (+) and minus (−) signs apply when the two points of measurement are in the same
+and different beam splitter outputs, respectively. We also assume, Ir = Io. Note that the information
+about the phase map of the object, which was lost in the intensity pattern [Eq. (2)], reappears in the
+intensity correlation [Eq. (3)].
+3
+
+%The 2nd-order intensity correlations map contains the full information required to optimally recon-
+struct φ(x, y) in the extreme case when only two photons are detected during the phase stability time.
+Our strategy of reconstructing the actual phase distribution in this scenario is optimal, which we prove
+rigorously using estimation theory tools, namely Fisher Information and Cram´er-Rao bound (see Sup-
+plementary S1 for detail).
+Laser
+I-sCMOS
+camera
+Calcite
+L1
+l/2
+L2
+l/4
+Sample
+j(x)/2
+l/4
+Delay line
+l/4 l/2
+PBS
+l/2
+Figure 2: Experimental setup for noise-resistant phase imaging. The incoming beam of Laser after pass-
+ing through a λ
+2 - half-wave plate, λ
+4 - quarter wave plate, PBS - polarization beam splitter, and another
+λ
+2 plate, the beam enters a Michelson type interferometer. Each of the two paths in the interferometer is
+encoded with orthogonal polarization. In one arm the spatial phase φ(x) is introduced next to the surface
+of the interferometer mirror. The interferometric mirror in the other arm is given a phase fluctuation by
+attaching it to a piezoelectric actuator. The two beams of the interferometric arms after combining at the
+PBS pass through L1, and L2 lenses. The calcite polarizer acts as a 50/50 Beamsplitter. The I-sCMOS
+- Intensified sCMOS camera records single photons at both outputs of the interferometer. The use of
+short exposure time of the I-sCMOS, in the single nanosecond timescale, gives it stability and resistance
+against fluctuations up to tens of MHz.
+Experimental setup
+The experimental setup is depicted in Fig.2. Light from a polarized, coherent source (780 nm laser) is
+attenuated, spatially filtered, and directed to two arms of a polarization-based Michelson interferometer.
+In order to prepare the object beam, in one of the arms, we place a phase mask to imprint spatially varying
+phase φ(x) to the beam. We perform experiments with three kinds of different phase masks applied to
+our object beam. We imprint a 1D quadratic local phase profile to the beam by placing a cylindrical lens
+of focal length, f = 1000 mm in proximity to the mirror (Fig. 2). Additionally, we also use a spatial
+light modulator (SLM) as a phase mask, as it can display any arbitrary phase profile. We imprint 1D
+4
+
+exponential and sinusoidal phases to our object beam by the SLM display (see supplementary S2 for
+detail).
+A time-dependent phase fluctuation is introduced in the other arm (the reference beam) to make
+it incoherent with the object beam. This is realized with a piezoelectric actuator driven by a RAMP.
+Light is combined on the PBS. Object and the reference planes are imaged onto two regions of an
+Intensified sCMOS (I-sCMOS) (26) camera with a 4f system using lenses L1 and L2. After the PBS,
+the object and the reference beams are distinguishable due to their orthogonal polarization. In order to
+observe interference we rotate their polarization by 45 degrees with a half-wave plate and we perform the
+projective measurement in the original bases with a calcite crystal. This mixes the light from both outputs
+and allows us to observe interference in both outputs of the beam splitter. The visibility is reduced due
+to imperfect imaging because of the path length difference in the calcite. In order to register very low
+photon flux and to minimize exposure time to circumvent fluctuations, we use an Intensified-sCMOS
+camera. We collect the data with a frame rate of 200 Hz. choosing a low exposure time Texp ∼ ns
+allows performing measurement under phase fluctuations with frequency up to (fn ∼ 1/Texp) tens of
+MHz.
+Results
+Data measured in our experiment consist of an average of 15 photons at both outputs of the interferome-
+ter per frame. We remove temporal correlations between subsequent frames by randomly permuting the
+order of frames before further processing—this process does not change the performance of our method
+but allows us to simulate the conditions, in which the global phase fluctuates faster than the camera
+frame rate. In such conditions, it is impossible to retrieve phases using standard interferometric methods.
+Averaging recorded intensities over multiple frames or increasing measurement time would result in a
+loss of the visibility of the interference fringes. In contrast, we average correlations of detected pho-
+tons’ positions without any loss of the phase information. Such averaging over multiple frames results
+in the reproduction of the correlation function (Eq.3), from which we can retrieve the phase profile us-
+ing the standard digital holography method, Fourier off-axis holography (27). The correlation function
+is measured from the coincidence map of the detected photons’ positions. This analyzing mechanism
+is the essence of our noise-resistant phase imaging technique. 1D quadratic phase measurement intro-
+duced by the cylindrical lens is shown in Fig. 3. The measured coincidence map (Fig. 3(a)) consists of
+approximately 107 registered photon pairs with the mean number of coincidences per pixel as 100. We
+estimate the phase profile shape using the collected data, and compute the Mean Squared Error (MSE)
+between the measured and real value. As we show in Fig. 3(c), the MSE drops down with the total
+number of measured photons, and eventually reaches the theoretical minimum, obtained with the help of
+the Cram´er-Rao bound (see Supplement 2 for details). This proves, that our method of phase estimation
+is optimal when at most two photons are measured during the phase stability time—notice, that this is
+the most extreme limit in which one can gain any information about the phase profile.
+SLM-encoded phase measurements shown in Fig. 4(a), (b), and (c) represent the measured hologram,
+the retrieved phase, and the error per pixel respectively when the sinusoidal phase is applied. Similarly,
+Figs. 4(e), (f), and (g) represent the measured hologram, the retrieved phase, and the error per pixel re-
+spectively when the 1D exponential phase is applied. Errors in the retrieved respective phases (Fig. 4(c),
+Fig. 4(g)) are due to a finite number of pixels on the SLM and discreet values of the displayed phases.
+5
+
+a
+b
+c
+Figure 3:
+(a) represents the measured coincidence map for a 1D quadratic phase profile, plotted with
+a solid line in (b). The reconstructed phase with error bars is also shown in (b). The visibility of
+the fringes in the correlation map (a) is equal to 0.62/2 (theoretical maximum with classical light is
+1/2). The total number of coincidences detected in the experiment is ∼ 107. By randomly removing
+a part of the collected signal, we can check how the Mean Squared Error (MSE) associated with the
+phase reconstruction scales with the mean number of photons detected in one pixel during the whole
+experiment (c). The MSE from the experiment is then compared with the MSE obtained using simulated
+hologram, with the same parameters as in the experiment. We calculate the fundamental Cram´er-Rao
+(C-R) lower bound on the MSE, assuming the visibility of hologram fringes to be equal to 0.62/2 (as in
+our experiment). When no noise apart from shot noise is present (as in simulation), our method allows to
+saturate this fundamental limit for large enough (∼ 5 · 104) number of photons detected per pixel. Other
+possible sources of noise (e.g.) dark counts may slightly affect the MSE obtained experimentally.
+Here we show that it is possible to retrieve complete phase profiles only with an average of two photons
+detected per frame which is an absolute minimum of detected photons per frame.
+Conclusion and Discussion
+In conclusion, we demonstrate a complete retrieval of phase patterns in the presence of high-frequency
+random phase fluctuations up to the order of tens of MHz when standard phase imaging techniques
+fail due to the scarcity of photons within a stable phase time interval. Our method is applicable to light
+sources described with different statistics, such as for example thermal light sources, and can be extended
+to interference between independent sources (21,28).
+6
+
+Figure 4: Experimental measurement of the spatial phases with the SLM - spatial light modulator. Mea-
+sured coincidence maps (correlation functions) between outputs of the interferometer for (a) sinusoidal,
+and (d) exponential phases set on SLM. Each axis of coincidence maps represents the positions of pho-
+tons detected along one output of the interferometer. (b) and (e) represent the aforementioned recon-
+structed phases. (c) and (f) show errors and square-root of the intensities.
+7
+
+We want to highlight, that the presented method optimality is proven using the Cramer-Rao bound –
+all the spatial phase information stored in the detected photons is retrieved (29).
+High temporal resolution (short gating time) is necessary for overcoming the problem of the rapidly
+fluctuating temporal phases. Such high temporal resolution in our experiment was obtained using an
+image-intensified camera, which allows us to collect data with short exposure times down to a few
+nanoseconds. However, our method is not limited to this camera type and can be implemented using
+various high-temporal resolution detection platforms. Because of high quantum efficiency, temporal
+resolution, and low noise level in recent single-photon avalanche diode (SPAD) array technology (30)
+development, our method can also be implemented by SPAD arrays in the future. We stress that the tech-
+nique can be implemented both in the photon counting regime and by employing less accurate intensity
+measurements, yet it is the most remarkable for cases where registering more than two photons per phase
+stability time is rare. Our method can be readily generalized to two-dimensional spatial phase profiles
+by creating higher-dimensional correlation maps. It also allows for implementation in different degrees
+of freedom, such as temporal or spectral, allowing the creation of joint probability maps both for photon
+detection times or their detected wavelengths. It is also possible to incorporate an additional degree of
+freedom to a measurement, measuring for instance joint temporal-spatial correlations maps.
+Additionally, this method could be expanded for different situations, in which multiple photons are
+detected or photons are registered at the same output. Each pair of photons can be treated as a separate
+coincidence, so the number of coincidences scales with a number of detected photons n as
+�n
+2
+�. We can
+also create such coincidence maps for multiple photons within each of the interferometer outputs as well
+as between them. Such holograms can build up much faster and shorten measurement time while the
+physics behind them is the same.
+This method is only valid when all values of the global phase Θ have the same probability of ap-
+pearing during the time interval in which the whole measurement is performed. To satisfy this condition
+for arbitrary temporal phase noise, it is enough to add random uniformly distributed signal oscillating
+between 0 and 2π to the unknown global phase fluctuations Θ(t). In fact, the added noise can be much
+slower than the rate of phase global phase fluctuations Θ(t).
+Our method opens up possible applications in wavefront sensing under low light conditions for mi-
+croscopy as well as fundamental research. Unbalanced interferometers, such as ones used in the time–
+bin encoding could be of particular interest, as our method enables using additional degrees of freedom
+(multi-dimensional information encoding) while filtering out phase fluctuations arising, for instance,
+from unmatched optical paths. In addition, because of the shorter wavelengths of X-rays (also of neu-
+trons or electrons), X-ray interferometry (31,32) requires much tighter alignment and better mechanical
+stability of the interferometer. We emphasize that because our technique is phase noise resistant, it holds
+a potential for phase-sensitive imaging using X-ray interferometry. In addition, analogous techniques
+might also find applications in matter-wave interferometry (33,34).
+Acknowledgments
+We acknowledge discussions with Piotr Wegrzyn, Lukasz Zinkiewicz, Michal Jachura, Wojciech Wasilewski,
+and Marek Zukowski. This work was supported by the Foundation for Polish Science under the FIRST
+TEAM project ’Spatiotemporal photon correlation measurements for quantum metrology and super-
+resolution microscopy’ co-financed by the European Union under the European Regional Development
+8
+
+Fund (POIR.04.04.00-00-3004/17-00), and by the National Laboratory for Photonics and Quantum Tech-
+nologies—NLPQT (POIR.04.02.00.00-B003/18).
+Supplementary materials
+S1 - Fundamental precision limits of phase imaging with fluctuating reference arm
+S2 - Experimental setup details
+References
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+10
+
+Supplementary materials and methods
+1
+S1: Fundamental precision limits of phase imaging with
+fluctuating reference arm
+1.1
+The measurement model
+Two cameras are set on the two outputs of the interferometer, each of them consists of
+the same number of pixels npix. The sample area giving the additional phase φi is imaged
+to the pixel number i on both cameras. Only two photons are received per the stability
+time of the interferometer phase. A single measurement consists of a detection of these
+two photons. The output of the single measurement is a pair (i+/−, j+/−). Numbers i, j
+stand for the numbers of pixels in which photons were detected, whereas indices + or
+− indicates in which of the two outputs the corresponding photon was measured. The
+probability of measuring a single photon in a pixel i+/− is:
+p(i+/−) = ˜NIi
+1
+2(1 ± v cos(φi + θ)),
+(1)
+where ˜N is a normalization factor, v is interferometer visibility, θ is an extra, global,
+fluctuating phase, and Ii is the intensity of the illuminating the phase mask in the are
+corresponding to pixel i. Phase θ is stable during the detection of each photon pair,
+its value for each pair is independently drawn from the continuous uniform probability
+distribution U(0, 2π). There is no information about θ value in each experiment, so the
+observed probability of obtaining pair (i+/−, j+/−) in every single frame is:
+p(i+/−, j+/−) =
+� 2π
+0
+p(i+/−, j+/−, θ)dθ
+(2)
+p(i+/−, j+/−, θ) is a joint probability distribution of measuring pair (i+/−, j+/−) with the
+fixed value of θ. From equation 2 we obtain the formulas:
+p(i+, j+) = p(i−, j−) = NIiIj(2 + v2 cos(φi − φj))
+(3)
+p(i+, j−) = p(i−, j+) = NIiIj(2 − v2 cos(φi − φj))
+(4)
+N is a new normalization factor. The above equations are our starting point to further
+inference about the maximal precision of the measurement. Full information about each
+single measurement is included in the dependendence of the probability p of the specific
+result of a measurement (i±, j±) on the estimated parameters φi.
+1
+arXiv:2301.11969v1  [physics.optics]  27 Jan 2023
+
+1.2
+Cramér-Rao bound
+In order to calculate maximal precision of estimation of the parameters φi, Fisher Infor-
+mation (FI) matrix will be calculated. There are 4 different types of events, which can
+occur during one experiment - two photons may be detected in one output (+ or −) or
+in different outputs ( we distinguish between +− and −+). We can distinguish between
+these 4 types, so the FI is the sum of FI matrices for all events’ types:
+Ftot = F++ + F−− + F+− + F−+
+(5)
+From equations 3 and 4 we can simply conclude, that F++ = F−− and F+− = F−+. In
+the next part of the article F++ matrix will be calculated.
+In order to simplify the formulas, the following notation will be used:
+p(i+, j+) ≡ p(i, j),
+∂
+∂φk
+≡ ∂k,
+F ≡ F++
+The element of the FI matrix can be written in the following form:
+Fkl =
+npix
+�
+i,j=1
+∂kp(i, j)∂lp(i, j)
+p(i, j)
+,
+(6)
+Subsequently:
+∂kp(i, j) = NIiIjv2(δjk − δik) sin(φi − φj)
+(7)
+∂kp(i, j)∂lp(i, j) = (δjk − δik)(δjl − δil)N2I2
+i I2
+j v4 sin2(φi − φj)
+(8)
+Consequently, the matrix element is:
+Fkl =
+npix
+�
+i,j=1
+(δjk − δik)(δjl − δil)NIiIjv4 sin2(φi − φj)
+2 + v2 cos(φi − φj)
+(9)
+If k ̸= l, then for any m we have δmkδml = 0, so (δjk − δik)(δjl − δil) = −δjkδil − δikδjl.
+That means, that non-diagonal matrix elements are:
+Fkl = −2NIkIlv4 sin2(φk − φl)
+2 + v2 cos(φk − φl)
+,
+k ̸= l
+(10)
+With the help of the equality (δjk − δik)2 = δjk + δik − 2δikδjk we can obtain diagonal
+terms of F:
+Fkk = 2NIkv4
+npix
+�
+i=1
+Ii sin2(φi − φk)
+2 + v2 cos(φi − φk)
+(11)
+For any function f:
+npix
+�
+i=1
+f(φi, Ii) = npix⟨f(φi, Ii)⟩i,
+(12)
+2
+
+where ⟨f(φi, Ii)⟩i is the mean value of the function over all pixels. In the next steps,
+the number of pixels is assumed to be big and each phase in the sample occurs with the
+same frequency. What is more, intensity of illuminating beam Ii is assumed to change
+slowly compared to the change of phase φi. In other words, many different phases occur
+in the region with approximately constant intensity. From these assumptions we obtain
+the equality:
+⟨f(φi, Ii)⟩i = 1
+2π
+� 2π
+0
+f(φ, ⟨I⟩)dφ,
+(13)
+where ⟨I⟩ stands for the mean intensity of the illuminating beam.
+Using the above
+assumptions, we can rewrite equation 11 as:
+Fkk = 2NIk⟨I⟩v4 npix
+2π
+� 2π
+0
+sin2(φ − φk)
+2 + v2 cos(φ − φk)dφ
+(14)
+Consequently all diagonal terms of F are the same:
+Fkk = 2N⟨I⟩Iknpix(2 −
+�
+4 − v4)
+(15)
+Now we need to calculate the value of a normalization factor N. We will use the fact,
+that sum of propabilities of all events must be equal to one:
+npix
+�
+i,j=1
+p(i+, j+) + p(i+, j−) + p(i−, j+) + p(i−, j−) = 1
+(16)
+Using equations 3 and 4 we obtain:
+8N
+npix
+�
+i,j=1
+IiIj = 1
+(17)
+We can rewrite the sum in the above equation as:
+npix
+�
+i,j=1
+IiIj =
+�npix
+�
+i=1
+Ii
+�2
+= n2
+pix⟨I⟩2
+(18)
+and obtain:
+N =
+1
+8n2
+pix⟨I⟩2
+(19)
+Finally F++ matrix can be written in the form:
+Fkl =
+�
+�
+�
+�
+�
+�
+�
+1
+4npix
+Ik
+⟨I⟩(2 −
+√
+4 − v4)
+for k = l
+−
+1
+4n2
+pix
+IkIl
+⟨I⟩2
+2v4 sin2(φk−φl)
+2+v2 cos(φk−φl)
+for k ̸= l
+(20)
+3
+
+We have calculated F++ matrix, which is obviously similar to F−− matrix, because
+formulas for propabilities in both cases are the same. Analogous calculation show, that
+also F+− = F−+ = F++. Using the FI additivity we obtain the terms of Ftot matrix:
+Ftot = 4F++
+(21)
+This is the FI matrix for a single measurement. If the whole experiment consists of nmes
+independent repetitions of the single measurement, we obtain the FI:
+F (nmes)
+tot
+= 4nmesF++ = 2nphotF++,
+(22)
+where nphot stands for the total number of measured photons during the experiment. In
+the next part F stands for the whole FI associated with detection of nphot number of
+photons. Terms of this matrix are:
+Fkl =
+�
+�
+�
+�
+�
+�
+�
+nphot
+npix
+Ik
+⟨I⟩(1 −
+�
+1 − v4/4)
+for k = l
+− nphot
+2n2
+pix
+IkIl
+⟨I⟩2
+2v4 sin2(φk−φl)
+2+v2 cos(φk−φl)
+for k ̸= l
+(23)
+From the Cramer-Rao bound, the minimal possible variance for estimating φk satisfy
+the inequality:
+∆2φk ≥ (F −1)kk
+(24)
+In general, the estimator which satisfy the above inequality may not exist, however, it is
+possible to get arbitrary close to the above bound if the number of measurement is big
+enough. That means, that the inequality becomes an equality if nphot → ∞. To simplify
+the calculations we also use the inequality:
+(F −1)kk ≥ (Fkk)−1,
+(25)
+which is true for all hermitian F. It’s clear, that in the general case the above inequality is
+not saturable. However, in our case the non-diagonal terms are asymptotically npix times
+smaller than diagonal terms. npix is also size of the F matrix. It may be proven, that for
+such scaling of non-diagonal terms with the size of matrix, the above inequality becomes
+saturable for npix → ∞. Using both of above inequalities, we obtain the following bound:
+∆φk ≥
+�
+npix⟨I⟩
+nphotIk
+1
+�
+1 −
+�
+1 − v4/4
+(26)
+The value nk = nphotIk
+npix⟨I⟩ may be interpreted as the expected value of photons detected in
+pixel number k ( in any output). The above bound may be rewritten in the intuitive
+form:
+∆φk ≥
+�
+1
+nk
+1
+�
+1 −
+�
+1 − v4/4
+(27)
+From this form of the inequality it’s clear, that the accuracy of measuring the value of the
+particular phase depends directly on the numer of photons interacting with the measured
+area.
+4
+
+1.3
+Comparison with long-stability-time interferometer
+Let’s compare our result with the phase estimation precision limit for an interferometer
+with slowly fluctuating phase θ. First of all, let’s notice that we can’t beat the accuracy
+achievable in the situation, in which extra phase θ is known for all the detected photons.
