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+Multi-video Moment Ranking with Multimodal Clue
+Danyang Hou1,2, Liang Pang1, Yanyan Lan4, Huawei Shen1,3, Xueqi Cheng2,3
+1 Data Intelligence System Research Center, Institute of Computing Technology, CAS, Beijing, China
+2 CAS Key Lab of Network Data Science and Technology,
+Institute of Computing Technology, CAS, Beijing, China
+3 University of Chinese Academy of Sciences, Beijing, China
+4 Institute for AI Industry Research, Tsinghua University, Beijing, China
+Abstract
+Video corpus moment retrieval (VCMR) is the task of re-
+trieving a relevant video moment from a large corpus of
+untrimmed videos via a natural language query. State-of-
+the-art work for VCMR is based on two-stage method. In
+this paper, we focus on improving two problems of two-stage
+method: (1) Moment prediction bias: The predicted mo-
+ments for most queries come from the top retrieved videos,
+ignoring the possibility that the target moment is in the
+bottom retrieved videos, which is caused by the incon-
+sistency of Shared Normalization during training and in-
+ference.
+(2) Latent key content: Different modalities of
+video have different key information for moment localiza-
+tion. To this end, we propose a two-stage model MultI-video
+raNking with mUlTimodal cluE (MINUTE). MINUTE uses
+Shared Normalization during both training and inference
+to rank candidate moments from multiple videos to solve
+moment predict bias, making it more efficient to predict tar-
+get moment. In addition, Mutilmdaol Clue Mining (MCM)
+of MINUTE can discover key content of different modali-
+ties in video to localize moment more accurately. MINUTE
+outperforms the baselines on TVR and DiDeMo datasets,
+achieving a new state-of-the-art of VCMR. Our code will be
+available at GitHub.
+1. Introduction
+The rise of video-sharing applications has led to a dra-
+matic increase in the number of videos on the Internet.
+Faced with such a huge video corpus, users need an accu-
+rate retrieval tool to meet the needs of fine-grained cross-
+modal information. We have the opportunity to address this
+challenge thanks to the recently proposed video corpus mo-
+ment retrieval (VCMR) [9, 16] task that requires retrieving
+a video moment via a natural language query from a collec-
+tion of untrimmed videos, where the moment is a temporal
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Number of retrieved videos
+Moment prediction accuracy
+Moment prediction
+Video retrieval
+Video retrieval accuracy
+Figure 1. Moment prediction bias: Video retrieval accuracy im-
+proves as the number of retrieved videos increases, indicating that
+the probability of predicting the correct moment also increases.
+However, when the number of retrieved videos exceeds 2, moment
+prediction accuracy hardly increases, which means that predicted
+moments for most queries come from the top 2 videos.
+segment of a video. VCMR consists of two sub-tasks: video
+retrieval (VR) and single video moment retrieval (SVMR).
+The goal of VR is to retrieve videos that may contain the
+target moment via a natural language query. And SVMR
+aims to use the query to localize the target moment in the
+retrieved videos.
+According to different strategies to learn two sub-tasks,
+existing methods can be divided into one-stage method and
+two-stage method. One-stage method [16,18,31,32] treats
+VCMR as a multi-task learning problem, using a shared
+backbone with two different heads to learn VR and SVMR.
+Whereas two-stage method [15] leverages a pipeline of two
+independent modules to learn the two sub-tasks. Specially,
+it first trains a video retriever by query-video pairs to learn
+VR, then takes advantage of Shared Normalization (Shared-
+Norm) [7] technique to train localizer to learn SVMR,
+where the negatives for Shared-Norm are from the training
+data sampled by the trained retriever. In inference, it first
+uses retriever to select the most relevant K videos from cor-
+1
+arXiv:2301.13606v1  [cs.CV]  29 Jan 2023
+
+00:47:1200:49:42
+House : or that these two 
+have green eyes?
+00:51:3100:55:14
+Foreman : You're not saying... 
+They're not brother and sister.
+00:43:1800:46:40
+House : Is it a coincidence 
+that your sister has great hair,
+Query: House shows a picture of the patient to his team and they have concluded that maybe the two are not related by blood.
+Figure 2. Latent key content: The images with a red border are
+visual key content because these are relevant to “House shows a
+picture of the patient to his team” in query. The highlighted subti-
+tle is textual key content, for it relates to ”they have concluded that
+maybe the two are not related by blood”.
+pus, then uses localizer to localize the candidate moments in
+the K videos. The final predicted moment depends on both
+retrieval score and localization score. Two-stage method is
+more suitable for VCMR because (1) Shared-Norm can en-
+hance the possibility of the target moment appearing in the
+correct video. (2) Two-stage method can select models with
+different query-video interaction modes in the two mod-
+ules. For example, it select late-fusion model as retriever
+for fast video retrieval, and leverage early-fusion model as
+localizer for accurate moment localization. State-of-the-art
+model [15] for VCMR is also based on two-stage method.
+However, two problems limit the performance of two-
+stage method. The first is Moment prediction bias: as
+shown in Fig. 1, the final predicted moments for most
+queries are from the top-ranked videos among the K re-
+trieved videos. This is counter-intuitive because the more
+videos retrieved, the more likely those videos contain the
+correct moment. This bias neglects the possibility that the
+target moment is in the bottom-ranked videos. The reason
+for this bias is that although two-stage method uses Shared-
+Norm to normalize the probability of correct moment across
+correct video and negative videos, it still only normalizes
+the probability of the candidate moments in a single video
+during inference. This inconsistency in training and infer-
+ence results in the incomparable localization scores of can-
+didate moments during inference. Since the final predicted
+moment depends on both video retrieval score and moment
+localization score, the incomparable localization scores will
+make the final moment mainly depend on video retrieval
+scores, resulting in the final predicted moment more tend-
+ing to come from videos with higher rankings. The sec-
+ond problem is Latent key content: the localizer of two-
+stage method neglects key content from different modali-
+ties for moment localization. Video is usually composed of
+multimodal information, such as images (vision) and subti-
+tles (text). As shown in Fig. 2, visual information and tex-
+tual information have different emphases, if we can find out
+the important visual information and textual information as
+clues, it will help better moment localization.
+In this paper, we propose MultI-video raNking with
+mUlTimodal cluE (MINUTE) to improve the two prob-
+lems of two-stage method. For the first problem, we keep
+the consistence of Shared-Norm between training and in-
+ference, which forces the localization scores of candidate
+moments among multiple videos retrieved by retriever to be
+comparable during inference. On this basis, we derive a
+new scoring function to rank the candidate moments, which
+can combine the scores of video retrieval and moment lo-
+calization more effectively. For the second problem, we
+propose an early-fusion localizer with a Multimodal Clue
+Mining (MCM) component which can discover key content
+from different modalities to help moment localization. Spe-
+cially, MCM first uses query to measure the importance of
+all images and subtitles in the video, then assigns weights
+to these elements according to their importance. The ele-
+ments with high importance can be seen as key clues to im-
+prove moment localization. Then we feed weighted video
+representation together with query representation to a mul-
+timodal Transformer that captures deeper interactions be-
+tween video and query to predict moments.
+We conduct extensive experiments on TVR and DiDeMo
+datasets. The experimental results show that our proposed
+MINUTE outperforms other baselines, achieving a new
+state-of-the-art result. Ablation experiments verify that our
+method improves the two problems of two-stage method.
+2. Related Work
+We first briefly introduce works related to two sub-tasks
+of VCMR. After that, we introduce recent works for VCMR
+in detail.
+Text-video retrieval is a cross-modal retrieval task whose
+goal is to retrieve relevant videos from a corpus through
+a natural language query.
+This task is similar to VR of
+VCMR task, but most content of the video in the former
+is relevant to the query, while only a small part of the con-
+tent of the video in the latter is relevant to the query. The
+works for text-video retrieval can be divided into two cat-
+egories depending on the interaction mode between query
+and video, e.g., late fusion and early fusion. Late-fusion
+methods [8,21,27] use two separated encoders to embed im-
+ages and videos into a shared semantic space. These models
+can be very efficient if we calculate and index each modal
+representation offline, for only the similarity between video
+and query should be applied in inference.
+Early-fusion
+methods [6,12,25] make fine-grained interactions between
+video and query with an attention mechanism [2,24] to im-
+prove retrieval accuracy.
+Temporal language grounding is a task similar to SVMR,
+which requires localizing a moment from a video given a
+natural language query. Temporal language grounding can
+be seen as a special case of VCMR, with only one video
+in the corpus for each query. According to the way of pre-
+dicting moment, the existing works for temporal language
+grounding can be divided into proposal-based and proposal-
+2
+
+free. Proposal-based method [3,5,13,19,26,34] first gener-
+ates several proposals as candidates and then ranks the pro-
+posals according to their matching degree with the query,
+and the proposal with the highest matching degree is re-
+garded as the answer. Unlike the proposal-based method,
+proposal-free method [4,17,29,30,33] directly predicts the
+start and end times of the moment without pre-extracting
+proposals as candidates.
+Video corpus moment retrieval is first proposed by [9],
+then [16] propose a new dataset TVR for VCMR who ex-
+tends the uni-modal video (image) in the previous dataset
+video to multiple modalities (image and subtitle). The ex-
+isting works for VCMR can be divided into two categories
+depending on how they learn the two sub-tasks, e,g., one-
+stage [16, 18, 31, 32] method and two-stage method [15].
+The one-stage method treats VCMR as a multi-task learn-
+ing problem , using a shared model with two different heads
+to learn VR and SVMR simultaneously. XML [16] is the
+first one-stage method for VCMR who uses a late-fusion
+model to encode video and query separately and then uses
+two different heads to learn the two tasks. ReLoCLNet [32]
+leverage contrastive learning to enhance the performance
+of XML. [18] also follows XML and proposes a video-
+language pre-train model HERO, which significantly im-
+proves the performance. HAMMER [31] is an early-fusion
+one-stage model that uses attention to make deep inter-
+actions between query and video for more accurate mo-
+ment retrieval. Two-stage method leverages two different
+modules to learn two sub-tasks.
+CONQUER [15] is the
+only two-stage method that uses video retrieval heads of
+HERO [18] as the retriever and proposes a model based on
+context-query attention (CQA) [28] as the localizer. CON-
+QUER achieves state-of-the-art results on VCMR. In train-
+ing, CONQUER uses Shared-Norm [7] technique to train
+localizer. In inference, CONQUER first uses a video re-
+triever to retrieve top-K videos, then uses a moment local-
+izer to localize the moment in the retrieved videos. Two-
+stage method is more suitable for VCMR, but it suffers from
+moment prediction bias and latent key content. In this
+paper, we focus on improving the two problems.
+3. Background
+We first formulate VCMR, then describe two-stage
+method, followed by analyzing moment prediction bias.
+3.1. Task Formulation
+We denote a corpus of videos V = {v1, v2, ..., v|V|}
+where |V| is the number of videos in corpus and vi =
+{f 1
+i , f 2
+i , ..., f |vi|
+i
+} the i-th video which contains |vi| frames.
+Each frame f j
+i consists of an image and a subtitle (Ij
+i , sj
+i).
+Note that if it contains no subtitle, sj
+i is set to empty.
+Given a natural language query q = {w1, w2, ..., w|w|}
+which consists of a sequence of words, the goal of VCMR
+is to retrieve most relevant moment m∗ from V. The target
+moment m∗ is a temporal segment (τ∗,st, τ∗,ed) in video v∗,
+where v∗ denotes the video that contains the target moment
+whose start and end timestamps are τ∗,st and τ∗,ed respec-
+tively.
+The goal of VCMR can be seen as maximizing the prob-
+ability of target moment m∗ given the query q and the video
+corpus V:
+m∗ = argmax
+m
+P(m|q, V).
+(1)
+According to the chain rule of conditional probability:
+P(m∗|q, V) = P(m∗|v∗, q) · P(v∗|q, V),
+(2)
+where P(v∗|q, V) and P(m∗|v∗, q) are the probabilities of
+retrieving a video v∗ from corpus V and localizing the target
+moment m∗ in the retrieved video respectively. The proba-
+bility of target moment depends on the probabilities of start
+and end timestamps:
+P(m∗|v∗, q) = Pst(τ∗,st|v∗, q) · Ped(τ∗,ed|v∗, q).
+(3)
+3.2. Two-stage Method
+Two-stage method uses a video retriever to model
+P(v∗|q, V) and a moment localizer to model P(m∗|v∗, q).
+In training, two-stage method use margin-based loss [10]
+to train video retriever, then use Shared-Norm to train mo-
+ment localizer. Specially, for a query, there is a positive
+video v+ whose moment (τ+,j, τ+,k) is ground truth and n
+negative videos {v−
+1 , v−
+2 , . . . , v−
+n } that do not contain target
+moment. Shared-Norm is leveraged to normalize the prob-
+abilities of τ∗,j as start time and τ∗,k as end time across all
+frames in positive video and negatives, such as:
+Pst(τ+,j|v+, q) =
+exp(lst
++,j)
+n+1
+�
+a=1
+|vb|
+�
+b=1
+exp(lst
+a,b)
+,
+(4)
+where lst
+a,b is the the logits that b-th frame in video va is start
+timestamp of ground truth moment, and |vb| is the number
+of frame in a video. Training with Shared-Norm enhances
+the possibility of the target moment existing in the correct
+video.
+In inference, the retriever first uses the query to retrieve
+top-K videos from the corpus, then the localizer localizes
+the target moment in the retrieved videos. The score of the
+final predicted moment (τi,j, τi,k) in video i with start time
+j and end time k depends on both retrieval score and local-
+ization score, the scoring function is:
+Si,jk = exp(α · SR
+i ) · SL
+i,jk,
+(5)
+where Si,jk is the final score of the predicted moment, SR
+i is
+the retrieval score of video vi , and SL
+i,jk is the localization
+3
+
+score of a moment in a video, and α is a hyper-parameter to
+encourage the target moment from top retrieved videos. The
+retrieval score is computed by cosine similarity between
+query representation and video representation. And the lo-
+calization score is computed by the probability of a moment
+in a single video:
+SL
+i,jk = Pst(τi,j|vi, q) · Ped(τi,k|vi, q),
+(6)
+where Pst(τi,j|vi, q) or Ped(τi,k|vi, q) is normalized across
+a single video :
+Pst(τi,j|vi, q) =
+exp(lst
+i,j)
+|vi|
+�
+b=1
+exp(lst
+i,b)
+,
+(7)
+3.3. Moment Prediction Bias
+As shown in Fig. 1, the final predicted moments of
+two-stage method for most queries come from top-ranked
+videos.
+This bias limits the performance of two-stage
+method on VCMR, because it neglects the possibility of the
+target moment existing in the bottom-ranked videos. We
+conjecture that this bias mainly comes from the inconsis-
+tency of normalization during training and inference, shown
+in Eq. (4) and Eq. (7).
+In training, it uses Shared-Norm to highlight the signif-
+icance of the correct moment being in the correct video.
+Nevertheless, in inference, this probability is based on ev-
+ery single video, resulting in the predicted candidate mo-
+ments from different videos being incomparable, so the sig-
+nificance no longer exists. Therefore, the score of the final
+predicted moment in Eq. (5) is more dependent on video
+retrieval score, making the final predicted moment more
+likely to be from the top-ranked videos.
+4. Method
+We first illustrate how we improve moment prediction
+bias. Then we introduce the proposed model MINUTE, we
+emphasize multimodal clue mining component. Finally, we
+describe the training of MINUTE.
+4.1. Multi-video Moment Ranking in Prediction
+We propose to adopt Shared-Norm in inference, so that
+the localization scores of candidate moments from multiple
+videos are comparable, which can enhance the influence of
+moment localization score SL
+i,jk on the final score Si,jk to
+improve moment prediction bias. Furthermore, we derive
+a new scoring function from Eq. (2) to combine the video
+retrieval and moment localization scores more effectively.
+Specially, to compute P(v∗|q, V), we obtain video repre-
+sentation vi = {f 1
+i , f 2
+i , ..., f |vi|
+i
+} and query representation
+q. In the following paper, we use bold notations to denote
+vectors. The j-th frame representation f j
+i consists of image
+representation and subtitle representation (Ij
+i , sj
+i). Query
+also has two representations (qI, qs) to compute similarity
+scores for images and subtitles respectively. The query and
+video representations details are in Sec. 4.2.1.
+Because only part of the content in the video is related to
+the query, the similarity score between the query and video
+SR
+i
+is the average of max-pooling of query-image scores
+and max-pooling of query-subtitle scores. We use the inner
+product as the similarity score sim():
+sim(qc, cj
+i) = qcT · cj
+i, c ∈ {I, s},
+φc =
+max
+1≤j≤|vi| sim(qc, cj
+i),
+SR
+i = φI + φs
+2
+.
+(8)
+The probability P(v∗|q, V) is computed by softmax nor-
+malized score across all query-video scores in corpus:
+P(v∗|q, V) =
+exp(SR
+∗ )
+�|V|
+j=1 exp(SR
+j )
+.
+(9)
+Computing the inner product between query and all videos
+in the corpus is computationally intensive, so we em-
+ploy Max Inner Product Search (MIPS) [22] to find top-K
+videos to approximate the probability. The calculation of
+P(v∗|q, V) in Eq. (9) can be approximated by P(v∗|q, V∗):
+P(v∗|q, V) ≈ P(v∗|q, V∗) =
+exp(SR
+∗ )
+�K
+j=1 exp(SR
+j )
+.
+(10)
+The probabilities of the rest videos in the corpus are con-
+sidered close to 0. The training of the retriever is to maxi-
+mize the log-likelihood of probability logP(v∗|q, V), which
+is different from the previous two-stage method who use
+margin-based loss.
+As for P(m∗|v∗, q), we use Shared-Norm in inference,
+which is consistent with that in training to improve moment
+prediction bias:
+P(m∗|v∗, q) ≈ P(m∗|V∗, q) =
+exp(lst
+∗,j)
+K
+�
+a=1
+|vi|
+�
+b=1
+exp(lst
+a,b)
+·
+exp(led
+∗,k)
+K
+�
+a=1
+|vi|
+�
+b=1
+exp(led
+a,b)
+.
+(11)
+A well-trained localizer should suppress the probability that
+the target moment appears in the wrong videos to close to
+zero, so P(m∗|V∗, q) approximately equals to P(m∗|v∗, q).
+The details of logits lst
+∗,j are introduced in Sec. 4.2.2.
+Combine Eq. (2), Eq. (10) and Eq. (11), the probability
+P(m∗|v∗, q) can be computed by:
+P(m∗|v∗, q) ≈
+exp(SR
+∗ )
+�K
+j=1 exp(SR
+j )
+exp(lst
+∗,j)
+K
+�
+a=1
+|vi|
+�
+b=1
+exp(lst
+a,b)
+exp(led
+∗,k)
+K
+�
+a=1
+|vi|
+�
+b=1
+exp(led
+a,b)
+, (12)
+4
+
+where the denominator is the same for all candidate mo-
+ments from K videos, so we can simplify this probability to
+a new scoring function:
+S∗ = SR
+∗ + lst
+∗,j + led
+∗,k,
+(13)
+where lst
+∗,j + led
+∗,k = SL
+∗,ij represents moment localization
+score. This scoring function is simpler than Eq. (5) and
+without hyper-parameter α which may greatly increase the
+weight of the top-ranked video retrieval score.
+In inference, we use scoring function in Eq. (13) to rank
+all moments in multiple retrieved videos.
+4.2. Model
+We propose a two-stage MINUTE model consisting of a
+late-fusion video retriever and an early-fusion moment lo-
+calizer.
+4.2.1
+Video Retriever
+The goal of video retriever is to select a small subset V∗
+from the corpus V given the query q, where videos in the
+subset may contain the target moment. The retriever of the
+proposed model is a late-fusion model that contains two en-
+coders, a query encoder and a video encoder, as shown in
+Fig. 3. The late-fusion architecture ensures retrieval effi-
+ciency if we index the representations of videos in advance.
+Video Encoder The video encoder encodes frames in the
+i-th video to frame representations vi = {f 1
+i , ..., f |vi|
+i
+},
+where the j-th frame f j
+i contains image representation Ij
+i
+and subtitle representation sj
+i. We first use RoBERTa [20]
+to extract sentence features of subtitle and use Slow-
+Fast [11] and ResNet [14] to extract image features. Then
+we feed subtitle features and image features to a one-
+layer multi-modal Transformer that simultaneously cap-
+tures intra-modal and inter-modal dependencies to output
+each image representation Ij
+i and subtitle representation sj
+i.
+Query Encoder The query encoder convert query q =
+{w1, w2, ..., w|q|} to query representation q. We first use
+RoBERTa to extract the feature wj of each word in the
+query. A one-layer Transformer is used to capture the con-
+textual representation of each word. We generate two query
+representations for query-image similarity score and query-
+subtitle similarity score, denoted as qI and qs. We adopt
+a modular pooling mechanism [16] to convert the sequence
+representations to the two vectors:
+oi = Wcwi, αi =
+exp(oi)
+|q|
+�
+j=1
+exp(oj)
+, qc =
+|q|
+�
+i=1
+αiwi,
+(14)
+where Wc is learnable parameters, c ∈ {I, s}. The modular
+mechanism can be regarded as a learnable pooling and is
+also used in previous works [16,18,32].
+Query: Foreman tells Enid
+why he had to sedate the
+patient.
+Transformer
+Corpus
+Enid : Did he need a
+sedative? I did.
+00:48:4500:52:12
+subtitles
+images
+SlowFast
+ResNet
+RoBERTa
+PE
+Multimodal Transformer
+RoBERTa
+ME
+ME
+PE
+Query Encoder
+Video Encoder
+Video
+𝑰1
+𝑰2
+𝑰|𝑣|
+𝒔1
+𝒔2
+𝒔|𝑣|
+Modular
+Pooling
+𝒒𝐼
+𝒒𝑠
+Figure 3. Video retriever consists of two encoders, video encoder
+and query encoder. ’ME’ and ’PE’ represent modality embedding
+and positional embedding, respectively.
+We also use the retrieval head of HERO [18] as retriever
+for a fair comparison with CONQUER [15]. The original
+HERO uses margin-based loss [10] to train video retrieval
+whose retrieval score only represents cosine similarity be-
+tween query and videos, so we re-train HERO in the same
+way as training the proposed retriever to model the proba-
+bility P(v∗|q, V) in Eq. (10). We use simple retriever to
+denote the proposed retriever and HERO retriever to de-
+note the retriever based on HERO.
+4.2.2
+Moment Localizer
+Moment localizer shown in Fig. 4 uses the query to localize
+the target moment m∗ in the top-K retrieved videos V∗. The
+proposed localizer is based on early-fusion architecture to
+explore deeper interactions between query and video. Be-
+cause the retrieved videos are narrowed down to a small
+range, the amount of computations is acceptable.
+The localizer first uses query encoder to get token rep-
+resentations { ¯
+w1, ..., ¯
+w|q|} and video encoder to get video
+representation ¯vi = { ¯
+f 1
+i , ..., ¯
+f |vi|
+i
+}, where ¯
+f j
+i contain an
+image representation and a subtitle representation (¯Ij
+i , ¯sj
+i).
+Video encoder and query encoder in localizer are same with
+those in retriever but do not share parameters.
+Our proposed localizer consists of two components:
+5
+
+query
+Transformer
+Modular
+Pooling
+Video 
+Encoder
+Multimodal Clue Mining
+Multimodal Transformer
+FC
+Query encoder
+ഥ𝒒𝐼
+ഥ𝒒𝑠
+ത𝑰
+ത𝒔
+෠𝑰
+ො𝒔
+෠𝒇
+ഥ𝒘
+video
+1D 
+Conv
+1D 
+Conv
+𝑙𝑠𝑡
+𝑙𝑒𝑑
+ഥ𝒘
+Figure 4. Moment localizer contains two components, multimodal
+clue mining and multimodal Transformer. For brevity, we omit the
+subscripts of the representations.
+multimodal clue mining and multi-modal Transformer.