+Indeed, the information we get in a situation with unknown θ is always smaller, even
+if the stability time if the interferometer is bigger. If θ values are known, each single
+photon detection could be treated as an independent event (which was not the case in
+the previous section). Let’s calculate the FI matrix for the single photon detection when
+θ is fixed. Single measurement is fully described by the probability distribution from
+equation 1. Further we obtain:
+∂kp(i+/−) = ∓1
+2δki ˜NIiv sin(φi + θ)
+(28)
+In this case FI matrix has the form:
+Fkl =
+npix
+�
+i=1
+∂kp(i+)∂lp(i+)
+p(i+)
++
+npix
+�
+i=1
+∂kp(i−)∂lp(i−)
+p(i−)
+(29)
+From equation 28 it’s clear, that all non-diagonal terms of the F matrix are equal to
+zero. This is because we obtain information about the φi phase only in case of detection
+a photon in the pixel i+/−. The diagonal terms are:
+Fkk = ˜NIi
+v2 sin2(φi + θ)
+1 − v2 cos2(φi + θ)
+(30)
+To make this case similar to the case descriped in the previous section let’s assume, that
+θ fluctuates and each value of θ appears with the same frequency ( the difference is that θ
+fluctuates slowly and we know it’s value). Then the mean FI for the single measurement
+is:
+⟨Fkk⟩θ = 1
+2π
+� 2π
+0
+Fkkdθ =
+Ii
+npix⟨I⟩
+�
+1 −
+�
+1 − v2
+�
+,
+(31)
+where formula ˜N =
+1
+npix⟨I⟩ obtained from the normalization condition was used. If nmes
+measurements were made, nphot photons were consumed. If we define nk = nphotIk
+npix⟨I⟩ as in
+the previous section, we obtain the best possible accuracy of measuring each phase φk:
+∆φk ≥
+�
+1
+nk
+1
+�
+1 −
+√
+1 − v2
+(32)
+Equation 32 is very similar to the equation 27- the only difference is that term v4
+4 is
+substitude by the term v2. That means, that having only two photons per phase fluc-
+tuations stability time, leads to decrease of the effective visibility of the interferometer
+from v to v2
+2 . As it was mentioned, it’s not possible to beat the bound from equation
+32 if θ value is not known in each measurement, even if the number of detected photons
+5
+
+in a phase stability time was increased. However, we can get close to that bound, if the
+phase stability time is big enough. Indeed, if we can measure many photons, when θ is
+stable, we don’t really need to care about its unknown value and obtain relative values
+of φk using the same method as in case of known θ ( it might be assumed to equal 0).
+This scheme is repeated independently for each θ . The bound from the equation 30 is
+saturated, because the number of measurements is big enough. That means, that we can
+also saturate the bound resulting from the mean FI (equation 32).
+2
+S2: Experimental setup details
+This is a polarization-based Michelson interferometer. As a light source, we use a diode
+laser at a wavelength of 780 nm coupled to a single-mode fiber. At the output of the
+fiber, for polarization control, the attenuated beam passes through a half-wave plate, a
+quarter-wave plate, and polarizing beam splitter (PBS), and another half-wave plate, and
+then enters a Michelson-type interferometer. Each of the two paths in the interferometer
+is encoded with orthogonal polarization. In order to prepare the object beam, in one
+of the arms of the interferometer, we build two kinds of slightly modified setups - one
+with a cylindrical lens placed in front of one of the mirror in the horizontally polarized
+light beam path in the Michelson interferometer while in the other setup we replace
+the mirror in the same path with a spatial light modulator (SLM), thereby introducing
+spatially varying phase φ(x) onto the beam in that path. In one arm the spatial phase
+φ(x) is introduced next to the surface of the interferometer mirror. The interferometeric
+mirror in the other arm is given a phase fluctuation by attaching it to a piezoelectric
+actuator.
+We perform experiments with three kinds of different phase masks applied to our
+object beam. Our first configuration is to imprint a one dimensional quadratic local phase
+profile to the beam by placing a cylindrical lens of focal length, f = 1000 mm in proximity
+to the mirror (Fig. 2 in the main text). Additionally, in our second configuration with
+SLM (from the HOLOEYE PLUTO) as a phase mask, we can display any arbitrary phase
+profile. As an example, we imprint one dimensional exponential and sinusoidal phases
+to our object beam by the SLM display.
+We introduce a time-dependent phase fluctuation is in the other arm (the reference
+beam - vertically polarized beam path in the interferometer) to make it incoherent with
+the object beam. This is realized with a piezoelectric actuator driven by a RAMP of 1.234
+Hz. This shouldn’t be confused with the maximal noise frequency for which our method
+works. Both of the object and reference beams are combined on the polarizing beam
+splitter (PBS). Afterthat, they are imaged onto two regions of an Intensified sCMOS
+(I-sCMOS - with the image intensifier from Hamamatsu V7090D-71-G272 and sCMOS
+from Andor Zyla) camera with a 4f system using lenses L3 and L4 of focal length 200 mm.
+To observe the interference, the orthogonally polarized object and the reference beam
+are required to be indistinguishable, and to do so, we rotate the polarization of both
+beams by 45 degrees with a half-wave plate and we perform projective measurement in
+the original bases with a calcite crystal. Here, the calcite acts as a 50/50 Beamsplitter.
+6
+
+I-sCMOS
+Camera
+Calcite
+λ/2
+λ/4
+Laser
+PBS
+λ/2
+L2
+PH
+M
+L1
+λ/4
+λ/4
+PBS
+λ/2
+Delay line
+M
+SLM
+L3
+L4
+M
+M
+Figure 1:
+Experimental setup for noise-resistant phase imaging. The incoming beam
+of Laser after passing through a λ/2 - half-wave plate, λ/4 - quarter wave plate, PBS
+- polarization beam splitter, and another λ/2 plate, the beam enters a Michelson type
+interferometer. Each of the two paths in the interferometer is encoded with orthogonal
+polarization. In one arm the spatial phase φ(x) is introduced by the spatial light modu-
+lator (SLM). The interferometric mirror in the other arm is given a phase fluctuation by
+attaching it to a piezoelectric actuator. The two beams of the interferometric arms after
+combining at the PBS pass through L3, and L4 lenses. The calcite polarizer acts as a
+50/50 Beamsplitter. The I-sCMOS - Intensified sCMOS camera records single photons
+at both outputs of the interferometer. The use of short exposure time of the I-sCMOS,
+in the single nanosecond timescale, gives it stability and resistance against fluctuations
+up to tens of MHz..
+This mixes the light from both outputs and allows us to observe interference in both
+outputs of the splitter. The I-sCMOS camera records single photons at both outputs
+of the interferometer. The use of short exposure time of the I-sCMOS, in the single
+nanosecond timescale, gives it stability and resistance against fluctuations up to tens of
+MHz. We collect the data with 200 Hz of frame rate.
+7
+

A9AzT4oBgHgl3EQf__9t/content/tmp_files/2301.01956v1.pdf.txt

@@ -0,0 +1,1542 @@
+High-level semantic feature matters few-shot unsupervised domain adaptation
+Lei Yu1, Wanqi Yang1*, Shengqi Huang1, Lei Wang2, Ming Yang1
+1School of Computer and Electronic Information, Nanjing Normal University, China
+2School of Computing and Information Technology, University of Wollongong, Australia
+[email protected], [email protected], [email protected], [email protected], [email protected]
+Abstract
+In few-shot unsupervised domain adaptation (FS-UDA), most
+existing methods followed the few-shot learning (FSL) meth-
+ods to leverage the low-level local features (learned from con-
+ventional convolutional models, e.g., ResNet) for classifica-
+tion. However, the goal of FS-UDA and FSL are relevant yet
+distinct, since FS-UDA aims to classify the samples in target
+domain rather than source domain. We found that the local
+features are insufficient to FS-UDA, which could introduce
+noise or bias against classification, and not be used to effec-
+tively align the domains. To address the above issues, we aim
+to refine the local features to be more discriminative and rele-
+vant to classification. Thus, we propose a novel task-specific
+semantic feature learning method (TSECS) for FS-UDA.
+TSECS learns high-level semantic features for image-to-class
+similarity measurement. Based on the high-level features, we
+design a cross-domain self-training strategy to leverage the
+few labeled samples in source domain to build the classi-
+fier in target domain. In addition, we minimize the KL diver-
+gence of the high-level feature distributions between source
+and target domains to shorten the distance of the samples be-
+tween the two domains. Extensive experiments on Domain-
+Net show that the proposed method significantly outperforms
+SOTA methods in FS-UDA by a large margin (i.e., ∼ 10%).
+keywords
+Few-shot unsupervised domain adaptation, image-to-class
+similarity, high-level semantic features, cross-domain self-
+training, cross-attention.
+Introduction
+Currently, a setting namely few-shot unsupervised domain
+adaptation (FS-UDA) (Huang et al. 2021)(Yang et al. 2022),
+which utilizes few labeled data in source domain to train
+a model to classify unlabeled data in target domain, owns
+its potential feasibility. Typically, a FS-UDA model could
+learn general knowledge from base classes during training
+to guide classification in novel classes during testing. It is
+known that both insufficient labels in source domain and
+large domain shift make FS-UDA as a challenging task.
+Previous studies, e.g., IMSE (Huang et al. 2021), first fol-
+lowed several few-shot learning (FSL) methods (Li et al.
+*The corresponding author is Wanqi Yang.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+Figure 1: A 5-way 1-shot task for FS-UDA where the sup-
+port set includes five classes and one sample for each class.
+The figure shows the similarity of query images to every
+support classes and the spatial similarity of query images
+to the predicted support class. We found using local fea-
+tures could cause some inaccurate regions of query images
+to match the incorrect classes, while our semantic features
+make the object region in query images similar with their
+true class, thus achieving correct classification.
+2019)(Tzeng et al. 2017) to learn the local features by us-
+ing convolutional models (e.g., ResNet) and then leveraged
+them to learn image-to-class similarity pattern for classifica-
+tion. However, we wish to clarify that the goal of FS-UDA
+and FSL are relevant yet distinct, since both of them suf-
+fer from insufficient labeled training data whereas FS-UDA
+aims to classify the samples in target domain rather than
+source domain. As shown in Fig. 1, by visualizing the spatial
+similarity of query images to predicted support classes, we
+found using local features causes the inaccurate regions of
+query images to match incorrect classes. This reason might
+be that few labeled samples and large domain shift between
+the support and query sets simultaneously result in the con-
+ventional local features in FSL to fail in classification. In this
+sense, the local features are insufficient to FS-UDA, which
+could introduce noise or bias against the classification in tar-
+get domain and not be used to effectively align the domains.
+To address this issue, we aim to refine the low-level local
+arXiv:2301.01956v1  [cs.CV]  5 Jan 2023
+
+support set in the source domain (sketch)
+sailboat
+bed
+glasses
+television
+snowman
+query set in the target domain (clipart)
+local features
+semantic features (ours)
+0.6
+0.6
+0.5
+0.5
+0.4
+0.4
+0.3
+0.3
+0.2
+0.2
+0.1
+bed
+local features
+semantic features (ours)
+0.6
+0.6
+0.5
+0.5
+0.4
+0.4
+0.3
+0.3
+0.2
+0.2
+0.1
+television
+as
+saFigure 2: Illustration of the process for cross-domain self-
+training in TSECS. Different shapes represent different do-
+mains. We first select the ‘confidence’ target samples (e.g.,
+a) that are very similar to support classes, and then regard
+them as the new class prototypes to further classify the other
+target samples (e.g., b, c). This process is executed itera-
+tively with using class matching loss to narrow the distance
+of query images and their most similar support classes.
+features to be more discriminative and relevant to classifica-
+tion, i.e., high-level semantic features, and meanwhile align
+the semantic features for domain adaptation. Therefore,
+we propose a novel task-specific semantic feature method
+(TSECS) that learns the semantic features for each task by
+clustering the local features of support set and query set. To
+obtain the related semantics from previous tasks, the cluster
+centroids of the current task are then fused by cross-attention
+with that of the previous task to generate high-level semantic
+features to boost classification performance.
+Moreover, for the domain shift between source and tar-
+get domains, many domain adaptation methods (Saito et al.
+2018)(Tzeng et al. 2017)(Tzeng et al. 2014) reduced the dis-
+tribution discrepancy between domains by using a discrim-
+inator to adverse against feature embedding. However, this
+way could fail in aligning the samples of the same class be-
+tween domains due to label missing in target domain, which
+could make the classes of two domains mismatched and thus
+affect the classification. Therefore, we aim to align the high-
+level semantic features by minimizing the KL divergence
+of the semantic feature distributions between domains, and
+meanwhile design a cross-domain self-training strategy to
+train the classifier in target domain.
+We hypothesis that there are usually several ‘confidence’
+samples in target domain that could be classified correctly by
+support set in source domain, in other words, they are very
+similar to their class prototypes in source domain. Mean-
+while, the target domain samples in the same class are more
+similar to each other than that of other classes. Based on this,
+we regard these ‘confidence’ samples in the target domain as
+new prototypes of the classes, which replace those from the
+support set of source domain. As shown in Fig. 2, several
+‘confidence’ samples (e.g., a) can be selected as prototypes
+of their similar classes for classification (e.g., b and c) in tar-
+get domain. Moreover, the process is conducted iteratively
+by using class matching loss for better domain alignment.
+In sum, we propose the novel method, namely TSECS,
+for FS-UDA. It refines the local features of convolutional
+network to generate specific semantic features of each task,
+and meanwhile perform cross-domain self-training to trans-
+port labels from support set in the source domain to query
+set in the target domain to effectively classify the samples in
+target domain. Our contributions can be summarized as:
+(1) A novel solution for FS-UDA. TSECS aims to learn
+high-level semantic features for classification and do-
+main alignment, which could be regarded as a more ef-
+fective and efficient way than using local features.
+(2) Task-specific semantic embedding for few-shot set-
+ting. It can be seamlessly add to existing FSL/FS-UDA
+models, which could alleviate the bias of classification.
+(3) Cross-domain self-training for domain alignment. It
+is designed to bring the samples of the same class close,
+which could guide effective domain alignment.
+We conduct extensive experiments on DomainNet. Our
+method significantly outperforms SOTA methods in FS-
+UDA by a large margin up to ∼ 10%.
+Related Works
+Unsupervised domain adaptation. The conventional UDA
+methods aim to reduce discrepancy between source domain
+and target domain in the feature space and utilize suffi-
+ciently labeled source domain data to classify data from tar-
+get domain. The difference between unsupervised domain
+adaptation methods often lies in the evaluation of domain
+discrepancy and the objective function of model training.
+Several researchers (Long et al. 2015)(Tzeng et al. 2014)
+minimize the feature discrepancy by using maximum mean
+discrepancy to measure the discrepancy between the distri-
+bution of domains. Moreover, adversarial training (Tzeng
+et al. 2017)(Ganin et al. 2016) to learn domain-invariant fea-
+tures is usually used to tackle domain shift. Several meth-
+ods (Tang, Chen, and Jia 2020)(Zou et al. 2019)(Zou et al.
+2018)(Kim et al. 2021)train the classifier in both source do-
+main and target domain and utilize pseudo-labels from target
+domain to calculate classification loss. Overall, these UDA
+methods all require sufficiently labeled source domain data
+to realize domain alignment and classification, but they per-
+form poor when labeled source domain data are insufficient.
+Few-shot learning. Few-shot learning has two main
+streams, metric-based and optimization-based approaches.
+Optimization-based methods (Bertinetto et al. 2019)(Finn,
+Abbeel, and Levine 2017)(Ravi and Larochelle 2017) usu-
+ally train a meta learner over auxiliary dataset to learn
+a general initialization model, which can fine-tune and
+adapt to new tasks very soon. The main purpose of metric-
+based methods (Li et al. 2019)(Snell, Swersky, and Zemel
+2017)(Vinyals et al. 2016)(Ye et al. 2020) is that learn a gen-
+eralizable feature embedding for metric learning, which can
+immediately adapt to new tasks without any fine-tune and
+retraining. Typically, ProtoNet (Snell, Swersky, and Zemel
+2017) learns the class prototypes in the support set and clas-
+sifies the query images based on the maximum similarity
+to these prototypes. Other than these metric-based methods
+on feature maps, many methods on local features have ap-
+peared. DN4 (Li et al. 2019) utilizes large amount of local
+features to measure the similarity between support and query
+
+select 'confidence" sanples
+use new prototypes for
+as new prototypes
+classification in target domain
+O
+b
+O
+0
+00
+00
+lass natching loss
+0
+00
+dims prototypes
+doeifiad query imega
+[sonmce dm ngin)
+(trt domain)
+at din)
+(trt domain)sets instead of flattening the feature map into a long vec-
+tor. Based on local features, DeepEMD (Zhang et al. 2020)
+adopts Earth Mover’s Distance distance to measure the re-
+lationship between query and support sets. Furthermore, a
+few recent works focus on the issue of cross-domain FSL in
+which domain shift exists between data of meta tasks and
+new tasks. The baseline models (Chen et al. 2019) are used
+to do cross-domain FSL. LFT (Tseng et al. 2020) performs
+adaptive feature transformation to tackle the domain shift.
+Few-shot unsupervised domain adaptation. Compared
+with UDA, FS-UDA is to deal with many UDA tasks by
+leveraging few labeled source domain samples for each. And
+compared with cross-domain FSL, FS-UDA are capable of
+handling the circumstances of no available labels in the tar-
+get domain, and large domain gap between the support and
+query sets in every task. For the one-shot UDA (Luo et al.
+2020), it deals with the case that only one unlabeled target
+sample is available, but does not require the source domain
+to be few-shot, which is different from ours. Recently, there
+are a few attempts in FS-UDA. PCS (Yue et al. 2021) per-
+forms prototype self-supervised learning in cross-domain,
+but they require enough unlabeled source samples to learn
+prototypes and ignore task-level transfer, which is also dif-
+ferent from ours. meta-FUDA (Yang et al. 2022) lever-
+ages meta learning-based optimization to perform task-level
+transfer and domain-level transfer jointly. IMSE (Huang
+et al. 2021) utilizes local features to learn similarity patterns
+for cross-domain similarity measurement. However, they did
+not consider that local features could bring the noise or bias
+to affect classification and domain alignment. Thus, we pro-
+pose task-specific semantic features to solve this problem.
+Methodology
+Problem Definition
+A N-way, K-shot FS-UDA task. Table 1 shows the main
+symbols used in this paper. The FS-UDA setting includes
+two domains: a source domain S and a target domain T.
+A N-way, K-shot FS-UDA task includes a support set XS
+from S and a query set QT from T. The support set XS
+contains N classes and K samples per class in the source
+domain. The query set QT contains the same N classes as
+in XS and Nq target domain samples per class. To classify
+query images in QT to the correct class in XS, it is popular
+to train a general model from base classes to adapt to handle
+new N-way, K-shot FS-UDA tasks for testing.
+Auxiliary dataset and episodic training. As in (Huang
+et al. 2021), the base classes are collected from an auxil-
+iary dataset Daux to perform episodic training to learn the
+general model. Note that the base classes in Daux are com-
+pletely different from new classes in testing tasks, which are
+unseen during episodic training. Moreover, Daux includes
+labeled source domain data and unlabeled target domain
+data for FS-UDA. We construct large amounts of episodes,
+each containing {XS, QS, QT } as in (Huang et al. 2021), to
+simulate the testing tasks for task-level generalization. Note
+that QS is introduced into episodic training to calculate clas-
+sification loss and perform domain alignment with QT .
+The flowchart of our method. Fig. 3 illustrates our
+Table 1: Notations
+Notations
+Descriptions
+N ∈ R
+The number of classes in the task.
+K ∈ R
+The number of samples per class in support set.
+XS, QS, QT
+Support set of source domain, and query sets
+of source domain and target domain.
+H, W, d ∈ R
+The height, width, and channel of feature map.
+L ∈ RHW ×d
+The local feature vectors in the feature map.
+k ∈ R
+The number of semantic clusters for an episode.
+C ∈ Rk×d
+The centroids of the clusters.
+F, ˆF,
+The semantic feature map, semantic features and
+ˆFXS, ˆFQS, ˆFQT
+the parts of support and query sets in both domains.
+M c
+q ∈ RH×W ×N
+The 3-D similarity matrix for classification.
+pc
+q ∈ RKHW
+Similarity pattern vectors of a query image q
+pi
+q ∈ RHW
+with a support class c and a support image i,
+ppos
+q
+, pneg
+q
+∈ RKHW
+and the most similar class and the second one for q.
+µA, µB ∈ RHW ×d
+The mean of semantic features or similarity patterns.
+ΣA, ΣB ∈ RHW ×HW
+Covariance matrix of semantic features
+or similarity patterns.
+λsfa, λspa, λclm
+Weight parameters of three loss terms in Eq. (6).
+method for 5-way, 1-shot FS-UDA tasks. In each episode,
+a support set (XS) and two query sets (QS and QT ) are first
+through the convolution network (e.g., ResNet) to extract
+their local features. Then, the task-specific semantic embed-
+ding module refines the local features to generate semantic
+features, which is computational efficient due to dimension
+reduction. Also, based on semantic features of QS and QT ,
+we leverage their similarity patterns (Huang et al. 2021) to
+calculate image-to-class similarity for classification with the
+loss Lcls. To improve its performance, cross-domain self-
+training module is performed to introduce the class proto-
+types of target domain and train a target domain classifier
+with a class matching loss Lclm. In addition, the seman-
+tic features and similarity patterns from both domains are
+further aligned by calculating their alignment losses Lsfa
+and Lspa, respectively. Finally, the losses above are back-
+propagated to update our model. After episodic training over
+all episodes, we utilize the learned model to test new FS-
+UDA tasks. Then, we calculate the averaged classification
+accuracy on these tasks for performance evaluation.
+Task-specific Semantic Feature Learning
+Most FSL methods and FS-UDA methods learned local fea-
+tures from convolutional networks for classification. How-
+ever, we found that the local features could introduce noise
+or bias that is valid for classification and domain alignment.
+Thus, we aim to refine the local features to generate high-
+level semantic features for each task. In the following, we
+will introduce our semantic feature embedding module.
+First of all, in each episode, all local features L
+∈
+R(|XS|+|QS|+|QT |)HW ×d are extracted from the convolu-
+tional network, where | · | is the number of samples in a
+set. Then, we cluster the local features to generate different
+semantic clusters for support set and query set, respectively,
+since clustering the two sets together could result in the clus-
+ters that relate to the domains due to the presence of large do-
+main gap. For simplification, we adopt K-means for cluster-
+ing, and meanwhile utilize the singular value decomposition
+(SVD) to adaptively take the number of eigenvalues greater
+than a certain threshold as the cluster number k (k ≪ d) for
+each task. Afterwards, we calculate the task-specific seman-
+
+Figure 3: Illustration of our method training per episode for 1-shot FS-UDA tasks. First, support classes and query images
+from both domains are through a convolution network to extract their local features, followed by the task-specific semantic
+embedding module to learn high-level semantic features. Then, these semantic features are fed into the cross-domain self-
+training module to update the class prototypes for target domain classification and calculate the class matching loss Lclm.
+Meanwhile, these semantic features are also used to generate similarity patterns in IMSE (Huang et al. 2021) for classification
+loss Lcls. In addition, both semantic features and similarity patterns from both domains are aligned by the domain alignment
+module with the alignment losses Lsfa and Lspa, respectively. Finally, all the losses are backpropagated to update our model.
+tic feature map F ∈ R(|XS|+|QS|+|QT |)HW ×k by measuring
+the Cosine similarity between the local features L and the
+centroids C ∈ Rk×d of all semantic clusters, i.e., F =
+L
+||L||2 ·
+C⊤
+||C||2 . Finally, we split F to 2×2 blocks based on height and
+weight dimension of the feature map, and then concatenate
+the four blocks together along the channel to generate se-
+mantic features ˆF ∈ R
+1
+4 (|XS|+|QS|+|QT |)HW ×4k. This is a
+simple yet effective way to maintain discriminative ability
+and spatial information of semantic features.
+Moreover, to leverage the semantics from previous tasks
+to guide the semantic feature learning of the current task, we
+utilize the centroids of previous clusters to update the initial-
+ization of clustering centroids by cross-attention (Li et al.
+2020). This makes K-means clustering converge rapidly.
+After obtaining the semantic features ˆF, we use them for
+domain alignment and classification. Firstly, ˆF is partitioned
+into ˆFXS, ˆFQS, ˆFQT along with the first dimension. Then,
+we align ˆFQS and ˆFQT by minimizing the KL divergence of
+their distributions that will be introduced later. Meanwhile,
+we utilize ˆFXS, ˆFQS and ˆFQT to build 3-D similarity matrix
+M c
+q (Huang et al. 2021) between support and query sets. Fi-
+nally, we calculate the similarity pattern pc
+q (measuring the
+similarity between query sample q and support class c) for
+classification (Huang et al. 2021). The classification loss us-
+ing cross-entropy can be written by:
+Lcls = −
+1
+|QS|
+�
+q∈QS
+log(
+exp(1 · pc
+q)
+�K
+i=1 exp(1 · piq)
+)
+(1)
+Cross-domain Self-training
+Since there is large domain shift between source and target
+domains, as well as label missing in target domain, adver-
+sarial domain adaptation on low-level local features cannot
+make samples of the same class between domains close, and
+thus could make the classes of two domains mismatched.
+To alleviate the mismatching issue, we aim to find the
+most similar ‘confidence’ samples in QT with XS to guide
+classification in target domain. We assume that it usually
+exists that the ‘confidence’ samples in QT could be clas-
+sified correctly by XS, when the distributions between do-
+mains are aligned. We iteratively select the ‘confidence’
+samples in QT as the new prototypes to replace that in XS
+for classification, as shown in Fig. 2. We call the process as
+cross-domain self-training. The process can find more ‘con-
+fidence’ samples from QT than that in XS for the same
+class, which could correct some misclassified samples in
+QT , thereby lightening the impact of domain gap.
+Moreover, to improve the performance of the target do-
+main classifier, we aim to make target domain samples q
+in QT closer to their most similar class and meanwhile far
+away from the other classes. Thus, we first calculate its sim-
+ilarity patterns ppos
+q
+(with the most similar class) and pneg
+q
+(with the second similar class), and then design the class
+matching loss with a margin m, which can be written by
+Lclm =
+�
+q∈QT
+max(softmax(pneg
+q
+)−softmax(ppos
+q
+)+m, 0),
+(2)
+where the similarity to the most similar class should be
+greater by m than the second similar class.
+Two-level Domain Alignment
+Conventional
+adversarial
+domain
+adaptation
+methods
+(Ganin et al. 2016)(Tzeng et al. 2017) iteratively train a
+discriminator to align the distribution of domains by adver-
+sarial training among tasks. However, they cannot be used
+to align the semantic features, because our semantic features
+are relevant to tasks, the semantics of the same channel
+
+Task-specific semantic embedding
+Local features
+Semantic feature maps
+High-level semantic features
+ Support class
+(Source domain)
+Similarity patterns in IMSE
+Qurey image
+MH
+(Target domain)
+conv
+Lcls
+Classification loss
+Query image
+(Source domain)
+Split into 2 x 2 blocks
+I Update the class
+ and concatenate them
+prototypes
+k
+Cross-domain self-training
+Domain alignment
+I Semantic features  Similarity paterns' I
+Centers of k clusters
+class prototype
+( confidence
+sample
+1
+KL(*，*)
+KL(*，*)
+Cp
+ Source domain in support set
+Clustering
+ Target domain in query set
+Lclm
+Lsfa
+Lspa
+Class matching loss
++ Source domain in query set
+Aligment loss
+Loss backpropagationcould be varied for different tasks. Meanwhile, symmetrical
+alignment could bring the inference information of the
+target domain to the source domain (Li et al. 2020). Thus,
+we use asymmetrical KL divergence to align the distribution
+of domains on both semantic features and similarity patterns
+within a task. Then, KL divergence can be calculated by:
+KL(A, B) =1
+2
+�
+tr(Σ-1
+AΣB) + ln(ΣA
+ΣB
+)
++(µA − µB)Σ-1
+A(µA − µB)⊤ − d
+�
+,
+(3)
+where µA, µB, ΣA and ΣB are the mean vectors and the co-
+variance matrices of sample matrix A and B, respectively.
+Thus, we minimize the KL divergence between semantic
+features ˆHQS and ˆHQT by
+Lsfa = KL( ˆFQS, ˆFQT ).
+(4)
+Meanwhile, we also minimize the KL divergence to align
+the similarity patterns {pc
+qS} of QS and {pc
+qT } of QT with
+class c, which can be written by
+Lspa =
+N
+�
+c=1
+KL({pc
+qS}, {pc
+qT }).
+(5)
+In sum, we combine all the above losses, w.r.t. classifi-
+cation (Eq. (1)), class matching (Eq. (2)) and KL-based do-
+main alignment (Eqs. (4) and (5)) to train our model on many
+episodes. The total objective function can be written by:
+min Lcls + λsfaLsfa + λspaLspa + λclmLclm,
+(6)
+where the hyper-parameters λsfa, λspa and λclm are intro-
+duced to balance the effect of different loss terms.
+Experiment
+DomainNet dataset. We conduct extensive experiments on a
+multi-domain benchmark dataset DomainNet to demonstrate
+the efficacy of our method. It was released in 2019 for the re-
+search of multi-source domain adaptation (Peng et al. 2019).
+It contains 345 categories and six domains per category, i.e.,
+quickdraw, clipart, real, sketch, painting and infograph do-
+mains. In our experiments, we follow the setting of IMSE
+in (Huang et al. 2021) to remove data insufficient domain
+infograph. There are 20 combinations totally for evaluation,
+and the dataset is split into 217, 43 and 48 categories for
+episodic training, model validation and testing new tasks,
+respectively. Note that in each split every category contains
+the five-domain images.
+Network architecture and setting. We employ ResNet-
+12 as the backbone of feature embedding network, which is
+widely used in few-shot learning (Huang et al. 2021) (Gi-
+daris et al. 2020). We obtain semantic features by first clus-
+tering the local features from each class of support set and
+two query sets and then concatenating them. During this pro-
+cess, we adopt cross-attention that consists of three convo-
+lution parameters to generate (Q, K, V ) for attention cal-
+culation. In cross-domain self-training module, we set the
+threshold 1.7 of similarity score to select the ‘confidence’
+samples in target domain. The margin m in Eq. (2) is empir-
+ically set to 1.5. In addition, we follow the setting of IMSE
+(Huang et al. 2021) to obtain similarity patterns. The hyper-
+parameters λsfa, λspa and λclm are set to 0.1, 0.05 and 0.01,
+by grid search, respectively.
+Model training, validation and testing. To improve the
+performance, before episodic training, the feature embed-
+ding network is pretrained by using source domain data in
+the auxiliary dataset, as in (Huang et al. 2021). Afterwards,
+we perform episodic training on 280 episodes, following the
+setting of (Huang et al. 2021). During episode training, the
+total loss in Eq. (6) is minimized to optimize the network
+parameters for each episode. Also, we employ Adam opti-
+mizer with an initial learning rate of 10-4, and meanwhile re-
+duce the learning rate by half every 280 episodes. For model
+validation, we compare the performance of different model
+parameters on 100 tasks, which is randomly sampled from
+the validate set containing 43 categories. Then, we select the
+model parameters with the best validation accuracy for test-
+ing. During the testing, we randomly select 3000 tasks to
+calculate the averaged top-1 accuracy on these tasks as the
+evaluation criterion.
+Comparison Experiments for FS-UDA
+We conduct extensive experiments on DomainNet to com-
+pare our method with five FSL methods (ProtoNet (Snell,
+Swersky, and Zemel 2017), DN4 (Li et al. 2019), ADM
+(Li et al. 2020), FEAT (Ye et al. 2020), DeepEMD (Zhang
+et al. 2020)), three UDA methods, (MCD (Saito et al. 2018),
+ADDA (Tzeng et al. 2017), DWT (Roy et al. 2019)), their
+combinations and the most related method IMSE (Huang
+et al. 2021). For fair comparison, the results of these above
+methods are all reported from (Huang et al. 2021) with the
+same setting. Moreover, we also modify IMSE by using
+our semantic features for classification and domain adver-
+sary, namely IMSE+TSE. For fair comparison, these com-
+pared methods also pretrain the embedding network before
+episodic training, and they are trained on 1000 episodes.
+Comparison analysis. Table 2 shows the results of all
+the compared methods for 20 cross-domain combinations,
+which records the averaged classification accuracy of tar-
+get domain samples over 3000 5-way 1-shot/5-shot FS-
+UDA tasks. As observed, our TSECS achieves the best per-
+formance for all combinations and their average. Specifi-
+cally, the UDA and FSL baselines in the first two parts per-
+form the worst. In the third part, the combination methods
+with ADDA (Tzeng et al. 2017) perform domain adversarial
+training each episode, thus generally better than the above
+two parts, but still inferior to IMSE (Huang et al. 2021)
+and our TSECS. This is because the combination methods
+only perform domain alignment based on original feature
+maps, not considering the alignment of similarity patterns
+(related to classification predictions). Also, IMSE is worse
+than IMSE+TSE, which indicates high-level semantic fea-
+tures are more effective for FS-UDA than local features.
+However, they are still much worse than our method, show-
+ing the efficacy of high-level semantic features and cross-
+domain self-training for FS-UDA.
+On the other hand, we can see that the 20 cross-domain
+combinations have considerably different performances.
+This is because several domains (e.g., quickdraw) are sig-
+nificantly different from other domains, while several other
+domains (e.g. real, clipart) are with the similar styles and
+features. Thus, for most compared methods, the perfor-
+
+Table 2: Comparison of our method with the related methods for 5-way 1-shot or 5-shot FS-UDA tasks. The first three blocks
+and IMSE are reported from (Huang et al. 2021), while the last two are the variant of IMSE we designed and ours, respectively.
+Each row represents the accuracy (%) of a compared method adapting between two domains, where the skt, rel, qdr, pnt, and
+cli denote the sketch, real, quickdraw, painting, and clipart domains in DomainNet, respectively. The best results are in bold.
+5-way, 1-shot
+Methods
+skt ←→ rel
+skt ←→ qdr
+skt ←→ pnt
+skt ←→ cli
+rel ←→ qdr
+rel ←→ pnt
+rel ←→ cli
+qdr ←→ pnt
+qdr ←→ cli
+pnt ←→ cli
+avg
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+-
+MCD
+48.07/37.74
+38.90/34.51
+39.31/35.59
+51.43/38.98
+24.17/29.85
+43.36/47.32
+44.71/45.68
+26.14/25.02
+42.00/34.69
+39.49/37.28
+38.21
+ADDA
+48.82/46.06
+38.42/40.43
+42.52/39.88
+50.67/47.16
+31.78/35.47
+43.93/45.51
+46.30/47.66
+26.57/27.46
+46.51/32.19
+39.76/41.24
+40.91
+DWT
+49.43/38.67
+40.94/38.00
+44.73/39.24
+52.02/50.69
+29.82/29.99
+45.81/50.10
+52.43/51.55
+24.33/25.90
+41.47/39.56
+42.55/40.52
+41.38
+ProtoNet
+50.48/43.15
+41.20/32.63
+46.33/39.69
+53.45/48.17
+32.48/25.06
+49.06/50.30
+49.98/51.95
+22.55/28.76
+36.93/40.98
+40.13/41.10
+41.21
+DN4
+52.42/47.29
+41.46/35.24
+46.64/46.55
+54.10/51.25
+33.41/27.48
+52.90/53.24
+53.84/52.84
+22.82/29.11
+36.88/43.61
+47.42/43.81
+43.61
+ADM
+49.36/42.27
+40.45/30.14
+42.62/36.93
+51.34/46.64
+32.77/24.30
+45.13/51.37
+46.8/50.15
+21.43/30.12
+35.64/43.33
+41.49/40.02
+40.11
+FEAT
+51.72/45.66
+40.29/35.45
+47.09/42.99
+53.69/50.59
+33.81/27.58
+52.74/53.82
+53.21/53.31
+23.10/29.39
+37.27/42.54
+44.15/44.49
+43.14
+DeepEMD
+52.24/46.84
+42.12/34.77
+46.64/43.89
+55.10/49.56
+34.28/28.02
+52.73/53.26
+54.25/54.91
+22.86/28.79
+37.65/42.92
+44.11/44.38
+43.46
+ADDA+ProtoNet
+51.30/43.43
+41.79/35.40
+46.02/41.40
+52.68/48.91
+37.28/27.68
+50.04/49.68
+49.83/52.58
+23.72/32.03
+38.54/44.14
+41.06/41.59
+42.45
+ADDA+DN4
+53.04/46.08
+42.64/36.46
+46.38/47.08
+54.97/51.28
+34.80/29.84
+53.09/54.05
+54.81/55.08
+23.67/31.62
+42.24/45.24
+46.25/44.40
+44.65
+ADDA+ADM
+51.87/45.08
+43.91/32.38
+47.48/43.37
+54.81/51.14
+35.86/28.15
+48.88/51.61
+49.95/54.29
+23.95/33.30
+43.59/48.21
+43.52/43.83
+43.76
+ADDA+FEAT
+52.72/46.08
+47.00/36.94
+47.77/45.01
+56.77/52.10
+36.32/30.50
+49.14/52.36
+52.91/53.86
+24.76/35.38
+44.66/48.82
+45.03/45.92
+45.20
+ADDA+DeepEMD
+53.98/47.55
+44.64/36.19
+46.29/45.14
+55.93/50.45
+37.47/30.14
+52.21/53.32
+54.86/54.80
+23.46/32.89
+39.06/46.76
+45.39/44.65
+44.75
+IMSE
+57.21/51.30
+49.71/40.91
+50.36/46.35
+59.44/54.06
+44.43/36.55
+52.98/55.06
+57.09/57.98
+30.73/38.70
+48.94/51.47
+47.42/46.52
+48.86
+IMSE+TSE
+60.71/56.15
+53.78/48.57
+56.50/48.59
+61.59/56.59
+45.48/49.45
+55.44/57.45
+59.60/59.52
+37.94/39.83
+58.83/56.22
+49.19/51.01
+52.79
+TSECS (ours)
+65.00/58.22
+62.25/51.97
+56.51/53.70
+69.45/64.59
+56.66/49.82
+58.76/63.18
+67.98/67.89
+38.26/46.15
+60.51/69.03
+54.40/52.76
+58.20
+5-way, 5-shot
+Methods
+skt ←→ rel
+skt ←→ qdr
+skt ←→ pnt
+skt ←→ cli
+rel ←→ qdr
+rel ←→ pnt
+rel ←→ cli
+qdr ←→ pnt
+qdr ←→ cli
+pnt ←→ cli
+avg
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+→ / ←
+-
+MCD
+66.42/47.73
+51.84/39.73
+54.63/47.75
+72.17/53.23
+28.02/33.98
+55.74/66.43
+56.80/63.07
+28.71/29.17
+50.46/45.02
+53.99/48.24
+49.65
+ADDA
+66.46/56.66
+51.37/42.33
+56.61/53.95
+69.57/65.81
+35.94/36.87
+58.11/63.56
+59.16/65.7
+723.16/33.50
+41.94/43.40
+55.21/55.86
+51.76
+DWT
+67.75/54.85
+48.59/40.98
+55.40/50.64
+69.87/59.33
+36.19/36.45
+60.26/68.72
+62.92/67.28
+22.64/32.34
+47.88/50.47
+49.76/52.52
+51.74
+ProtoNet
+65.07/56.21
+52.65/39.75
+55.13/52.77
+65.43/62.62
+37.77/31.01
+61.73/66.85
+63.52/66.45
+20.74/30.55
+45.49/55.86
+53.60/52.92
+51.80
+DN4 (Li et al. 2019)
+63.89/51.96
+48.23/38.68
+52.57/51.62
+62.88/58.33
+37.25/29.56
+58.03/64.72
+61.10/62.25
+23.86/33.03
+41.77/49.46
+50.63/48.56
+49.41
+ADM
+66.25/54.20
+53.15/35.69
+57.39/55.60
+71.73/63.42
+44.61/24.83
+59.48/69.17
+62.54/67.39
+21.13/38.83
+42.74/58.36
+56.34/52.83
+52.78
+FEAT
+67.91/58.56
+52.27/40.97
+59.01/55.44
+69.37/65.95
+40.71/28.65
+63.85/71.25
+65.76/68.96
+23.73/34.02
+42.84/53.56
+57.95/54.84
+53.78
+DeepEMD
+67.96/58.11
+53.34/39.70
+59.31/56.60
+70.56/64.60
+39.70/29.95
+62.99/70.93
+65.07/69.06
+23.86/34.34
+45.48/53.93
+57.60/55.61
+53.93
+ADDA+ProtoNet
+66.11/58.72
+52.92/43.60
+57.23/53.90
+68.44/61.84
+45.59/38.77
+60.94/69.47
+66.30/66.10
+25.45/41.30
+46.67/56.22
+58.20/52.65
+54.52
+ADDA+DN4
+63.40/52.40
+48.37/40.12
+53.51/49.69
+64.93/58.39
+36.92/31.03
+57.08/65.92
+60.74/63.13
+25.36/34.23
+48.52/51.19
+52.16/49.62
+50.33
+ADDA+ADM
+64.64/54.65
+52.56/33.42
+56.33/54.85
+70.70/63.57
+39.93/27.17
+58.63/68.70
+61.96/67.29
+21.91/39.12
+41.96/59.03
+55.57/53.39
+52.27
+ADDA+FEAT
+67.80/56.71
+60.33/43.34
+57.32/58.08
+70.06/64.57
+44.13/35.62
+62.09/70.32
+57.46/67.77
+29.08/44.15
+49.62/63.38
+57.34/52.13
+55.56
+ADDA+DeepEMD
+68.52/59.28
+56.78/40.03
+58.18/57.86
+70.83/65.39
+42.63/32.18
+63.82/71.54
+66.51/69.21
+26.89/42.33
+47.00/57.92
+57.81/55.23
+55.49
+IMSE
+70.46/61.09
+61.57/46.86
+62.30/59.15
+76.13/67.27
+53.07/40.17
+64.41/70.63
+67.60/71.76
+33.44/48.89
+53.38/65.90
+61.28/56.74
+59.60
+IMSE+TSE
+72.75/62.24
+64.49/55.04
+62.86/61.10
+77.39/69.87
+53.88/54.48
+63.97/72.46
+69.86/72.49
+37.43/51.66
+64.43/67.46
+63.40/57.89
+62.76
+TSECS (ours)
+78.23/70.44
+77.90/55.77
+66.70/68.03
+83.82/74.28
+64.33/55.16
+68.40/79.74
+78.23/77.69
+39.74/63.02
+67.99/80.31
+73.67/61.63
+69.25
+Table 3: Ablation study (%) of the modules designed in
+TSECS, where the FS-UDA tasks are evaluated from a do-
+main (sketch) to the other four domains in DomainNet.