+Multimodal Clue Mining (MCM) solves late key content
+problem by discovering important content from multiple
+modalities of video to help moment localization.
+MCM
+first uses query to measure the importance of each image
+and subtitle in video, then assigns weights to these elements
+from different modalities according to importance.
+Specially, we leverage modular pooling to obtain query
+representations ¯qI and ¯qs to measure image importance and
+subtitle importance respectively. The importance is com-
+puted by:
+pj
+c = ( ¯
+Wc¯cj) ⊙ ¯qc, c ∈ {I, s},
+(15)
+where ¯
+Wc is learnable parameters, and pj
+c is the importance
+of j-th image or subtitle. Then we use the importance to
+weight the image and subtitle representations:
+ˆcj = norm(pj
+c) ⊙ ¯cj, c ∈ {I, s},
+(16)
+where ˆcj is weighted image representation or subtitle rep-
+resentation and norm is L2-normalization which makes the
+model converge better.
+MCM can be seen as an amplifier that allows localizer to
+focus on important content which we call clues from multi-
+ple modalities.
+We fuse the weighted representations ˆIj and ˆsj in a
+frame by a fully-connect layer:
+ˆ
+f j = FC([ˆIj; ˆsj]),
+(17)
+where [; ] is concatenation and ¯f j is the fused represen-
+tation the j-th frame.
+The fused video representation is
+ˆvi = { ˆ
+f 1
+i , ..., ˆ
+f |vi|
+i
+} and are fed to a multimodal Trans-
+former together with query token representations.
+Multimodal Transformer (MMT) We use a three-layer
+multi-modal Transformer to make deep interactions be-
+tween fused video representation and token representations.
+In addition, two 1D-convolution layers are leveraged to cap-
+ture dependencies between adjacent frames and output log-
+its lst
+i,j, led
+i,k of the start and end times of the target moment.
+4.3. Training
+We first train retriever by text-video pairs, then use the
+trained retriever to sample negative videos as hard negatives
+to train localizer.
+Training retriever To maximize the log-likelihood of prob-
+ability logP(v∗|q, V) in Eq. (9), we adopt InfoNCE [23]
+loss with in-batch negative sampling to train retriever. Spe-
+cially, let d = {(v1, q1), ..., (vb, qb)} denote training data in
+a batch, where b is batch size. Each pair (vi, qi) in d has
+b − 1 negative samples for query-to-video loss or video-to-
+query loss, such (vz, qi)z̸=i and (vi, qz)z̸=i:
+Lv = −log
+exp(SR
+i,i)
+b�
+z=1
+exp(SR
+z,i)
+, Lq = −log
+exp(SR
+i,i)
+b�
+z=1
+exp(SR
+i,z)
+,
+(18)
+where Lv and Lq are query-to-video loss and video-to-
+query loss, respectively. We use the sum of the two losses
+to train retriever.
+Training localizer We use the well-trained retriever to re-
+trieve top-ranked videos from training data and sample n
+videos as hard negatives to train the localizer with Shared-
+Norm technique.
+Lst = −log
+exp(lst
++,j)
+n+1
+�
+a=1
+|vb|
+�
+b=1
+exp(lst
+a,b)
+, Led = −log
+exp(led
++,k)
+n+1
+�
+a=1
+|vb|
+�
+b=1
+exp(led
+a,b)
+,
+(19)
+The sum of Lst and Led are used to train localizer.
+5. Experiment
+We first introduce datasets and metrics. Then we de-
+scribe implementation details. After that, we introduce ex-
+6
+
+Table 1. Comparisons of VCMR results(IoU=0.7) with baselines
+on TVR validation set and testing set.’SR’ denotes simple re-
+triever, and ’HR’ denotes HERO retriever.
+Model
+Validation
+Testing
+R1
+R10
+R100
+R1
+R10
+R100
+XML
+2.62
+9.05
+22.47
+3.32
+13.41
+30.52
+ReLoCLNet
+4.15
+14.06
+32.42
+-
+-
+-
+HAMMER
+5.13
+11.38
+16.71
+-
+-
+-
+HERO
+5.13
+16.26
+24.55
+6.21
+19.34
+36.66
+CONQUER
+7.76
+22.49
+35.17
+9.24
+28.67
+41.98
+MINUTE(SR)
+8.17
+23.38
+37.93
+9.59
+28.96
+45.23
+MINUTE(HR)
+10.70
+29.37
+45.09
+12.60
+33.72
+50.23
+Table 2.
+Comparisons of VCMR results with baselines on
+DiDeMo testing set.
+Model
+IoU=0.5
+IoU=0.7
+R1
+R5
+R10
+R1
+R5
+R10
+XML
+2.36
+-
+10.42
+1.59
+-
+6.77
+HERO
+3.37
+8.97
+13.26
+2.76
+7.73
+11.78
+CONQUER
+3.31
+9.27
+13.99
+2.79
+8.04
+11.90
+MINUTE(HR)
+3.44
+9.62
+14.62
+2.81
+7.89
+12.03
+perimental results comparison with baselines. Then we il-
+lustrate ablation studies of the proposed model. Finally, we
+present the case study.
+5.1. Datasets
+TVR [16] is built on TV Shows whose videos consist of
+images and subtitles. TVR contains 17435, 2179, and 1089
+videos on the training, validation, and testing sets. The av-
+erage length of the videos is 76.2 seconds, while the average
+length of the moments is 9.1 secs.
+DiDeMo [1] is a dataset whose videos are from the real
+world, with only images and no subtitles in the video.
+DiDeMo contains 8395, 1065, and 1004 training, valida-
+tion, and testing videos, respectively. The average duration
+of videos and moments is 54 secs and 6.5 secs, respectively.
+5.2. Evaluation Metrics
+We follow the metrics in [16] as evaluation metrics of ex-
+periments. For VCMR task, the evaluation metric is R@K,
+IoU=p that represents the percentage that at least one pre-
+dicted moments whose Intersection over Union(IoU) with
+the ground truth exceed p in the top-K retrieved moments.
+The two sub-tasks are also evaluated. The metric of SVMR
+task is the same as that of VR task, but the evaluation is
+conducted in only ground truth video for each query. As for
+VR task, the metric is R@K which denotes the percentage
+that correct video is in the top-K ranked videos.
+5.3. Implementation Details
+Training We train simple retriever for 100 epochs with the
+batch size 256. As for localizer, we sample 4 and 2 negative
+Table 3. Comparisons of VR results with baselines on TVR vali-
+dation set.
+Model
+R@1
+R@5
+R@10
+R@100
+XML
+16.54
+38.11
+50.41
+88.22
+ReLoCLNet
+22.13
+45.85
+57.25
+90.21
+HERO
+29.01
+52.82
+63.07
+89.91
+SR
+23.12
+46.86
+57.83
+90.22
+HR
+32.88
+55.62
+65.35
+91.26
+Table 4. Comparisons of SVMR results with baselines on TVR
+Validation set.
+Model
+IoU=0.5
+IoU=0.7
+R1
+R10
+R100
+R1
+R10
+R100
+XML
+31.43
+-
+-
+13.89
+-
+-
+ReLoCLNet
+31.88
+-
+-
+15.04
+-
+-
+HERO
+32.22
+60.08
+80.66
+15.30
+40.84
+63.45
+CONQUER
+43.63
+-
+-
+22.84
+-
+-
+MINUTE(SR)
+44.49
+78.62
+93.57
+23.98
+61.30
+80.13
+MINUTE(HR)
+44.74
+78.90
+93.80
+24.08
+62.10
+80.45
+videos for each query from top-100 ranked videos on TVR
+and DiDeMo respectively, and train it for 10 epochs with the
+batch size 32. Both simple retriever and localizer are trained
+by AdamW with the learning rate 0.0001 and the weight
+decay of 0.01 in a single 3090 GPU. For HERO retriever,
+we retrain it with InfoNCE loss in 8 3090 GPUs with the
+same setting as the original HERO [18].
+Inference The localizer localizes the target moment in the
+top-10 retrieved videos. The length of predicted moments is
+limited to [1, 24] and [1, 7] for TVR and DeDiMo, respec-
+tively. We use non-maximum suppression(NMS) with the
+IoU 0.7 to post-process the predicted moments.
+5.4. Comparison with Baselines
+We compare the proposed model with baselines on
+VCMR task including four one-stage models XML [16],
+ReLoCLNet [32], HAMMER [31], HERO [18] and a two-
+stage model CONQUER [15].
+TVR As shown in Tab. 1, the proposed models outperform
+all baseline methods.
+Compared with the best previous
+method CONQUER who also uses HERO to address the
+VR task, our proposed model with HERO retriever achieves
+36% improvement at R@1 on the testing set. We also re-
+port the results on two sub-task in Tab. 3 and Tab. 4. For
+VR, HERO retriever trained by InfoNCE loss has better re-
+trieval accuracy than the original HERO. For SVMR, our
+proposed models also achieve the best results. It is worth
+noting that the proposed model with simple retriever out-
+performs CONQUER on VCMR even though the perfor-
+mance of VR(R@1 23.12) is much worse than that in CON-
+QUER(R@1 29.01). This is because moment prediction
+bias limits the performance of CONQUER.
+DiDeMo We report the VCMR results on DiDeMo testing
+7
+
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Number of retrieved videos
+7
+8
+9
+10
+R@1, IOU=0.7
+MINUTE(HR)
+CONQUER
+CONQUER*
+Figure 5. The performances of VCMR of our model and COU-
+QUER under different numbers of the retrieved videos, where
+’CONQUER*’ denotes CONQUER with our retriever and scoring
+function.
+Table 5. Performances of VCMR and SVMR (R@1, IOU=0.5,0.7)
+when remove two componets in localizer. MCM denotes multi-
+modal clue mining, and MMT represents multimodal Transformer.
+Model
+VCMR
+SVMR
+0.5
+0.7
+0.5
+0.7
+MINUTE(HR)
+19.22
+10.70
+44.74
+24.08
+w/o MCM
+18.21
+10.17
+43.41
+23.46
+w/o MMT
+16.71
+8.66
+40.5
+20.97
+set in Tab. 2. The performance of the proposed model is still
+better than others. All the methods perform worse than the
+results on TVR because the DiDeMo dataset is designed for
+temporal language grounding, so the difficulty of retrieving
+video is not considered. The query of DiDeMo is not as spe-
+cific as that of TVR, such as ”a girl is playing ball”, making
+it hard to retrieve the correct video.
+5.5. Moment Prediction Bias
+As shown in Fig. 5, when the number of retrieved videos
+increases, the performance of our model improves, but the
+CONQUER does not change much, which indicates that
+moment prediction bias limits its performance. This bias
+is from the inconsistency of Shared-Norm in training and
+inference.
+Our prediction based on the scoring function
+in Eq. (13) addresses this prediction bias by ranking mo-
+ments in multiple retrieved videos in inference. When we
+replace CONQUER’s retriever and scoring function with
+ours, CONQUER* in Fig. 5 can also improve moment pre-
+diction bias, showing the proposed model’s effectiveness.
+5.6. Multimodal Clue Mining
+We perform ablation studies on the effectiveness of two
+components of localizer in Tab. 5. When removing MCM,
+the accuracy drops, which shows that discovering key con-
+tent from images and subtitles as clue is helpful for moment
+localization. When we only use MCM, the accuracy drops
+a lot, indicating that using clues is not enough, fine-grained
+cross-modal interactions are also needed.
+37.5s
+MINUTE
+38.01s
+40.91s
+24.00s
+27.00s
+Ground Truth
+CONQUER
+43.50s
+Query: Amy and Bernadette spin around on their bar seats to face the other way.
+00:35:0100:37:39
+Bandleader : Mr. and 
+Mrs. Chandler Bing.
+00:30:5100:34:51
+Bandleader : Ladies and gentlemen, it gives 
+me great pleasure to introduce to you : 
+31.46s
+43.40s
+Ground Truth
+30.00s
+42.00s
+MINUTE
+Query: The bandleader announces Chandler and Monica and they walk into the room. 
+30.00s
+36.00s
+CONQUER
+Figure 6. Two cases on TVR from the proposed model and CON-
+QUER.
+5.7. Case Study
+We show two cases of VCMR in Fig. 6. In the first case,
+two models retrieve the correct video first, the moment pre-
+dicted by the proposed model is closer to the ground truth.
+The proposed model captures key images related to ”they
+walk into the room” to help localize the moment, indicating
+the effectiveness of MCM in our model. In the second case,
+both models rank the wrong video first because the scenario
+in this video is similar to that in the correct video. CON-
+QUER fails to predict correct moment from correct video,
+for it places too much emphasis on top-ranked videos. Our
+proposed model can predict correct moment, which verifies
+that our prediction improves moment prediction bias.
+6. Conclusion
+In this paper, we propose a model MultI-video raNking
+with mUlTimodal cluE (MINUTE) improving two prob-
+lems of two-stage method on video corpus moment retrieval
+task, moment prediction bias and latent key content. We
+first analyze the reason for moment prediction bias that in-
+consistency of Shared-Norm in training and inference, then
+we adopt Shared-Norm in inference and rank moments in
+multiple videos based on our derived scoring function to
+improve moment prediction bias. As for latent key con-
+tent, we propose a multimodal clue mining component to
+discover important content from two modalities of video as
+clue for better moment localization. Extensive experiments
+on two datasets TVR and DiDeMo show that our proposed
+model improves two problems and achieves a new state-of-
+the-art of video corpus moment retrieval task.
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+JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, NOVEMBER 2022
+1
+Domain Generalization via Ensemble Stacking for Face Presentation
+Attack Detection
+Usman Muhammad1, Djamila Romaissa Beddiar1, and Mourad Oussalah1, Fellow, IEEE
+1 Center for Machine Vision and Signal Analysis, University of Oulu, Finland
+Face presentation attack detection (PAD) plays a pivotal role in securing face recognition systems against spoofing attacks. Although
+great progress has been made in designing face PAD methods, developing a model that can generalize well to an unseen test domain
+remains a significant challenge. Moreover, due to different types of spoofing attacks, creating a dataset with a sufficient number
+of samples for training deep neural networks is a laborious task. This work addresses these challenges by creating synthetic data
+and introducing a deep learning-based unified framework for improving the generalization ability of the face PAD. In particular,
+synthetic data is generated by proposing a video distillation technique that blends a spatiotemporal warped image with a still image
+based on alpha compositing. Since the proposed synthetic samples can be generated by increasing different alpha weights, we train
+multiple classifiers by taking the advantage of a specific type of ensemble learning known as a stacked ensemble, where each such
+classifier becomes an expert in its own domain but a non-expert to others. Motivated by this, a meta-classifier is employed to learn
+from these experts collaboratively so that when developing an ensemble, they can leverage complementary information from each
+other to better tackle or be more useful for an unseen target domain. Experimental results using half total error rates (HTERs) on
+four PAD databases CASIA-MFSD (6.97%), Replay-Attack (33.49%), MSU-MFSD (4.02%), and OULU-NPU (10.91%)) demonstrate
+the robustness of the method and open up new possibilities for advancing presentation attack detection using ensemble learning
+with large-scale synthetic data.
+Index Terms—Face Anti-Spoofing, Ensemble Learning, Deep Learning, Synthetic Data, LSTM.
+I. Introduction
+O
+VER the past few decades, facial recognition (FR)
+technology has been frequently used in numerous real-
+world applications, such as mobile payments, access control,
+immigration, education, surveillance, and healthcare [1]. The
+accuracy of FR is no longer a major concern and the error
+rate has dropped to 0.08%, according to tests conducted by
+the National Institute of Standards and Technology (NIST)
+[2]. Despite great success, a simple FR system might be
+vulnerable to spoofing, known as a presentation attack. For
+instance, print attacks, video replay, and 3D masks are the
+most common attacks reported recently in the face anti-
+spoofing domain [3], [4]. Thus, a number of hand-crafted and
+deep representation methods have been proposed to protect FR
+systems against presentation attacks [5], [6], [7], [8], [9], [10],
+[11]. Many of them report promising performance in intra-
+domain testing scenario. However, the performance remains
+limited in cross-dataset testing scenario due to distributional
+discrepancy between source domain and the target domain.
+One of the major reasons that deep-learning-based models
+are prone to overfitting due to the lack of availability of a
+sufficient amount of training samples in the source domain.
+Another possible reason might be that many face PAD methods
+assume that training and testing data come from the same
+target distribution. However, if a model was trained on cut
+photo attack images, would it work on mask attack images?
+What if the model trained only on replay attack images and
+tested in warped photo attacks? Is it possible to deploy a model
+that is trained using different illumination conditions and
+background scenes under control lighting systems? Answers to
+Manuscript received January 1, 2022; revised August 26, 2022. Correspond-
+ing author: M. Usman (email: Muhammad.usman@oulu.fi)
+all these questions depend on how a machine learning model
+can deal with this domain shift problem. Thus, to alleviate
+this issue, domain adaptation (DA) techniques are used to
+leverage a source dataset and maintain a good accuracy on
+the target dataset by using unlabeled target data. However, in
+many applications, it is difficult to collect sufficient target data.
+For instance, in face PAD, hackers are using different types
+of spoofing attacks which makes it impractical to collect each
+type of new attack sample in advance.
+To overcome the domain shift problem, domain generaliza-
+tion (DG) methods have been introduced to improve the gener-
+alization [9], [10], [11]. However, the generalization capability
+of PAD methods remains challenging because either the deep
+feature-based methods or low-level feature-based methods may
+not generalize well into new applications. Generalizability
+refers to the performance difference of a model when the PAD
+models are trained and tuned on one or multiple databases and
+then tested on a completely unseen database. As shown in
+Fig.1, the goal of domain generalization is to use the training
+samples from one or several different source domains but
+related domains (i.e., diverse training datasets) that perform
+well when evaluated on a completely unseen target domain.
+To improve the generalization, the majority of recent ap-
+proaches in face PAD such as adversarial learning [12],
+meta pattern learning [13], generative domain adaptation [14],
+hypothesis verification [15], or cross-adversarial learning [16],
+address the domain generalization issue by exploiting a com-
+mon feature space from multiple source domains, but the
+performance remains limited due to a substantial distribution
+difference among source domains. For instance, research in
+[17] relies on a shared feature space and assumes that it
+would also be invariant to domain shift. This assumption has a
+flaw because when the source domains become more diverse,
+arXiv:2301.02145v1  [cs.CV]  5 Jan 2023
+
+JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, NOVEMBER 2022
+2
+Fig. 1: The source domains are trained with diverse sets of
+synthetic images where the meta-learner seeks complementary
+information to generalize well to unseen target distribution.
+learning a domain-invariant model becomes more difficult
+[18]. For instance, instead of concentrating on some domain-
+specific differentiation cues such as cut photo texture cues
+available in the CASIA database, models can be benefited from
+generalized feature space if more generalized cues are shared
+by all source domains [11]. In addition, spoofing attacks have
+been launched physically by malicious hackers (i.e., outside
+the control of the biometric system). Therefore, building new
+datasets to collect large samples of fake faces, especially for
+each type of new attack remain infeasible in the face anti-
+spoofing domain. Although the dominant approaches such as
+Generative adversarial networks (GANs) [19], Bidirectional
+GANs [20], the DCGAN [21], can be applied to mitigate
+the gap between the target domain and the source domain by
+generating synthetic faces, these models require careful tuning
+of their parameters.
+In this paper, rather than proposing a specific model
+suited for the intra-database testing scenario, a novel unified
+framework is introduced based on the idea of stacking-based
+ensemble learning to improve the generalization of the face
+PAD. We first generate different sets of synthetic training
+samples and then train different sub-models on each of the
+synthetic sets to specialize in their own domain. More specif-
+ically, our goal is to understand the relationship between the
+spatiotemporal artifacts that appear in synthetic samples. Con-
+sequently, we train three sub-models in which we investigate
+the characteristics of these spatiotemporal artifacts. By doing
+this, we assume that sub-models that are trained on specific
+source domains would be experts in domain-specific sources
+but non-expert in all other source domains as well as the
+target domain. Motivated by this, we train a meta-learner that
+minimizes the cross-domain generalization error by combining
+the input predictions of all experts (sub-models). Thus, our
+key idea is to train the sub-models separately so that when
+forming stacking, a meta-learner can leverage complementary
+information in order to better approach the target domain.
+To achieve our goal, we first introduce a video distillation
+technique to generate synthetic samples. This is inspired by
+our previous works [8], [22] that claim estimation of global
+motion is important for face PAD. Specifically, a 2D image
+morphing technique is proposed with a combination of a warp
+and a cross dissolve. The main idea is to blend the encoded
+spatiotemporal warped images with the still images using
+alpha blending. By doing so, we generate multiple sets of
+2D synthetic images with different alpha weights and expand
+the training samples significantly. Several synthetic examples
+are shown in Fig.2. We then train different recurrent neural
+networks with each subset of synthetic data and use the
+prediction of each subset to train the meta-classifier. Moreover,
+the interpretability methods are employed to further assess how
+robust is the model, by revealing that the most significant areas
+for determining the deep learning model decision on the PAD
+task are consistent with motion cues associated with the arti-
+facts, i.e., screen sloping, hand movement, material reflection,
+and expression changes. Overall, the main contributions of this
+study are five-fold:
+• A video distillation technique is proposed to train a
+2D CNN on a still image, where “still” encodes both
+appearance and temporal information from the video
+sequence into a single RGB image.
+• 2D image morphing is introduced to create large-scale
+synthetic training samples that greatly promote the per-
+formance of the face anti-spoofing model.
+• Stacked recurrent neural networks are utilized to predict
+spatiotemporal inconsistencies and then those predictions
+are employed to form the deep architecture (meta-model).
+• Techniques of interpretation are provided for exploring
+the decisions made by the employed model. The model
+revealed that the motion cues are the most important
+factors for distinguishing whether an input image is
+spoofed or not.
+• Experiments on four benchmark datasets, consisting
+of CASIA-MFSD, Replay-Attack, MSU-MFSD, and
+OULU-NPU databases, show that our proposed method
+is significantly superior on three databases in comparison
+with other state-of-the-art generalization methods used
+now.
+The rest of this work is organized as follows. Section II
+discusses the recent developments and related past works.
+Section III explains all the steps of the proposed method.
+Section IV shows the implementation details, ablation study,
+and comparison against several public benchmark datasets.
+Section V concludes the entire work and gives suggestions
+for future research.
+II. Literature Review
+Over the past few years, face PAD methods have re-
+ceived considerable attention from both academia and in-
+dustry. In general, these methods can be roughly classified
+into appearance-based methods and temporal-based methods.
+Appearance-based methods: Traditional appearance-based
+methods usually extract hand-crafted features such as LBP
+[23] and SIFT [24] based on various appearance cues. The
+authors in [5] claimed that color information is crucial and
+luminance-chrominance color spaces improve the detection
+
+Source domain
+Domain
+1
+α = 0.5
+α=1.0
+α=1.5
+Target domain
+Meta-
+Domain
+learner
+2
+α=0.5
+α=1.0
+α=1
+Domain
+3
+α = 0.5
+α=1.0JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, NOVEMBER 2022
+3
+Fig. 2: 2D synthetic samples from CASIA-MFSD. Left col-
+umn: Video sequence used to generate synthetic samples.
+Right column: Spatiotemporal encoded images morphed with
+the still image using alpha values of 0.5 (Synt 1), 1.0 (Synt 2),
+and 1.5 (Synt 3), respectively. These synthetic samples can be
+used for ensemble stacking to significantly improve the face
+anti-spoofing performance.
+performance of face PAD in comparison to the RGB and
+the gray-scale image representations. The multiscale filtering
+approach proposed in [25] was found to be effective where
+LBP-based multiscale features provide improved performance.