+Components
+Target Domains
+TSE
+catt
+CS
+cli
+rel
+qdr
+pnt
+✓
+61.98
+60.00
+52.21
+51.62
+✓
+57.07
+53.31
+41.93
+46.66
+✓
+✓
+62.74
+60.54
+53.64
+54.23
+✓
+✓
+68.25
+61.15
+58.31
+53.34
+✓
+✓
+✓
+69.45
+65.00
+62.25
+56.51
+mance becomes relatively low when the domain gap is large.
+For example, from quickdraw to painting, it performs the
+worst in all the other combinations because of larger domain
+gap, but our TSECS outperforms IMSE and the other com-
+pared methods by 8% and 12%, respectively. We found that
+our method has the larger performance improvement over
+IMSE, for these combinations containing quickdraw, which
+shows the efficacy of our method for large domain gap. Also,
+like TSECS, IMSE+TSE performs much better than IMSE
+for large domain gap, which indicates the high-level seman-
+tic features could conduct domain adaptation better than lo-
+cal features. In sum, these results reflect the advantages of
+our TSECS to deal with domain shift and task generaliza-
+tion in FS-UDA, no matter how large the domain gap is.
+Ablation study of our method. We conduct various ex-
+periments on DomainNet to evaluate the effect of our mod-
+ules: task-specific semantic embedding (TSE), cross-domain
+self-training (CS) and cross-attention in TSE (catt). The ac-
+curacies on the four target domains are reported in Table
+3. As seen, our method achieve the best performance when
+three modules are all used. The performance of the single
+CS is the worst that shows that local features cannot align
+the distributions of the two domains, thus affecting cross-
+domain self-training. The module TSE is introduced into
+four combinations, all improving the performance, which
+validates the efficacy of our task-specific semantic features
+for FS-UDA again. Also, the addition of cross-attention into
+TSE will further improve the performance, which can help
+discover more semantics from previous tasks.
+Ablation study of different losses. We conduct various
+experiments on DomainNet to further evaluate the effect of
+different losses in Eq. (6). Besides the classification loss
+(Lcls), we combine the remaining three loss terms: 1) se-
+mantic features alignment loss (Lsfa), 2) similarity pattern
+alignment loss (Lspa), and 3) class matching loss (Lclm).
+We evaluate 5-way 1-shot FS-UDA tasks from sketch to the
+other four domains, respectively, and their accuracies are re-
+ported in Table 4. As observed, the more the number of loss
+terms involved, the higher the accuracy. The combination of
+all the three losses is the best. For the single loss, both Lsfa
+
+Table 4: Ablation study (%) of the three losses designed in
+TSECS, where the FS-UDA tasks are evaluated from a do-
+main (sketch) to the other four domains in DomainNet.
+Components
+Target Domains
+Lsfa
+Lspa
+Lclm
+cli
+rel
+qdr
+pnt
+✓
+66.67
+58.84
+56.91
+43.28
+✓
+64.28
+57.32
+52.11
+42.46
+✓
+66.83
+58.29
+56.51
+44.25
+✓
+✓
+66.64
+62.64
+57.41
+53.40
+✓
+✓
+68.04
+63.98
+59.13
+55.39
+✓
+✓
+67.61
+62.47
+53.07
+54.14
+✓
+✓
+✓
+69.45
+65.00
+62.25
+56.51
+Figure 4: Comparison of introducing our TSE module or not
+into two FSL methods with ADDA (Tzeng et al. 2017) com-
+bined, i.e., ADDA+ProtoNet and ADDA+DN4.
+and Lclm perform better than Lspa, and their combination is
+also considerably better than the other paired combinations,
+showing the efficacy of semantic feature domain alignment
+and class matching in target domain. Based on the above,
+adding Lspa further improves the performance, indicating
+positive effect of aligning the similarity patterns.
+Evaluation on the effect of our task-specific se-
+mantic embedding module on two FSL methods with
+ADDA (Tzeng et al. 2017) combined. Compared with
+ADDA+DN4 and ADDA+ProtoNet, we add our semantic
+embedding module (TSE) with the loss Lsfa into their fea-
+ture embedding models, and test them on 3000 new 5-way
+1/5-shot FS-UDA tasks. For simplification and clarification,
+we calculate the averaged accuracies from every domain to
+the other four domains and show them in Fig. 4. As seen,
+the methods using TSE generally perform better than that
+without it, which validates that the semantic embedding in
+TSE could generate more discriminative semantic features
+for classification than original local features. In addition, the
+performances of these methods are still far from our method
+because using ADDA is insufficient to align the domains and
+could result in class mismatching, but our method can effec-
+tively solve it by cross-domain self-training.
+Evaluation of dataset generalization. We evaluate the
+generalization of our model trained on DomainNet to adapt
+to a substantially different dataset miniImageNet. We mod-
+ify miniImageNet by transferring a half of real images (rel)
+into sketch images (skt) by MUNIT (Huang et al. 2018) to
+Table 5: Evaluation (%) of dataset generalization for 5-way
+1-shot FS-UDA tasks between domains real and sketch, per-
+forming episodic training on DomainNet and testing on ex-
+panded dataset miniImageNet.
+Methods
+skt → rel
+rel → skt
+ADDA+DN4
+44.01 ± 0.87
+40.61 ± 0.90
+ADDA+DeepEMD
+46.14 ± 0.82
+45.91 ± 0.77
+IMSE
+48.78 ± 0.78
+48.52 ± 0.81
+TSECS (ours)
+53.33 ± 1.08
+49.83 ± 0.96
+Figure 5: The tSNE visualization of our TSECS using cross-
+domain self-training or not for a 5-way 5-shot FS-UDA task
+from sketch to clipart. The samples with different colors be-
+long to different classes, and the stars in the left and right
+figures represent the class centroids of support set and se-
+lected target domain query samples, respectively.
+produce two domains for FS-UDA. We compare our method
+with ADDA+DN4, ADDA+DeepEMD and IMSE for 5-way
+1-shot FS-UDA tasks for rel ↔ skt. The results are shown
+as Table 5. As observed, our method outperforms other
+methods, specially for ske → rel. For rel → skt, our method
+is slightly better than IMSE, because the style of sketch im-
+ages in miniImageNet is relatively different from that in Do-
+mainNet, which could effect the learned semantic features.
+Visualization of our method using cross-domain self-
+training or not. We illustrate the tSNE results of a 5-way 5-
+shot FS-UDA task from sketch to clipart in Fig. 5. Note that
+the class prototypes in the left subfigure belong to the sup-
+port set in source domain, while those in the right subfigure
+are generated by ‘confidence’ samples in target domain. It
+is obvious that two class prototypes in the left subfigure are
+fully overlapped so that many samples could not be correctly
+classified. In contrast, the right subfigure has the better class
+prototypes, and samples from different classes are more dis-
+tinguishable. This shows the efficacy of our cross-domain
+self-training that finds ‘confidence’ samples to train the tar-
+get domain classifier and uses class matching loss Lclm to
+shorten the distance of samples of the same class.
+Conclusion
+In this paper, we propose a novel method TSECS for FS-
+UDA. We extract high-level semantic features than local fea-
+tures to measure the similarity of query images in target do-
+main to support classes in source domain. Moreover, we de-
+sign cross-domain self-training to train a target domain clas-
+sifier. In addition, asymmetrical KL-divergence is used to
+align the semantic features between domains. Extensive ex-
+periments on DomainNet show the efficacy of our TSECS,
+significantly improving the performance for FS-UDA.
+
+ProtoNet/1-shot
+.-
+DN4/1-shot
+701
+701
+60
+60
+50
+50
+40
+40
+30
+30
+skt
+cli rel qdr
+pnt
+skt
+clirel qdr
+pnt
+ProtoNet/5-shot
+DN4/5-shot
+70
+70
+60
+60
+50
+50
+40
+40
+30
+30
+skt
+clirel
+Ipb
+pnt
+skt
+clirel qdr
+pnt
+ADDA+ProtoNet
+ADDA+DN4
+ADDA+TSE+ProtoNet
+ADDA+TSE+DN4
+ours
+oursTSECS (no CS)
+TSECS
+1.0
+1.0
+0.8
+0.8
+0.6
+0.6
+0.4
+0.4
+.
+★
+0.2 
+.
+0.2
+.
+：
+*
+.
+0.0
+.
+0.0
+.
+.
+.
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0Acknowledgments
+Wanqi Yang and Ming Yang are supported by Na-
+tional Natural Science Foundation of China (Grant Nos.
+62076135, 62276138, 61876087). Lei Wang is supported
+by an Australian Research Council Discovery Project (No.
+DP200101289) funded by the Australian Government.
+References
+Bertinetto, L.; Henriques, J. F.; Torr, P.; and Vedaldi,
+A. 2019.
+Meta-learning with differentiable closed-form
+solvers. In International Conference on Learning Represen-
+tations, 1–8.
+Chen, W.-Y.; Liu, Y.-C.; Kira, Z.; Wang, Y.-C. F.; and
+Huang, J.-B. 2019. A Closer Look at Few-shot Classifica-
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+PERCEPTUAL–NEURAL–PHYSICAL SOUND MATCHING
+Han Han, Vincent Lostanlen, and Mathieu Lagrange
+Nantes Universit´e, ´Ecole Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000 Nantes, France
+ABSTRACT
+Sound matching algorithms seek to approximate a target waveform
+by parametric audio synthesis. Deep neural networks have achieved
+promising results in matching sustained harmonic tones. However,
+the task is more challenging when targets are nonstationary and inhar-
+monic, e.g., percussion. We attribute this problem to the inadequacy
+of loss function. On one hand, mean square error in the parametric
+domain, known as “P-loss”, is simple and fast but fails to accommo-
+date the differing perceptual significance of each parameter. On the
+other hand, mean square error in the spectrotemporal domain, known
+as “spectral loss”, is perceptually motivated and serves in differen-
+tiable digital signal processing (DDSP). Yet, spectral loss has more
+local minima than P-loss and its gradient may be computationally
+expensive; hence a slow convergence. Against this conundrum, we
+present Perceptual-Neural-Physical loss (PNP). PNP is the optimal
+quadratic approximation of spectral loss while being as fast as P-loss
+during training. We instantiate PNP with physical modeling synthesis
+as decoder and joint time–frequency scattering transform (JTFS) as
+spectral representation. We demonstrate its potential on matching
+synthetic drum sounds in comparison with other loss functions.
+Index Terms— auditory similarity, scattering transform, deep
+convolutional networks, physical modeling synthesis.
+1. INTRODUCTION
+Given an audio synthesizer g, the task of sound matching [1] consists
+in retrieving the parameter setting θ that “matches” a target sound
+x; i.e., such that a human ear judges the generated sound g(θ) to
+resemble x. Sound matching has applications in automatic music
+transcription, virtual reality, and audio engineering [2, 3]. Of particu-
+lar interest is the case where g(θ) solves a known partial differential
+equation (PDE) whose coefficients are contained in the vector θ. In
+this case, θ reveals some key design choices in acoustical manufac-
+turing, such as the shape and material properties of the resonator.
+Over the past decade, the renewed interest for deep neural net-
+works (DNN’s) in audio content analysis has led researchers to formu-
+late sound matching as a supervised learning problem [4]. Intuitively,
+the goal is to optimize the synaptic weights W of a DNN f W so
+that f W(xn) = ˜θn approximates θn over a training set of pairs
+(xn, θn). Because g automates the mapping from parameter θn to
+sound xn, this training procedure incurs no real-world audio acquisi-
+tion nor human annotation. However, prior publications have pointed
+out that the approximation formula ˜θn ≈ θn lacks a perceptual mean-
+ing: depending on the choice of target xn, some deviations (˜θn−θn)
+may be judged to have a greater effect than others [5, 6, 7].
+The paradigm of differentiable digital signal processing (DDSP)
+has brought a principled methodology to address this issue [8]. The
+key idea behind DDSP is to chain the learnable encoder f W with
+the known decoder g and a non-learnable but differentiable feature
+map Φ. In DDSP, f W is trained to minimize the perceptual distance
+θ
+Parametric
+domain
+x
+original
+Audio
+domain
+S
+Perceptual
+domain
+˜θ
+˜x
+reconstruction
+˜S
+DDSP
+spectral ≈
+loss
+PNP
+quadratic
+form
+θ
+x
+˜θ
+M(θ)
+Riemannian
+metric
+g
+Φ
+f W
+g
+Φ
+g
+f W
+∇(Φ◦g)
+Fig. 1. Graphical outline of the proposed method. Given a known
+synthesizer g and feature map Φ, we train a neural network f W to
+minimize the “perceptual–neural–physical” (PNP) quadratic form
+⟨˜θ − θ
+��M(θ)|˜θ − θ⟩ where M is the Riemannian metric associated
+to (Φ◦g). Hence, PNP approximates DDSP spectral loss yet does not
+need to backpropagate ∇(Φ◦g)(˜θ) at each epoch. Transformations in
+solid (resp. dashed) lines can (resp. cannot) be cached during training.
+between vectors Φ(˜xn) = (Φ◦g◦f W)(xn) and Φ(xn) on average
+over samples xn. Yet, a practical shortcoming of DDSP is that it
+requires to evaluate the Jacobian ∇(Φ◦g) over each DNN prediction
+˜θn; and so at every training step, as W is updated by stochastic
+gradient descent (SGD).
+In this article, we propose a new learning objective for sound
+matching, named perceptual–neural–physical (PNP) autoencoding.
+The main contribution of PNP is to compute the Riemannian met-
+ric M associated to the Jacobian ∇(Φ◦g) over each sample θn (see
+Section 2.1). The PNP encoder is penalized in terms of a first-order
+Taylor expansion of the spectral loss ∥Φ(˜x) − Φ(x)∥2, making
+it comparable to DDSP. Yet, unlike in DDSP, the computation of
+∇(Φ◦g) is independent from the encoder f W: thus, it may be par-
+allelized and cached during DNN training. A second novelty of
+our paper resides in its choice of application: namely, differentiable
+sound matching for percussion instruments. This requires not only
+a fine characterization of the spectral envelope, as in the DDSP of
+sustained tones; but also of attack and release transients. For this
+purpose, we need g and Φ to accommodate sharp spectrotemporal
+modulations. Specifically, we rely on a differentiable implementation
+of the functional transformation method (FTM) for g and the joint
+time–frequency scattering transform (JTFS) for Φ.
+2. METHODS
+2.1. Approximating spectral loss with Riemannian geometry
+We assume the synthesizer g and the feature map Φ to be contin-
+uously differentiable. Let us denote by LDDSP the “spectral loss”
+arXiv:2301.02886v1  [cs.SD]  7 Jan 2023
+
+associated to the triplet (Φ, f W, g). Its value at a parameter set θ is:
+LDDSP
+θ
+(W) = 1
+2∥Φ(˜x) − Φ(x)∥2
+2
+= 1
+2
+��(Φ ◦ g ◦ f W ◦ g)(θ) − (Φ ◦ g)(θ)
+��2
+2
+(1)
+by definition of ˜x and x. Using ˜θ as shorthand for (f W ◦ g)(θ), we
+conduct a first-order Taylor expansion of (Φ ◦ g) near θ. We obtain:
+Φ(˜x) = Φ(x) + ∇(Φ◦g)(θ) · (˜θ − θ) + O(∥˜θ − θ∥2
+2),
+(2)
+where the Jacobian matrix ∇(Φ◦g)(θ) contains P = dim Φ(x) rows
+and J = dim θ columns. The differentiable map (Φ ◦ g) induces a
+weak Riemannian metric M onto the open set U ⊂ RJ of parameters
+θ, whose matrix values derive from ∇(Φ◦g)(θ):
+M(θ)j,j′ =
+P
+�
+p=1
+�
+∇(Φ◦g)(θ)p,j
+� �
+∇(Φ◦g)(θ)p,j′�
+.
+(3)
+The square matrix M(θ) ∈ GLJ(R) defines a positive semidefinite
+kernel which, once plugged into Equation 2, serves to approximate
+LDDSP
+θ
+(W) in terms of a quadratic form over (˜θ − θ):
+∥Φ(˜x) − Φ(x)∥2
+2 =
+�˜θ − θ
+��M(θ)
+��˜θ − θ
+�
++ O
+�
+∥˜θ − θ∥3
+2
+�
+. (4)
+The advantage of the approximation above is that the metric
+M may be computed over the training set once and for all. This is
+because Equation 3 is independent of the encoder f W. Furthermore,
+since θ is low-dimensional, we may store M(θ) on RAM. From
+this perspective, we define the perceptual–neural–physical loss (PNP)
+associated to (Φ, f W, g) as the linearization of spectral loss at θ:
+LPNP
+θ
+(W) = 1
+2
+�
+(f W ◦ g)(θ) − θ
+��M(θ)
+��(f W ◦ g)(θ) − θ
+�
+= LDDSP
+θ
+(W) + O
+�
+∥(f W ◦ g)(θ) − θ∥3
+2
+�
+.
+(5)
+According to the chain rule, the gradient of PNP loss at a given
+training pair (xn, θn) with respect to some scalar weight Wi is:
+∂LPNP
+θ
+∂Wi (θn) =
+�
+f W(xn) − θn
+���M(θn)
+���∂f W
+∂Wi (xn)
+�
+.
+(6)
+Observe that replacing M(θn) by the identity matrix in the equation
+above would give the gradient of parameter loss (P-loss); that is,
+the mean squared error between the predicted parameter ˜θ and the
+true parameter θ. Hence, we may regard PNP as a perceptually
+motivated extension of P-loss, in which parameter deviations are
+locally recombined and rescaled so as to simulate a DDSP objective.
+The matrix M(θ) is constant in W. Hence, its value may be
+cached across training epochs, and even across hyperparameter set-
+tings of the encoder. In comparison with P-loss, the only computa-
+tional overhead of PNP is the bilinear form in Equation 6. However,
+this computation is performed in the parametric domain, i.e., in low
+dimension (J = dim θ). Hence, its cost is negligible in front of the
+forward (f W) and backward pass (∂f W/∂Wi) of DNN training.
+2.2. Damped least squares
+The principal components of the Jacobian ∇(Φ◦g)(θ) are the eigen-
+vectors of M(θ). We denote them by vj and the corresponding
+eigenvalues by σ2
+j : for each of them, we have M(θ)vj = σ2
+j vj. The
+vj’s form an orthonormal basis of RJ, in which we can decompose
+the parameter deviation (˜θ − θ). Recalling Equation 5, we obtain an
+alternative formula for PNP loss:
+LPNP
+θ
+(W) = 1
+2
+J
+�
+j=1
+σ2
+j
+��⟨(f W ◦ g)(θ) − θ
+��vj⟩
+��2 v2
+j
+(7)
+The eigenvalues σ2
+j stretch and compress the error vector along their
+associated direction vj, analogous to the magnification and suppres-
+sion of perceptually relevant and irrelevant parameter deviations. In
+practice however, when σ2
+j cover drastic ranges or contain zeros, as
+presented below in Section 4.3, the error vector is subject to extreme
+distortion and potential instability due to numerical precision errors.
+These scenarios, commonly referred to as M being ill-conditioned,
+can lead to intractable learning objective LPNP
+θ
+.
+Reminiscent of the damping mechanism introduced in Levenberg-
+Marquardt algorithm when solving nonlinear optimization problems,
+we update Equation 5 as
+LPNP
+θ
+(W) = 1
+2
+�˜θ − θ
+��M(θ) + λI
+��˜θ − θ
+�
+(8)
+The damping term λI up-shifts all eigenvalues of M by a constant
+positive amount λ, thereby changing its condition number. When
+λ is huge, M(θ) + λI is close to an identity matrix with uniform
+eigenvalues, LPNP
+θ
+is optimizing in parameter loss regime. On the
+other hand when λ is small, small correctional effects keeps LPNP
+θ
+in the spectral loss regime. Alternatively, Equation 8 may also be
+viewed as a L2 regularization with coefficient λ, which allows smooth
+transition between spectral and parameter loss regimes.
+To further address potential convergence issues, λ may be sched-
+uled or adaptively changed according to epoch validation loss. We
+adopt delayed gratification mechanism to decrease λ by a factor of 5
+when loss is going down, and fix λ otherwise.
+3. APPLICATION TO DRUM SOUND MATCHING
+3.1. Perceptual: Joint time–frequency scattering (JTFS)
+The joint time–frequency scattering transform (JTFS) is a nonlinear
+convolutional operator which extracts spectrotemporal modulations
+in the constant-Q scalogram [9]. Its kernels proceed from a separable
+product between two complex-valued wavelet filterbanks, defined
+over the time axis and over the log-frequency axis respectively. After
+convolution, we apply pointwise complex modulus and temporal
+averaging to each JTFS coefficient. These coefficients are known as
+scattering “paths” p. We apply a logarithmic transformation to the
+feature vector JTFS(xn) corresponding to each sound xn, yielding
+Sn,p = (Φ ◦ g)(θn)p = log
+�
+1 + JTFS(xn)p
+ε
+�
+,
+(9)
+where we have set the hyperparameter ε = 10−3 of the order of the
+median value of JTFS across all examples xn and paths p.
+The multiresolution structure of JTFS is reminiscent of spec-
+trotemporal receptive fields (STRF), and thus may serve as a bio-
+logically plausible predictor of neurophysiological responses in the
+primary auditory cortex [10]. At a higher level of music cognition, a
+recent study has shown that Euclidean distances in Φ space predict
+auditory judgments of timbre similarity within a large vocabulary
+of instrumental playing techniques, as collected from a group of
+professional composers and non-expert music listeners [11].