+Wen et al [26] utilize image distortion analysis (IDA) and
+develop an ensemble classifier, where multiple SVM classifiers
+are implemented. In particular, the features are selected based
+on specular reflection, blurriness, chromatic moment, and color
+diversity to provide input to SVM classifiers. A component-
+based coding framework is proposed to encode different
+components of the face in [27]. To deploy secure face locking
+on a smartphone, a method is developed based on extracting
+color distortion, Moiré-pattern analysis, surface reflection, and
+shape deformation [24]. The LBP features are combined with
+the feature maps of a deep learning model to improve the
+detection of face PAD in [28]. The authors show that the need
+for large training samples in face PAD can be mitigated by
+using convolutional feature maps. Moreover, a hybrid deep
+learning method is introduced in [29] to encode appearance
+information from two CNNs where the SVM classifier is used
+to discriminate live and spoofed images. Although appearance-
+based methods provide improved performance in an intra-
+database testing scenario, the performance remains limited
+when evaluated on a completely unseen testing domain.
+Temporal-based methods: The study reported in [8] es-
+timates global motion and amplifies motion cues such as
+hand movements or head rotation where BiLSTM is used to
+predict the motion. Since global estimation leaves the artifacts
+such as black framing at the border of the encoded images
+in [8], this issue was solved by using dense sampling with
+similarity transformation [22]. Moreover, in order to encode
+head movements, eye-blinking, and lip movements, a dynamic
+mode decomposition (DMD) method is introduced to capture
+the temporal cues from frame sequences [30]. Eulerian motion
+magnification is used to magnify the facial expressions in [31].
+Then, local descriptors such as HOOF and LBP are utilized
+to improve the classification performance. Photoplethysmogra-
+phy (rPPG) signal was found to be crucial to improve the face
+PAD performance [32]. A unified framework based on CNN-
+BiLSTM is used to capture both appearance and temporal cues
+in [29]. A study conducted in [33] shows that the spontaneous
+blinking of a person provides an intrinsic detection cue to
+improve live face detection. A dense optical flow scheme is
+proposed to estimate the motion of two successive frames
+in [34]. The authors claimed that real and attack videos
+have different optical flow motion patterns which help to
+improve the PAD performance. A 3D CNN model is employed
+to capture both spatial and temporal information in [35].
+A combined CNN-RNN model is developed to capture the
+auxiliary information (i.e., the depth map and rPPG signals)
+for improving the detection performance [36]. However, when
+the temporal and appearance-based methods are employed in
+a cross-dataset scenario, the detection performance remains
+vulnerable to degradation due to real-world variations (such
+as user demographics, input cameras, and variations in illu-
+mination). Therefore, domain generalization that aims to learn
+from several source domains becomes significant while dealing
+with presentation attack detection.
+Deep Domain Generalization methods: Several deep do-
+main generalization methods have been introduced to im-
+prove the generalization ability of face PAD. For instance,
+a domain adaptation method that generates pseudo-labeled
+samples named cyclically disentangled feature translation net-
+work (CDFTN) is proposed in [37]. Chuang et al proposed
+to improve the generalization based on one-side triplet loss
+[38]. A two-stream network is utilized to fuse the input RGB
+image and meta-pattern learning was proposed to improve
+the generalization [13]. A cross-adversarial training scheme
+is proposed to improve the generalization by minimizing
+the correlation among two sets of features [16]. The work
+reported in [14], aims to learn a generalized feature space
+by designing the target data to the source-domain style and
+called Generative Domain Adaptation (GDA). A hypothesis
+verification framework is proposed in [15] where two hy-
+pothesis verification modules are utilized for improving the
+generalization. A novel Shuffled Style Assembly Network
+(SSAN) is introduced by aligning multiple source domains
+into a stylized feature space and domain generalization was
+improved by a contrastive learning strategy [39]. To select
+common features space, adversarial learning is proposed and
+aggregation of live faces is performed to achieve a generalized
+feature space in [12]. However, there is no consensus that the
+pre-defined distributions can be considered the optimal ones
+for the feature space. Thus, we argue that a model can un-
+derstand faces much better by simply aligning multiple source
+domains based on the idea of collaborative ensemble learning.
+In particular, the generalized feature space can automatically
+capture spatiotemporal inconsistencies based on the knowledge
+provided by multiple source domains.
+III. The Proposed Method
+Figure.3 illustrates the overall framework. Firstly, we
+present a method to show how to synthesize training samples.
+
+Video sequence
+Encoded clip
+Synt 1
+Synt 2
+Synt 3JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, NOVEMBER 2022
+4
+Fig. 3: Flow chart of our proposed method. A video of length V is divided into non-overlapping segments of smaller length
+v. For each segment, global motion is estimated and the stabilized sequence is accumulated to obtain a spatiotemporal warped
+image. Then, the encoded spatiotemporal warped image is morphed with a still image (i.e., the first frame of the segment) by
+using alpha compositing. Since different alpha values are used to create multiple synthetic images, we build multiple classifiers
+on these synthetic images to form stacking-based ensemble learning for improving the generalization of face PAD.
+The purpose of synthesis is to bring spatiotemporal artifacts
+that can be used to train multiple individual models for
+understanding the relationship between them. Secondly, a
+unified CNN-RNN network is proposed due to the fact that
+mainstream 2D CNN frameworks cannot deal with sequential
+data (i.e., sequences to sequences). Then, model stacking is
+designed in such a way that it can minimize the weakness and
+maximize the strengths of every individual model based on
+the meta-learner. Lastly, the model interpretation is provided to
+investigate the contribution of synthetic data on which the deep
+model mainly relies. Each step is explained in the following
+sub-sections.
+A. 2D Virtual Synthesis
+To generate synthetic samples, a video V is equally divided
+into P non-overlapping segments, i.e., V = {Sk}P
+s=1, where
+Sk is the k-th segment. The length of each segment is set
+to be (w = 40) frames. For each segment, features are
+extracted from the fixed (first) and moving (second) image
+of the segment. In particular, the FAST feature detector [40]
+is utilized to detect interest points and then FREAK descriptor
+[41] extracts the features to collect points of interest from both
+frames. Since salient image features are extracted, the next step
+is interest points matching where Hamming distance (HD) is
+utilized in our work. The inter-frame parameters are estimated
+throughout the whole length of the segment (since the first
+frame) by using the rigid (Euclidean space) transformation. As
+the name suggests, rigid transformation preserves the distance
+and angles (i.e, distance between two points remains the same).
+The rigid transformation matrix M is a 3×3 matrix. We find
+the 2D pixel coordinates in Cartesian coordinate system by
+estimating the translation map from M. Let [a, b, 1]T illustrate
+the homogeneous coordinates in moving image and [a′, b′, 1]T
+define the coordinates in the fixed image, we have
+�
+�
+a′
+b′
+1
+�
+� =
+�
+�
+d11
+d12
+d13
+d21
+d22
+d23
+d31
+d32
+d33
+�
+�
+�
+�
+a
+b
+1
+�
+�
+(1)
+and pixel shift can be calculated as
+�∆a
+∆b
+�
+=
+�a′ − a
+b′ − b
+�
+(2)
+To eliminate false-matching points and robust estimation of the
+geometric transformation between the frames, we use the M-
+estimator Sample Consensus (MSAC) algorithm [42] to detect
+outliers and remove false matching points. To obtain warped
+images, we simply average the stabilized frame sequences
+using the following aggregation function:
+ev = 1
+w
+w
+�
+k=1
+evk,
+(3)
+
+Moving 2D Image
+Input video
+Convert to
+grayscale
+Still image
+(Target)
+Segment i
+Blending
+Grayscale Image Points detection Feature's matching
+Calculate rigid
+Warp the
+Segment 2
+Fixed2D Image
+transformation
+moving
+matrix T
+image with T
+Convert to
+Synthetic
+grayscale
+images
+Segment N
+02
+Model 1
+ve
+Model 2
+Stacking
+RNN
+RNN
+RNN
+Spoof
+Model 3
+f3
+Training set
+Convolutional layers
+Recurrent neural networks
+Stacking ensembleJOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, NOVEMBER 2022
+5
+where w denotes the total number of selected frames in
+segment k for video V . By the above aggregation, the average
+over frames directly merges temporal information, and the
+image registration combines available spatial reference infor-
+mation. Figure.4 shows the effectiveness of the proposed video
+distillation scheme. The results demonstrate that the removal
+of global motion must be taken into account before the feature
+extraction step during the development of a face PAD model.
+Since our target is to predict the temporal inconsistencies,
+a synthetic image is generated in such a way that every
+spatiotemporal encoded image acquired from Eq.3 is blended
+into the first (still) image of the segment to obtain a synthetic
+image. By doing this, we make sure that the synthetic image
+would never leave the space of the human face (see Fig.2).
+Thus, the proposed blending process involves two steps: 1)
+obtain a source image (i.e., a spatiotemporal encoded image
+from a video distillation technique), and 2) target image:
+choosing a first (still) image of each segment to blend into
+a source image (usually known as cross dissolving). Let’s
+assume that we blend source image (P1) over target image
+(P2) as:
+Pmorph(a, b) = αP1(a, b) + (1 − α)P2(a, b)
+(4)
+where α is the morphing weight (0 < α ≤ 1). Thus, a
+synthetic image is obtained at new location Pmorph(a, b) gets
+α percentage from αP1(a, b) and (1 − α) from P2(a, b) [43].
+It is worthwhile to mention that the proposed video dis-
+tillation scheme is inspired by our previous works [8], [22]
+that estimate global motion. Thus, benefiting from the video
+distillation nature of the previous methods, we extend our
+previous works to generate synthetic samples by introducing
+a cross-dissolve. Moreover, we use the FREAK descriptor
+and rigid transformation to estimate inter-frame motion. By
+doing this, the computation cost of the method is significantly
+reduced (We further discuss this argument in section IV).
+B. Recurrent Neural Network (RNN)
+Deep learning methods based on 2D Convolutional Neural
+Networks (CNNs) have shown an improved performance than
+classical machine learning approaches [9], [6], [7]. However,
+the mainstream 2D CNN frameworks focus on spatial infor-
+mation, thus lacking the capacity to understand sequential
+data. Specifically, CNNs do not have a memory mechanism in
+order to capture the temporal relations. Motivated by the fact
+that recurrent neural networks (RNNs) can deal with temporal
+information, we develop a unified framework consisting of
+CNN-RNN to encode complementary information between
+frames. In particular, a CNN is fine-tuned on the labeled
+dataset in the first stage. Then, the fine-tuned features are
+extracted from the pooling layer and used as input to train
+a Long-short-term memory (LSTM) [44] network.
+The LSTM is the most popular RNN architecture and
+capable of learning long-term dependencies. It is composed of
+memory cell (Ce), an input gate (ie), an output gate (oe) and a
+forget gate (ge). The input gate governs the information flow
+into the cell by multiplying the cell’s non-linear transformation
+of inputs me. The output gate decides how much information
+Fig. 4: (a) We computed the mean of the raw video frames to
+visualize the global motion that shows a great deal of distor-
+tion in the encoded image. (b) The proposed spatiotemporal
+encoded images after removing the global motion.
+from the cell is used to compute the output activation of the
+LSTM unit. The forget gate regulates the extent to which a
+value remains in the cell. The LSTM unit updates for time
+step e are:
+�
+���
+ge
+ie
+me
+oe
+�
+��� =
+�
+���
+σ
+σ
+tanh
+σ
+�
+��� H · [pe−1, xe]
+(5)
+Ce = ge ⊙ Ce−1 + me ⊙ ie
+(6)
+pe = tanh(Ce) ⊙ oe
+(7)
+where xe is the input at the current time-step, ie is the
+current cell state, g, i, and m represent input gate activation,
+forget gate activation and output gate activation, respectively.
+σ illustrates the logistic sigmoid function and ⊙ represents
+element-wise multiplication. The fully connected and softmax
+layer is used for detecting real and fake images.
+C. Model Stacking
+Ensemble learning has been supported by multiple ap-
+proaches like bagging, boosting, or stacking which results
+in a better generalization of the learning models [45]. Es-
+pecially, stacking is one of the integration techniques that
+involves combining the predictions based on the different
+weak models’ predictions wherein the meta-learning model
+is used to integrate the output of base models [46]. One of
+the common approaches in stacked ensemble learning is to
+develop a bench of T Tier-1 classifiers S1, S2, S3, ..., SN based
+on cross-validation to the training sample [47].
+Rather than focusing on the prediction of a single model,
+we train diverse RNN-based sub-models in our work with
+different synthetic training samples to predict the temporal
+inconsistencies from the data. In particular, the LSTM [44]
+and the Bidirectional LSTM (BiLSTM) [48] with different
+hidden layers are trained on three synthetic sets where each
+sub-model works independently to specialize in its own source
+domain. To better understand the learning of sub-models, Fig.5
+represents the proposed validation scheme, where each RNN
+is trained with k-1 folds, k-2 folds, and k-3 folds to get the
+
+(a) Encoded images with global motion
+(b) Encoded images after removing global motionJOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, NOVEMBER 2022
+6
+Fig. 5: The proposed validation for ensemble learning.
+TABLE I: BiLSTM architectures and parameters.
+First Architecture
+Second Architecture
+Third Architecture
+No. of layers
+1
+1
+1
+Layers type
+LSTM
+BiLSTM
+LSTM
+No. of units
+500
+20
+100
+Optimizer
+ADAM
+ADAM
+ADAM
+learning rate
+0.0001
+0.0001
+0.001
+Cost function
+cross entropy
+cross entropy
+cross entropy
+TABLE II: Meta model architecture and parameters.
+No. of layers
+1
+Layers type
+LSTM
+No. of units
+20
+Optimizer
+ADAM
+learning rate
+0.0001
+Cost function
+cross entropy
+most out of the stacking. Thus, by making experts on different
+training subsets, we reinforce each model to concentrate on
+different aspects of data (i.e., temporal inconsistencies), such
+as one model can focus on certain type of features using a
+subset of synthetic data. Similarly, another model can perform
+better on the others. We then combine the predictions from
+these experts (sub-models) models by running another model
+called a meta-learner (meta-classifier). By doing this, the meta-
+learner helps to maximize the strengths of every individual
+model and reduce generalization errors.
+Table I shows the architectures and parameters of the base
+models, while Table II depicts the meta-model architecture. It
+is worth mentioning here that we accumulate the output of the
+three base models’ validation sets as the new validation set for
+training the meta-model. This way, the meta-model will make
+the final test prediction on the test set.
+D. Interpretation of a deep neural network
+Interpretation is essential to observe what learning patterns
+in data are important but there is no clear consensus that
+how interpretability should be best defined in the context
+of machine learning. Although explanation methods intend
+to make neural networks more trustworthy and interpretable,
+the question arises of how some features favor deep learning
+to make such a valuable prediction. For instance, synthetic
+samples in our work are found to be more useful to train a deep
+model and it shows better interpretability in comparison to the
+same model trained without synthetic samples. This is due to
+the fact that the motion cues which are naturally available in
+the frame sequences are "easy to learn" for the model, and play
+an important role in model optimization. Thus, the importance
+of interpretation is becoming increasingly popular and leads
+to useful or promising findings.
+In our work, Gradient-weighted class activation mapping
+(denoted as Grad-CAM) [49], Occlusion sensitivity maps
+(denoted as OCC-SEN) [50], Gradient Attribution map using
+Guided Backpropagation (denoted as Grad-ATT) [51], and
+locally interpretable model-agnostic explanations (denoted as
+LIME) [52] are utilized to understand what patterns in data
+are deemed important or make the contributions to the final
+decision. In particular, this enables us to trust the behavior
+of the developed deep learning model, and/or further tune
+the model by observing its interpretations. In particular, we
+extract visualization maps from pretrained DenseNet-201 [57]
+convolutional neural network for all of the methods above
+in our experiments. In Fig.6, we visualize diverse sets of
+synthetic images from the CASIA datasets. The first four rows
+show print attack images while the next four rows show replay
+attack images. Each visualization method captures the class
+discriminative region thanks to the proposed video distillation
+and synthetic data generation scheme that force the network to
+use more subtle cues for its correct classification. In particular,
+the first row shows that the neurons in the deep convolutional
+layers focus on the paper’s texture, and hand movement cues.
+However, Grad-ATT [51] interpretation shows that the model
+also takes background as context to make the prediction.
+Surprisingly, this issue is eliminated by the proposed synthetic
+data generation scheme where the second, third, and fourth
+row shows that the model only considers motion cues, the
+surface edges and barely touches the background context.
+In case of a replay attack, the remaining rows show that
+the tablet screen and hand movement provide discriminative
+information for the model prediction. Since we cannot present
+this for every image from the dataset, we observed that the
+mouth information, eye blinking, or head rotation contribute
+positively to distinguishing live and spoofed images. Thus,
+interpretation from the above methods demonstrates that the
+proposed learning model is focusing on the correct features of
+the input data, and the model’s decision can be viewed in a
+human-understandable way. Moreover, the proposed synthetic
+data generation method provides informative RGB images and
+helps the model to make the features of spoofed faces more
+dispersed which allows a better class boundary to generalize
+well to the target domain.
+IV. Experimental analysis of using open datasets
+To assess the effectiveness of the synthesized face im-
+ages, four publicly available databases are used: OULU-NPU
+database [58] (denoted as O), CASIA Face Anti-Spoofing
+database (denoted as C) [59], Idiap Replay-Attack database
+[60] (denoted as I), and MSU Mobile Face Spoofing database
+[26] (denoted as M). The performance is evaluated in-terms
+of Half Total Error Rate (HTER) (half of the summation of
+false acceptance rate and false rejection rate) and Area Under
+Curve on the target testing dataset.
+
+Fold 0
+Fold 1
+Fold 2
+Fold 3
+Train set
+Synthetic set 1
+Synthetic set 2
+Synthetic set 3JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, NOVEMBER 2022
+7
+Fig. 6: Visualization of feature maps. The types of images are labelled in the first column. The second column shows the
+original encoded and synthetic images. The third column illustrates the feature maps from Grad-CAM [49] while the fourth
+column shows the feature maps from occlusion sensitivity maps [50]. Similarly, the fifth and sixth column visualize the features
+maps from Gradient Attribution map using Guided Backpropagation [51], and locally interpretable model-agnostic explanations
+[52], respectively. The last column shows the masked images obtained from LIME predictions.
+A. Implementation details
+All the images are resized to 224 × 224 according to
+the input requirement of pretrained DenseNet-201 [57] ar-
+chitecture. The CNN model is fine-tuned by using Stochastic
+Gradient Descent (SGD) optimizer with a validation frequency
+of 30, and mini-batch size of 32. We set the learning rate
+up to 0.0001, and do not use fixed size epochs because an
+early stopping function [61] is utilized to stop the model
+automatically to prevent overfitting.
+During the ensemble learning stage, the CNN model is fine-
+tuned with original encoded video clips and three different
+synthetic sets separately. Then, the features from each fine-
+tuned model are used as input to train three diverse RNN
+models. In particular, the Adam optimizer is utilized with a
+validation frequency of 30. The learning rate is set to 0.0001,
+and the weights are initialized with He initializer [62] for the
+first LSTM (Sub-model 1) model. We do not set the fixed
+epochs because an early stopping function [61] was used to
+prevent overfitting. For training the second sub-model, the
+BiLSTM is trained with the hidden layer dimension of 20.
+The other parameters were kept the same as sub-model 1.
+For the third sub-model, the LSTM model is trained with the
+hidden layer dimension of 100 by decreasing the learning rate
+of 0.001. For the data synthetic method, we generate three
+synthetic samples, in which 0.5, 1.0, and 1.5 alpha values
+are used to expand the training images. In order to train the
+meta-model, the epochs size 80, the Gradient threshold 1, and
+a hidden layer dimension of 20 was used to train the meta-
+
+Image Type
+Grad-CAM
+OCC-SEN
+Grad-ATT
+LIME
+Masked
+Encoded image
+Synthetic sample
+Synthetic sample
+2
+Synthetic sample
+3
+Encoded image
+Synthetic sample
+Synthetic sample
+2
+Synthetic sampleJOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, NOVEMBER 2022
+8
+TABLE III: Performance evaluation using MSU-MFSD (M), CASIA-MFSD (C), replay-attack (I), and OULU-NPU (0)
+databases. Comparison results are obtained directly from the corresponding papers.
+O&C&I to M
+O&M&I to C
+O&C&M to I
+I&C&M to O
+Method
+HTER(%)
+AUC(%)
+HTER(%)
+AUC(%)
+HTER(%)
+AUC(%)
+HTER(%)
+AUC(%)
+MADDG [11]
+17.69
+88.06
+24.50
+84.51
+22.19
+84.99
+27.89
+80.02
+DAFL [10]
+14.58
+92.58
+17.41
+90.12
+15.13
+95.76
+14.72
+93.08
+SSDG-R [17]
+07.38
+97.17
+10.44
+95.94
+11.71
+96.59
+15.61
+91.54
+DR-MD [9]
+17.02
+90.10
+19.68
+87.43
+20.87
+86.72
+25.02
+81.47
+MA-Net [6]
+20.80
+-
+25.60
+-
+24.70
+-
+26.30
+-
+RFMetaFAS [7]
+13.89
+93.98
+20.27
+88.16
+17.30
+90.48
+16.45
+91.16
+FAS-DR-BC(MT) [53]
+11.67
+93.09
+18.44
+89.67
+11.93
+94.95
+16.23
+91.18
+ADL [12]
+05.00
+97.58
+10.00
+96.85
+12.07
+94.68
+13.45
+94.43
+ResNet-BiLSTM w/DS [3]
+04.12
+99.93
+07.04
+99.87
+13.48
+97.42
+41.33
+88.48
+HFN + MP [13]
+05.24
+97.28
+09.11
+96.09
+15.35
+90.67
+12.40
+94.26
+Cross-ADD [16]
+11.64
+95.27
+17.51
+89.98
+15.08
+91.92
+14.27
+93.04
+ASGS [22]
+05.91
+99.88
+10.21
+99.86
+45.84
+76.09
+13.54
+99.73
+GDA [14]
+09.20
+98.00
+12.20
+93.00
+10.00
+96.00
+14.40
+92.60
+SSAN-R [39]
+06.67
+98.75
+10.00
+96.67
+08.88
+96.79
+13.72
+93.63
+FG +HV [15]
+09.17
+96.92
+12.47
+93.47
+16.29
+90.11
+13.58
+93.55
+Ensemble (CNN-RNN)
+04.02
+99.95
+06.97
+99.97
+33.49
+93.16
+10.91
+99.89
+TABLE IV: The results of cross-dataset testing on limited source domains. The comparison results are obtained directly from
+the corresponding papers.
+O&I to M
+M&I to C
+O&I to C
+O&M to I
+C&M to O
+Method
+HTER(%) AUC(%) HTER(%) AUC(%) HTER(%) AUC(%) HTER(%) AUC(%) HTER(%) AUC(%)
+Supervised [54]
+12.1
+94.2
+30.4
+77.0
+18.0
+90.1
+16.8
+93.8
+17.9
+89.5
+Mean-Teacher [55]
+19.6
+86.5
+31.1
+76.6
+23.7
+84.9
+18.4
+86.0
+23.5
+84.9
+USDAN [56]
+15.8
+88.1
+35.6
+69.0
+33.3
+72.7
+19.8
+87.9
+20.2
+88.3
+EPCR-labeled [54]
+12.5
+95.3
+18.9
+89.7
+18.9
+89.7
+14.0
+92.4
+17.9
+90.9
+EPCR-unlabeled [54]
+10.4
+94.5
+25.4
+83.8
+16.7
+91.4
+12.4
+94.3
+17.8
+91.3
+Ensemble (CNN-RNN) 07.8
+98.5
+17.1
+94.3
+12.5
+97.1
+15.1
+93.1
+14.7
+93.4
+learner. For reproducibility of our results, we keep the same
+parameter settings for conducting the experiments on all the
+databases.
+B. Comparison against the state-of-the-art methods
+To compare the performance with the recently introduced
+domain generalization methods, we conduct cross-dataset test-
+ing where the model is trained on three source databases
+and evaluated on a completely unseen database using the
+leave-one-out (LOO) strategy. In particular. the testing sets
+of source databases are used as a validation set for computing
+the equal error rate. Thus, the HTER is calculated directly
+on the target (unseen) dataset for a fair comparison with
+the previous methods. As shown in Table III, the proposed
+ensemble learning provides the best results on three proto-
+cols of O&C&I to M, O&M&I to C, I&C&M to O, and
+demonstrates that the model can extract more generalized
+differentiation cues for face PAD. This is due to the recently
+proposed countermeasures paying more attention by exploring
+a common feature space from multiple source domains that
+only fit data in the source domains [17]. In contrast to the
+existing approaches where adversarial learning [12], generative
+domain adaptation [14] and meta-learning [13] has been used,
+the proposed ensemble learning improves the generalization by
+exploiting the relationship of multiple trained models which
+are expert in their own source domain, but ensure that meta-
+learner can take complementary information from them to
+improve the generalization of face PAD model.
+C. Experiment on Limited Source Domains.
+We also consider the scenario of a limited source domain
+by training the model on two source domains instead of three
+as shown in Table IV. The model continues to achieve the
+best performance on all the target domains. In particular, the
+lowest HTER in four protocols and the highest AUC show
+that limited source data does not degrade the generalization
+capability of our network in a challenging testing scenario.
+D. Ablation study
+To verify the superiority of our proposed ensemble learning
+and the contributions of each sub-model, we conduct exper-
+iments for multi-source domains and limited-source domains
+separately. Table V reports the numerical results for multi-
+source domain settings. The baseline results represent the
+performance of the ResNet-BiLSTM model without synthetic
+data. These results are based on encoded spatiotemporal im-
+ages obtained from the proposed video distillation scheme.
+Sub-model 1 represents the results when one set of synthetic
+images were added with spatiotemporal encoded images by
+using the value of alpha (0.5). The numerical results of CNN
+and CNN-RNN show that synthetic images start improving the
+model’s performance on all datasets. In particular, the RNN
+improves the performance significantly. Similarly, sub-model
+2 represents the results with a different set of synthetic images
+(i.e., alpha value was increased to 1.0). The proposed model
+experienced a slight drop in performance for CNN predictions
+but continues to improve the performance of RNN on M, I
+and O. Moreover, when we further evaluate the performance
+
+JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, NOVEMBER 2022
+9
+TABLE V: Ablation study using cross-database evaluation.
+O&C&I to M
+O&M&I to C
+O&C&M to I
+I&C&M to O
+Method
+HTER(%)
+AUC(%)
+HTER(%)
+AUC(%)
+HTER(%)
+AUC(%)
+HTER(%)
+AUC(%)
+Baseline w/o synthetic data
+19.02
+86.12
+19.52
+87.63
+31.66
+76.22
+35.44
+85.54
+Sub-model 1 (CNN)
+18.11
+87.63
+18.66
+86.22
+29.00
+78.39
+36.55
+84.25
+Sub-model 1 (CNN-RNN)
+09.97
+99.26
+07.31
+99.98
+36.87
+76.05
+19.90
+99.55
+Sub-model 2 (CNN)
+18.82
+91.09
+22.20
+84.49
+35.63
+77.24
+34.01
+74.91
+Sub-model 2 (CNN-RNN)
+08.40
+98.64
+10.14
+97.04
+34.44
+77.01
+12.41
+98.55
+Sub-model 3 (CNN)
+17.21
+90.87
+23.22
+84.49
+37.33
+75.05
+33.45
+76.14
+Sub-model 3 (CNN-RNN)
+06.05
+98.53
+06.08
+99.11
+39.33
+76.41
+14.40
+98.95
+Ensemble (CNN)
+08.95
+97.79
+15.80
+95.47
+35.66
+90.19
+12.89
+98.94
+Ensemble (CNN-RNN)
+04.02
+99.95
+06.97
+99.97
+33.49
+93.16
+10.91
+99.89
+TABLE VI: Ablation study with limited open-source databases using cross-database evaluation.
+O&I to M
+M&I to C
+O&I to C
+O&M to I
+C&M to O
+Method
+HTER(%) AUC(%) HTER(%) AUC(%) HTER(%) AUC(%) HTER(%) AUC(%) HTER(%) AUC(%)
+Sub-model 1 19.6
+86.5
+31.1
+76.6
+23.7
+84.9
+22.4
+84.0
+23.5
+84.9
+Sub-model 2 15.8
+88.1
+35.6
+69.0
+33.3
+72.7
+19.8
+87.9
+20.2
+88.3
+Sub-model 3 12.5
+95.3
+18.9
+89.7
+18.9
+89.7
+18.0
+92.4
+17.9
+90.9
+Ensemble
+07.8
+98.5
+17.1
+94.3
+12.5
+97.1
+15.1
+93.1
+14.7
+93.4
+(a)
+(b)
+(c)
+Fig. 7: The T-SNE visualization of feature distributions on cross-testing scenarios. (a) shows the feature distribution of the
+original encoded video clips, (b) reflects the feature distribution of encoded video clips with a subset of synthetic samples, (c)
+shows the feature distribution of meta-learner.
+TABLE VII: Average execution time (in seconds)
+Dataset
+Optical flow [63] ASGS method [22] TSS method [8] Ours
+CASIA-FASD
+1560
+1487
+1140
+1023
+REPLAY-ATTACK
+1082
+1003
+780
+641
+on the third set of synthetic (α = 1.5) images, sub-model 3
+shows that further improvement can be achieved with synthetic
+images. When we combine the prediction of these sub-models
+and train the meta-learner, we achieve remarkable performance
+on three datasets in comparison to state-of-the-art methods
+[53],[6], [7],[8],[9],[10],[11]. The quantitative results indicate
+that the ensemble learning guided by video distillation scheme
+is beneficial to improve the performance for cross-domain face
+PAD.
+Analysis of limited source domains: In Table VI, we com-
+pare the domain generalization ability of our proposed method
+when limited source domain databases are accessible (i.e. only
+two source datasets). The results indicate that the proposed
+method is effective even in challenging cases. We hypothesize
+that this improvement is due to the fact that encoded RGB
+images with synthetic samples are almost as descriptive as the
+entire video.
+Comparisons of execution times: We analyze the execution
+times of the proposed video distillation technique with the
+previous global motion estimation methods [8], [22] and
+optical flow[63]. Table VII reports the numerical results in the
+total number of seconds used to generate the training samples
+on two datasets. All these comparison results were reported
+by using a MATLAB environment based on a workstation
+with 3.5 GHz Intel Core i7-5930k and 64 GB RAM. One
+can see that the proposed global motion estimation technique
+is computationally less expensive than the previous motion
+estimated methods reported recently in the literature.
+E. Visualization and Analysis
+To intuitively show the contribution of each sub-model, we
+visualize the feature distribution of different features using
+t-SNE [64], as illustrated in Fig. 7. The model is trained
+on 0+C+I source domains without synthetic samples and
+shows a trivial distribution in Fig. 7 (a) with an unclear
+interface between live and spoofed samples. One can see
+these overlapped areas can be easily misclassified and cause to
+degrade the performance. After adding synthetic samples to the
+sub-model, as represented in Fig. 7 (b), the feature distribution
+improves and provides a relatively clear interface than the
+baseline model, that is because the synthetic samples force
+
+Attack
+RealAttack
+RealAttack
+RealJOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, NOVEMBER 2022
+10
+(a)
+(b)
+(c)
+(d)
+Fig. 8: The Receiver Operating Characteristics (ROC) curves. (a) O&C&I to M, (b) O&M&I to C, (c) O&C&M to I, and (d)
+I&C&M to O are developed by plotting the true positive rate (TPR) against the false positive rate (FPR).
+the model to predict the spatiotemporal artifacts. Nonetheless,
+when the meta-model is introduced, a well-structured and
+compact distribution with a clear interface can be seen in
+Fig 7 (c). Thus, our proposed ensemble learning shows good
+generalizability on unseen target data.
+In Fig.8, we visualize ROC curves to show how much the
+model is capable of distinguishing real and attack classes. As
+illustrated in Fig.8, the meta-model on all datasets achieves
+more than 90% AUC which is a very impressive performance
+on unseen testing sets. The ROC curve is plotted with TPR
+against the FPR where FPR is on the x-axis and TPR is on the
+y-axis. In particular, the meta-model (ensemble) drag curves
+closer to the top-left corner indicate better performance.
+V. Conclusions
+In this paper, we show that ensemble learning represents an
+interesting research direction for improving the generalization
+of face PAD. In particular, the model is comprised of multiple
+synthetic source domains, and each sub-model predicts the
+spatiotemporal inconsistencies based on their similarity to each
+training domain. Besides, a meta-learner is introduced to take
+the complementary information from each sub-model. Based
+on the experimental results on four benchmark datasets, the
+proposed method exhibits better performance than a single
+model trained only on original training data. Thus, using
+ensemble stacking is shown to outperform the existing state-
+of-the-art generalization methods. Finally, the interpretation of
+the model shows that capturing the motion information is quite
+helpful to improve the generalization ability of the proposed
+method. Our future work will focus on the development of
+robust motion estimation methods in end-to-end learning to
+improve the generalization of face PAD. .
+VI. Declaration of Competing Interest
+The authors have no conflict of interest that could have
+appeared to influence the work reported in this paper.
+VII. Acknowledgments
+This work is supported by the Center for Machine Vision
+and Signal Analysis (CMVS) in the Faculty of Information
+Technology and Electrical Engineering (ITEE) at University
+of Oulu, Finland.
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+Open-Set Face Identification
+on Few-Shot Gallery by Fine-Tuning
+Hojin Park, Jaewoo Park, and Andrew Beng Jin Teoh
+School of Electrical and Electronics Engineering
+College of Engineering, Yonsei University
+Seoul, Korea
+[email protected], [email protected], [email protected]
+Abstract—In this paper, we focus on addressing the open-
+set face identification problem on a few-shot gallery by fine-
+tuning. The problem assumes a realistic scenario for face iden-
+tification, where only a small number of face images is given
+for enrollment and any unknown identity must be rejected
+during identification. We observe that face recognition models
+pretrained on a large dataset and naively fine-tuned models
+perform poorly for this task. Motivated by this issue, we propose
+an effective fine-tuning scheme with classifier weight imprinting
+and exclusive BatchNorm layer tuning. For further improvement
+of rejection accuracy on unknown identities, we propose a novel
+matcher called Neighborhood Aware Cosine (NAC) that computes
+similarity based on neighborhood information. We validate the
+effectiveness of the proposed schemes thoroughly on large-scale
+face benchmarks across different convolutional neural network
+architectures. The source code for this project is available at:
+https://github.com/1ho0jin1/OSFI-by-FineTuning
+I. INTRODUCTION
+Recently face recognition (FR) has achieved astonishing
+success attributed to three factors in large. Deep convolutional
+neural network (CNN) architectures [2], [3] that have strong
+visual prior were developed and leveraged as FR embedding
+models. Large-scale face datasets [4], [5] that cover massive
+identities with diverse ethnicity and facial variations became
+available. On top of these, various metric learning losses
+[6]–[9] elevated the performance of deep FR models to an
+unprecedented level.
+The majority of FR embedding models have been evaluated
+on numerous benchmarks with closed-set identification [7]–
+[11]. The closed-set identification protocol assumes all probe
+identities present in the gallery. However, in a realistic sce-
+nario, an unknown identity that is not enrolled may be en-
+countered. Another important but practical aspect to consider
+is the scarcity of intra-class samples for the gallery identities
+to be registered; namely, due to the expensive data acquisition
+cost and privacy issue, only a very small number of samples
+might be available for each gallery identity to register. In this
+respect, open-set face identification (OSFI) with the small-
+sized gallery is closer to a real scenario as it needs to perform
+both known probe identity identification and unknown probe
+identity rejection based on the limited information from the
+small gallery set. Despite its versatile practical significance,
+however, OSFI with a small gallery has been rarely explored.
+Devising a model specific to OSFI with a small gallery
+can be challenging in the following aspects: Firstly, an OSFI
+(a)
+(b)
+Fig. 1.
+(a) Full fine-tuning all parameters severely degrades the OSFI
+performance, while our method significantly improves the pre-trained model.
+Detection & Identification Rate (DIR) [1] quantifies both correct identification
+of the known probe identities and detection of the unknown. (b) An outline of
+our proposed fine-tuning scheme: Given a model pretrained on a large-scale
+face database, we initialize the gallery set classifier by weight imprinting,
+and then fine-tune the model on a few-shot gallery set by training only the
+BatchNorm layers. In the evaluation stage, a given probe is either accepted as
+known or rejected as an unknown identity based on novel similarity matcher
+dubbed Neighborhood Aware Cosine (NAC) matcher.
+model performs both identifications of a known probe identity
+but also correct rejection of unknown probe identity. Hence,
+conventional FR embedding models devised mainly for closed-
+set identification can perform poorly at the rejection of the
+unknown. In fact, as observed in Fig. 1 (a), FR embedding
+models pretrained on a large-scale public face database are
+not effective for open-set identification, leaving room for
+improvement. This suggests the need for fitting the pretrained
+arXiv:2301.01922v1  [cs.CV]  5 Jan 2023
+
+IJB-C
+CASIA-WebFace
+0.8
+0.9
+Rate
+Pretrained
+Pretrained
+0.7
+Full finetuning
+0.8
+Full finetuning
+&Identification
+Ours
+Ours
+0.6
+0.7
+0.5
+0.6
+etection
+0.4
+0.5
+0.3
+0.4
+0.01
+0.1
+1.0
+0.01
+0.1
+1.0
+FalseAlarmRate
+FalseAlarmRatePretrain Set
+Evaluation Set (disjoint from pretrain set)
+Known
+Unknown
+Gallery (Few-shot)
+Known Query
+Unknown Query
+Weight Imprinting
+Probe
+Accept
+M
+NAC
+2Q
+Reject
+Evaluation
+Pretraining
+BatchNorm-only
+Fine-Tuningmodel to be more specific to the given gallery set.
+Secondly, due to the few-shot nature of the small-sized
+gallery set, there is a high risk of overfitting for fine-tuning
+the pretrained model. As shown in Fig. 1 (a), full fine-tuning
+(i.e. updating all parameters) of the pretrained model results
+in severe performance degradation. This drives us to devise an
+overfitting-resilient parameter tuning scheme.
+Moreover, an ordinary cosine similarity matcher used in the
+closed-set identification might have a large tradeoff between
+the known probe identity identification and unknown probe
+identity rejection. As will be observed in Sec. III-D, the simple
+cosine matcher has a severe drawback for the task at hand. This
+motivates us to devise a robust matcher for OSFI.
+Based on these observations, we propose an efficient fine-
+tuning scheme and a novel similarity-based matcher for OSFI
+constrained on a small gallery set. Our fine-tuning scheme
+consists of weight initialization of the classifier governed by
+weight imprinting (WI) [12] and training only BatchNorm
+(BN) layers [13] for overfitting-resilient adaptation on the
+small gallery set. Moreover, for both effective detection of
+the unknown and identification of the known probe identities,
+a novel Neighborhood Aware Cosine (NAC) matcher that
+respects the neighborhood information of the learned gallery
+features, and hence better calibrates the rejection score is
+proposed. Our contributions are summarized as follows:
+1) To effectively solve the OSFI problem constrained on a
+small gallery set, we propose to fine-tune the pretrained
+face embedding model. Since full fine-tuning deterio-
+rates the embedding quality, we search for the optimal
+method.
+2) We demonstrate that the combination of weight imprint-
+ing and exclusive BatchNorm layer fine-tuning excels
+other baselines.
+3) We recognize that the commonly used cosine similarity
+is a sub-optimal matcher for rejection. We propose a
+novel matcher named NAC that significantly improves
+the rejection accuracy.
+II. RELATED WORKS
+A. Open Set Face Identification (OSFI)
+[14], one of the earliest works in OSFI, used their proposed
+Open-set TCM-kNN on top of features extracted by PCA and
+Fisher Linear Discriminant. [15] proposed their own OSFI
+protocol and showed that an extreme value machine [16]
+trained on the gallery set performs better than using cosine
+similarity or linear discriminant analysis for matchers. [17]
+trained a matcher composed of locality sensitive hashing [18]
+and partial least squares [19]. [20] applied OpenMax [21]
+and PROSER [22], two methods for open-set recognition of
+generic images, on top of extracted face features.
+All previous works propose to train an open-set classifier
+(matcher) of some form, but all of them use a fixed
+encoder. To the best of our knowledge, we are the first to
+propose an effective fine-tuning scheme as a solution to OSFI.
+B. Cosine Similarity-based Loss Functions
+[23] proposed to l2-normalize the features such that the
+train loss is only determined by the angle between the feature
+and the classifier weights. [7] further extended this idea by
+applying a multiplicative margin to the angle between a feature
+and its corresponding weight vector. This penalized the intra-
+class features to be gathered while forcing inter-class centers
+(prototypes) to be separated. A number of follow-up papers
+such as [8]–[11] modify this angular margin term in different
+ways, but their motivations and properties are generally simi-
+lar. Therefore, in our experiments we only use CosFace loss [8]
+as a representative method. For comprehensive understanding
+of these loss functions, refer to [24].
+III. APPROACH
+Our proposed approach is two-fold: fine-tuning on the
+gallery and open-set identification evaluation. In the fine-
+tuning stage, the classifier is initialized by weight imprinting
+to initiate learning from optimal discriminative features, and
+the model is fine-tuned by updating only the BatchNorm layers
+to avoid overfitting on the few-shot gallery data. In evaluation,
+we utilize a novel matcher NAC that computes a neighborhood
+aware similarity for better-calibrated rejection of the unknown.
+We demonstrate that the combination of these three methods
+significantly outperforms all other baselines.
+A. Problem Definition and Metrics
+Formally, in an OSFI problem, we assume the availability
+of an encoder φ pretrained on a large-scale face database (FR
+embedding model), which is disjoint from the evaluation set
+with respect to identity. The evaluation set consists of a gallery
+G = {(xG
+i , yG
+i )}Cm
+i=1 and a probe set Q. The probe set Q is
+further divided into the known probe set K = {(xK
+i , yK
+i )}
+and the unknown probe set U = {(xU
+i , yU
+i )}. G and K has no
+overlapping images x but shares same identities y ∈ {1, ..., C}
+whereas U has disjoint identities, i.e., YU ∩ {1, ..., C} = Ø.
+m refers to the number of images per identity in G, which
+we fix to 3 to satisfy the few-shot constraint. We allow the
+encoder to be fine-tuned over the gallery set.
+The evaluation of OSFI performance uses the detection
+and identification rate at some false alarm rate (DIR@FAR).
+FAR=1 means we do not reject any probe. Note that unlike
+the general case shown in [1], here we only consider rank-
+1 identification rate for DIR. Therefore, DIR@FAR=1 is the
+rank-1 closed-set identification accuracy.
+B. Classifier Initialization by Weight Imprinting
+Due to the few-shot nature of the gallery set where we
+fine-tune on, the initialization of model parameters and, in
+particular, of classifier weights is crucial to avoid overfitting.
+The most naive option is a random initialization of the
+classifier weight matrix W. Another commonly used strategy
+is linear probing [25], i.e., finding an optimized weight W
+that minimizes the classification loss over the frozen encoder
+embeddings φ(x).
+
+We experimentally find that, as seen in Fig. 2, both of these
+initialization schemes do not induce discriminative structure
+for the encoder embedding φ(x). In particular, during fine-
+tuning, each weight vector wc in the classifier acts as a center
+(or prototype) for the c-th class (i.e. identity). Fig. 2 shows that
+neither random initialization nor linear probing of wc derives
+optimally discriminative weight vectors wc, resulting in low
+quality of class separation of gallery features.
+Motivated from this issue, we propose to initialize by weight
+imprinting (WI), which induces the optimal discriminative
+quality for the gallery features:
+wc =
+�
+wc
+∥�
+wc∥2
+,
+�
+wc = 1
+m
+�
+yG
+i =c
+φ(xG
+i )
+(1)
+where ∥·∥2 is the l2 norm, and the embedding feature φ(x) is
+unit-normalized such that ∥φ(x)∥2 = 1.
+As expected, Fig. 2 verifies that fine-tuning from the weight
+imprinted initialization achieves a much higher discriminative
+quality. This shows the superiority of weight imprinting com-
+pared to random initialization and linear probing.
+Note that weight imprinting has been frequently used in FR
+embedding models [8], [9]. However, the critical difference
+is that those models utilize weight imprinting only to prepare
+templates before evaluation. In our case, the WI initialization
+is utilized particularly for fine-tuning.
+C. BatchNorm-only Fine-Tuning
+Choosing the appropriate layer to tune is another important
+issue for fine-tuning. Moreover, due to the extremely small
+number of samples for each gallery identity, there is a risk
+of overfitting as suggested by the classical theory on the vc
+dimension [26]. In fact, a recent study [25] suggests that full
+fine-tuning hurts the pretrained filters including the useful
+convolutional filters learned from a large-scale database.
+To minimize the negative effect of this deterioration, we
+fine-tune only the BatchNorm (BN) layers along with the
+classifier weight:
+min
+W, θBN L(W T φθ(x), y),
+θ = [θBN, θrest]
+(2)
+where θ refers to all parameters in the encoder φ = φθ and
+θBN and θrest respectively refers to BatchNorm parameters
+and the rest. During fine-tuning, θrest is fixed with no gradient
+flow. The loss function L can be a softmax cross-entropy, or
+widely used FR embedding model losses such as ArcFace [9]
+and CosFace [8].
+Due to selective fine-tuning of only the BN layers (and clas-
+sifier weight), the convolutional filters learned from the large-
+scale pre-train database are simply transferred. The BN-only
+training is thus computationally efficient as it occupies only
+0.1-0.01% of the total parameters in the CNN. Nevertheless,
+its model complexity is sufficient to learn a general image task
+as guaranteed by [27].
+Fig. 2.
+The Intra-class variance (left) and inter-class separation (right) of
+classifiers that are initialized by different schemes. NormFace [23], CosFace
+[8] and ArcFace [9] loss are used for linear probing initialization. The weight
+imprinting initialization does not require training, thus stays constant.
+Fig. 3. An unknown
+feature u placed be-
+tween gallery proto-
+types of class i and
+j. ϵ is some small
+positive constant.
+TABLE I
+AVERAGE ANGLE (DEGREES) BETWEEN IJB-C
+PROBE FEATURE VECTORS AND THEIR TOP-K
+CLOSEST GALLERY PROTOTYPES. THE THIRD
+COLUMN REFERS TO THE AVERAGE OF TOP-2 TO
+TOP-16.
+Encoder
+top-1
+top-2
+2∼16
+Res50
+K
+50.7◦
+64.0◦
+69.1◦
+U
+63.8◦
+66.0◦
+69.7◦
+VGG19
+K
+53.4◦
+66.2◦
+71.4◦
+U
+65.9◦
+68.2◦
+72.1◦
+D. Neighborhood Aware Cosine Similarity
+The cosine similarity function is the most predominant
+matcher for contemporary face verification and identification.
+Denoting the probe feature vector as p and the gallery pro-
+totypes as {gj}C
+j=1, where gj :=
+1
+m
+�
+yG
+i =j φ(xG
+i ) is the
+mean of all the normalized gallery feature vectors of class
+j, identification is performed by finding the maximum class
+index c = arg maxj=1,...,C cos(p, gj). On the other hand, in
+the extension to OSFI, the decision of accepting as known or
+rejecting as unknown can be formulated:
+max
+j=1,...,C cos(p, gj)
+Accept
+≷
+Reject
+τ
+(3)
+where cos(p, q) =
+p
+∥p∥2 ·
+q
+∥q∥2 is the cosine similarity between
+two feature vectors, τ is the rejection threshold.
+Now, consider an example illustrated in Fig. 3. The cosine
+matcher will assign the probe u to the identity i with the
+acceptance score 0.866, which is fairly close to the maximum
+score 1. This value alone might imply that the probe is a
+known sample as it is close to the gallery identity i. However,
+the probe feature vector is placed right in the middle of the
+identities i and j. The in-between placement of u suggests
+that the probe can be possibly unknown and thus should be
+assigned with a lesser value of the acceptance score.