+We compute JTFS with same parameters as [11]: Q1 = 12,
+Q2 = 1, and Qfr = 1 filters per octave respectively. We set the
+
+temporal averaging to T = 3 seconds and the frequential averaging
+to F = 2 octaves; hence a total of P = 20762 paths. We run Φ and
+∇(Φ◦g) in PyTorch on GPU via the implementation of [12, 13].
+3.2. Neural: Deep convolutional network (convnet)
+EfficientNet is a convolutional neural network architecture that bal-
+ances the scaling of the depth, width and input resolution of con-
+secutive convolutional blocks [14]. Achieving state-of-the-art per-
+formance on image classification with significantly less trainable
+parameters, its most light-weight version EfficientNet-B0 also suc-
+ceeded in benchmarking audio classification tasks [15]. We adopt
+EfficientNet-B0 as our encoder f W, resulting in 4M learnable pa-
+rameters. We append a linear dense layer of J = dim θ neurons
+and a 1D batch normalization before tanh activation. The goal of
+batch normalization is to gaussianize the input, such that the activated
+output is capable of uniformly cover the normalized prediction range.
+The input to f W is the log-scaled CQT coefficients of each example,
+computed with a filterbank spanning 10 octaves with 12 filters per
+octave.
+3.3. Physical: Functional transformation method (FTM)
+We are interested in the perpendicular displacement X(t, u) on a
+rectangular drum face, which can be solved from the following partial
+differential equation defined in the Cartesian coordinate system u =
+(u1, u2).
+�∂2X
+∂t2 (t, u) − c2∇2X(t, u)
+�
++ S4�
+∇4X(t, u)
+�
++ ∂
+∂t
+�
+d1X(t, u) + d3∇2X(t, u)
+�
+= Y(t, u)
+(10)
+In addition to the standard traveling wave equation in the first above
+parenthesis, the fourth-order spatial and first-order time derivatives
+incorporate damping factors induced by stiffness, internal friction
+in the drum material and air friction in the external environment,
+rendering the solution a closer simulation to reality. Specifically,
+α, S, c, d1, d3 designate respectively the side length ratio, stiffness,
+traveling wave speed, frequency-independent damping and frequency-
+dependent damping of the drum. Even though real world drums
+are mostly circular, a rectangular drum model is equally capable of
+eliciting representative percussive sounds in real world scenarios. The
+circular drum model simply requires a conversion of Equation 10
+into the Polar coordinate system. We bound the four sides of this l
+by lα rectangular drum at zero at all time. Moreover, we assume its
+excitation function to be separable and localized in space and time
+Y(t, u) = yu(u)δ(t).
+We implement generator g as a PDE solver to this high-order
+damped wave equation, namely the functional transformation method
+(FTM) [16, 17]. FTM solves the PDE by transforming the equation
+into its Laplace and functional space domain, where an algebraic
+solution can be obtained. It then finds the time-space domain solution
+via inverse functional transforms, expressed in an infinite modal
+summation form
+x(t) = X(t, u) =
+�
+m∈N2
+Km(u, t) exp(σmt) sin(ωmt)
+(11)
+The coefficients Km(u, t), σm, ωm are derived from the original
+PDE parameters in the following ways.
+ω2
+m = (S4 − d2
+3
+4 )Γ2
+m1,m2 + (c2 + d1d3
+2
+)Γm1,m2 − d2
+1
+4
+(12)
+Fig. 2. Distributions of the sorted eigenvalues of M(θn). For the
+sake of comparison between PNP and P-loss, the dashed line indicates
+the eigenvalues of the identity matrix (see Equation 6).
+σm = d3
+2 Γm1,m2 − d1
+2
+(13)
+Km(u, t) = ym
+u δ(t) sin(πm1u1
+l
+) sin
+�πm2u2
+lα
+�
+(14)
+where Γm1,m2 = π2m2
+1/l2 + π2m2
+2/(lα)2, and ym
+u is the mth coef-
+ficient associated to the eigenfunction sin(πmu/l) that decomposes
+yu(u).
+Without losing connections to the acoustical manufacturing of
+the drum yet better relating g’s input with perceptual dimensions,
+we reparametrize the PDE parameters {S, c, d1, d3, α} into θ =
+{log ω1, τ1, log p, log D, α}, detailed in Section 3.4 of [18]. We pre-
+scribe sonically-plausible ranges for each parameter in θ, normalize
+them between −1 and 1, uniformly sample in the hyper-dimensional
+cube, and obtain a dataset of 100k percussive sounds sampled at
+22050 HZ. The train/test/validation split is 8 : 1 : 1.
+In particular, fundamental frequency ω1, duration τ1 falls into
+ranges [40, 1000] Hz and [0.4, 3] seconds respectively. Inhomoge-
+neous damping rate p, frequential dispersion D and aspect ratio α
+ranges are [10−5, 0.2], [10−5, 0.3], and [10−5, 1].
+4. RESULTS
+4.1. Baselines
+We train fW with 3 different losses - multi-scale spectral loss [19],
+parameter loss, and PNP loss. We use a batch size of 64 samples
+for spectral loss, and 256 samples for parameter and PNP loss. The
+training proceeds for 70 epochs, where around 20% of the training
+set is seen at each epoch. We use Adam optimizer with learning rate
+10−3. Table 1 reports the training time per epoch on a single Tesla
+V100 16GB GPU.
+4.2. Evaluation with JTFS-based spectral loss
+We propose to use the L2 norm of JTFS coefficients error averaged
+over test set for evaluation. As a point of reference, we also include
+the average multi-scale spectral error, implemented as in Section
+4.1. One of the key distinctions between Euclidean JTFS distance
+and multi-scale spectral error is the former’s inclusion of spectro-
+temporal modulations information. Meanwhile unlike mean squared
+parameter error, both metrics reflect the perceptual closeness instead
+of parametric retrieval accuracy for each proposed model.
+4.3. Discussion
+Despite being the optimal quadratic approximation of spectral loss,
+it is nontrivial to apply the bear PNP loss form as Equation 5 in
+
+OCD
+15
+10
+5
+0
+5
+log1o(on,j)Pitch
+JTFS distance
+(avg. on test set)
+MSS
+(avg. on test set)
+Training time
+per epoch
+P-loss
+Known
+22.23 ± 2.17
+0.31 ± 0.013
+49 minutes
+DDSP with MSS loss
+Known
+31.86 ± 0.332
+0.335 ± 0.005
+54 minutes
+PNP with JTFS loss
+Known
+23.58 ± 0.877
+0.335 ± 0.005
+49 minutes
+DDSP with JTFS loss
+—
+—
+—
+est., > 1 day
+P-loss
+Unknown
+61.91 ± 6.26
+1.02 ± 0.094
+53 minutes
+DDSP with MSS loss
+Unknown
+138.95 ± 37.12
+1.59 ± 0.307
+59 minutes
+PNP with JTFS loss
+Unknown
+61.21 ± 1.207
+0.97 ± 0.019
+49 minutes
+Table 1. Report of average JTFS distance and MSS metrics evaluated on test set. Six models are trained with two modalities: 1. the inclusion
+of pitch retrieval i.e.regressing θ = {τ, log p, log D, α} vs. θ = {log ω1, τ, log p, log D, α}, and 2. the choice of loss function: P-loss, MSS
+loss, or PNP loss with adaptive damping mechanism. The best performing models with known and unknown pitch are P-loss and PNP loss
+respectively. Training with MSS loss is more time consuming than training with P-loss or PNP loss. Training with differentiable JTFS loss is
+unrealistic in the interest of time.
+experimental settings. On one hand, Φ◦g potentially has undesirable
+property that exposes the Riemannian metric calculations to numeri-
+cal precision errors. On the other hand, extreme deformation of the
+optimization landscape may lead to the same numerical instability
+facing stochastic gradient descent with spectral loss. We report on a
+few remedies that helped stabilize learning with PNP loss, and offer
+insights on future directions to take.
+First and foremost, our preliminary experiments show that train-
+ing PNP loss without damping λ = 0 subjects to serious convergence
+issues. Recalling Section 2.2, indeed our empirical Ms suffer from
+high condition numbers. Fig. 2 shows the sorted eigenvalue distribu-
+tion of all Ms in test set, where Ms are rank-2,3 or 4 matrices with
+eigenvalues ranging from 0 to 1020. This could be an implication
+that entries of θ contain implicit linear dependencies in generator g,
+or that local variations of certain θ fail to linearize differences in the
+output of g or Φ ◦ g. As an example, the aspect ratio α influences the
+modal frequencies and decay rates via [18, Equations 12–13], where
+in fact its variant 1/α + 1/α2 could be a better choice of variable
+that linearizes g.
+To address Ms’ ill conditions we attempted at numerous damping
+mechanisms to update λ, namely constant λ, scheduled λ decay, and
+adaptive λ decay. The intuition is to have LPNP
+θ
+start in the parameter
+loss regime and move towards the spectral loss regime while training.
+The best performing model is achieved with adaptive λ decay, as
+described in Section 2.2. We initialize λ to be 1020 to match the
+largest empirical σ2
+j , which later gets adaptively decayed to 3 × 1014
+in 20 epochs, breaking records 7 times. This indicates that f W is
+able to learn with damped PNP loss, under the condition that λ being
+large enough to simulate parameter loss regime and compensate for
+deficiency in M.
+The diagonal elements of M(θ) can be regarded as both the ap-
+plied weights’ magnitudes and proxies for θ’s perceptual significance.
+To gain further insights on how each model regresses different pa-
+rameters, we visualize in Fig.3 pairs of (|˜θ − θ|2
+j, M(θ)j,j). Three
+trends can be observed: First, τ and ω are regressed with the best
+accuracy across all learning objectives. Second, spectral loss particu-
+larly struggles in pitch ω and inharmonicity p retrievals. Third, we
+may interpret x-axis as describing from left to right samples with
+increasing perceptual significance. We observe that in Fig.3(b), PNP
+loss is able to suppress more errors in samples with high M(θ)j,j
+than parameter loss, by a nonnegligible margin.
+We believe that more of PNP loss’ mathematical potential can
+be exploited in the future, notably its ability to interpolate between
+various loss regimes and its use in hybrid optimization schemes. To
+start with, we plan to resort to a simpler differentiable synthesizer g
+Fig. 3. X-axis: weight assigned by PNP to one of the physical
+parameters in θn. Y-axis: log squared estimation error for that same
+parameter. α is omitted due to its poor retrieval results from all
+models.
+that guarantees a well-conditioned Riemannian metric M(θ). More-
+over, we plan to explore other damping schemes and optimizers. The
+current update mechanism, originated from the Leverberg-Marquardt
+Algorithm, aims to improve the conditioning of a matrix inversion
+problem in the Gauss-Newton algorithm. However when used jointly
+with stochastic gradient descent, each λ update may change the opti-
+mization landscape drastically. The resulting optimization behavior
+is thus not fully understood. We consider interfacing nonlinear least
+squares solver with SGD and forming a hybrid learning scheme in
+future work.
+5. CONCLUSION
+In this article we have presented Perceptual-Neural-Physical (PNP)
+autoencoding, a bilinear form learning objective for sound matching
+task. In our application, PNP optimizes the retrieval of physical
+parameters from sounds in a perceptually-motivated metric space,
+enabled by differentiable implementations of domain knowledge in
+physical modeling and computational proxy of neurophysiological
+construct of human auditory system.
+We demonstrated PNP’s mathematical proximity to spectral loss
+and its generalizability to parameter loss. Using this formulation,
+we motivated and established one way of enabling smooth transition
+between optimizing in parameter and spectral loss regimes. We
+have presented damping mechanisms to facilitate its learning under
+ill-conditioned empirical settings and discussed its mathematical
+potential.
+
+w - w]2 vs. M[0,0]
+ - T|2 vs. M[1, 1]
+100
+10~2
+10-3
+10~5
+Ploss
+10~6
+10-8
+Spec
+109
+PNP
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0
+100000
+200000
+300000
+400000
+(a)
+le10
+(b)
+Ip -p]2 vs. M[2,2]
+ID - D2 vs. M[3,3]
+101
+100
+10~3 
+10~2
+10~6,
+10-5
+10-9
+10-8
+0
+50000
+100000
+150000
+200000
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+(c) 
+(d)
+le106. REFERENCES
+[1] Andrew Horner, “Wavetable matching synthesis of dynamic
+instruments with genetic algorithms,” Journal of the Audio
+Engineering Society, vol. 43, no. 11, pp. 916–931, 1995.
+[2] Jordie Shier, Kirk McNally, George Tzanetakis, and Ky Grace
+Brooks,
+“Manifold learning methods for visualization and
+browsing of drum machine samples,” Journal of the Audio
+Engineering Society, vol. 69, no. 1/2, pp. 40–53, 2021.
+[3] Philippe Esling, Naotake Masuda, Adrien Bardet, Romeo De-
+spres, Axel Chemla, et al., “Universal audio synthesizer control
+with normalizing flows,” in Proceedings of the International
+Conference on Digital Audio Effects (DAFX), 2019.
+[4] Leonardo Gabrielli, Stefano Tomassetti, Carlo Zinato, and
+Francesco Piazza,
+“End-to-end learning for physics-based
+acoustic modeling,” IEEE Transactions on Emerging Topics in
+Computational Intelligence, vol. 2, no. 2, pp. 160–170, 2018.
+[5] Naotake Masuda and Daisuke Saito, “Synthesizer sound match-
+ing with differentiable DSP.,” in Proceedings of the Interna-
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+[6] Martin Roth and Matthew Yee-king, “A comparison of para-
+metric optimization techniques for musical instrument tone
+matching,” Journal of the Audio Engineering Society, May
+2011.
+[7] Matthew Yee-King, Leon Fedden, and Mark d’Inverno, “Au-
+tomatic programming of vst sound synthesizers using deep
+networks and other techniques,” IEEE Transactions on Emerg-
+ing Topics in Computational Intelligence, vol. 2, pp. 150–159,
+04 2018.
+[8] Jesse Engel, Lamtharn (Hanoi) Hantrakul, Chenjie Gu, and
+Adam Roberts, “DDSP: Differentiable Digital Signal Process-
+ing,” in Proceedings of the International Conference on Learn-
+ing Representations (ICLR), 2020.
+[9] Joakim And´en, Vincent Lostanlen, and St´ephane Mallat, “Joint
+time–frequency scattering,” IEEE Transactions on Signal Pro-
+cessing, vol. 67, no. 14, pp. 3704–3718, 2019.
+[10] Taishih Chi, Powen Ru, and Shihab A Shamma, “Multiresolu-
+tion spectrotemporal analysis of complex sounds,” The Journal
+of the Acoustical Society of America, vol. 118, no. 2, pp. 887–
+906, 2005.
+[11] Vincent Lostanlen, Christian El-Hajj, Mathias Rossignol,
+Gr´egoire Lafay, Joakim And´en, and Mathieu Lagrange, “Time–
+frequency scattering accurately models auditory similarities
+between instrumental playing techniques,” EURASIP Journal
+on Audio, Speech, and Music Processing, vol. 2021, no. 1, pp.
+1–21, 2021.
+[12] Mathieu Andreux, Tom´as Angles, Georgios Exarchakis,
+Roberto Leonarduzzi, Gaspar Rochette, Louis Thiry, John
+Zarka, St´ephane Mallat, Joakim And´en, Eugene Belilovsky,
+Joan Bruna, Vincent Lostanlen, Muawiz Chaudhary, Matthew J.
+Hirn, Edouard Oyallon, Sixin Zhang, Carmine Cella, and
+Michael Eickenberg,
+“Kymatio: Scattering transforms in
+Python.,”
+Journal of Machine Learning Research, vol. 21,
+no. 60, pp. 1–6, 2020.
+[13] John Muradeli, Cyrus Vahidi, Changhong Wang, Han Han, Vin-
+cent Lostanlen, Mathieu Lagrange, and George Fazekas, “Dif-
+ferentiable time-frequency scattering in kymatio,” in Proceed-
+ings of the International Conference on Digital Audio Effects
+(DAFX), 2022.
+[14] Mingxing Tan and Quoc Le, “EfficientNet: Rethinking model
+scaling for convolutional neural networks,” in Proceedings
+of the International conference on Machine Learning (ICML).
+PMLR, 2019, pp. 6105–6114.
+[15] Neil Zeghidour, Olivier Teboul, F´elix de Chaumont Quitry, and
+Marco Tagliasacchi, “Leaf: A learnable frontend for audio
+classification,” ICLR, 2021.
+[16] L. Trautmann and Rudolf Rabenstein, Digital Sound Synthesis
+by Physical Modeling Using the Functional Transformation
+Method, 01 2003.
+[17] Maximilian Sch¨afer, Manuel Werner, and Rudolf Rabenstein,
+“Physical modeling in sound synthesis: Vibrating plates,” 05
+2019.
+[18] Han Han and Vincent Lostanlen, “wav2shape: Hearing the
+Shape of a Drum Machine,” in Proceedings of Forum Acusticum,
+2020, pp. 647–654.
+[19] Christian J. Steinmetz and Joshua D. Reiss, “auraloss: Audio
+focused loss functions in PyTorch,” in Digital Music Research
+Network One-day Workshop (DMRN+15), 2020.
+

DdE1T4oBgHgl3EQfEAP6/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf,len=312
+page_content='PERCEPTUAL–NEURAL–PHYSICAL SOUND MATCHING  Han Han, Vincent Lostanlen, and Mathieu Lagrange  Nantes Universit´e, ´Ecole Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000 Nantes, France  ABSTRACT  Sound matching algorithms seek to approximate a target waveform  by parametric audio synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Deep neural networks have achieved  promising results in matching sustained harmonic tones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' However,  the task is more challenging when targets are nonstationary and inhar-  monic, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=', percussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We attribute this problem to the inadequacy  of loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' On one hand, mean square error in the parametric  domain, known as “P-loss”, is simple and fast but fails to accommo-  date the differing perceptual significance of each parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' On the  other hand, mean square error in the spectrotemporal domain, known  as “spectral loss”, is perceptually motivated and serves in differen-  tiable digital signal processing (DDSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Yet, spectral loss has more  local minima than P-loss and its gradient may be computationally  expensive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' hence a slow convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Against this conundrum, we  present Perceptual-Neural-Physical loss (PNP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' PNP is the optimal  quadratic approximation of spectral loss while being as fast as P-loss  during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We instantiate PNP with physical modeling synthesis  as decoder and joint time–frequency scattering transform (JTFS) as  spectral representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We demonstrate its potential on matching  synthetic drum sounds in comparison with other loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  Index Terms— auditory similarity, scattering transform, deep  convolutional networks, physical modeling synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' INTRODUCTION  Given an audio synthesizer g, the task of sound matching [1] consists  in retrieving the parameter setting θ that “matches” a target sound  x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=', such that a human ear judges the generated sound g(θ) to  resemble x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Sound matching has applications in automatic music  transcription, virtual reality, and audio engineering [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Of particu-  lar interest is the case where g(θ) solves a known partial differential  equation (PDE) whose coefficients are contained in the vector θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' In  this case, θ reveals some key design choices in acoustical manufac-  turing, such as the shape and material properties of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  Over the past decade, the renewed interest for deep neural net-  works (DNN’s) in audio content analysis has led researchers to formu-  late sound matching as a supervised learning problem [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Intuitively,  the goal is to optimize the synaptic weights W of a DNN f W so  that f W(xn) = ˜θn approximates θn over a training set of pairs  (xn, θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Because g automates the mapping from parameter θn to  sound xn, this training procedure incurs no real-world audio acquisi-  tion nor human annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' However, prior publications have pointed  out that the approximation formula ˜θn ≈ θn lacks a perceptual mean-  ing: depending on the choice of target xn, some deviations (˜θn−θn)  may be judged to have a greater effect than others [5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  The paradigm of differentiable digital signal processing (DDSP)  has brought a principled methodology to address this issue [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' The  key idea behind DDSP is to chain the learnable encoder f W with  the known decoder g and a non-learnable but differentiable feature  map Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' In DDSP, f W is trained to minimize the perceptual distance  θ  Parametric  domain  x  original  Audio  domain  S  Perceptual  domain  ˜θ  ˜x  reconstruction  ˜S  DDSP  spectral ≈  loss  PNP  quadratic  form  θ  x  ˜θ  M(θ)  Riemannian  metric  g  Φ  f W  g  Φ  g  f W  ∇(Φ◦g)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Graphical outline of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Given a known  synthesizer g and feature map Φ, we train a neural network f W to  minimize the “perceptual–neural–physical” (PNP) quadratic form  ⟨˜θ − θ  ��M(θ)|˜θ − θ⟩ where M is the Riemannian metric associated  to (Φ◦g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Hence, PNP approximates DDSP spectral loss yet does not  need to backpropagate ∇(Φ◦g)(˜θ) at each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Transformations in  solid (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' dashed) lines can (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' cannot) be cached during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  between vectors Φ(˜xn) = (Φ◦g◦f W)(xn) and Φ(xn) on average  over samples xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Yet, a practical shortcoming of DDSP is that it  requires to evaluate the Jacobian ∇(Φ◦g) over each DNN prediction  ˜θn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' and so at every training step, as W is updated by stochastic  gradient descent (SGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  In this article, we propose a new learning objective for sound  matching, named perceptual–neural–physical (PNP) autoencoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  The main contribution of PNP is to compute the Riemannian met-  ric M associated to the Jacobian ∇(Φ◦g) over each sample θn (see  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' The PNP encoder is penalized in terms of a first-order  Taylor expansion of the spectral loss ∥Φ(˜x) − Φ(x)∥2, making  it comparable to DDSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Yet, unlike in DDSP, the computation of  ∇(Φ◦g) is independent from the encoder f W: thus, it may be par-  allelized and cached during DNN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' A second novelty of  our paper resides in its choice of application: namely, differentiable  sound matching for percussion instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' This requires not only  a fine characterization of the spectral envelope, as in the DDSP of  sustained tones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' but also of attack and release transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' For this  purpose, we need g and Φ to accommodate sharp spectrotemporal  modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Specifically, we rely on a differentiable implementation  of the functional transformation method (FTM) for g and the joint  time–frequency scattering transform (JTFS) for Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' METHODS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Approximating spectral loss with Riemannian geometry  We assume the synthesizer g and the feature map Φ to be contin-  uously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Let us denote by LDDSP the “spectral loss”  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='02886v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='SD]  7 Jan 2023  associated to the triplet (Φ, f W, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Its value at a parameter set θ is:  LDDSP  θ  (W) = 1  2∥Φ(˜x) − Φ(x)∥2  2  = 1  2  ��(Φ ◦ g ◦ f W ◦ g)(θ) − (Φ ◦ g)(θ)  ��2  2  (1)  by definition of ˜x and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Using ˜θ as shorthand for (f W ◦ g)(θ), we  conduct a first-order Taylor expansion of (Φ ◦ g) near θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We obtain:  Φ(˜x) = Φ(x) + ∇(Φ◦g)(θ) · (˜θ − θ) + O(∥˜θ − θ∥2  2),  (2)  where the Jacobian matrix ∇(Φ◦g)(θ) contains P = dim Φ(x) rows  and J = dim θ columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' The differentiable map (Φ ◦ g) induces a  weak Riemannian metric M onto the open set U ⊂ RJ of parameters  θ, whose matrix values derive from ∇(Φ◦g)(θ):  M(θ)j,j′ =  P  �  p=1  �  ∇(Φ◦g)(θ)p,j  � �  ∇(Φ◦g)(θ)p,j′�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  (3)  The square matrix M(θ) ∈ GLJ(R) defines a positive semidefinite  kernel which, once plugged into Equation 2, serves to approximate  LDDSP  θ  (W) in terms of a quadratic form over (˜θ − θ):  ∥Φ(˜x) − Φ(x)∥2  2 =  �˜θ − θ  ��M(θ)  ��˜θ − θ  �  + O  �  ∥˜θ − θ∥3  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' (4)  The advantage of the approximation above is that the metric  M may be computed over the training set once and for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' This is  because Equation 3 is independent of the encoder f W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Furthermore,  since θ is low-dimensional, we may store M(θ) on RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' From  this perspective, we define the perceptual–neural–physical loss (PNP)  associated to (Φ, f W, g) as the linearization of spectral loss at θ:  LPNP  θ  (W) = 1  2  �  (f W ◦ g)(θ) − θ  ��M(θ)  ��(f W ◦ g)(θ) − θ  �  = LDDSP  θ  (W) + O  �  ∥(f W ◦ g)(θ) − θ∥3  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  (5)  According to the chain rule, the gradient of PNP loss at a given  training pair (xn, θn) with respect to some scalar weight Wi is:  ∂LPNP  θ  ∂Wi (θn) =  �  f W(xn) − θn  ���M(θn)  ���∂f W  ∂Wi (xn)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  (6)  Observe that replacing M(θn) by the identity matrix in the equation  above would give the gradient of parameter loss (P-loss);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' that is,  the mean squared error between the predicted parameter ˜θ and the  true parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Hence, we may regard PNP as a perceptually  motivated extension of P-loss, in which parameter deviations are  locally recombined and rescaled so as to simulate a DDSP objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  The matrix M(θ) is constant in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Hence, its value may be  cached across training epochs, and even across hyperparameter set-  tings of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' In comparison with P-loss, the only computa-  tional overhead of PNP is the bilinear form in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' However,  this computation is performed in the parametric domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=', in low  dimension (J = dim θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Hence, its cost is negligible in front of the  forward (f W) and backward pass (∂f W/∂Wi) of DNN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Damped least squares  The principal components of the Jacobian ∇(Φ◦g)(θ) are the eigen-  vectors of M(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We denote them by vj and the corresponding  eigenvalues by σ2  j : for each of them, we have M(θ)vj = σ2  j vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' The  vj’s form an orthonormal basis of RJ, in which we can decompose  the parameter deviation (˜θ − θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Recalling Equation 5, we obtain an  alternative formula for PNP loss:  LPNP  θ  (W) = 1  2  J  �  j=1  σ2  j  ��⟨(f W ◦ g)(θ) − θ  ��vj⟩  ��2 v2  j  (7)  The eigenvalues σ2  j stretch and compress the error vector along their  associated direction vj, analogous to the magnification and suppres-  sion of perceptually relevant and irrelevant parameter deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' In  practice however, when σ2  j cover drastic ranges or contain zeros, as  presented below in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='3, the error vector is subject to extreme  distortion and potential instability due to numerical precision errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  These scenarios, commonly referred to as M being ill-conditioned,  can lead to intractable learning objective LPNP  θ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  Reminiscent of the damping mechanism introduced in Levenberg-  Marquardt algorithm when solving nonlinear optimization problems,  we update Equation 5 as  LPNP  θ  (W) = 1  2  �˜θ − θ  ��M(θ) + λI  ��˜θ − θ  �  (8)  The damping term λI up-shifts all eigenvalues of M by a constant  positive amount λ, thereby changing its condition number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' When  λ is huge, M(θ) + λI is close to an identity matrix with uniform  eigenvalues, LPNP  θ  is optimizing in parameter loss regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' On the  other hand when λ is small, small correctional effects keeps LPNP  θ  in the spectral loss regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Alternatively, Equation 8 may also be  viewed as a L2 regularization with coefficient λ, which allows smooth  transition between spectral and parameter loss regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  To further address potential convergence issues, λ may be sched-  uled or adaptively changed according to epoch validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We  adopt delayed gratification mechanism to decrease λ by a factor of 5  when loss is going down, and fix λ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' APPLICATION TO DRUM SOUND MATCHING  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Perceptual: Joint time–frequency scattering (JTFS)  The joint time–frequency scattering transform (JTFS) is a nonlinear  convolutional operator which extracts spectrotemporal modulations  in the constant-Q scalogram [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Its kernels proceed from a separable  product between two complex-valued wavelet filterbanks, defined  over the time axis and over the log-frequency axis respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' After  convolution, we apply pointwise complex modulus and temporal  averaging to each JTFS coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' These coefficients are known as  scattering “paths” p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We apply a logarithmic transformation to the  feature vector JTFS(xn) corresponding to each sound xn, yielding  Sn,p = (Φ ◦ g)(θn)p = log  �  1 + JTFS(xn)p  ε  �  ,  (9)  where we have set the hyperparameter ε = 10−3 of the order of the  median value of JTFS across all examples xn and paths p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  The multiresolution structure of JTFS is reminiscent of spec-  trotemporal receptive fields (STRF), and thus may serve as a bio-  logically plausible predictor of neurophysiological responses in the  primary auditory cortex [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' At a higher level of music cognition, a  recent study has shown that Euclidean distances in Φ space predict  auditory judgments of timbre similarity within a large vocabulary  of instrumental playing techniques, as collected from a group of  professional composers and non-expert music listeners [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  We compute JTFS with same parameters as [11]: Q1 = 12,  Q2 = 1, and Qfr = 1 filters per octave respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We set the  temporal averaging to T = 3 seconds and the frequential averaging  to F = 2 octaves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' hence a total of P = 20762 paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We run Φ and  ∇(Φ◦g) in PyTorch on GPU via the implementation of [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Neural: Deep convolutional network (convnet)  EfficientNet is a convolutional neural network architecture that bal-  ances the scaling of the depth, width and input resolution of con-  secutive convolutional blocks [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Achieving state-of-the-art per-  formance on image classification with significantly less trainable  parameters, its most light-weight version EfficientNet-B0 also suc-  ceeded in benchmarking audio classification tasks [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We adopt  EfficientNet-B0 as our encoder f W, resulting in 4M learnable pa-  rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We append a linear dense layer of J = dim θ neurons  and a 1D batch normalization before tanh activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' The goal of  batch normalization is to gaussianize the input, such that the activated  output is capable of uniformly cover the normalized prediction range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  The input to f W is the log-scaled CQT coefficients of each example,  computed with a filterbank spanning 10 octaves with 12 filters per  octave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Physical: Functional transformation method (FTM)  We are interested in the perpendicular displacement X(t, u) on a  rectangular drum face, which can be solved from the following partial  differential equation defined in the Cartesian coordinate system u =  (u1, u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  �∂2X  ∂t2 (t, u) − c2∇2X(t, u)  �  + S4�  ∇4X(t, u)  �  + ∂  ∂t  �  d1X(t, u) + d3∇2X(t, u)  �  = Y(t, u)  (10)  In addition to the standard traveling wave equation in the first above  parenthesis, the fourth-order spatial and first-order time derivatives  incorporate damping factors induced by stiffness, internal friction  in the drum material and air friction in the external environment,  rendering the solution a closer simulation to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Specifically,  α, S, c, d1, d3 designate respectively the side length ratio, stiffness,  traveling wave speed, frequency-independent damping and frequency-  dependent damping of the drum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Even though real world drums  are mostly circular, a rectangular drum model is equally capable of  eliciting representative percussive sounds in real world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' The  circular drum model simply requires a conversion of Equation 10  into the Polar coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We bound the four sides of this l  by lα rectangular drum at zero at all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Moreover, we assume its  excitation function to be separable and localized in space and time  Y(t, u) = yu(u)δ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  We implement generator g as a PDE solver to this high-order  damped wave equation, namely the functional transformation method  (FTM) [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' FTM solves the PDE by transforming the equation  into its Laplace and functional space domain, where an algebraic  solution can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' It then finds the time-space domain solution  via inverse functional transforms, expressed in an infinite modal  summation form  x(t) = X(t, u) =  �  m∈N2  Km(u, t) exp(σmt) sin(ωmt)  (11)  The coefficients Km(u, t), σm, ωm are derived from the original  PDE parameters in the following ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  ω2  m = (S4 − d2  3  4 )Γ2  m1,m2 + (c2 + d1d3  2  )Γm1,m2 − d2  1  4  (12)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Distributions of the sorted eigenvalues of M(θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' For the  sake of comparison between PNP and P-loss, the dashed line indicates  the eigenvalues of the identity matrix (see Equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  σm = d3  2 Γm1,m2 − d1  2  (13)  Km(u, t) = ym  u δ(t) sin(πm1u1  l  ) sin  �πm2u2  lα  �  (14)  where Γm1,m2 = π2m2  1/l2 + π2m2  2/(lα)2, and ym  u is the mth coef-  ficient associated to the eigenfunction sin(πmu/l) that decomposes  yu(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  Without losing connections to the acoustical manufacturing of  the drum yet better relating g’s input with perceptual dimensions,  we reparametrize the PDE parameters {S, c, d1, d3, α} into θ =  {log ω1, τ1, log p, log D, α}, detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='4 of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We pre-  scribe sonically-plausible ranges for each parameter in θ, normalize  them between −1 and 1, uniformly sample in the hyper-dimensional  cube, and obtain a dataset of 100k percussive sounds sampled at  22050 HZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' The train/test/validation split is 8 : 1 : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  In particular, fundamental frequency ω1, duration τ1 falls into  ranges [40, 1000] Hz and [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='4, 3] seconds respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Inhomoge-  neous damping rate p, frequential dispersion D and aspect ratio α  ranges are [10−5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='2], [10−5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='3], and [10−5, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' RESULTS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Baselines  We train fW with 3 different losses - multi-scale spectral loss [19],  parameter loss, and PNP loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We use a batch size of 64 samples  for spectral loss, and 256 samples for parameter and PNP loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' The  training proceeds for 70 epochs, where around 20% of the training  set is seen at each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We use Adam optimizer with learning rate  10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Table 1 reports the training time per epoch on a single Tesla  V100 16GB GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Evaluation with JTFS-based spectral loss  We propose to use the L2 norm of JTFS coefficients error averaged  over test set for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' As a point of reference, we also include  the average multi-scale spectral error, implemented as in Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' One of the key distinctions between Euclidean JTFS distance  and multi-scale spectral error is the former’s inclusion of spectro-  temporal modulations information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Meanwhile unlike mean squared  parameter error, both metrics reflect the perceptual closeness instead  of parametric retrieval accuracy for each proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Discussion  Despite being the optimal quadratic approximation of spectral loss,  it is nontrivial to apply the bear PNP loss form as Equation 5 in  OCD  15  10  5  0  5  log1o(on,j)Pitch  JTFS distance  (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' on test set)  MSS  (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' on test set)  Training time  per epoch  P-loss  Known  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='23 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='013  49 minutes  DDSP with MSS loss  Known  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='332  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='335 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='005  54 minutes  PNP with JTFS loss  Known  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='877  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='335 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='005  49 minutes  DDSP with JTFS loss  —  —  —  est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=', > 1 day  P-loss  Unknown  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='91 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='26  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='094  53 minutes  DDSP with MSS loss  Unknown  138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='95 ± 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='307  59 minutes  PNP with JTFS loss  Unknown  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='21 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='207  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='019  49 minutes  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Report of average JTFS distance and MSS metrics evaluated on test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Six models are trained with two modalities: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' the inclusion  of pitch retrieval i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='regressing θ = {τ, log p, log D, α} vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' θ = {log ω1, τ, log p, log D, α}, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' the choice of loss function: P-loss, MSS  loss, or PNP loss with adaptive damping mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' The best performing models with known and unknown pitch are P-loss and PNP loss  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Training with MSS loss is more time consuming than training with P-loss or PNP loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Training with differentiable JTFS loss is  unrealistic in the interest of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' On one hand, Φ◦g potentially has undesirable  property that exposes the Riemannian metric calculations to numeri-  cal precision errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' On the other hand, extreme deformation of the  optimization landscape may lead to the same numerical instability  facing stochastic gradient descent with spectral loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We report on a  few remedies that helped stabilize learning with PNP loss, and offer  insights on future directions to take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  First and foremost, our preliminary experiments show that train-  ing PNP loss without damping λ = 0 subjects to serious convergence  issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Recalling Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='2, indeed our empirical Ms suffer from  high condition numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' 2 shows the sorted eigenvalue distribu-  tion of all Ms in test set, where Ms are rank-2,3 or 4 matrices with  eigenvalues ranging from 0 to 1020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' This could be an implication  that entries of θ contain implicit linear dependencies in generator g,  or that local variations of certain θ fail to linearize differences in the  output of g or Φ ◦ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' As an example, the aspect ratio α influences the  modal frequencies and decay rates via [18, Equations 12–13], where  in fact its variant 1/α + 1/α2 could be a better choice of variable  that linearizes g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  To address Ms’ ill conditions we attempted at numerous damping  mechanisms to update λ, namely constant λ, scheduled λ decay, and  adaptive λ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' The intuition is to have LPNP  θ  start in the parameter  loss regime and move towards the spectral loss regime while training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  The best performing model is achieved with adaptive λ decay, as  described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We initialize λ to be 1020 to match the  largest empirical σ2  j , which later gets adaptively decayed to 3 × 1014  in 20 epochs, breaking records 7 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' This indicates that f W is  able to learn with damped PNP loss, under the condition that λ being  large enough to simulate parameter loss regime and compensate for  deficiency in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  The diagonal elements of M(θ) can be regarded as both the ap-  plied weights’ magnitudes and proxies for θ’s perceptual significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  To gain further insights on how each model regresses different pa-  rameters, we visualize in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='3 pairs of (|˜θ − θ|2  j, M(θ)j,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Three  trends can be observed: First, τ and ω are regressed with the best  accuracy across all learning objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Second, spectral loss particu-  larly struggles in pitch ω and inharmonicity p retrievals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Third, we  may interpret x-axis as describing from left to right samples with  increasing perceptual significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We observe that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='3(b), PNP  loss is able to suppress more errors in samples with high M(θ)j,j  than parameter loss, by a nonnegligible margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  We believe that more of PNP loss’ mathematical potential can  be exploited in the future, notably its ability to interpolate between  various loss regimes and its use in hybrid optimization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' To  start with, we plan to resort to a simpler differentiable synthesizer g  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' X-axis: weight assigned by PNP to one of the physical  parameters in θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Y-axis: log squared estimation error for that same  parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' α is omitted due to its poor retrieval results from all  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  that guarantees a well-conditioned Riemannian metric M(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' More-  over, we plan to explore other damping schemes and optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' The  current update mechanism, originated from the Leverberg-Marquardt  Algorithm, aims to improve the conditioning of a matrix inversion  problem in the Gauss-Newton algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' However when used jointly  with stochastic gradient descent, each λ update may change the opti-  mization landscape drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' The resulting optimization behavior  is thus not fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We consider interfacing nonlinear least  squares solver with SGD and forming a hybrid learning scheme in  future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' CONCLUSION  In this article we have presented Perceptual-Neural-Physical (PNP)  autoencoding, a bilinear form learning objective for sound matching  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' In our application, PNP optimizes the retrieval of physical  parameters from sounds in a perceptually-motivated metric space,  enabled by differentiable implementations of domain knowledge in  physical modeling and computational proxy of neurophysiological  construct of human auditory system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  We demonstrated PNP’s mathematical proximity to spectral loss  and its generalizability to parameter loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' Using this formulation,  we motivated and established one way of enabling smooth transition  between optimizing in parameter and spectral loss regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' We  have presented damping mechanisms to facilitate its learning under  ill-conditioned empirical settings and discussed its mathematical  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='  w - w]2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' M[0,0]  T|2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' M[1, 1]  100  10~2  10-3  10~5  Ploss  10~6  10-8  Spec  109  PNP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='0  0  100000  200000  300000  400000  (a)  le10  (b)  Ip -p]2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' M[2,2]  ID - D2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content=' M[3,3]  101  100  10~3  10~2  10~6,  10-5  10-9  10-8  0  50000  100000  150000  200000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
+page_content='6  (c)  (d)  le106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'}
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DdE3T4oBgHgl3EQfUwqw/content/tmp_files/2301.04454v1.pdf.txt

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+Allo-centric Occupancy Grid Prediction for Urban Traffic Scene Using
+Video Prediction Networks
+Rabbia Asghar1, Lukas Rummelhard1, Anne Spalanzani1, Christian Laugier1
+Abstract— Prediction of dynamic environment is crucial to
+safe navigation of an autonomous vehicle. Urban traffic scenes
+are particularly challenging to forecast due to complex interac-
+tions between various dynamic agents, such as vehicles and
+vulnerable road users. Previous approaches have used ego-
+centric occupancy grid maps to represent and predict dynamic
+environments. However, these predictions suffer from blurri-
+ness, loss of scene structure at turns, and vanishing of agents
+over longer prediction horizon. In this work, we propose a
+novel framework to make long-term predictions by representing
+the traffic scene in a fixed frame, referred as allo-centric
+occupancy grid. This allows for the static scene to remain fixed
+and to represent motion of the ego-vehicle on the grid like
+other agents’. We study the allo-centric grid prediction with
+different video prediction networks and validate the approach
+on the real-world Nuscenes dataset. The results demonstrate
+that the allo-centric grid representation significantly improves
+scene prediction, in comparison to the conventional ego-centric
+grid approach.
+Index Terms— Scene Prediction, Deep Learning, Autonomous
+Vehicles
+I. INTRODUCTION
+Prediction of traffic scene evolution is essential to an
+autonomous vehicle for planning as well as detecting dan-
+gerous situations. In urban traffic scenarios, the vehicles not
+only interact with other vehicles, but also share space with
+vulnerable road users such as pedestrians and cyclists. Key
+challenges involve the uncertainty and multi-modality of the
+behaviour of agents in the environment, and complex multi-
+agents interactions [1]. While human drivers show superior
+ability to forecast the agents’ behaviour and interactions in
+such traffic scenes, it remains a challenge for autonomous
+vehicles.
+Data-driven methods provide powerful tools to solve pre-
+diction problems, particularly dealing with complex social
+interactions [2]. Most conventional approaches are object
+or agent-based and rely on heavily pre-processed data [3],
+[4]. Dynamic Occupancy Grip Maps (DOGMs), on the other
+hand, allow for end-to-end learning due to their discretized
+spatial representation, without higher-level segmentation [5].
+Additionally, DOGMs are versatile in terms of sensor depen-
+dency, and can be generated from a variety of raw sensor
+data, such as Lidar or camera images. In our work, we
+use Bayesian-filter-based DOGM [6] that provide us with
+a spatially-dense model representation of static and dynamic
+space, as well as free and unknown space in the environment,
+as shown in Fig1.