+Motivated by this intuition, we propose the Neighborhood
+Aware Cosine (NAC) matcher that respects all top-k surround-
+ing gallery features:
+NAC(p, gi) = exp(cos(p, gi)) · 1[i ∈ Nk]
+�
+j∈Nk exp(cos(p, gj))
+(4)
+
+Intra-classvariance
+Inter-class separation
+。06
+104 °
+NormFace
+CosFace
+80 °
+103 °
+ArcFace
+70 °
+Weight Imprinting
+102 °
+Angle
+60 °
+101 °
+。09
+100 °
+40 °
+。66
+98 °
+0
+5
+10
+15 
+20 
+0
+5
+10
+15 
+20 
+Epochs
+Epochs9i
+30°
+30° + E
+.9jFig. 4. The distributions of scores for known (K) and unknown (U) probes
+of IJB-C dataset using cosine similarity (left) and NAC with k = 16 (right).
+The scores are min-max normalized and τ is set such that FAR=0.01 for both
+cases. DIR=48.05% (left) vs DIR=54.53% (right). ResNet-50 was used as the
+encoder.
+Here, Nk is the index set of k gallery prototypes that are
+nearest to the probe feature p, and 1 is the indicator function.
+The main goal of the NAC matcher is to improve the unknown
+rejection. Table I shows that known probe features are much
+closer to their closest prototype than the second-closest proto-
+type, unlike unknown probes. By exploiting this phenomenon,
+the NAC matcher is able to assign a much smaller score to
+unknown probe, as shown in Fig. 4.
+IV. EXPERIMENTS
+A. Datasets
+We use VGGFace2 [4] dataset for pretraining the encoders,
+and CASIA-WebFace [28] and IJB-C [29] for evaluation. Us-
+ing MTCNN [30], we align and crop every images to 112x112
+with equal parameters for all datasets. For VGGFace2, we
+remove all identities overlapping with the evaluation datasets.
+The evaluation datasets are equally split into two groups
+such that the number of known and unknown identities are
+equal. Then we randomly choose m=3 images of the known
+identities to create the gallery (G), and the rest are known
+probes (K). All images of unknown identities are unknown
+probes (U). Table II summarizes the statistics of the datasets
+we use. Note that we chose every known identity to have more
+than 10 images such that there can be at least 7 probe samples.
+Also note that IJB-C dataset consists of still images and video
+frames (video frames typically have poorer image quality). We
+sample the gallery from still images and probes from video
+frames, which makes this dataset much challenging. We note
+that the protocol devised here can be regarded as an extension
+of that in [15].
+B. Baselines
+1) Classifier Initialization: Along with Weight Imprinting
+(denoted WI), we report the results of using random ini-
+tialization and linear probing initialization as described in
+Sec. III-B.
+2) Encoder Layer Fine-Tuning: Along with BatchNorm-
+only fine-tuning (denoted as BN), we explore tuning other
+layers of the encoder. The simplest one is tuning every layer
+(i.e. all parameters of a model), which we denote as full. The
+second is freezing the early layers and training only the deeper
+ones, which we denote as partial. We also consider the parallel
+residual adapter [31], which adds additional 1x1 convolutional
+TABLE II
+DATASET STATISTICS. THE NUMBER INSIDE THE PARENTHESES REFERS
+TO THE AVERAGE NUMBER OF IMAGES PER IDENTITY. FOR EVALUATION
+DATASETS, KNOWN IDENTITIES CONSIST OF THE GALLERY (G) AND
+KNOWN PROBE (K), WHERE THE GALLERY HAS 3 IMAGES PER IDENTITY.
+Pretrain
+# IDs (images / ID)
+VGGFace2
+7,689 (354.0)
+Evaluation
+Known (G + K)
+Unknown (U)
+CASIA-WebFace
+5,287 (3+20.0)
+5,288 (16.5)
+IJB-C
+1,765 (3+15.3)
+1,765 (13.9)
+TABLE III
+THE TOTAL NUMBER OF PARAMETERS AND NUMBER OF FINE-TUNED
+PARAMETERS FOR EACH ENCODER FINE-TUNING SCHEME. ‘+’ REFERS TO
+THE NUMBER OF ADDED PARAMETERS FOR THE PARALLEL ADAPTER.
+# Params (million)
+VGG19
+Res50
+Pretrained
+32.88
+43.58
+Full fine-tuning
+32.88
+43.58
+Partial fine-tuning
+4.72
+4.72
+Parallel Adapter
++2.22
++3.39
+BN-only fine-tuning
+0.01
+0.03
+filters to the original convolutional layers. During fine-tuning,
+only these additional filters are trained to capture the subtle
+difference in the new dataset. Note that the authors in [31]
+apply this technique to ResNet [3], hence the name residual
+parallel adapter. But this idea can be generally applied to
+CNNs without residual connection, hence we also apply this
+to a VGG-style network. We denote this as PA, referring to
+Parallel Adapter.
+3) Matcher: During OSFI evaluation, the vanilla cosine
+similarity matcher is adopted as the baseline matcher. When
+the NAC matcher is used, we denote by NAC. For comparison,
+we also use the extreme value machine (EVM) proposed by
+[15]. We train the EVM on the gallery set with the best
+parameters found by the authors.
+In summary, classifier initialization methods we consider are
+{Random, Linear probing, WI}, fine-tuning layer configu-
+rations are {Full, Partial, PA, BN}, and matchers are {cos,
+EVM, NAC}. We test the OSFI performances among different
+combinations of these three components. Our proposed OSFI
+scheme is to use WI+BN+NAC jointly.
+C. Training Details
+We choose VGG19 [2] and ResNet-50 [3] for the encoders
+with the feature dimension 512. We pretrain these encoders
+on the VGGFace2 dataset with CosFace with scale=32, mar-
+gin=0.4 as loss function until convergence.
+Then we fine-tune the encoder with different classifier
+initialization schemes and encoder layer configurations. When
+using the linear probing initialization, we train the classifier
+until the training accuracy reaches 95%.
+We follow the encoder layer finetuning in Sec. IV-B. For
+the partial fine-tuning, we only train the last 2 convolutional
+layers (Conv-BN-ReLU-Conv-BN-ReLU). Table III shows the
+number of total and updated parameters for each fine-tuning
+scheme.
+
+COS
+NAC (k=16)
+T
+Known
+Known
+Unknown
+Unknown
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00Fig. 5. The OSFI performance of cosine similarity and NAC with different values of k on IJB-C dataset, using VGGNet-19 (left) and ResNet-50 (mid) as the
+encoder. The square markers refer to cosine similarity and star marks the optimal k for different layer fine-tuning methods. To summarize the OSFI performance
+into a single number, we used the area under the curve (AUC, %) of DIR@FAR curve. (Right) DIR@FAR curve of Pretrained and BN configuration using
+cosine similarity and NAC (k=16) as the matcher. Numbers in the legend show the AUC values. When k = 1, NAC is replaced by cos.
+We fix the number of epochs to 20 and batch size to 128
+for every method. We again use CosFace loss for consistency.
+For the optimizer we use Adam [32] with cosine annealing.
+The initial learning rate is set to 1e-4 for full and PA, and 1e-
+3 for partial and BN, which we find as the optimal learning
+rate for each method. For data augmentation, we use random
+horizontal flipping and random cropping with the random scale
+from 0.7 to 1.0. The cropped images are resized to the original
+size.
+D. Optimal k for NAC
+Since the gallery set is too small, we cannot afford a separate
+validation set to individually optimize k for each dataset.
+Instead, we attempt to find a global value that has optimal
+performance regardless of the fine-tuning method, if one exists.
+We first fine-tune the encoders with different layer con-
+figurations, which gives us five different encoders includ-
+ing one without any fine-tuning; pretrained, full, partial,
+PA, and BN. Then we search the best parameter k for the
+NAC matcher by grid search strategy, where the grid is
+[2,4,8,16,32,128,256,512,1024,C], and C is the total number
+of identities. Note that k = 1 refers to using cosine similarity
+instead of NAC, which we added for comparison. Since a
+single-value objective is preferred, we use the area under the
+curve (AUC) of the DIR@FAR curve instead of DIR value
+at different FAR values. We repeat this process with different
+datasets and encoder architectures.
+The results are shown in Fig. 5. We did not include
+the results of CASIA-WebFace as it shows a similar trend.
+Excluding k = 1 which is not NAC, the results show a smooth
+unimodal curve with a peak at k = 16 or 32. This shows
+that the NAC matcher indeed has a globally optimal k value
+that is robust against different datasets, encoders, and fine-tune
+methods. Thus we choose k = 16 (k = 32 also gives similar
+results) as the global parameter throughout this paper.
+Note that when k = C, NAC becomes equivalent to softmax
+function with cosine similarity logits. However, this is notably
+inferior compared to k = 16, which implies that considering
+only the k-nearest is superior to considering every gallery
+prototype.
+E. Comparison of Fine-Tuning Methods
+We compare the OSFI performances of the pretrained model
+(non-fine-tuned) with six different combinations of classifier
+initialization schemes and layer finetuning configurations: ran-
+dom+full, linear probing+full, WI+full, WI+partial, WI+PA,
+WI+BN. The matcher is fixed to cosine similarity. These
+correspond to row 4-9 in Table IV.
+First, to compare different classifier initialization schemes,
+we fix the fine-tuning scheme to full. When using random
+initialization, rejection accuracy (DIR@FAR=0.001,0.01,0.1)
+and closed-set accuracy (DIR@FAR=1) severely drops. For
+linear probing, rejection accuracy improves while closed-
+set accuracy drops. Only WI clearly improves the encoder
+performance, supporting the superiority of weight imprinting.
+Now we fix the classifier initialization to WI and compare
+different layer finetuning configurations. full clearly has the
+worst performance. While PA is better than partial in closed-
+set accuracy, partial clearly outperforms PA in rejection ac-
+curacy. BN outperforms all others in closed-set accuracy with
+a large margin but sometimes falls behind partial in rejection
+accuracy.
+With the aid of the NAC matcher, our method WI+BN+NAC
+outperforms all other methods in every aspect. Compared
+to original, this gains 4.60%, 8.11%, 4.57%, 1.68% higher
+DIR in average with respect to FAR of 0.001, 0.01, 0.1, 1.0,
+respectively.
+F. Analysis on Discriminative Quality of Different Fine-tuning
+Methods
+How do different layer finetuning configurations affect the
+final OSFI performance? To analyze this, we adopt three
+different metrics; inter-class separation, intra-class variance,
+and Davies-Bouldin Index (DBI) [33]. The definitions of the
+first two metrics are identical to that of Fig. 2. DBI is a metric
+for evaluating the clustering quality, where DBI ≈ 0 means
+perfect clustering. We compute these metrics on the gallery
+features after fine-tuning, and the results are shown in Table
+V.
+Here we can easily separate these configurations into two
+groups: full and partial vs PA and BN. The first group has
+
+VGG19
+ResNet-50
+ResNet-50
+75.5
+0.8
+70.5
+75.0
+74.5
+0.6 
+(%)
+70.0
+74.0
+AUC
+69.5
+0.4 
+Pretrained
+73.5
+Full
+-
+Pretrained+cos:72.36%
+69.0
+Partial
+73.0
+0.2 
+Pretrained+nac:73.42%
+PA
+72.5
+Cosine
+68.5
+WI+BN+cos: 75.02%
+BN
+★
+NAC (best)
+72.0
+0.0
+WI+BN+nac: 75.41%
+2
+8
+16
+32
+64
+128 256512 1024C
+2
+8
+16
+32
+64
+128 256512 1024C
+0.0001
+0.0010
+0.0100
+0.1000
+1.0000
+k
+k
+False Alarm RateTABLE IV
+DIR@FAR OF DIFFERENT METHODS ON CASIA-WEBFACE DATASET AND IJB-C DATASET, USING VGGNET-19 AND RESNET-50 AS THE ENCODER.
+DIR@FAR=1 (100%) IS THE CLOSED-SET ACCURACY. THE HIGHEST VALUE IN EACH COLUMN IS MARKED IN BOLD. FOR THE FIRST THREE ROWS THE
+ENCODER IS NOT FINE-TUNED AND ONLY THE MATCHERS ARE CHANGED. THE LAST ROW (WI+BN+NAC) IS OUR PROPOSED METHOD.
+Encoder
+Method
+CASIA-WebFace
+IJB-C
+Classifier
+initialization
+Fine-tuning
+layers
+Matcher
+DIR @ FAR (%)
+DIR @ FAR (%)
+0.1
+1.0
+10.0
+100.0
+0.1
+1.0
+10.0
+100.0
+VGG19
+None
+None
+cos
+25.23
+52.97
+70.07
+80.89
+28.35
+45.55
+61.71
+73.80
+None
+None
+EVM
+37.57
+57.75
+71.03
+80.78
+35.03
+53.64
+63.34
+73.70
+None
+None
+NAC
+25.15
+55.68
+71.41
+80.89
+36.73
+51.92
+64.27
+73.80
+Random
+Full
+cos
+23.95
+43.19
+59.03
+70.94
+17.18
+32.62
+46.90
+60.23
+Linear probing
+Full
+cos
+28.82
+55.64
+70.44
+79.84
+30.80
+45.91
+59.63
+70.09
+WI
+Full
+cos
+27.63
+57.58
+72.02
+80.94
+35.49
+50.52
+63.56
+73.53
+WI
+Partial
+cos
+28.91
+57.31
+72.29
+81.16
+34.81
+51.98
+64.53
+73.89
+WI
+PA
+cos
+26.29
+57.90
+72.82
+81.82
+31.74
+50.21
+64.26
+74.50
+WI
+BN
+cos
+25.39
+56.65
+72.54
+82.14
+32.19
+48.74
+63.87
+74.43
+WI
+BN
+NAC
+25.94
+58.01
+72.92
+82.14
+38.09
+53.08
+65.30
+74.43
+Res50
+None
+None
+cos
+23.85
+58.06
+74.15
+83.69
+32.11
+48.05
+65.31
+76.96
+None
+None
+EVM
+39.44
+61.61
+75.02
+83.57
+38.12
+38.12
+66.81
+76.96
+None
+None
+NAC
+21.24
+60.23
+75.31
+83.69
+36.67
+54.53
+68.14
+76.96
+Random
+Full
+cos
+25.31
+45.43
+60.80
+72.44
+14.88
+32.05
+49.39
+61.88
+Linear probing
+Full
+cos
+28.35
+60.11
+74.63
+82.73
+30.35
+46.42
+61.90
+72.34
+WI
+Full
+cos
+26.73
+63.92
+77.49
+84.65
+39.05
+56.00
+67.83
+76.94
+WI
+Partial
+cos
+25.98
+64.66
+78.07
+85.02
+44.31
+57.11
+69.13
+77.49
+WI
+PA
+cos
+24.89
+63.85
+77.58
+85.01
+36.69
+54.86
+68.30
+77.63
+WI
+BN
+cos
+25.70
+65.83
+79.66
+86.73
+40.29
+55.71
+69.29
+78.74
+WI
+BN
+NAC
+23.65
+67.72
+80.34
+86.73
+40.25
+58.25
+70.40
+78.74
+TABLE V
+INTER-CLASS SEPARATION, INTRA-CLASS VARIANCE, DBI, AND AUC
+GAIN BY USING NAC (REFER TO FIG. 5) FOR EACH LAYER FINETUNING
+CONFIGURATION. THESE VALUES ARE AVERAGED ACROSS DATASETS AND
+ENCODER ARCHITECTURES. ↑ MEANS THAT LARGER QUANTITY IS BETTER
+AND VICE VERSA.
+Inter (↑)
+Intra (↓)
+DBI (↓)
+∆AUC (↑)
+Pretrained Model
+106.3◦
+34.5◦
+1.52
+0.740
+Full finetuning
+106.7◦
+24.2◦
+0.87
+0.025
+Partial finetuning
+106.4◦
+24.5◦
+0.90
+0.058
+Parallel Adapter
+107.0◦
+31.8◦
+1.32
+0.135
+BN-only finetuning
+107.3◦
+33.6◦
+1.46
+0.335
+similar inter-class separation with Pretrained and significantly
+smaller intra-class variance, which leads to small DBI. This is
+in stark contrast with the second group.
+With this observation, we can conjecture the different opti-
+mization strategies of each group. The first group was able to
+easily reduce the training loss by collapsing the gallery fea-
+tures into a single direction (shown by the small angle between
+intra-class features). This was possible because both full and
+partial directly updated the parameters of the convolutional
+filters. On the other hand, all convolutional filters were frozen
+for both PA and BN. This constraint may have prevented these
+methods from taking the shortcut, i.e. simply collapsing the
+gallery features, and instead led to separating the embeddings
+of different identities. This explains why PA and BN have
+higher closed-set accuracy.
+This can also explain the AUC gain (∆AUC) when using
+NAC instead of cosine similarity. Features become redundant
+when they collapse, and so does the prototype. Therefore the
+information from neighboring prototypes becomes less helpful
+in rejecting unknown samples, leading to the marginal gain
+from using NAC. This is why full and partial do not benefit
+from using NAC matcher.
+Fig. 6. The performance of our method against the baseline w.r.t. different
+gallery size. AUC of DIR@FAR curve is used as the performance measure.
+G. Performance with respect to Different Gallery Size
+Fig. 6 shows the OSFI performance of our method against
+the baseline (pretrained encoder with cos matcher) with respect
+to different gallery size. We can see that our method consis-
+tently improves upon the baseline, except for the extreme case
+where only one image is provided for each identity.
+V. CONCLUSION AND FUTURE WORKS
+In this work we showed that combining weight-imprinted
+classifier and BatchNorm-only tuning of the encoder effec-
+tively improves the encoder’s OSFI performance without suf-
+fering from overfitting. We further facilitated the performance
+by our novel NAC matcher instead of the commonly used
+cosine similarity. Future works will explore extending this idea
+to the open-set few-shot recognition of generic images.
+Acknowledgements:
+This work was supported by the National Research Foundation
+of Korea (NRF) grant funded by the Korea government (MSIP)
+(NO. NRF-2022R1A2C1010710)
+
+IJB-C, ResNet-50
+CASIA-WebFace.ResNet-50
+80
+Pretrained
+90
+Pretrained
+Ours
+Ours
+AUC(%)
+80
+70
+70
+60
+60
+50
+50
+2
+NumberofImagesperGalleryIdentity
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+High Dimensional Analysis of Variance in Multivariate Linear
+Regression
+Zhipeng Lou1, Xianyang Zhang2 and Wei Biao Wu3
+January 12, 2023
+Abstract
+In this paper, we develop a systematic theory for high dimensional analysis of variance in multivariate
+linear regression, where the dimension and the number of coefficients can both grow with the sample
+size. We propose a new U type test statistic to test linear hypotheses and establish a high dimensional
+Gaussian approximation result under fairly mild moment assumptions.
+Our general framework and
+theory can be applied to deal with the classical one-way multivariate ANOVA and the nonparametric
+one-way MANOVA in high dimensions. To implement the test procedure in practice, we introduce a
+sample-splitting based estimator of the second moment of the error covariance and discuss its properties.
+A simulation study shows that our proposed test outperforms some existing tests in various settings.
+Keywords: Data-splitting; Gaussian approximation; Multivariate analysis of variance; One-way
+layout; U statistics
+1
+Introduction
+In statistical inference of multivariate linear regression, a fundamental problem is to investigate the rela-
+tionships between the covariates and the responses. In this article, we aim to test whether a given set of
+covariates are associated with the responses by multivariate analysis of variance (MANOVA). To fix the idea,
+we build the multivariate linear regression model with p predictors as
+Yi = B⊤Xi + Vi (i = 1, . . . , n),
+(1.1)
+where Yi = (Yi1, . . . , Yid)⊤ and Xi = (Xi1, . . . , Xip)⊤ are respectively the response vector and the predictor
+vector respectively for the ith sample, B⊤ = (B1, . . . , Bp) is the unknown coefficient matrix with Bk ∈ Rd
+consisting of coefficients on the kth covariate, and the innovation vectors V1, . . . , Vn ∈ Rd are independent
+and identically distributed random vectors with E(V1) = 0 and cov(V1) = Σ. The first element of Xi can be
+set to be 1 to reflect an intercept term. Equivalently we can write (1.1) in compact matrix form as
+Y = XB + V,
+(1.2)
+1Department of Operations Research and Financial Engineering, Princeton, NJ 08544.
+2Department of Statistics, Texas A&M University, College Station, TX 77843.
+3Department of Statistics, University of Chicago, Chicago, IL, 60637.
+1
+arXiv:2301.04209v1  [stat.ME]  10 Jan 2023
+
+where Y = (Y1, . . . , Yn)⊤, X = (X1, . . . , Xn)⊤ and V = (V1, . . . , Vn)⊤. Let C ∈ Rm×p be a matrix of rank m,
+where m ∈ {1, . . . , p}. We are interested in testing a collection of linear constraints on the coefficient matrix
+H0 : CB = 0 versus H1 : CB ̸= 0.
+(1.3)
+This testing problem has been extensively studied in the low dimensional setting where both the number
+of predictors and the dimension of the response are relatively small compared to the sample size. A natural
+and popular choice is the classical likelihood ratio test when the errors are normally distributed; see Chapter
+8 in Anderson (2003) for a review of theoretical investigations. In recent years, high dimensional data are
+increasingly encountered in various applications. Over the past decade, there have been tremendous efforts
+to develop new methodologies and theories for high dimensional regression. The paradigm where d is 1
+or small and p can increase with n has received considerable attention, while on the other hand the one
+where d is very large and p is relatively small has been less studied. The model (1.2) in the latter setting
+has been applied to a number of research problems involving high-dimensional data types such as DNA
+sequence data, gene expression microarray data, and imaging data; see for example Zapala and Schork
+(2006), Wessel and Schork (2006) and Zapala and Schork (2012). Those related studies typically generate
+huge amounts of data (responses) that, due to their expense and sophistication, are often collected on a
+relatively small number of individuals, and investigate how the data can be explained by a certain number
+of predictor variables such as the ages of individuals assayed, clinical diagnoses, strain memberships, cell
+line types, or genotype information (Zapala and Schork, 2006). Owing to inappropriateness of applying the
+standard MANOVA strategy and shortage of high-dimensional MANOVA theory, biological researchers often
+considered some form of data reduction such as cluster analysis and factor analysis, which can suffer from
+many problems, as pointed out by Zapala and Schork (2012). In the works Zapala and Schork (2006, 2012),
+the authors incorporated a distance matrix to modify the standard MANOVA, but they commented that
+there is very little published material that can be used to guide a researcher as to which distance measure is
+the most appropriate for a given situation. Motivated by these real-world applications, we aim to develop a
+general methodology for high dimensional MANOVA and lay a theoretical foundation for assessing statistical
+significance.
+The testing problem (1.3) for model (1.2) is closely related to a group of high dimensional hypothesis
+tests. Two-sample mean test, for testing H0 : µ1 = µ2 where µ1 ∈ Rd and µ2 ∈ Rd are mean vectors of two
+different populations, is a special case with p = 2, B = (µ1, µ2)⊤ and C = (1, −1). There is a large literature
+accommodating the Hotelling T 2 type statistic into the high-dimensional situation where d is large; see for
+example, Bai and Saranadasa (1996), Chen and Qin (2010), Srivastava et al. (2013) among many others.
+It can be generalized to test the equality of multiple mean vectors in high dimensions. Some notable work
+includes Schott (2007), Cai and Xia (2014), Hu et al. (2017), Li et al. (2017), Zhang et al. (2017) and Zhou
+et al. (2017). In most existing work, the random samples were assumed to be Gaussian or follow some linear
+structure as that of Bai and Saranadasa (1996). In contrast, the testing problem we are concerned is much
+more general. For one thing, all the aforementioned high dimensional mean test problems can be fitted into
+our framework, apart from which, we can deal with the more general multivariate linear regression in the
+presence of an increasing number of predictor variables. For another, we do not assume the Gaussianity or
+any particular structure of the error vectors {Vi}n
+i=1.
+Throughout the paper, we assume that p < n and the design matrix X is of full column rank such that
+2
+
+X⊤X is invertible. The conventional MANOVA test statistic for (1.3) is given by
+Qn = |PY |2
+F =
+n
+�
+i=1
+n
+�
+j=1
+PijY ⊤
+i Yj,
+(1.4)
+where | · |F stands for the Frobenius norm and
+P = X(X⊤X)−1C⊤{C(X⊤X)−1C⊤}−1C(X⊤X)−1X⊤ = (Pij)n×n
+is the orthogonal projection matrix onto the column space of the matrix X(X⊤X)−1C⊤. We shall reject the
+null hypothesis H0 if Qn is larger than some critical value. In the univariate case where d = 1, the asymptotic
+behavior of Qn has been extensively studied in literature; see G¨otze and Tikhomirov (1999) and G¨otze and
+Tikhomirov (2002) for detailed discussions. The validity to perform a test for (1.3) using Qn when d is large
+has been open for a long time. The first goal of the paper is to provide a solution to this open problem by
+rigorously establishing a distributional approximation of the traditional MANOVA test statistic when d is
+allowed to grow with n. Our key tool is the Gaussian approximation for degenerate U type statistics: under
+fairly mild moment conditions, quadratic functionals of non-Gaussian random vectors can be approximated
+by those of Gaussian vectors with the same covariance structure. It is worth mentioning that Chen (2018)
+established a Gaussian approximation result for high dimensional non-degenerate U statistics by Stein’s
+method, which can not be applied to the degenerate case here. From a technical point of view, we employ
+completely different arguments to bound distance between the distribution functions of the test statistic and
+its Gaussian analogue.
+The main contributions of this paper are three-fold. Firstly, we develop a systematic theory for the
+conventional MANOVA test statistic Qn in the high dimensional setting. More specifically, we shall establish
+a dichotomy result: Qn can be approximated either by a linear combination of independent chi-squared
+random variables or by a normal distribution under different conditions; see Theorem 2.1. While this reveals
+the interesting theoretical properties of the test statistics, it causes difficulties in applications as one may
+not know which asymptotic distribution to use in practice. To overcome this difficulty, as the second main
+contribution of our paper, we propose using a new U type test statistic. Using the modified test statistic,
+such a dichotomy does not appear; see Theorem 2.5 for the asymptotic result. Thirdly, we will propose a
+new estimator for the second spectral moment of the covariance matrix via a data-splitting technique. To
+the best of our knowledge, it is the first work concerning an unbiased and ratio consistent estimator in the
+multivariate linear regression model.
+We now introduce some notation. Let I{·} denote the indicator function. For random variables X ∈ R
+and Y ∈ R, the Kolmogorov distance is defined by ρ(X, Y ) = supz∈R |P(X ≤ z) − P(Y ≤ z)|. For q > 0,
+we write ∥X∥q = (E|X|q)1/q if E|X|q < ∞.
+For two matrices A = (aij)i≤I,j≤J and B = (bij)i≤I,j≤J,
+A ◦ B = (aijbij)i≤I,j≤J denotes their Hardmard product. For any positive integer m, we use Im to denote
+m × m identity matrix. For two sequences of positive numbers (an) and (bn), we write an ≲ bn if there
+exists some constant C such that an ≤ Cbn for all large n. We use C, C1, C2, . . . to denote positive constants
+whose value may vary at different places.
+3
+
+2
+Theoretical results
+We start with some notational definitions and basic assumptions. Let λ1(Σ) ≥ . . . ≥ λd(Σ) ≥ 0 denote the
+eigenvalues of Σ = cov(V1) and let ς = |Σ|F = {�d
+k=1 λ2
+k(Σ)}1/2. For q ≥ 2, we define
+Mq = E
+����
+V ⊤
+1 V2
+ς
+����
+q
+and Lq = E
+����
+V ⊤
+1 ΣV1
+ς2
+����
+q/2
+.
+(2.1)
+Assumption 2.1. Recall that P11, . . . , Pnn are diagonal elements of the matrix P. Assume that
+1
+m
+n
+�
+i=1
+P 2
+ii → 0 as n → ∞.
+Remark 1. Assumption 2.1 is quite natural and mild for testing (1.3). For instance, it automatically holds
+for one sample test of mean vector as m−1 �n
+i=1 P 2
+ii = 1/n. Additionally, in the context of K-sample test, as
+discussed in Section 3.1, Assumption 2.1 is satisfied as long as the minimum sample size goes to infinity. More
+generally, since �n
+i=1 Pii = m, a simple sufficient condition for Assumption 2.1 would be max1≤i≤n Pii → 0.
+Further discussions on this condition will be given in Remark 6 and Example 2.1.
+2.1
+Asymptotic distribution of the conventional MANOVA test statistics
+Under the null hypothesis CB = 0, PXB = X(X⊤X)−1C⊤{C(X⊤X)−1C⊤}−1CB = 0 and hence Qn =
+|PXB + PV |2
+F
+H0
+= |PV |2
+F, which can be further decomposed as
+Qn
+H0
+=
+n
+�
+i=1
+n
+�
+j=1
+PijV ⊤
+i Vj =
+n
+�
+i=1
+PiiV ⊤
+i Vi +
+n
+�
+i=1
+�
+j̸=i
+PijV ⊤
+i Vj =: Dn + Q⋆
+n.
+(2.2)
+Observe that Dn is a weighted sum of i.i.d. random variables and Q⋆
+n is a second order non-degenerate U -
+statistic of high dimensional random vectors. These two terms can be differently distributed under the high
+dimensional setting. More specifically, since Dn and Q⋆
+n are uncorrelated, we have var(Qn) = var(Dn) +
+var(Q⋆
+n), where
+var(Dn) =
+n
+�
+i=1
+P 2
+ii∥E0(V ⊤
+1 V1)∥2
+2 and var(Q⋆
+n) = 2
+�
+m −
+n
+�
+i=1
+P 2
+ii
+�
+ς2,
+where E0(V ⊤
+1 V1) = V ⊤
+1 V1 − E(V ⊤
+1 V1). When the dimension d increases with the sample size n, the mag-
+nitudes of var(Dn) and var(Q⋆
+n) can be quite different for non-Gaussian {Vi}n
+i=1; cf. Example 4.1. As a
+consequence, Qn can exhibit different asymptotic null distributions. More precisely, to asymptotically quan-
+tify the discrepancy between var(Dn) and var(Q⋆
+n), under Assumption 2.1, we define
+Λ2 =
+�n
+i=1 P 2
+ii∥E0(V ⊤
+1 V1)∥2
+2
+mς2
+.
+Before presenting the distributional theory for Qn, we first define its Gaussian analogue. Let Z1, . . . , Zn be
+i.i.d. N(0, Σ) Gaussian random vectors and write Z = (Z1, . . . , Zn)⊤. Then the Gaussian analogue of Qn is
+defined as the same quadratic functional of {Zi}n
+i=1,
+Gn = |PZ|2
+F =
+n
+�
+i=1
+n
+�
+j=1
+PijZ⊤
+i Zj.
+(2.3)
+4
+
+Theorem 2.1. Let q = 2 + δ, where 0 < δ ≤ 1. Suppose Assumption 2.1 holds and
+∆q =
+�n
+i=1
+�
+j̸=i |Pij|q
+mq/2
+Mq +
+�n
+i=1 P q/2
+ii
+mq/2
+Lq → 0.
+(2.4)
+1. Assume Λ → 0. Then, under (2.4) and the null hypothesis, we have
+ρ(Qn, Gn) ≤ C1Λ2/5 + Cq∆1/(2q+1)
+q
++ C2
+�
+1
+m
+n
+�
+i=1
+P 2
+ii
+�1/5
+→ 0.
+2. Assume Λ → ∞ and the Lindeberg condition holds for Wi = E0(PiiV ⊤
+i Vi)/(Λς√m), that is, �n
+i=1 E(W 2
+i I{|Wi| >
+ϵ}) → 0 for any ϵ > 0. Then, under the null hypothesis, we have
+Qn − mtr(Σ)
+Λς√m
+⇒ N(0, 1).
+(2.5)
+Remark 2. Theorem 2.1 illustrates an interesting dichotomy: the conventional MANOVA test statistic
+Qn can have one of the two different asymptotic null distributions, depending on the magnitude of the
+unknown quantity Λ.
+This nature of dichotomy poses extra difficulty for utilizing Qn to test (1.3) in
+practical implementation as we need to predetermine which asymptotic distribution to use. Any subjective
+choice may lead to unreliable conclusion.
+To illustrate this, suppose now Λ → 0.
+For α ∈ (0, 1), let
+G−1
+n (α) denote the (1 − α)th quantile of Gn. Based on Theorem 2.1, an α level test for (1.3) is given by
+Φ0 = I{Qn > G−1
+n (α)}. However, if one implements Φ0 under the case where Λ → ∞, then the type I error
+of Φ0 satisfies that P(Φ0 = 1 | H0) → 1/2, which implies that Φ0 in this scenario (Λ → ∞) is no better than
+random guessing.
+Remark 3. Recently much attention has been paid to studying the dichotomy and similar phase transition
+phenomenon of the asymptotic distribution of classical tests under the high dimensional setting. For instance,
+Xu et al. (2019) studied the Pearson’s chi-squared test under the scenario where the number of cells can
+increase with the sample size and demonstrated that the corresponding asymptotic distribution can be either
+chi-squared or normal. He et al. (2021) derived the phase transition boundaries of several standard likelihood
+ratio tests on multivariate mean and covariance structures of Gaussian random vectors. In addition to these
+tests, we suspect similar phenomenon can occur for many other traditional tests as the dimension increases
+with the sample size. More importantly, as in our paper, investigating these phase transition phenomena
+of classical tests not only contributes to the theoretical development but also motivates us to propose new
+test procedure or more advanced approximation distributional theory which are suitable under the high
+dimensional scenario.
+The following lemma establishes an upper bound for ∆q.
+Lemma 2.2. Assuming that Mq < ∞, then we have
+∆q < 2
+� 1
+m max
+1≤i≤n Pii
+�δ/2
+Mq.
+Remark 4. Condition (2.4) can be viewed as the Lyapunov-type condition for high dimensional Gaussian
+approximation of Qn. It is quite natural and does not impose any explicit restriction on the relation between
+5
+
+the dimension d and the sample size n directly. In particular, (2.4) can be dimension free for some commonly
+used models, namely, (2.4) holds for arbitrary dimension d ≥ 1 as long as n → ∞. For instance, suppose
+that {Vi}n
+i=1 follow the linear process model
+Vi = Aξi (i = 1, . . . , n),
+(2.6)
+where A is a d × L matrix for some integer L ≥ 1, ξi = (ξi1, . . . , ξiL)⊤ and {ξiℓ}i,ℓ∈N are independent zero-
+mean random variables with uniformly bounded qth moment E|ξiℓ|q ≤ C < ∞. Applying the Burkholder
+inequality leads to Mq ≤ (1 + δ)q max1≤ℓ≤L ∥ξiℓ∥2q
+q .
+Consequently, Lemma 2.2 reveals that a sufficient
+condition for ∆q → 0 is
+1
+m max
+1≤i≤n Pii → 0.
+(2.7)
+It is worth mentioning that (2.7) depends only on the projection matrix P and does not impose any re-
+striction on the dimension d. Moreover, under Assumption 2.1, (2.7) is automatically satisfied in view of
+max1≤i≤n(Pii/m)2 ≤ m−2 �n
+i=1 P 2
+ii → 0.
+2.2
+Modified U type test statistics
+The dichotomous nature of the asymptotic null distribution makes Qn unsuitable for testing (1.3) in the high
+dimensional setting. This motivates us to propose a modified U type test statistic of Qn for which such a
+dichotomy does not occur. To fix the idea, let B0 ∈ Rp×d denote the coefficient matrix of model (1.2) under
+the null hypothesis such that CB0 = 0 and Y
+H0
+= XB0 + V . Motivated by Theorem 2.1, a natural candidate
+of the test statistic Qn would be
+Qn,0 = Qn −
+n
+�
+k=1
+Pkk(Yk − B⊤
+0 Xk)⊤(Yk − B⊤
+0 Xk),
+(2.8)
+which coincides with Q⋆
+n in (2.2) under the null hypothesis. However, B0 is unknown in practice and hence
+Qn,0 is infeasible. The primary goal of this section is to propose a consistent empirical approximation Un for
+Qn,0. In particular, motivated by the discussions in Section 2.1, the modified test statistic Un should satisfy
+that
+Un
+H0
+=
+n
+�
+i=1
+�
+j̸=i
+KijV ⊤
+i Vj and
+Un − Qn,0
+√var(Qn,0)
+H0
+= oP(1),
+for some symmetric matrix K = (Kij)n×n. The latter ensures that Un is asymptotically equivalent to Qn,0
+in (2.8). Towards this end, let �B0 be the least square estimator of B under the constraint CB = 0. Then
+Y − X �B0 = (In − P0)Y , where P0 = X(X⊤X)−1X⊤ − P is the projection matrix of model (1.2) under the
+null hypothesis. In view of (2.8), the modified U type test statistic is then defined by
+Un = Qn −
+n
+�
+k=1
+θk(Yk − �B⊤
+0 Xk)⊤(Yk − �B⊤
+0 Xk)
+H0
+=
+n
+�
+i=1
+�
+Pii −
+n
+�
+k=1
+θk ¯P 2
+ik,0
+�
+V ⊤
+i Vi +
+n
+�
+i=1
+�
+j̸=i
+�
+Pij −
+n
+�
+k=1
+θk ¯Pik,0 ¯Pjk,0
+�
+V ⊤
+i Vj
+=
+n
+�
+i=1
+�
+j̸=i
+�
+Pij −
+n
+�
+k=1
+θk ¯Pik,0 ¯Pjk,0
+�
+V ⊤
+i Vj,
+(2.9)
+6
+
+where ¯P0 = In − P0 = ( ¯Pij,0)n×n and the last equality follows by taking θ1, . . . , θn to be the solutions of the
+following linear equations
+n
+�
+k=1
+¯P 2
+ik,0θk = Pii (i = 1, . . . , n).
+(2.10)
+It is worth mentioning that typically θk in (2.9) are not Pkk, as one would naturally like to use in view
+of (2.8). We can view (2.10) as a detailed balanced condition as it removes the diagonals in (2.9). Denote
+θ = (θ1, . . . , θn)⊤ and rewrite (2.10) in the more compact matrix form
+( ¯P0 ◦ ¯P0)θ = (P11, . . . , Pnn)⊤.
+(2.11)
+Let Pθ = P − ¯P0Dθ ¯P0 = (Pij,θ)n×n, where Dθ = diag(θ1, . . . , θn) is a diagonal matrix. Then Pii,θ = 0 for
+all i = 1, . . . , n in view of (2.11) and
+Un
+H0
+= tr(V ⊤PθV ) =
+n
+�
+i=1
+�
+j̸=i
+Pij,θV ⊤
+i Vj.
+Before proceeding, we first introduce a sufficient condition such that Un exists and is well defined.
+Lemma 2.3. Assume that there exists a positive constant ϖ0 < 1/2 such that
+max
+1≤i≤n Pii,0 ≤ ϖ0.
+(2.12)
+Then the matrix ¯P0◦ ¯P0 is strictly diagonally dominant and |Pθ|2
+F = m−�n
+i=1 θiPii. Moreover, if max1≤i≤n Pii ≤
+ϖ1ζ for some positive constant ϖ1 < 1/2, where ζ = (1 − 2ϖ0)(1 − ϖ0), then we have max1≤i≤n |θi| ≤ ϖ1 <
+1/2.
+Remark 5. Condition (2.12) ensures the matrix ¯P0 ◦ ¯P0 is invertible. Consequently the solution θ of (2.11)
+exists and is unique. It is worth noting that θ is independent of the dimension d and only depends on the
+projection matrices P and P0. Moreover, as shown in the proof of Lemma 2.3,
+n
+�
+i=1
+θiPii ≤ 1
+ζ
+n
+�
+i=1
+P 2
+ii and
+max
+1≤i≤n |θi| ≤ 1
+ζ max
+1≤i≤n Pii,
+which are essential to upper bound the quantity ∆q,θ in Lemma 2.6 below. Consequently, under Assump-
+tion 2.1, suppose �n
+i=1 P 2
+ii ≤ mζ/2 for sufficiently large n, we obtain
+var(Un) = 2|Pθ|2
+Fς2 = 2
+�
+m −
+n
+�
+i=1
+θiPii
+�
+ς2 > mς2,
+which ensures the proposed test statistic Un is non-degenerate and well defined.
+Remark 6. Since col(X(X⊤X)−1C⊤) ⊂ col(X), where col(·) denotes the column space, P0 = X(X⊤X)−1X⊤−
+P defined above is also a projection matrix.
+Hence max{Pii, Pii,0} ≤ X⊤
+i (X⊤X)−1Xi uniformly for
+i ∈ {1, . . . , n} and a sufficient condition for Lemma 2.3 would be
+max
+1≤i≤n X⊤
+i (X⊤X)−1Xi ≤ min{ϖ0, (1 − 2ϖ0)(1 − ϖ0)ϖ1},
+(2.13)
+7
+
+which is fairly mild on the design matrix X.
+More specifically, it is commonly assumed (Huber, 1973,
+Portnoy, 1985, Wu, 1986, Shao and Wu, 1987, Shao, 1988, Mammen, 1989, Navidi, 1989, Lahiri, 1992)
+for the linear regression model that max1≤i≤n X⊤
+i (X⊤X)−1Xi → 0, which ensures a kind of “robustness of
+design” (Huber, 1973). It also implies Assumption 2.1 in view of Remark 1 and can be viewed as a imbalance
+measure of model (1.2) (Shao and Wu, 1987).
+Example 2.1. Suppose X1, . . . , Xn are independent Gaussian random vectors N(0, Γ), where the covariance
+matrix Γ ∈ Rp×p has minimal eigenvalue λmin(Γ) > 0. Then, with probability at least 1−2 exp(−n/2)−n−1,
+we have
+max
+1≤i≤n X⊤
+i (X⊤X)−1Xi ≤ 9p + 18√2p log n + 36 log n
+n
+.
+(2.14)
+Consequently, condition (2.13) holds with high probability as long as p/n is sufficiently small.
+Proposition 2.4. Under the conditions of Lemma 2.3, we have E(Un) ≥ 0. In particular,
+E(Un) = 0 if and only if CB = 0.
+2.3
+Asymptotic distribution of the modified test statistics
+The primary goal of this section is to establish a Gaussian approximation for the modified test statistic Un.
+Following (2.3), the Gaussian analogue of Un is defined by
+Gn = tr(Z⊤PθZ) =
+n
+�
+i=1
+�
+j̸=i
+Pij,θZ⊤
+i Zj.
+The following theorem establishes a non-asymptotic upper bound of the Kolmogorov distance between the
+distribution functions of Un and its Gaussian analogue Gn. Compared with Theorem 2.1, it reveals that
+the modification of the test statistic Qn in (2.9) removes the dichotomous nature of its asymptotic null
+distribution.
+Theorem 2.5. Let q = 2 + δ, where 0 < δ ≤ 1. Assume that (2.12) holds and that
+∆q,θ =
+�n
+i=1
+�
+j̸=i |Pij,θ|q
+mq/2
+Mq +
+�n
+i=1(�
+j̸=i P 2
+ij,θ)q/2
+mq/2
+Lq → 0.
+Then, under Assumptions 2.1 and the null hypothesis, we have
+ρ(Un, Gn) ≤ Cq∆1/(2q+1)
+q,θ
++ C
+�
+1
+m
+n
+�
+i=1
+P 2
+ii
+�1/5
+→ 0.
+Similar to Lemma 2.2, we establish a similar upper bound for ∆q,θ in the following lemma.
+Lemma 2.6. Under condition (2.12), we have
+∆q,θ ≲
+� 1
+m max
+1≤i≤n Pii
+�δ/2
+Mq.
+8
+
+For α ∈ (0, 1), Proposition 2.4 and Theorem 2.5 motivate an α level test for (1.3) as follows,
+Φθ = I
+�
+Un
+ς|Pθ|F
+√2 > c1−α
+�
+,
+(2.15)
+where c1−α is the (1 − α)th quantile of the standardized Gn/√var(Gn).
+Remark 7. It is worth mentioning that the approximating distribution Gn may or may not be asymptotically
+normal.
+Let λ1(Pθ), . . . , λn(Pθ) denote the eigenvalues of the symmetric matrix Pθ.
+Being a quadratic
+functional of Gaussian random vectors {Zi}n
+i=1, Gn is distributed as a linear combination of independent
+chi-squared random variables,
+Gn
+D=
+d
+�
+k=1
+n
+�
+i=1
+λk(Σ)λi(Pθ)ηik(1) =
+d
+�
+k=1
+n
+�
+i=1
+λk(Σ)λi(Pθ){ηik(1) − 1},
+where {ηik(1)}i,k∈N are independent χ2
+1 random variables and the last equality follows from the fact that
+�n
+i=1 λi(Pθ) = �n
+i=1 Pii,θ = 0. More specifically, the Lindeberg-Feller central limit theorem and Lemma 2.3
+imply that Gn/√var(Gn) ⇒ N(0, 1) if and only if
+λ1(Σ)
+ς√m → 0.
+(2.16)
+Consequently, c1−α in (2.15) is asymptotically equal to the standard normal quantiles whenever (2.16) holds.
+When m → ∞, condition (2.16) automatically holds for arbitrary dimension d ≥ 1 as λ1(Σ) ≤ ς.
+Otherwise, (2.16) is equivalent to tr(Σ4)/ς4 → 0, which is a common assumption to ensure the asymptotic
+normality of high dimensional quadratic statistics; see, for example, Bai and Saranadasa (1996), Chen and
+Qin (2010), Cai and Ma (2013), Yao et al. (2018) and Zhang et al. (2018) among others. In particular, it
+reveals that the asymptotic null distribution of Un can be non-normal if (2.16) is violated. For example, let
+Y1, . . . , Yn ∈ Rd be i.i.d. random vectors with mean vector µY = E(Y1) and consider testing whether µY = 0.
+Assume that Σ = cov(Y1) = (Σjk)d×d has entries Σjk = ϑ + (1 − ϑ)I{j = k} for some constant ϑ ∈ (0, 1).
+Then λ1(Σ)/(ς√m) → 1 and it follows from Theorem 2.5 that
+Un
+√var(Un) =
+�n
+i=1
+�
+j̸=i Y ⊤
+i Yj
+ς√{2n(n − 1)}
+⇒ χ2
+1 − 1
+√2
+.
+The simulation study in Section 5 shows that our Gaussian multiplier bootstrap approach have a satisfactory
+performance regardless of whether Un is asymptotically normal or not.
+3
+Applications
+As mentioned in the introduction, our paradigm (1.3) is fairly general and it can be applied to many
+commonly studied hypothesis testing problems. In this section, we consider two specific examples to illustrate
+the usefulness of the proposed U type test statistic and the corresponding asymptotic distribution theory.
+3.1
+High dimensional one-way MANOVA
+Let {Yij}ni
+j=1, i = 1, . . . , K, be K ≥ 2 independent samples following the model
+Yij = µi + Vij (j = 1, . . . , ni; i = 1, . . . , K),
+9
+
+where µ1, . . . , µK ∈ Rd are unknown mean vectors of interest, {Vij}j∈N are i.i.d. d-dimensional random
+vectors with E(Vi1) = 0 and cov(Vi1) = Σ. We are interested in testing the equality of the K mean vectors,
+namely, testing the hypotheses
+H0 : µ1 = . . . = µK versus H1 : µi ̸= µl for some 1 ≤ i ̸= l ≤ K.