+1 Univ. Grenoble Alpes, Inria, 38000 Grenoble, France, email: First-
+[email protected]
+As the DOGM is generated using data from the vehicle-
+mounted sensors, the grid is traditionally ego-centric,i.e. the
+position of ego-vehicle is fixed in the grid. While this is an
+effective method in scene representation, it complicates the
+long-term prediction problem. For a dynamic ego-vehicle,
+the complete scene translates and/or rotates around the ego-
+vehicle, even the static components in the scene. Therefore,
+the prediction network must transform every cell in the
+grid, leading to blurry and vanishing static scene at longer
+prediction time horizons.
+To address this, we instead generate DOGMs with respect
+to a fixed reference frame, referred as allo-centric grid.
+While the observed space around the ego-vehicle remains
+the same, the static scene structure in the allo-centric grid
+remains fixed. This is illustrated in Fig. 1 where the ego-
+vehicle is encircled, the vehicle moves like other agents in
+the scene.
+We approach the long-term multi-step predictions of allo-
+centric DOGM as a video prediction problem due to the
+inherent similarities between an image and an occupancy
+grid, and both being a spatio-temporal problem [7]. Results
+incorporating different video prediction networks are stud-
+ied, including state-of-the-art recurrent neural networks and
+memory-augmented network approaches. We compare and
+evaluate the prediction results of allo-centric and ego-centric
+grids for identical scenes and demonstrate the superior per-
+formances of the allo-centric grid predictions.
+The proposed approach is validated with the real-world
+NuScenes dataset [3] of urban traffic scenes. We show that
+allo-centric grids significantly improve the prediction results
+and demonstrate the ability to retain the scene structure and
+learn behaviours.
+The paper is organized as follows. Section II discusses
+related work to video and scene predictions. Section III
+describes the system overview. Section IV and V present
+implementations, results and analysis. Finally conclusions
+are drawn in section VI.
+II. RELATED WORK
+A. Video Prediction
+Spatio-temporal deep-learning methods have been ef-
+fectively used for video prediction problems. Commonly,
+combinations of Convolutional Neural Networks (CNNs)
+and Recurrent Neural Networks (RNNs) are incorporated.
+CNNs are capable of extracting spatial information and
+capturing inter-dependencies of the surrounding pixels while
+RNNs, such as long short-term memory (LSTM) blocks,
+arXiv:2301.04454v1  [cs.CV]  11 Jan 2023
+
+Fig. 1: Overview of our proposed approach. The allo-centric DOGM is represented as an image. Each channel red, green and
+blue represent unknown, dynamic and static cells respectively. The black space represents known free space. The ego-vehicle
+is circled in dotted line in both input and target output sequences.
+capture the sequential or temporal dependencies. Lotter et
+al. proposed Predictive Coding Network (PredNet), a deep
+learning network architecture that comprises of vertically-
+stacked Convolutional LSTMs (ConvLSTMs) where the local
+error and the prediction signal are propagated bottom-up
+and top-down respectively [8]. Wang et al. addresses the
+video prediction challenges of capturing short-term and long-
+term dynamics with the PredRNN architecture [9]. Building
+on their original approach [10], they introduce memory-
+decoupled spatio-temporal LSTM (ST-LSTM) blocks, fea-
+ture zigzag memory flow and a novel curriculum learning
+strategy to improve prediction results. Kim et al. takes in-
+spiration from memory-augmented networks to use external
+memory (LMC-Memory) to learn and store long-term motion
+dynamics and propose a memory query decomposition to ad-
+dress the high-dimensionality of motions in video predictions
+[11].
+B. Occupancy Grid Prediction
+Jeon et al. proposed conventional ConvLSTM to predict
+interaction-intensive traffic scenes on occupancy grids [12].
+The approach represents only vehicles in the occupancy grid,
+their states extracted from camera inputs. Desquaire et al.
+[13], proposed an end-to-end object-tracking approach by
+incorporating directly Lidar sensor data to predict the binary
+grid, using recurrent neural network. To incorporate ego-
+vehicle motion, they utilize a spatial transformer to allow
+internal memory of RNNs to learn environment of the state.
+Mohajerin et al. [14] suggested an RNN-based architecture
+with a difference learning method, and makes OGM pre-
+diction in the field of view of ego-vehicle front camera.
+Schreiber et al. [15] proposed an encoder-decoder network
+architecture, along with skip connections, to make long-term
+DOGM predictions. While they collect the sensor data from
+an autonomous vehicle, the vehicle remains stationary and
+only acts as the sensor collection point at different inter-
+sections. Itkina et al. proposed to use evidential occupancy
+grid and implement PredNet architecture for the prediction
+[16]. The approach is then carried forward to develop the
+double-pronged architecture [17] and attention-augmented
+ConvLSTM [18]. The latter work is able to make long-term
+predictions, however at turns the predictions still lose the
+scene structure. Mann et al. [19] addressed the problem of
+OGM prediction in urban scenes by incorporating vehicles
+semantics in the environment. Their proposed method de-
+pends on the annotated vehicle data labels available in the
+dataset.
+Contrary to the conventional Occupancy Grid Prediction,
+we present an allo-centric DOGM representation to predict
+the urban traffic scene with respect to a fixed reference frame.
+Apart from the conventional recurrent representation learning
+approaches, we also use memory-augmented learning-based
+video-prediction method, in relevance to learning long-term
+motion context of the dynamic agents.
+III. SYSTEM OVERVIEW
+We discuss here the overall proposed approach for allo-
+centric DOGM prediction, the pipeline is summarized in Fig.
+1.
+A. Dynamic Occupancy Grid Maps
+Dynamic occupancy grid maps provide a discretized rep-
+resentation of environment in a bird’s eye view, where every
+cell in the grid is independent and carries information about
+the associated occupancy and velocity.
+To generate DOGMs, we incorporate the Conditional
+Monte Carlo Dense Occupancy Tracker (CMCDOT) [6].
+This approach associates four occupancy states to the grid.
+Each cell carries the probabilities of the cell being i) occupied
+and static, ii) occupied and dynamic, iii) unoccupied or free
+and iv) if the occupancy is unknown. The probabilities of
+these four states sum to one. In our work, we make use
+of three of these states and represent the grid as an RGB
+image. The channels Red, Green and Blue represent the
+unknown state, dynamic state and static state respectively.
+The associated probabilities of the cell in the 3-channel
+DOGM grid are interpreted as the pixel values of the RGB
+images. The RGB grid images can be seen in Fig. 1-2. Low
+probabilities in all three channels leave the grid-image black,
+therefore, representing free space.
+For allo-centric grid generation, we define the grid in the
+world frame, close to the initial position of ego-vehicle.
+The state probabilities are initially computed in an ego-
+centric grid, since we use the on-board sensor data. To ensure
+that we have cell information for the complete allo-centric
+grid dimensions when the vehicle is dynamic and moving
+away from the world frame origin, a much larger ego-centric
+
+Allo-centric
+Video
+DOGM
+Prediction
+Generation
+NetworkDOGM is computed. This information is then fused to update
+every cell states in the allo-centric grid in the world frame.
+We compare the allo-centric and ego-centric grids at 4
+time instants for the same scene and same grid dimensions in
+Figure 2. In the allo-centric grid, the ego-vehicle (illustrated
+in the pink box) can be seen moving with respect to the grid,
+while it remains fixed in the ego-centric grid. It is important
+to note that the observable space around the ego-vehicle
+remains the same for both grids. However, since they are
+defined in different frames, the two cover different spaces in
+the scene at a given time. We illustrate the common space
+covered by both grids since the start of the sequence, marked
+by yellow boundary.
+Fig. 2: Visualization of allo-centric and ego-centric grids,
+generated for the same scene. The area marked by yellow
+lines is the common region covered by both grids up until
+the t-th sequence. The ego-vehicle is boxed in pink grid and
+the bus passing by is encircled in white.
+B. Problem Formulation
+We formally define the task of predicting the scene in
+allo-centric DOGM representation, as sequence-to-sequence
+learning, see Fig. 1. A sequence comprises of a set of
+sequential grid images that capture the evolution of a given
+scene. Let Xt ∈ R3xW xH and Yt ∈ R3xW xH be the t-th frame
+of the 3-channel grid-image where W and H denote the width
+and height respectively. The input sequence for the grid-
+image is denoted by Xt−N:t, representing N consecutive
+frames. Given a set of input sequence, the task of the
+network is to predict future grid images, i.e. output sequence.
+The target and predicted output sequences are denoted by
+Yt+1:t+P and ˆYt+1:t+P where P is the prediction horizon.
+For training and testing data, the DOGMs can be generated
+for both the input and the target sequences, leaving behind
+no additional need for labelled data or human intervention.
+Since the input sequences, Xt−N:t, and output sequences,
+Yt+1:t+P , are represented as images, this prediction task can
+be considered a video prediction problem.
+C. Deep Learning Prediction Architectures
+To study and compare the scene prediction with ego-
+centric and allo-centric grids, we train our datasets with
+different video prediction networks. We consider 3 networks,
+briefly discussed in section II-A: PredNet, PredRNN, LMC-
+Memory with memory alignment learning (here on referred
+as LMC-Memory).
+PredNet [8], inspired from predictive coding, makes pre-
+dictions based on how the predicted frames deviate from the
+target [20]. The original work tests the network on vehicle
+mounted camera images from Kitti dataset [21] and demon-
+strates the ability to capture both egocentric motion as well as
+motion of objects in camera images. We consider PredRNN
+[9] and LMC-Memory architecture [11] as the state of the
+art video prediction networks that aim to capture long-term
+dependencies and motion context. PredRNN implements
+novel ST-LSTM units with a zigzag internal memory flow
+and proposes memory decoupling loss to discourage learning
+redundant features. LMC-Memory architecture, on the other
+hand, proposes an external memory block with its own
+parameters to store various motion contexts. The approach
+also offers an efficient computation method since the motion
+context for long-term multi-step predictions is computed only
+once for a given input sequence.
+We study these networks capabilities to retain the occu-
+pancy of the static region, and the ability to predict motion
+of dynamic agents in DOGM.
+D. Unknown Channel and Loss functions
+In both ego-centric and allo-centric grids, a significant
+part of the scene remains unobserved, see Fig. 2 (unknown
+channel is represented in red). This is more pronounced in
+the initial frames of the allo-centric grid, where the Lidar is
+unable to detect the farthest area from the ego-vehicle.
+While it is more relevant to learn the evolution of static
+and dynamic components in the scene, inclusion of unknown
+channel is useful for our prediction task. A Lidar based grid
+is often unable to capture the full shape of a vehicle. For
+example, we can see in Fig. 2 how the occupied cells by the
+bus vary in different time steps on the grid. It is only in the
+2.0s time step that a rectangular shape is observed, otherwise
+different parts of the bus remain unknown. The unknown
+channel at different instants also carries spatial information
+of the agents with respect to the ego-vehicle. Thus, with the
+sequential frames and the unknown channel, we assist the
+network to be able to extract spatial information and learn
+scene representation.
+The inclusion of unknown channel and emphasis on
+learning static and dynamic components is addressed in the
+loss function. Loss function L in the implemented video
+prediction networks is modified to carry the weighted sum
+of the RGB channels:
+L = αLR + β(LG + LB)
+(1)
+where,
+LR, LG and LB represent the loss for unknown (red),
+dynamic (green) and static channels (blue) respectively. In
+order to encourage the network to learn and improve the
+prediction of the static and dynamic channels, α is always
+kept smaller than β.
+
+IV. EXPERIMENTS
+A. Dataset
+We study the prediction performance on the real-world
+NuScenes dataset [3]. The original dataset consists of 850
+scenes for training and 150 scenes for testing, each scene
+is approximately 20s long. We generate the DOGM grid-
+based on the Lidar pointcloud and available odometry. For
+allo-centric grid, we represent the scene with respect to a
+fixed reference frame and a grid dimension of 60 x 60m,
+with a resolution of 0.1m per cell. Each sequence starts with
+the ego-vehicle heading facing up, capturing the scene 10m
+behind and 50m ahead of it. The initial pose was selected to
+ensure that the ego-vehicle remains within the grid for the
+total sequence length, even when running at a high speed. For
+egocentric grid, we generate a grid of the same dimensions
+and resolution, and the ego-vehicle fixed in the center. Each
+sequence is comprised of 35 frames, a time duration of 3.5s
+with DOGM grid images generated every 0.1s. In total, we
+have 4,250 training and 750 testing sequences respectively.
+B. Training
+The input sequence Xt−9:t consists of 10 frames (1.0s).
+Each network is trained to make predictions ˆYt+1:t+25 for
+25 future frames (2.5s). Both the allo-centric and ego-
+centric datasets are trained with the original parameters of
+the respective video prediction network. For training with
+PredRNN and LMC Memory networks, both allo-centric
+and ego-centric grid images are resized to 192x192 pixels.
+PredRNN is trained with a batch size of 4 and a learning
+rate of 10−4. The number of channels of each hidden state is
+set to 64. The loss function is the sum of L2 and decoupling
+loss, and the values of α and β in Eq. (1) are set to 0.2 and
+0.8. LMC-Memory is trained with a learning rate of 2x10−4,
+memory slot is set to 100 and ConvLSTM to 4 layers for
+frame predictions. The loss function is the sum of L1 and
+L2 losses. The values of α and β are set to 0.2 and 0.8.
+For training with PredNet, the grid images are resized to
+160x160 pixels. The network is set to 4 hierarchical layers
+with an initial learning rate of 10−3. The loss function is the
+L1 loss of only the first layer, the values of α and β are set
+to 0.05 and 0.8. All models are trained on Adam optimizer
+for 30 epochs.
+V. EVALUATION
+For evaluation, we are particularly interested in static and
+dynamic agents in the scene. We discussed in section III-D,
+the utility of unknown regions in learning scene representa-
+tion. But the unknown region occupies a big portion of the
+grid and, thus, in evaluation, overshadows the performance of
+more interesting and relevant segments: static and dynamic
+regions. For this reason, we evaluate the dataset and network
+performances based on two channels of the predicted images,
+the blue and green channels representing static and dynamic
+components in the scene. We encourage the readers to refer
+to the video1 for a better visualization of the results.
+1https://youtu.be/z-0BVM93X8c
+A. Quantitative Evaluation
+The allo-centric and ego-centric grids at any instant ob-
+serve different parts of the scene, see Fig. 2. For fair
+comparison between them, we modify the test dataset and
+crop out the part of each t-frame that has not been observed
+until the t-th sequence by both grids. Thus, for example, the
+part of the grids outside of the yellow dotted boxes in Fig.
+2 are blacked out for the input sequence frames Xt−N:t as
+well as the target frames in the output sequence Yt+1:t+P .
+We measure the performances using three metrics: MSE
+(Mean Square Error), SSIM (Structured Similarity Indexing
+Method), and LPIPS (Learned Perceptual Image Patch Sim-
+ilarity) [22]. MSE is calculated by the pixel-wise difference
+between the ground truth and the predicted frame per channel
+and per cell. However, with MSE, the slightest error in
+predicted motion can result in large errors in the ego-
+centric grids dataset. The SSIM and LPIPS metrics evaluate
+the prediction results based on the structural similarity and
+perception similarity respectively. Lower values are better for
+MSE and LPIPS while higher values are better for SSIM.
+Table I shows average results for the complete 2.5s pre-
+diction horizons. The MSE score of allo-centric grids is
+significantly lower compared to the one of ego-centric grids.
+Since the complete scene transforms with respect to the
+ego-vehicle, the MSE is always higher in the ego-centric
+grid. The SSIM and LPIPS scores are also significantly
+superior for the allo-centric grid, due to the tendency of ego-
+centric grids to get increasingly blurry for higher prediction
+horizons.
+Network
+MSE x 10−2(���)
+SSIM(↑)
+LPIPS(↓)
+Allo-centric grid
+LMC-Memory
+0.894
+0.895
+0.167
+PredRNN
+0.882
+0.904
+0.167
+PredNet
+0.905
+0.888
+0.172
+Ego-centric grid
+LMC-Memory
+1.302
+0.856
+0.217
+PredRNN
+1.138
+0.845
+0.234
+PredNet
+1.335
+0.847
+0.225
+TABLE I: Average results with allo-centric and ego-centric
+grids for prediction horizon of 2.5s. The allocentric grid
+outperforms the other in all three video prediction networks.
+In Fig. 3, we plot scores of the metrics for every 0.5s
+prediction step. The results with allo-centric grid (shown
+in blue) always perform better than the ego-centric grids.
+Among the three prediction networks, overall PredRNN
+performs the best with allo-centric grids. However, with the
+ego-centric grids (results shown in orange), PredRNN offers
+a good MSE score but the SSIM and LPIPS performances
+drop after 1.0s. This is because PredRNN tends to make
+blurry and diffused predictions in the output frames; this
+helps reduce the MSE but the scene loses its structures. This
+is further seen in the qualitative results discussed in section
+V-B and illustrated in Fig. 4.
+B. Qualitative Evaluation
+The prediction results between the allo-centric and ego-
+centric grids differ drastically when the ego-vehicle is turning
+
+Fig. 3: Results with the MSE(↓), SSIM(↑) and LPIPS(↓) metrics with allo-centric and ego-centric grids for input sequences
+of 1.0s and prediction horizon up to 2.5s. For fair comparison, all test sequence frames were modified to only contain the
+scene observable in both the allo-centric and ego-centric grids. The allo-centric grid (results plotted in blue) outperforms
+the other with all three video prediction networks.
+(a) Allo-centric grids
+(b) Ego-centric grids
+Fig. 4: Qualitative results for the ego-vehicle leaving a roundabout on both allo-centric (4a) and ego-centric grids (4b). The
+input sequence consists of 10 frames (1.0s) and output predicted sequence of up to 25 frames (2.5s). The prediction results
+are shown at 0.5s, 1.5s and 2.5s instants and are magnified at the interesting spaces, marked by red box in the target(ground
+truth) frames. The best results can be observed with LMC-Memory network with the allo-centric grid that retains the scene
+structure and predicts the motion of the ego-vehicle best.
+at an intersection or driving on a curved road. Figure 4 shows
+results for a sequence where the ego-vehicle is exiting a
+roundabout. In this scene, while there are no other dynamic
+agents, the network needs to predict the behaviour of the ego-
+vehicle, when it is driving along the curved static segment
+(alluding to road structure) and is headed towards static
+objects/obstacles. For the allo-centric grids, the challenge is
+to predict the ego-vehicle pose while the scene remains static.
+The best results are achieved with the LMC-Memory. The
+vehicle pose is well-predicted up to 2.5s, its orientation is
+adjusted so that it does not hit the static components. For
+the same grid, the PredRNN fails to learn and predict the
+behaviour resulting in false prediction of collisions. The ego-
+vehicle, while getting more blurry, diffuses into the static
+obstacles on the road. With the PredNet, the ego vehicle
+is almost already lost at 1.5s prediction horizon. This is
+expected behaviour since PredNet is ideally not aimed at
+long-term video predictions. With all three networks, the ego-
+vehicle gets more blurry, however with PredRNN, the static
+scene also tends to get blurry at larger prediction horizon.
+
+洋LMC-ego
+PredRNN-ego
+PredNet-ego
+LMC-allo
+PredRNN-allo
+PredNet-alloIn the ego-centric grid, the whole scene rotates around the
+ego-vehicle. LMC-memory and PredNet significantly lose
+the static components ahead of the vehicle. The rotation
+results in increasing blurriness at every time step. PredRNN
+predictions are more diffused and faint blurry cells are
+still visible ahead of the vehicle, even at 2.5s prediction
+horizon. In context of planning and safe navigation, this
+high uncertainty in the environment structure renders the
+prediction results unreliable.
+VI. DISCUSSION AND FUTURE WORK
+In this work, we presented a novel allo-centric dynamic
+ocuupancy grid approach for long-term prediction of urban
+traffic scene, and compared it to the conventional ego-
+centric DOGM approach. We trained and tested various
+video prediction networks to show that allo-centric DOGM
+representation has superior ability to predict the same scene.
+The most significant improvement is the allo-centric grid’s
+ability to retain the static scene structure, especially when the
+vehicle turns. The ego-centric grid, on the other hand tends
+to lose the static scene, and hence the crucial information
+about whether the given space is occupied or free.
+The results of allo-centric grids prediction with state-of-
+the-art PredRNN and LMC-Memory approaches have shown
+complementary benefits. PredRNN predictions, though dif-
+fuse and get more blurry, are capable of maintaining agents
+longer. We observe that LMC-memory shows better tendency
+at learning behaviours in comparison to the PredRNN.
+It is pertinent to mention here that the two grids are still
+very similar. In both scenarios, the observable space updates
+relative to the position of the vehicle in the scene. Thus,
+in allo-centric grid while the grid is no more fixed to the
+ego-vehicle, the ego-vehicle bias remains.
+All three video prediction networks tested in this work
+address the prediction problem as deterministic. However,
+the behaviour of agents in urban traffic scene tends to be
+multimodal. For future work, the addition of multimodal
+prediction capabilities in the network architecture would be
+interesting. Additionally, the incorporation of semantics in
+the occupancy grid such as agent type and offline road infor-
+mation could assist in learning behaviours and interactions.