+Following the construction of (2.9), we propose the U type test statistic
+UnK =
+K
+�
+i=1
+Pii,K
+ni
+�
+j=1
+�
+k̸=j
+Y⊤
+ijYik +
+K
+�
+i=1
+�
+l̸=i
+Pil,K
+ni
+�
+j=1
+nl
+�
+k=1
+Y⊤
+ijYlk,
+(3.1)
+where n = �K
+i=1 ni is the total sample size,
+Pii,K =
+1
+n − 2
+� n
+ni
+− n + K − 2
+n − 1
+�
+and Pil,K =
+1
+n − 2
+� 1
+ni
++ 1
+nl
+− n + K − 2
+n − 1
+�
+.
+In the context of two sample test for mean vectors where K = 2, UnK in (3.1) reduces to
+UnK =
+�n1
+i=1
+�
+j̸=i
+�n2
+k=1
+�
+l̸=k(Y1i − Y2k)⊤(Y1j − Y2l)
+(n − 1)(n − 2)n1n2/n
+,
+which coincides with the commonly used U type test statistic (Chen and Qin, 2010).
+For each i ∈ {1, . . . , K}, let {Zij}j∈N be i.i.d. centered Gaussian random vectors with covariance matrix
+cov(Zij) = Σ. Following (2.3), the Gaussian analogue of UnK is defined by
+GnK =
+K
+�
+i=1
+Pii,K
+ni
+�
+j=1
+�
+k̸=j
+Z⊤
+ijZik +
+K
+�
+i=1
+�
+l̸=i
+Pil,K
+ni
+�
+j=1
+nl
+�
+k=1
+Z⊤
+ijZlk.
+Let nmin = min1≤l≤K nl. Since max1≤i≤n Pii ≤ n−1
+min, Assumption 2.1 holds as long as nmin → ∞. The
+following proposition establishes a non-asymptotic upper bound on the Kolmogorov distance between the
+distribution functions of UnK and GnK.
+Proposition 3.1. Let q = 2 + δ for some 0 < δ ≤ 1. Assume that nmin → ∞ and
+�
+Mq =
+max
+1≤l,l′≤K E
+����
+V⊤
+l1Vl′2
+ς
+����
+q
+< ∞, where ς = |Σ|F.
+Then, under the null hypothesis, we have
+ρ(UnK, GnK) ≤ Cq
+�
+�
+Mqn−δ/2
+min
+�1/(2q+1)
+→ 0.
+Remark 8. It is worth mentioning that both the dimension d and the number of groups K can grow with the
+total sample size n. In particular, as discussed in Remark 4, if all the K samples follow the linear process
+model in (2.6), ρ(UnK, GnK) → 0 as long as nmin → ∞.
+10
+
+3.2
+High dimensional nonparametric one-way MANOVA
+For each i ∈ {1, . . . , K}, let Fi denote the distribution function of Yi1. We consider testing whether these
+K independent samples are equally distributed, namely, testing the hypotheses
+H0 : F1 = . . . = FK versus H1 : Fi ̸= Fl for some 1 ≤ i ̸= l ≤ K.
+(3.2)
+Being fundamental and important in statistical inference, (3.2) has been extensively studied; see, for example,
+Kruskal and Wallis (1952), Akritas and Arnold (1994), Brunner and Puri (2001), Rizzo and Sz´ekely (2010)
+and Thas (2010) among many others. However, all the aforementioned works mainly focus on the traditional
+low dimensional scenario and testing (3.2) for high dimensional random vectors has been much less studied.
+In this section, we propose a new U type test statistic for (3.2) following the intuition of (2.9) and establish
+the corresponding distributional theory. In particular, our asymptotic framework is fairly general and allows
+both the dimension d and the number of groups K to grow with n.
+To begin with, for each i ∈ {1, . . . , K}, let φi(t) = E{exp(ıt⊤Yij)} denote the characteristic function of
+Yij, where ı stands for the imaginary unit. Then it is equivalent to test the hypotheses
+H0 : φ1 = . . . = φK versus H1 : φi ̸= φl for some 1 ≤ i ̸= l ≤ K.
+(3.3)
+Denote Yij(t) = exp(ıt⊤Yij). Similar to (3.1), our test statistic for (3.3) is defined by
+�UnK =
+K
+�
+i=1
+Pii,K
+ni
+�
+j=1
+�
+k̸=j
+�
+Yij(t)Yik(t)w(t)dt +
+K
+�
+i=1
+�
+l̸=i
+Pil,K
+ni
+�
+j=1
+nl
+�
+k=1
+�
+Yij(t)Ylk(t)w(t)dt,
+where w(t) ≥ 0 is a suitable weight function such that the integrals above are well defined. Discussions of
+some commonly used weight functions are given in Remark 9 below.
+Before proceeding, we first define the Gaussian analogue of �UnK under the null hypothesis that the K
+samples are equally distributed. Define the covariance function of Y11(t) as
+Σ(t, s) = E{Y11(t) − φ1(t)}{Y11(s) − φ1(s)} = φ1(t − s) − φ1(t)φ1(−s) (t, s ∈ Rd).
+Throughout this section, by Mercer’s theorem, we assume that the covariance function above admits the
+following eigendecomposition
+Σ(t, s) =
+∞
+�
+m=1
+λmϕm(t)ϕm(s) (t, s ∈ Rd),
+where λ1 ≥ λ2 ≥ . . . ≥ 0 are eigenvalues and ϕ1, ϕ2, . . ., are the corresponding eigenfunctions. We now
+apply the Karhunen–Lo`eve theorem. Let {Zijk}i,j,k∈N be independent standard normal random variables
+and define Gaussian processes
+Zij(t) =
+∞
+�
+m=1
+√λmZijmϕm(t) (t ∈ Rd).
+Then, following (2.3), the Gaussian analogue of �UnK is defined by
+�GnK =
+K
+�
+i=1
+Pii,K
+ni
+�
+j=1
+�
+k̸=j
+�
+Zij(t)Zik(t)w(t)dt +
+K
+�
+i=1
+�
+l̸=i
+Pil,K
+ni
+�
+j=1
+nl
+�
+k=1
+�
+Zij(t)Zlk(t)w(t)dt.
+11
+
+Proposition 3.2. Let q = 2 + δ for some 0 < δ ≤ 1. Assume that nmin → ∞ and
+�
+Mq = E
+�����
+�
+Rd E{Y11(t)}E0{Y12(t)}w(t)dt
+F
+�����
+q
+< ∞, where F2 =
+∞
+�
+m=1
+λ2
+m.
+Then, under the null hypothesis that these K independent samples are equally distributed, we have
+ρ(�UnK, �GnK) ≤ Cq
+�
+�
+Mqn−δ/2
+min
+�1/(2q+1)
+→ 0.
+Remark 9. It is worth mentioning that the proposed test statistic �UnK contains high dimensional integral
+over t ∈ Rd, which can be computational intractable in practice. To make �UnK well defined and facilitate
+the computation, we shall choose suitable weight function w(t) such that �UnK has a simple closed-form
+expression. In the literature, various kinds of weight functions have been proposed such as the Gaussian
+kernel function (Gretton et al., 2012), the Laplace kernel function (Gretton et al., 2012) and the energy
+kernel function (Sz´ekely et al., 2007, Rizzo and Sz´ekely, 2010). For instance, let w(t) denote the density
+function of the random vector Xκ/√η for some κ > 0, where X ∼ N(0, Id) and η ∼ χ2
+1 are independent
+(equivalently Xκ/√η is a Cauchy random variable with location parameter 0 and scale parameter κ). Then
+it is straightforward to verify that
+�
+Yij(t)Ylk(t)w(t)dt =
+�
+cos{t⊤(Yij − Ylk)}w(t)dt = exp(−κ|Yij − Ylk|),
+which is the same as the Laplace kernel function with 1/κ being its bandwidth, where | · | stands for the
+Euclidean distance. A more general result can be derived using Bochner’s Theorem, see e.g., Theorem 3.1
+of Gretton et al. (2009). Consequently, the proposed test statistic �UnK reduces to
+�UnK =
+K
+�
+i=1
+Pii,K
+Ni
+�
+j=1
+�
+k̸=j
+exp(−κ|Yij − Yik|) +
+K
+�
+i=1
+�
+l̸=i
+Pil,K
+Ni
+�
+j=1
+Nl
+�
+k=1
+exp(−κ|Yij − Ylk|),
+which is fairly convenient to compute in practice. Moreover, suitable choice of the weight function w(t) also
+facilitate the analysis of the quantities Mq and F.
+4
+Practical implementation
+In this section, we propose an unbiased estimator for ς2, which is ratio-consistent under fairly mild moment
+conditions. To begin with, since E(V ⊤
+i Vj)2 = ς2 for any i ̸= j, a natural unbiased U type estimator for ς2
+based on {Vi}n
+i=1 would be
+�ς2
+o =
+1
+n(n − 1)
+n
+�
+i=1
+�
+j̸=i
+(V ⊤
+i Vj)2.
+(4.1)
+Let ¯P1 = In − X(X⊤X)−1X⊤ = (Pij,1)n×n and �V = ¯P1Y = (�V1, . . . , �Vn)⊤. It is worth noting that directly
+substituting the residual vectors {�Vi}n
+i=1 into (4.1) yields a feasible but generally biased estimator for ς2.
+More specifically, for any i ̸= j,
+E(�V ⊤
+i �Vj)2 = ( ¯Pii,1 ¯Pjj,1 + ¯P 2
+ij,1)ς2 + ¯P 2
+ij,1E(V ⊤
+1 V1)(V ⊤
+2 V2) +
+n
+�
+k=1
+( ¯Pik,1 ¯Pjk,1)2 �
+∥E0(V ⊤
+1 V1)∥2
+2 − 2ς2�
+,
+12
+
+which reveals that (�V ⊤
+i �Vj)2 is no longer unbiased of ς2 even after proper scaling. This motivates us to
+propose a new unbiased estimator for ς2 via data-splitting, which excludes the bias terms (V ⊤
+i Vi)2 and
+(V ⊤
+i Vi)(V ⊤
+j Vj). Without loss of generality, we assume that the sample size n is even in what follows.
+1. Randomly split {1, . . . , n} into two halves A and Ac. Denote MA = {(Xi, Yi), i ∈ A} and MAc =
+{(Xi, Yi), i ∈ Ac}.
+2. For both MA and MAc, fit model (1.1) with the least squares estimates and compute
+�ΣA =
+1
+n/2 − p
+�V ⊤
+A �VA and �ΣAc =
+1
+n/2 − p
+�V ⊤
+Ac �VAc,
+where �VA and �VAc are the residual matrices of MA and MAc, respectively.
+3. Compute the estimator �ς2
+A = tr(�ΣA�ΣAc).
+Since �ΣA and �ΣAc are independent and both of them are unbiased estimators of Σ, �ς2
+A is unbiased for ς2 as
+E(�ς2
+A) = tr{E(�ΣA)E(�ΣAc)} = tr(Σ2) = ς2.
+Theorem 4.1. Assume that p/n < ϖ2 for some positive constant ϖ2 < 1/2 and that the least squares
+estimates are well defined for both MA and MAc. Then we have
+E
+����
+�ςA
+ς − 1
+����
+2
+≲ M4
+n2 + p × tr(Σ4)
+n2ς4
++ ∥E0(V ⊤
+1 ΣV1)∥2
+2
+nς4
+.
+Remark 10. The proof of Theorem 4.1 is given in Section 7.2, where a more general upper bound on
+E|�ςA/ς −1|τ is established for 1 < τ ≤ 2. Theorem 4.1 reveals that �ςA is ratio consistent under mild moment
+conditions. Suppose now {Vi}i∈N follow the linear process model (2.6) with max1≤ℓ≤L E|ξiℓ|4 ≤ C < ∞.
+Then M4 is bounded and ∥E0(V ⊤
+1 ΣV1)∥2
+2 ≲ tr(Σ4). Consequently,
+E
+����
+�ςA
+ς − 1
+����
+2
+≲ n−2 + tr(Σ4)
+nς4 .
+In this case, �ςA is ratio consistent for arbitrary dimension d ≥ 1 as long as n → ∞.
+Remark 11. There are totally
+� n
+n/2
+�
+different ways of splitting {1, . . . , n} into two halves. To reduce the
+influence of randomness of an arbitrary splitting, we can repeat the procedure independently for multiple
+times and then take the average of the resulting estimators. We refer to Fan et al. (2012) for more discussions
+about data-splitting and repeated data-splitting.
+Remark 12. Let �Σ = (n − p)−1 �V ⊤ �V . Observe that E(�V ⊤
+i �Vj) = ¯Pij,1tr(Σ). We can estimate ς2 via
+�ς2
+S =
+�n
+i,j=1 |�V ⊤
+i �Vj − ¯Pij,1tr(�Σ)|2
+(n − p + 2)(n − p − 1)
+=
+(n − p)2
+(n − p + 2)(n − p − 1)
+�
+|�Σ|2
+F − {tr(�Σ)}2
+n − p
+�
+,
+which is same as the estimator proposed in Srivastava and Fujikoshi (2006), where {Vi}n
+i=1 are assumed to
+be Gaussian random vectors. See also Bai and Saranadasa (1996). However, for non-Gaussian {Vi}n
+i=1 such
+that ∥E0(V ⊤
+1 V1)∥2
+2 ̸= 2ς2, this estimator is generally biased as
+E(�ς2
+S) − ς2 =
+�n
+i=1 ¯P 2
+ii,1
+(n − p)(n − p + 2)
+�
+∥E0(V ⊤
+1 V1)∥2
+2 − 2ς2�
+.
+In particular, the bias of �ς2
+S can diverge when ∥E0(V ⊤
+1 V1)∥2
+2 is much larger than ς2. Below we provide an
+example that typifies the diverging bias.
+13
+
+G
+G
+G
+G
+G
+G
+2
+4
+6
+8
+10
+12
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+d × 100
+G
+Split
+SF
+Oracle
+G
+G
+G
+G
+G
+G
+2
+4
+6
+8
+10
+12
+0.0
+0.5
+1.0
+1.5
+2.0
+d × 100
+Figure 1: Empirical averages of the values of |�ς/ς − 1|
+Example 4.1. Let {ξi}i∈N and {ξ′
+i}i∈N be two sequences of independent Gaussian random vectors N(0, Σ),
+where Σ = (Σij)n×n has entries Σij = ϑ|i−j| for some ϑ ∈ (0, 1). Following Wang et al. (2015), we draw
+i.i.d. innovations {Vi}n
+i=1 from a scale mixture of two independent multivariate Gaussian distributions as
+follows,
+Vi = νi × ξi + 3(1 − νi) × ξ′
+i (i = 1, . . . , n),
+where {νi}i∈N are independent Bernoulli random variables with P(νi = 1) = 0.9. A simulation study is given
+in Section 5 by setting ϑ = 0.3 and 0.7. We report in Figure 1 the average values of |�ς/ς −1| for �ςA, �ςo and �ςS,
+based on 1000 replications with the numerical setup (n, p, m) = (100, 20, 10) and d = 200, 400, 800, 1000, 1200.
+For both cases of ϑ, |�ςA/ς −1| and |�ςo/ς −1| are very close to 0, while |�ςS/ς −1| is quite large. More precisely,
+we can derive that ∥E0(V ⊤
+1 V1)∥2
+2 ≈ (18 + d)ς2.
+Substituting the ratio-consistent estimator �ς2
+A into var(Un) = 2|Pθ|2
+Fς2 yields Un/(�ςA|Pθ|F) ⇒ N(0, 2)
+under (2.16). Then, for α ∈ (0, 1), an asymptotic α level test is given by
+ΦZ = I
+�
+Un
+�ςA|Pθ|F
+√2 > z1−α
+�
+,
+(4.2)
+where z1−α is the (1 − α)th quantile of the standard normal distribution.
+5
+A simulation study
+In this section, we conduct a Monte Carlo simulation study to assess the finite sample performance of
+the proposed tests.
+In the model (1.1), we write Xi = (1, x⊤
+i )⊤ ∈ Rp to include an intercept.
+Here
+x1, . . . , xn ∈ Rp−1 are i.i.d. N(0, Ip−1) random vectors. Let m < p. For all k ∈ {1, . . . , p − m}, all entries
+of the coefficient vector Bk are i.i.d. uniform random variables in the interval (1, 2). After those Bk’s are
+generated, we keep their values throughout the simulation. Our goal is to identify the zero Bk’s by testing
+H0 : Bp−m+1 = Bp−m+2 = · · · = Bp = 0.
+14
+
+In our simulation, we set (p, m) = (20, 10), n = 100, 200 and d = 400, 800, 1200. We consider two different
+designs of the innovations (Vi): the one introduced in Example 4.1 and the one in Example 5.1 below. In
+both examples, the parameter ϑ is set to be 0.3 and 0.7.
+Example 5.1. Let {ξij}i,j∈N be i.i.d. random variables with E(ξ11) = 0 and var(ξ11) = 1. In particular, we
+consider two cases for (ξij); they are drawn from the standardized t5 distribution and the standardized χ2
+5
+distribution, respectively. For some ϑ ∈ (0, 1), we generate
+Vi = √(1 − ϑ) × ξi + √ϑ × (ξi0, ξi0, . . . , ξi0)⊤, i ∈ N.
+We shall apply a Gaussian multiplier bootstrap approach to implement our proposed test. The procedure
+is as follows.
+1. Compute the residual matrix �V = (�V1, . . . , �Vn)⊤ = ¯P1Y . Generate i.i.d. N(0, 1) random variables
+{ωij}i,j∈N and compute the bootstrap residuals V ⋆ = (V ⋆
+1 , . . . , V ⋆
+n )⊤, where
+V ⋆
+i =
+1
+√(n − p)
+n
+�
+j=1
+ωij �Vi (i = 1, . . . , n).
+2. Use V ⋆ to compute �ς⋆
+A and the bootstrap test statistic U ⋆
+n = tr(V ⋆⊤PθV ⋆).
+3. Repeat the first two steps independently B times and collect U ⋆
+nk and �ς⋆
+Ak, k = 1, . . . , B.
+4. Let �c1−α be the (1 − α)th quantile of {U ⋆
+nk/(�ς⋆
+Ak|Pθ|F
+√2)}k=1,...,B. The our test is
+ΦB = I
+�
+Un
+�ςA|Pθ|F
+√2 > �c1−α
+�
+,
+(5.1)
+and we shall reject the null hypothesis whenever ΦB = 1.
+Similar to Gn, U ⋆
+n is a quadratic functional of i.i.d. Gaussian random vectors conditional on {X, Y } and
+is distributed as a linear combination of independent chi-squared random variables. To justify the validity
+of the proposed Gaussian multiplier bootstrap approach, it suffices to bound the distance between the
+distribution functions of these two quadratic functionals, which can be established by verifying the normalized
+consistency (Xu et al., 2014) of the corresponding covariance matrix. However, this can be highly non-trivial
+in the high dimensional setting and is beyond the scope of current paper. Hence we leave it for future work.
+In our simulation, we set the bootstrap size B = 1000. As comparison, we also perform the test suggested
+in (4.2) based on the central limit theorem and the one proposed in Srivastava and Kubokawa (2013) which
+we denote by SK. For each test, we report the empirical size based on 2000 replications as displayed in
+Table 1 and Table 2. The results suggest that our proposed test by using the bootstrap procedure provides
+the best size accuracy in general as the empirical sizes are close to the nominal level α.
+For Example 4.1, both of the test by CLT and our Gaussian multiplier bootstrap method have better
+performance than the SK test since the latter is too conservative as d is large.
+As expected from our
+theoretical results, normal approximation can work reasonably well in this design.
+For Example 5.1, the Gaussian multiplier bootstrap method outperforms other two procedures in size
+accuracy for all cases. The SK test suffers from size distortion. The test by CLT inflates the size more than
+15
+
+Table 1: Empirical sizes for Example 4.1 with α = 0.05
+θ = 0.3
+θ = 0.7
+n
+d
+CLT
+GMB
+SK
+CLT
+GMB
+SK
+100
+400
+0.057
+0.047
+0.041
+0.059
+0.051
+0.036
+800
+0.049
+0.045
+0.033
+0.063
+0.056
+0.026
+1200
+0.062
+0.055
+0.021
+0.048
+0.045
+0.028
+200
+400
+0.056
+0.052
+0.042
+0.052
+0.047
+0.037
+800
+0.052
+0.049
+0.037
+0.053
+0.050
+0.033
+1200
+0.045
+0.044
+0.029
+0.050
+0.046
+0.035
+Table 2: Empirical sizes for Example 5.1 with α = 0.05
+t5
+χ2
+5
+θ
+n
+d
+CLT
+GMB
+SK
+CLT
+GMB
+SK
+0.3
+100
+400
+0.068
+0.058
+0.023
+0.083
+0.065
+0.036
+800
+0.082
+0.066
+0.023
+0.074
+0.058
+0.016
+1200
+0.082
+0.068
+0.015
+0.067
+0.053
+0.011
+200
+400
+0.073
+0.059
+0.022
+0.067
+0.054
+0.018
+800
+0.071
+0.057
+0.012
+0.074
+0.058
+0.014
+1200
+0.076
+0.059
+0.011
+0.077
+0.058
+0.011
+0.7
+100
+400
+0.074
+0.055
+0.002
+0.082
+0.062
+0.002
+800
+0.084
+0.066
+0.001
+0.085
+0.071
+0.000
+1200
+0.073
+0.057
+0.000
+0.076
+0.062
+0.001
+200
+400
+0.083
+0.067
+0.001
+0.080
+0.064
+0.000
+800
+0.068
+0.050
+0.000
+0.075
+0.062
+0.000
+1200
+0.070
+0.051
+0.001
+0.074
+0.056
+0.000
+16
+
+the GMB method, which can be explained by the fact that condition (3.1) does not hold and the CLT for
+Un fails. More specifically, for both θ = 0.3 and θ = 0.7, elementary calculations show that λ1(Σ)/ς → 1.
+As a result, (2.16) is violated as m = 10; see also the comment at the end of Section 2.2 for discussion on
+the non-normality of Un. To have more insight, we display in Figure 2 the density plots of Un/√var(Un) for
+n = 100 as well as the density of N(0, 1). As we can see from the plots, the distribution of Un/√var(Un) is
+skewed to the right for all cases, which explains the inflated sizes of the CLT test.
+More simulation studies on power comparison of these three tests are conducted in Section 7.1.
+Figure 2: Density plots of Un/√var(Un) and N(0, 1)
+−4
+−2
+0
+2
+4
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Density
+d=400
+d=800
+d=1200
+Normal
+−4
+−2
+0
+2
+4
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+−4
+−2
+0
+2
+4
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+x
+Density
+−4
+−2
+0
+2
+4
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+x
+density
+6
+Data analysis
+We apply the proposed method to two data sets. Our first dataset came from a study of the impact of the
+gut microbiome on host serum metabolome and insulin sensitivity in non-diabetic Danish adults (Pedersen
+et al., 2016). It consists of measurements of 1201 metabolites (325 serum polar metabolites and 876 serum
+molecular lipids) on 289 serum samples using mass spectrometry.
+The cleaned dataset was downloaded
+from https://bitbucket.org/hellekp/clinical-micro-meta-integration (Pedersen et al., 2018). We use this data
+set to identify insulin resistance (IR)-associated metabolites. IR was estimated by the homeostatic model
+assessment (Pedersen et al., 2016). Body mass index (BMI) is a confounder for this dataset since it is highly
+correlated with IR (Spearman’s ρ = 0.67) and is known to affect the serum metabolome. Two samples
+without IR measurement were excluded. For metabolites with zero measurements, zeros were replaced by
+half of the minimal nonzero value. Log transformation was performed to make the data more symmetrically
+distributed before analysis. The p-values associated with the three methods (GLT, GMB, and SK) are all
+17
+
+very close to zero, indicating a strong dependence between metabolites and IR. We further perform a linear
+regression analysis on each metabolite using IR and BMI as the covariates. Figure 3 (left panel) presents
+the histogram of the p-values on testing the significance of the coefficients associated with IR. We see a
+high peak close to zero, which provides strong evidence on the association between metabolites and IR. We
+further apply the Holm–Bonferroni procedure to the p-values to control the family-wise error rate at the 5%
+level, resulting in 164 discoveries.