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+

DdE3T4oBgHgl3EQfUwqw/content/tmp_files/load_file.txt

@@ -0,0 +1,493 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf,len=492
+page_content='Allo-centric Occupancy Grid Prediction for Urban Traffic Scene Using  Video Prediction Networks  Rabbia Asghar1, Lukas Rummelhard1, Anne Spalanzani1, Christian Laugier1  Abstract— Prediction of dynamic environment is crucial to  safe navigation of an autonomous vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Urban traffic scenes  are particularly challenging to forecast due to complex interac-  tions between various dynamic agents, such as vehicles and  vulnerable road users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Previous approaches have used ego-  centric occupancy grid maps to represent and predict dynamic  environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' However, these predictions suffer from blurri-  ness, loss of scene structure at turns, and vanishing of agents  over longer prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' In this work, we propose a  novel framework to make long-term predictions by representing  the traffic scene in a fixed frame, referred as allo-centric  occupancy grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' This allows for the static scene to remain fixed  and to represent motion of the ego-vehicle on the grid like  other agents’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' We study the allo-centric grid prediction with  different video prediction networks and validate the approach  on the real-world Nuscenes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The results demonstrate  that the allo-centric grid representation significantly improves  scene prediction, in comparison to the conventional ego-centric  grid approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Index Terms— Scene Prediction, Deep Learning, Autonomous  Vehicles  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' INTRODUCTION  Prediction of traffic scene evolution is essential to an  autonomous vehicle for planning as well as detecting dan-  gerous situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' In urban traffic scenarios, the vehicles not  only interact with other vehicles, but also share space with  vulnerable road users such as pedestrians and cyclists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Key  challenges involve the uncertainty and multi-modality of the  behaviour of agents in the environment, and complex multi-  agents interactions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' While human drivers show superior  ability to forecast the agents’ behaviour and interactions in  such traffic scenes, it remains a challenge for autonomous  vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Data-driven methods provide powerful tools to solve pre-  diction problems, particularly dealing with complex social  interactions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Most conventional approaches are object  or agent-based and rely on heavily pre-processed data [3],  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Dynamic Occupancy Grip Maps (DOGMs), on the other  hand, allow for end-to-end learning due to their discretized  spatial representation, without higher-level segmentation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Additionally, DOGMs are versatile in terms of sensor depen-  dency, and can be generated from a variety of raw sensor  data, such as Lidar or camera images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' In our work, we  use Bayesian-filter-based DOGM [6] that provide us with  a spatially-dense model representation of static and dynamic  space, as well as free and unknown space in the environment,  as shown in Fig1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  1 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Grenoble Alpes, Inria, 38000 Grenoble, France, email: First-  Name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='LastName@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='fr  As the DOGM is generated using data from the vehicle-  mounted sensors, the grid is traditionally ego-centric,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' the  position of ego-vehicle is fixed in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' While this is an  effective method in scene representation, it complicates the  long-term prediction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' For a dynamic ego-vehicle,  the complete scene translates and/or rotates around the ego-  vehicle, even the static components in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Therefore,  the prediction network must transform every cell in the  grid, leading to blurry and vanishing static scene at longer  prediction time horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  To address this, we instead generate DOGMs with respect  to a fixed reference frame, referred as allo-centric grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  While the observed space around the ego-vehicle remains  the same, the static scene structure in the allo-centric grid  remains fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' 1 where the ego-  vehicle is encircled, the vehicle moves like other agents in  the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  We approach the long-term multi-step predictions of allo-  centric DOGM as a video prediction problem due to the  inherent similarities between an image and an occupancy  grid, and both being a spatio-temporal problem [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Results  incorporating different video prediction networks are stud-  ied, including state-of-the-art recurrent neural networks and  memory-augmented network approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' We compare and  evaluate the prediction results of allo-centric and ego-centric  grids for identical scenes and demonstrate the superior per-  formances of the allo-centric grid predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  The proposed approach is validated with the real-world  NuScenes dataset [3] of urban traffic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' We show that  allo-centric grids significantly improve the prediction results  and demonstrate the ability to retain the scene structure and  learn behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Section II discusses  related work to video and scene predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Section III  describes the system overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Section IV and V present  implementations, results and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Finally conclusions  are drawn in section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' RELATED WORK  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Video Prediction  Spatio-temporal deep-learning methods have been ef-  fectively used for video prediction problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Commonly,  combinations of Convolutional Neural Networks (CNNs)  and Recurrent Neural Networks (RNNs) are incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  CNNs are capable of extracting spatial information and  capturing inter-dependencies of the surrounding pixels while  RNNs, such as long short-term memory (LSTM) blocks,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='04454v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='CV]  11 Jan 2023  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' 1: Overview of our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The allo-centric DOGM is represented as an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Each channel red, green and  blue represent unknown, dynamic and static cells respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The black space represents known free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The ego-vehicle  is circled in dotted line in both input and target output sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  capture the sequential or temporal dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Lotter et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' proposed Predictive Coding Network (PredNet), a deep  learning network architecture that comprises of vertically-  stacked Convolutional LSTMs (ConvLSTMs) where the local  error and the prediction signal are propagated bottom-up  and top-down respectively [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' addresses the  video prediction challenges of capturing short-term and long-  term dynamics with the PredRNN architecture [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Building  on their original approach [10], they introduce memory-  decoupled spatio-temporal LSTM (ST-LSTM) blocks, fea-  ture zigzag memory flow and a novel curriculum learning  strategy to improve prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' takes in-  spiration from memory-augmented networks to use external  memory (LMC-Memory) to learn and store long-term motion  dynamics and propose a memory query decomposition to ad-  dress the high-dimensionality of motions in video predictions  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Occupancy Grid Prediction  Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' proposed conventional ConvLSTM to predict  interaction-intensive traffic scenes on occupancy grids [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  The approach represents only vehicles in the occupancy grid,  their states extracted from camera inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Desquaire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  [13], proposed an end-to-end object-tracking approach by  incorporating directly Lidar sensor data to predict the binary  grid, using recurrent neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' To incorporate ego-  vehicle motion, they utilize a spatial transformer to allow  internal memory of RNNs to learn environment of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Mohajerin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' [14] suggested an RNN-based architecture  with a difference learning method, and makes OGM pre-  diction in the field of view of ego-vehicle front camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Schreiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' [15] proposed an encoder-decoder network  architecture, along with skip connections, to make long-term  DOGM predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' While they collect the sensor data from  an autonomous vehicle, the vehicle remains stationary and  only acts as the sensor collection point at different inter-  sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Itkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' proposed to use evidential occupancy  grid and implement PredNet architecture for the prediction  [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The approach is then carried forward to develop the  double-pronged architecture [17] and attention-augmented  ConvLSTM [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The latter work is able to make long-term  predictions, however at turns the predictions still lose the  scene structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' [19] addressed the problem of  OGM prediction in urban scenes by incorporating vehicles  semantics in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Their proposed method de-  pends on the annotated vehicle data labels available in the  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Contrary to the conventional Occupancy Grid Prediction,  we present an allo-centric DOGM representation to predict  the urban traffic scene with respect to a fixed reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Apart from the conventional recurrent representation learning  approaches, we also use memory-augmented learning-based  video-prediction method, in relevance to learning long-term  motion context of the dynamic agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' SYSTEM OVERVIEW  We discuss here the overall proposed approach for allo-  centric DOGM prediction, the pipeline is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Dynamic Occupancy Grid Maps  Dynamic occupancy grid maps provide a discretized rep-  resentation of environment in a bird’s eye view, where every  cell in the grid is independent and carries information about  the associated occupancy and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  To generate DOGMs, we incorporate the Conditional  Monte Carlo Dense Occupancy Tracker (CMCDOT) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  This approach associates four occupancy states to the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Each cell carries the probabilities of the cell being i) occupied  and static, ii) occupied and dynamic, iii) unoccupied or free  and iv) if the occupancy is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The probabilities of  these four states sum to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' In our work, we make use  of three of these states and represent the grid as an RGB  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The channels Red, Green and Blue represent the  unknown state, dynamic state and static state respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  The associated probabilities of the cell in the 3-channel  DOGM grid are interpreted as the pixel values of the RGB  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The RGB grid images can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Low  probabilities in all three channels leave the grid-image black,  therefore, representing free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  For allo-centric grid generation, we define the grid in the  world frame, close to the initial position of ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  The state probabilities are initially computed in an ego-  centric grid, since we use the on-board sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' To ensure  that we have cell information for the complete allo-centric  grid dimensions when the vehicle is dynamic and moving  away from the world frame origin, a much larger ego-centric  Allo-centric  Video  DOGM  Prediction  Generation  NetworkDOGM is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' This information is then fused to update  every cell states in the allo-centric grid in the world frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  We compare the allo-centric and ego-centric grids at 4  time instants for the same scene and same grid dimensions in  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' In the allo-centric grid, the ego-vehicle (illustrated  in the pink box) can be seen moving with respect to the grid,  while it remains fixed in the ego-centric grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' It is important  to note that the observable space around the ego-vehicle  remains the same for both grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' However, since they are  defined in different frames, the two cover different spaces in  the scene at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' We illustrate the common space  covered by both grids since the start of the sequence, marked  by yellow boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' 2: Visualization of allo-centric and ego-centric grids,  generated for the same scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The area marked by yellow  lines is the common region covered by both grids up until  the t-th sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The ego-vehicle is boxed in pink grid and  the bus passing by is encircled in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Problem Formulation  We formally define the task of predicting the scene in  allo-centric DOGM representation, as sequence-to-sequence  learning, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' A sequence comprises of a set of  sequential grid images that capture the evolution of a given  scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Let Xt ∈ R3xW xH and Yt ∈ R3xW xH be the t-th frame  of the 3-channel grid-image where W and H denote the width  and height respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The input sequence for the grid-  image is denoted by Xt−N:t, representing N consecutive  frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Given a set of input sequence, the task of the  network is to predict future grid images, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' output sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  The target and predicted output sequences are denoted by  Yt+1:t+P and ˆYt+1:t+P where P is the prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  For training and testing data, the DOGMs can be generated  for both the input and the target sequences, leaving behind  no additional need for labelled data or human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Since the input sequences, Xt−N:t, and output sequences,  Yt+1:t+P , are represented as images, this prediction task can  be considered a video prediction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Deep Learning Prediction Architectures  To study and compare the scene prediction with ego-  centric and allo-centric grids, we train our datasets with  different video prediction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' We consider 3 networks,  briefly discussed in section II-A: PredNet, PredRNN, LMC-  Memory with memory alignment learning (here on referred  as LMC-Memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  PredNet [8], inspired from predictive coding, makes pre-  dictions based on how the predicted frames deviate from the  target [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The original work tests the network on vehicle  mounted camera images from Kitti dataset [21] and demon-  strates the ability to capture both egocentric motion as well as  motion of objects in camera images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' We consider PredRNN  [9] and LMC-Memory architecture [11] as the state of the  art video prediction networks that aim to capture long-term  dependencies and motion context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' PredRNN implements  novel ST-LSTM units with a zigzag internal memory flow  and proposes memory decoupling loss to discourage learning  redundant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' LMC-Memory architecture, on the other  hand, proposes an external memory block with its own  parameters to store various motion contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The approach  also offers an efficient computation method since the motion  context for long-term multi-step predictions is computed only  once for a given input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  We study these networks capabilities to retain the occu-  pancy of the static region, and the ability to predict motion  of dynamic agents in DOGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Unknown Channel and Loss functions  In both ego-centric and allo-centric grids, a significant  part of the scene remains unobserved, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' 2 (unknown  channel is represented in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' This is more pronounced in  the initial frames of the allo-centric grid, where the Lidar is  unable to detect the farthest area from the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  While it is more relevant to learn the evolution of static  and dynamic components in the scene, inclusion of unknown  channel is useful for our prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' A Lidar based grid  is often unable to capture the full shape of a vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' For  example, we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' 2 how the occupied cells by the  bus vary in different time steps on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' It is only in the  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='0s time step that a rectangular shape is observed, otherwise  different parts of the bus remain unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The unknown  channel at different instants also carries spatial information  of the agents with respect to the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Thus, with the  sequential frames and the unknown channel, we assist the  network to be able to extract spatial information and learn  scene representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  The inclusion of unknown channel and emphasis on  learning static and dynamic components is addressed in the  loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Loss function L in the implemented video  prediction networks is modified to carry the weighted sum  of the RGB channels:  L = αLR + β(LG + LB)  (1)  where,  LR, LG and LB represent the loss for unknown (red),  dynamic (green) and static channels (blue) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' In  order to encourage the network to learn and improve the  prediction of the static and dynamic channels, α is always  kept smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' EXPERIMENTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Dataset  We study the prediction performance on the real-world  NuScenes dataset [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The original dataset consists of 850  scenes for training and 150 scenes for testing, each scene  is approximately 20s long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' We generate the DOGM grid-  based on the Lidar pointcloud and available odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' For  allo-centric grid, we represent the scene with respect to a  fixed reference frame and a grid dimension of 60 x 60m,  with a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='1m per cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Each sequence starts with  the ego-vehicle heading facing up, capturing the scene 10m  behind and 50m ahead of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The initial pose was selected to  ensure that the ego-vehicle remains within the grid for the  total sequence length, even when running at a high speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' For  egocentric grid, we generate a grid of the same dimensions  and resolution, and the ego-vehicle fixed in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Each  sequence is comprised of 35 frames, a time duration of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s  with DOGM grid images generated every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' In total, we  have 4,250 training and 750 testing sequences respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Training  The input sequence Xt−9:t consists of 10 frames (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='0s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Each network is trained to make predictions ˆYt+1:t+25 for  25 future frames (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Both the allo-centric and ego-  centric datasets are trained with the original parameters of  the respective video prediction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' For training with  PredRNN and LMC Memory networks, both allo-centric  and ego-centric grid images are resized to 192x192 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  PredRNN is trained with a batch size of 4 and a learning  rate of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The number of channels of each hidden state is  set to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The loss function is the sum of L2 and decoupling  loss, and the values of α and β in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' (1) are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='2 and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' LMC-Memory is trained with a learning rate of 2x10−4,  memory slot is set to 100 and ConvLSTM to 4 layers for  frame predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The loss function is the sum of L1 and  L2 losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The values of α and β are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  For training with PredNet, the grid images are resized to  160x160 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The network is set to 4 hierarchical layers  with an initial learning rate of 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The loss function is the  L1 loss of only the first layer, the values of α and β are set  to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' All models are trained on Adam optimizer  for 30 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' EVALUATION  For evaluation, we are particularly interested in static and  dynamic agents in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' We discussed in section III-D,  the utility of unknown regions in learning scene representa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' But the unknown region occupies a big portion of the  grid and, thus, in evaluation, overshadows the performance of  more interesting and relevant segments: static and dynamic  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' For this reason, we evaluate the dataset and network  performances based on two channels of the predicted images,  the blue and green channels representing static and dynamic  components in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' We encourage the readers to refer  to the video1 for a better visualization of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  1https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='be/z-0BVM93X8c  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Quantitative Evaluation  The allo-centric and ego-centric grids at any instant ob-  serve different parts of the scene, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' For fair  comparison between them, we modify the test dataset and  crop out the part of each t-frame that has not been observed  until the t-th sequence by both grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Thus, for example, the  part of the grids outside of the yellow dotted boxes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  2 are blacked out for the input sequence frames Xt−N:t as  well as the target frames in the output sequence Yt+1:t+P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  We measure the performances using three metrics: MSE  (Mean Square Error), SSIM (Structured Similarity Indexing  Method), and LPIPS (Learned Perceptual Image Patch Sim-  ilarity) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' MSE is calculated by the pixel-wise difference  between the ground truth and the predicted frame per channel  and per cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' However, with MSE, the slightest error in  predicted motion can result in large errors in the ego-  centric grids dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The SSIM and LPIPS metrics evaluate  the prediction results based on the structural similarity and  perception similarity respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Lower values are better for  MSE and LPIPS while higher values are better for SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Table I shows average results for the complete 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s pre-  diction horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The MSE score of allo-centric grids is  significantly lower compared to the one of ego-centric grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Since the complete scene transforms with respect to the  ego-vehicle, the MSE is always higher in the ego-centric  grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The SSIM and LPIPS scores are also significantly  superior for the allo-centric grid, due to the tendency of ego-  centric grids to get increasingly blurry for higher prediction  horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Network  MSE x 10−2(↓)  SSIM(↑)  LPIPS(↓)  Allo-centric grid  LMC-Memory  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='894  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='895  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='167  PredRNN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='882  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='904  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='167  PredNet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='905  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='888  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='172  Ego-centric grid  LMC-Memory  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='302  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='856  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='217  PredRNN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='138  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='845  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='234  PredNet  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='335  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='847  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='225  TABLE I: Average results with allo-centric and ego-centric  grids for prediction horizon of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The allocentric grid  outperforms the other in all three video prediction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' 3, we plot scores of the metrics for every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s  prediction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The results with allo-centric grid (shown  in blue) always perform better than the ego-centric grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  Among the three prediction networks, overall PredRNN  performs the best with allo-centric grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' However, with the  ego-centric grids (results shown in orange), PredRNN offers  a good MSE score but the SSIM and LPIPS performances  drop after 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='0s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' This is because PredRNN tends to make  blurry and diffused predictions in the output frames;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' this  helps reduce the MSE but the scene loses its structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' This  is further seen in the qualitative results discussed in section  V-B and illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Qualitative Evaluation  The prediction results between the allo-centric and ego-  centric grids differ drastically when the ego-vehicle is turning  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' 3: Results with the MSE(↓), SSIM(↑) and LPIPS(↓) metrics with allo-centric and ego-centric grids for input sequences  of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='0s and prediction horizon up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' For fair comparison, all test sequence frames were modified to only contain the  scene observable in both the allo-centric and ego-centric grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The allo-centric grid (results plotted in blue) outperforms  the other with all three video prediction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  (a) Allo-centric grids  (b) Ego-centric grids  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' 4: Qualitative results for the ego-vehicle leaving a roundabout on both allo-centric (4a) and ego-centric grids (4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The  input sequence consists of 10 frames (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='0s) and output predicted sequence of up to 25 frames (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The prediction results  are shown at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s instants and are magnified at the interesting spaces, marked by red box in the target(ground  truth) frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The best results can be observed with LMC-Memory network with the allo-centric grid that retains the scene  structure and predicts the motion of the ego-vehicle best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  at an intersection or driving on a curved road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Figure 4 shows  results for a sequence where the ego-vehicle is exiting a  roundabout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' In this scene, while there are no other dynamic  agents, the network needs to predict the behaviour of the ego-  vehicle, when it is driving along the curved static segment  (alluding to road structure) and is headed towards static  objects/obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' For the allo-centric grids, the challenge is  to predict the ego-vehicle pose while the scene remains static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  The best results are achieved with the LMC-Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The  vehicle pose is well-predicted up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s, its orientation is  adjusted so that it does not hit the static components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' For  the same grid, the PredRNN fails to learn and predict the  behaviour resulting in false prediction of collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The ego-  vehicle, while getting more blurry, diffuses into the static  obstacles on the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' With the PredNet, the ego vehicle  is almost already lost at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' This is  expected behaviour since PredNet is ideally not aimed at  long-term video predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' With all three networks, the ego-  vehicle gets more blurry, however with PredRNN, the static  scene also tends to get blurry at larger prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  洋LMC-ego  PredRNN-ego  PredNet-ego  LMC-allo  PredRNN-allo  PredNet-alloIn the ego-centric grid, the whole scene rotates around the  ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' LMC-memory and PredNet significantly lose  the static components ahead of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The rotation  results in increasing blurriness at every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' PredRNN  predictions are more diffused and faint blurry cells are  still visible ahead of the vehicle, even at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='5s prediction  horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' In context of planning and safe navigation, this  high uncertainty in the environment structure renders the  prediction results unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' DISCUSSION AND FUTURE WORK  In this work, we presented a novel allo-centric dynamic  ocuupancy grid approach for long-term prediction of urban  traffic scene, and compared it to the conventional ego-  centric DOGM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' We trained and tested various  video prediction networks to show that allo-centric DOGM  representation has superior ability to predict the same scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  The most significant improvement is the allo-centric grid’s  ability to retain the static scene structure, especially when the  vehicle turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' The ego-centric grid, on the other hand tends  to lose the static scene, and hence the crucial information  about whether the given space is occupied or free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  The results of allo-centric grids prediction with state-of-  the-art PredRNN and LMC-Memory approaches have shown  complementary benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' PredRNN predictions, though dif-  fuse and get more blurry, are capable of maintaining agents  longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' We observe that LMC-memory shows better tendency  at learning behaviours in comparison to the PredRNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  It is pertinent to mention here that the two grids are still  very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' In both scenarios, the observable space updates  relative to the position of the vehicle in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Thus,  in allo-centric grid while the grid is no more fixed to the  ego-vehicle, the ego-vehicle bias remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content='  All three video prediction networks tested in this work  address the prediction problem as deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' However,  the behaviour of agents in urban traffic scene tends to be  multimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' For future work, the addition of multimodal  prediction capabilities in the network architecture would be  interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
+page_content=' Additionally, the incorporation of semantics in  the occupancy grid such as agent type and offline road infor-  mation could assist in learning behaviours and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'}
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