+Our second dataset is from the study of the smoking effect on the human upper respiratory tract (Charlson
+et al., 2010). The original data set contains samples from both throat and nose microbiomes and both body
+sides. Here we focus on the throat microbiome of the left body side, which includes 60 subjects consisting of 32
+nonsmokers and 28 smokers. More precisely, the data set is presented as a 60×856 abundance table recording
+the frequencies of detected operational taxonomic units (OTUs) in the samples using the 16S metagenomics
+approach, together with a metadata table capturing the sample-level information, including the smoking
+status and sex. We transform the OTU abundance using center log-ratio (CLR) transformation after adding
+a pseudo-count of 0.5 to the zero counts. Our goal is to test the association of throat microbiomes with
+smoking status adjusting for sex. The proposed method using either the normal approximation or bootstrap
+approximation detects a strong association between the throat microbiomes with smoking status. In contrast,
+the SK method fails to discover the association.
+We further perform an OTU-wise linear regression analysis using each OTU (after the CLR transfor-
+mation) as the response and the smoking status and sex as covariates. Figure 3 (right panel) presents the
+histogram of the p-values for testing the association between each OTU and smoking status after adjusting
+sex in each linear regression. Interestingly, adjusting the multiplicity using either the Holm–Bonferroni pro-
+cedure or the BH procedure at the 5% level gives zero discovery (Zhou et al., 2021). These results suggest
+that the association between individual OTU and smoking status is weak. However, after aggregating the
+weak effects from all the OTUs, the combined effect is strong enough to be detected by the proposed method.
+Table 3: P-values of the three methods applying to the metabolomics and microbiome data sets.
+Metabolomics
+Microbiome
+CLT
+GMB
+SK
+CLT
+GMB
+SK
+p-value
+0.00
+0.00
+0.00
+9.7 × 10−6
+0.002
+0.13
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+22
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@@ -0,0 +1,1475 @@
+Estimation of thermal load on the nozzle base plate from small
+plumes at high temperature
+Kamal Khemani1, Pradeep Kumar1*, Ganesh Natarajan2
+1 Numerical Experiment Laboratory (Radiation & Fluid Flow Physics)
+Indian Institute of Technology Mandi, Himachal Pradesh, 175075, India
+2 Discipline of Mechanical Engineering,
+Indian Institute of Technology Palakkad, Palakkad, Kerala, 678557, India
+Abstract
+A numerical study is performed to estimate thermal load on the nozzle base plate, which is in the
+upstream direction to the flow, from three hot plumes of pure (CO2), (H2O) and 50-50 (%) composition of
+(CO2) and (H2O) expanding through a convergent-divergent (CD) nozzle in a quiescent medium at 1.1 bar
+pressure and 298K temperature. The base plate of the nozzle heats up due to thermal radiation, emitting
+from the hot gases in the form of plumes. The spectral radiative properties of major participating gases such
+as (CO2), (H2O) are calculated from HITEMP-2010 database. A small CD nozzle which is designed for the
+perfect expansion of air by 1D calculation with nozzle throat diameter 1.98 mm and area ratio 1.5942, is
+considered as the design of nozzle for present study [1]. All three plumes are in the under-expanded state for
+this CD nozzle and hence expands rapidly at supersonic speed as the plumes exit from the nozzle and forms
+a series of expansion and compression waves. The hot plumes emanating from the nozzle develop very high
+temperature in a small vicinity around the base plate, due to diffusion and develop very high temperature
+on the base plate. Barring this region, the maximum amount of radiative flux on base plate for these three
+plumes, i.e., CO2 plume, mixture plume and H2O plume are 4000 W/m2, 2300 W/m2 and 1300 W/m2,
+respectively and the maximum temperature developed due to these corresponding fluxes are 323 K, 312 K
+and 308 K, respectively.
+Keywords: Compressible flow, gas radiation, thermal load, underexpanded
+URL: [email protected] (Pradeep Kumar1*)
+arXiv:2301.04855v1  [physics.comp-ph]  12 Jan 2023
+
+NOMENCLATURE
+English Symbols
+c1, c2
+First and second radiation constants
+cp
+Specific heat at constant pressure
+e
+Internal energy
+h
+Enthalpy
+k
+Thermal conductivity, turbulent kinetic energy
+ˆn
+Unit normal vector
+p
+Pressure
+q
+Heat flux
+s
+Direction vector
+t
+Time
+u
+Velocity
+x
+Cartesian coordinate coordinate
+Ar
+Area ratio
+Iη
+Spectral intensity
+Ibη
+Planck function
+R
+Universal gas constant
+Y
+Species mass-fraction
+Greek Symbols
+2
+
+βη
+Spectral extinction coefficient
+ϵ
+Emissivity, turbulent dissipation rate
+η
+Wavenumber
+κη
+Spectral absorption coefficient
+µ
+Dynamic viscosity
+∇ · q
+Divergence of radiative heat flux
+Ω
+Solid angle
+φ
+Azimuthal angle
+Φ
+Scattering phase function
+ρ
+Density of fluid
+σsη
+Spectral scattering coefficient
+θ
+Polar angle
+τ
+Viscous stress tensor, transmissivity of gas, optical thickness
+Subscript
+b
+Blackbody
+c
+Conduction
+cv
+Convection
+eff
+Effective
+η
+Spectral
+g
+Gas
+k
+Turbulent kinetic energy
+r
+Radiation
+t
+Turbulent, total
+w
+Wall
+3
+
+1. Introduction
+The exhaust plume from the nozzle is a product of high temperature and high pressure gases exiting from
+the combustion chamber. These gases expand rapidly in the convergent divergent (CD) nozzle at supersonic
+velocities because of the conversion of thermal energy into kinetic energy, which generates the thrust to lift
+off the rocket. The structure of the plume is non uniform, containing different flow regimes and supersonic
+shock patterns. It appear as bright luminous flame which emits radiation in the visible, ultraviolet (UV)
+and infrared (IR) parts of the electromagnetic spectrum [2]. The major part of plume radiation comes from
+participating gases like CO2, CO and H2O which show strong emission of thermal radiation in the infrared
+region of the spectrum [3]. This heats up the base plate of the rocket and becomes the source of tracking by
+enemies in the case of missiles, fighter jets and combat aircrafts.
+Tien and Abu-Romia [4] used analytical method to estimated the amount of radiative heat flux on the
+rocket base plate from exhaust CO2 and H2O gas plume with idealised physical models. They evaluated
+apparent emissivity at base plate from semi infinite cylinder shape for H2O gas plume for a temperature
+of 2000oR, pressure 1 atm and CO2 gas plume for a temperature of 2500oR. Nelson [5] used backward
+Monte Carlo method to estimate radiative heat flux on rocket base plate from exhaust plume. They further
+studied the effect of cone angle of exhaust plume and scattering albedo on the base plate heating from plume.
+The increase in cone angle increased the heat flux on the base plate whereas increase of albedo decreased
+the heat flux. However, increase in albedo increased the searchlight emission from plume. Baek and Kim
+[6] calculated the heat load on the base plate from both exhaust plume and searchlight emission from the
+particles. They used finite volume method to solve radiative transfer equation. Tan et al. [7] conducted a
+study in which they changed the temperature distribution of plume from isothermal to non-isothermal and
+concluded that the thermal load on thebase plate reduced 2-3 times for non-isothermal plume. They also
+observed that by increasing optical thickness of medium the amount of radiative flux on the wall increased.
+Everson and Nelson [8] developed reverse Monte Carlo method to predict base plate heating from plume
+due to radiation and found that, reverse Monte Carlo was computationally more efficient than forward
+Monte Carlo method. This was owing to the fact that only the rays that strikes the target point was only
+considered. For calculations they used band models for gas spectrum and Henyey-Greenstein function for
+particle scattering. They performed reverse Monte Carlo calculations for four different cases which included
+pure scattering plume, gas only emission for main engine plume, solid rocket motor plume and a plume with
+non-uniform temperature which absorbs, emits and scatters, and finally found that majority of emission
+is due to alumina particles coming from the centre. While, H2O and Al2O2 emitted radiation from the
+4
+
+center of the plume and moreover major contribution of emission came from Al2O3 particles. Kumar and
+Ramamurthy [9] estimated radiative heat load on the rocket base plate using forward Monte-Carlo technique
+for gray conical plume with axial and radial temperature variations. They found that the radiative heat
+flux changed drastically with the change in radial temperature profile also the amount of radiative heat flux
+decreased with the increase in altitude as plume cools down faster. Similar arguments were given by Gu and
+Baek [10] as they examined radiative heat flux from WSGGM method for a solid rocket motor from which
+the thermal load was estimated by long plumes of 5 and 10 km.
+Accurate modelling of heat transfer due to radiation is very necessary for safe and efficient designing of
+rocket. Estimation of radiative properties of gases is crucial and the most important part in determining
+heat transfer due to radiation accurately. The radiative properties of participating gases can be calculated
+using some of the most popular spectral database like High Resolution Transmission Spectroscopic Molecular
+Absorption database (HITRAN) [11], Carbon-Dioxide Spectroscopic Database (CDSD) [12], High Temperature
+spectroscopic absorption parameter (HITEMP) [13] etc.
+The spectral absorption coefficients are highly
+erratic in nature containing millions of spectral lines which attain same value multiple times. This unnecessarily
+increases the computational cost required to solve the radiation transfer equation (RTE) as the line-by-line
+method considers calculation for each and every line on the spectrum and is therefore, mostly used only for
+benchmarking purposes [14].
+Many methods are proposed to reduce the computation resource requirements such as Full spectrum
+scaled and correlated k-Distribution (FSSK/FSCK) [14], Lookup based Full spectrum K-Distribution [15],
+Spectral line weight sum of gray gases [16] etc. The accuracy of the above methods is well demonstrated for
+uniform composition of gases [17, 18], however, the variation in composition of gaseous and their mixture
+poses another level of challenge and further modelling is required [19]. In order to use look up table based
+FSK method, some interpolation techniques should be adopted for the properties for current thermodynamic
+states of gases in the domain. It is evident from the above literature that only a few work is available to
+calculate the heat load on the rocket base plate, that to with fixed conical plume shape and radiative
+properties of gases. The general heat transfer applications like, combustion, rocket propulsion, gasification
+contain numerous thermodynamic states, thus it is useful to generate a database for absorption coefficient
+at different temperatures, pressures and mole-fractions. The present case is optically thin thus, the RTE
+is solved using the Planck mean absorption coefficient at different thermodynamic states, from look-up
+table. The thermal load on the nozzle base plate has been calculated from the accurate solution of flow
+and temperature fields by solving complete set of governing equation. The radiative property is obtained
+5
+
+from the HITEMP-2010 database, stored in the form of lookup table for range of thermodynamic states
+of gases and utilized during the solution of radiative transfer equation.
+The thermodynamic states for
+which data is available can directly be used. Further, the Planck mean absorption coefficient for unavailable
+thermodynamic states can easily be calculated by using multidimensional linear interpolation technique.
+The fvDOM numerical method is used for solution of RTE coupled with fluid flow using a pressure based
+compressible flow application sonicRadFoam, modified from sonicFoam application of OpenFOAM [20].
+Finally it includes the work done due to viscous forces, species transfer equation and RTE with Planck mean
+absorption-emission model.
+The manuscript is organised as section 2 describing the problem statement, and section 3 describing the
+mathematical models and governing differential equations followed by validation in section 4, results and
+discussions in section 5, and finally the present work is concluded in section 6.
+2. Problem description
+The convergent-divergent (CD) nozzle has throat diameter and an area-ratio of 1.98 mm and 1.5942,
+respectively, and the length of convergent and divergent section is 7 mm and 14 mm, respectively as shown
+in Fig. 1 which also include the buffer zone for emanating the jet in the atmosphere. The base plate is
+attached at the end and the fluid expands from a stagnation pressure and temperature of 7.11 bar and 2000
+K, respectively, to a quiescent medium at the atmospheric condition of 1 atm pressure and 298K. The present
+CD nozzle designed for perfect expansion of air by one dimensional calculation, has been considered for the
+flow of three plumes whose constituents are pure CO2, pure water vapour and 50-50(%) CO2 and H2O from
+above pressure and temperature. Initially whole domain is filled with N2 gas at 1 atm pressure and 298 K
+temperature. The following assumptions have been considered for in the present study.
+1. Reynolds-averaged Navier-Stokes assumption is used to model turbulent flow.
+2. The participating medium only absorbs or emits the thermal radiation but does not scatters.
+3. Refractive index of medium and walls are equal to one.
+4. Turbulence radiation interaction is neglected.
+5. Constant turbulent Prandtl number assumption has been used in the present study:
+6
+
+Figure 1: Schematic diagram of geometry for the calculation of the thermal load on the nozzle base plate from the hot plume
+2.1. Governing equations
+The density and temperature fluctuations must be accounted for compressible flow of a fluid along with
+velocity and pressure fluctuations. To account for these factors, the mass based averaging commonly known
+as Favre averaging [21, 22], is used to describe the flow and energy transfer for compressible turbulent fluids.
+which is defined as,
+�φ = ρφ
+ρ
+(1)
+where, ρ is the density of fluid. φ is a scalar and the averaging of density is defined below,
+ρ = 1
+T
+� T
+0
+ρ dT
+(2)
+∂ρ
+∂t + ∂ρ �ui
+∂xi
+= 0
+(3)
+∂ρ �ui
+∂t
++ ∂ρ �ui �uj
+∂xj
+= − ∂p
+∂xi
++ ∂�
+τij
+∂xj
+(4)
+7
+
+Outlet
+BasePlate
+ww
+5
+Wall
+7mm
+Inlet
+Axis
+14mm
+7 mm
+28mmwhere,
+�
+τij = µeff
+� ∂ �ui
+∂xj
++ ∂ �uj
+∂xi
+− 2
+3 δij
+∂�
+uk
+∂xk
+�
+− 2
+3ρkδij
+(5)
+where, µeff is the effective dynamic viscosity of fluid which is the summation of molecular and turbulent
+dynamic viscosity of fluid i.e (µ + µt) and the molecular viscosity of gases is given by Sutherland
+µ = As T 3/2
+T + Ts
+(6)
+As and Ts are Sutherland’s constants and depend on the type of gas and it’s molecules, and µt is the turbulent
+viscosity which is calculated as,
+µt = ρ Cµ
+k2
+ϵ
+(7)
+where k is turbulent kinetic energy and ϵ is turbulent dissipation rate and Cµ is the closure constant and
+these are modelled by two equation (k.ϵ) turbulence model and given as
+∂ρκ
+∂t + ∂ρ �ujκ
+∂xj
+=
+∂
+∂xi
+��
+µ + µt
+σκ
+� ∂κ
+∂xi
+�
++ Pκ − ρϵ
+(8)
+where, k = 1
+2
+�3
+i=1
+ρu′′
+i u′′
+i
+ρ
+is the turbulent kinetic energy, Pk is the production of kinetic energy.
+∂ρϵ
+∂t + ∂ρ �ujϵ
+∂xj
+=
+∂
+∂xi
+��
+µ + µt
+σϵ
+� ∂ϵ
+∂xi
+�
++ Cϵ1
+ϵ
+κPκ − Cϵ2ρϵ2
+κ Pκ
+(9)
+where, ϵ = ν
+�
+∂u′′
+i ∂u′′
+i
+∂xjxj
+is the turbulent disspation rate and the value of closure constants are as below. Cµ =
+0.09, σk = 1, sigmaϵ = 1.3, Cϵ1 = 1.44, C2 = 1.92 The pressure is calculated from equation of state for ideal
+gas law as,
+p = ρR �T
+(10)
+where, R is universal gas constant and T is temperature. The distribution of species is calculated by species
+transport equation as below
+∂ρi �Yi
+∂t
++ ∂ρi �ui �Yi
+∂xi
+=
+∂
+∂xi
+�
+−ρµeff
+∂ �Yi
+∂xi
+�
+(11)
+where, Yi is species mass-fraction and is given as,
+Yi = ρi
+ρ
+(12)
+8
+
+The distribution of temperature field is calculated from the energy equation as below
+∂ρ �E
+∂t
++ ∂ρ �uj �E
+∂xj
++ ∂ �ujp
+∂xj
+= − ∂ �qj
+∂xj
++ ∂ �uj �
+τij
+∂xj
+(13)
+where, E is the total energy which includes internal energy e, kinetic energy K and turbulent kinetic energy
+k. The heat flux is defined as,
+qj = −cpµeff
+Pr
+∂T
+∂xi
++ �qr
+(14)
+cp depends on temperature and are taken from JANAF table of thermodynamics and given as below,
+cp = R((((a4T + a3)T + a2)T + a1)T + a0)
+(15)
+a0, a1, a2, a3, a4 are constants of polynomial,
+qr =
+� ∞
+0
+�
+4π
+Iη(ˆs) |ˆn · ˆs| dΩ dη
+(16)
+where qr is the radiative heat flux which can be calculated on the wall, ˆn is the surface normal vector,
+∂qr/∂xj is the divergence of radiative heat flux and can be calculated as,
+∇ · q =
+� ∞
+0
+κη
+�
+4πIbη −
+�
+4π
+Iη dη
+�
+dη
+or
+∇ · q =
+� ∞
+0
+κη (4πIbη − Gη) dη
+(17)
+where η is the wavenumber, Ibη is the Planck function and κη is the spectral absorption coefficient, Gη is
+spectral irradiation, Iη(ˆs) is the intensity field which is obtained by solving the radiative transfer equation
+(RTE) as explained in the subsequent paragraph. The above equations are subject to boundary conditions
+as given in table 1.
+The intensity field in equation 17 is obtained by solving the spectral radiative transfer equation (s-RTE)
+for absorbing emitting (not scattering) medium as,
+dIη
+ds = κηIbη − κηIη
+(18)
+9
+
+the above equation is subjected to boundary condition,
+Iη(rw, ˆs) = ϵwηIbη(rw) + 1 − ϵwη
+π
+�
+ˆn·ˆs>0
+Iη(rw, ˆs) |ˆn · ˆs| dΩ
+(ˆn · ˆs < 0)
+(19)
+where, ϵwη is the spectral wall emissivity, Iη is the spectral intensity along ˆsi, Ibη is the Planck function, κη
+is the spectral absorption coefficient, η is the wavenumber, and Ω is the solid angle. The length scale of the
+current problem is very small, i.e., the optical length τ = κηL << 1, this means that the absorptivity of
+the medium is far less than 1, therefore, the most of the radiation energy will escape the medium without
+getting absorbed. Thus, the radiative source term (Eq. 17)
+� ∞
+0
+κη4πIbηdη <<
+� ∞
+0
+κηGηdη
+The radiative source term Eq. 17 becomes
+∇ · q =
+� ∞
+0
+κη4πIbηdη
+� ∞
+0
+Ibηdη
+� ∞
+0
+Ibηdη = 4κpσT 4
+where, κp is the Planck mean absorption coefficient. Therefore, the solution for the present case can be
+Table 1: Boundary conditions for plume with thermal radiation simulation
+Fields
+Inlet
+Outlet
+Wall
+Pressure (p)
+totalPressure
+Po = P + 0.5 ρ U 2
+Po = 7.11 bar
+fixedValue
+P=1 atm
+zeroGradient
+∇P = 0
+Velocity (U)
+pressureInletOutletVelocity
+Po = P + 0.5 ρ U 2
+inflow: U = (0,0,0)
+outflow: ∇U = 0
+inletOutlet
+inflow: U = (0,0,0)
+outflow: ∇U = 0
+noSlip
+U = (0,0,0)
+Temperature (T)
+fixedValue T = 2000 K
+zeroGradient
+∇T = 0
+qc + qr = 0 [23]
+Species (x)
+fixedValue x = 1
+for pure H2O plume
+zeroGradient
+∇x = 0
+zeroGradient
+∇x = 0
+10
+
+obtained by Planck Mean absorption coefficient based radiation property model. Thus, the RTE becomes,
+dIp
+ds = κp · (Ib − Ip) ,
+(20)
+with boundary conditions,
+Ip = ϵwIb + 1 − ϵw
+π
+�
+ˆn·ˆs>0
+Ip |ˆn · ˆs| dΩ
+(ˆn · ˆs < 0)
+(21)
+The Planck mean absorption coefficients are calculated for the range of thermodynamic states of gases in
+the certain intervals as mentioned in ([18]) and stored in the form of lookup table. Furthermore, interpolation
+techniques are employed to calculate the absorption coefficient which are not available in the lookup table.
+The radiative heat transfer, work done due to viscous forces and species transport models have been
+added into the existing application ”sonicFOAM” of the OpenFOAM and named as ”radSonicFOAM”. The
+algorithms of the new application is described below and has been extensively verified and validated as
+explained in the subsequent section and finally, has been used for the estimating the thermal load on the
+nozzle base plate.
+2.2. Numerical Procedure and solution algorithm for solving plume flow with radiation
+The above mass, momentum, species, energy and radiation transfer equation are discretized using finite
+volume method [24]. Further second order upwind scheme is used for the face value interpolation and final set
+of algebraic equation is solved iteratively, by the SIMPLE algorithm till the residual for mass, momentum,
+species, energy and radiation reaches to 10−5 level. The algorithm of above solution method is stated below,
+1. Initialize pressure, velocity, species and temperature field.
+2. Solve mass, momentum, species transport and energy equations without radiation till convergence.
+3. Using converged field, initialize intensity field.
+4. Calculate Planck mean absorption coefficient from the converged field of temperature, pressure and
+mole-fraction of species using the Planck mean look-up table and solve RTE till convergence.
+5. Compute divergence of radiative heat flux.
+6. Update the temperature field with radiation sink term.
+11
+
+7. Repeat 2 to 6 until all the fields reach at steady state. furthermore, the flow diagram of the above
+algorithm is shown in fig 2.
+3. Verification and validation studies
+The above mathematical modelling and solution algorithm are verified in three steps
+• The calculated radiative properties are verified.
+• The incompressible flow solution is verified with the published result.
+• The radiative heat flux on the base plate is verified from the assumed shape of the plume in the sections
+below.
+3.1. Verification of Planck mean absorption coefficient of pure H2O and CO2
+The Planck mean absorption coefficients obtained for H2O and CO2 for various temperatures from
+HITEMP-2010 using in-house C++ code [25, 26, 27], match with good agreement from Chu et al. [28]as
+in Figure 3.
+The Planck mean absorption coefficient of H2O decreases exponentially with increase in
+temperature, whereas it first increases up to a temperature of 750 K then decreases till 2000 K for CO2.
+The Planck mean absorption coefficient of H2O is higher than CO2 at lower temperatures, however, this is
+opposite for higher temperature. This difference, decreases with increase in temperatures of compressible
+flow.
+3.2. Validation of compressible flow field
+Darwish et al. [1] have designed a convergent divergent (C-D) nozzle using one dimensional flow isentropic
+relations for perfect expansion conditions for air. The designed C-D nozzle has an exit diameter of 2.5 mm
+and throat diameter of 1.98 mm, thus the area ratio Ar = 1.5942. The schematic diagram of C-D nozzle
+with buffer section where flow eminates is shown in Fig. 1. They simulated the flow using OpenFOAM for
+axisymmetric geometry for this nozzle along with the buffer zone. They further performed experiments to
+visualize the flow using shadow-graphic technique. In the present study, we will be using the same nozzle
+to validate pressure based compressible flow application ”sonicFOAM”. The air is allowed to expand from
+7.1 atm pressure and 288 K to a quiescent medium at 1 atm pressure. The boundary conditions used for
+this case is same as given in Table 1 except the temperature at the inlet is 288 K and the walls are at
+zeroGradient (∇ · T = 0) boundary condition for temperature.
+The flow is simulated for axisymmetric
+12
+
+Figure 2: Flow chart for the solution of high temperature and pressure plume flow with radiation
+13
+
+Start
+Converged p,T,u,x without radiation
+Time loop
+Initialize intensity field
+Obtain absorption coefficient from look-up
+table and solve RTE to obtain V.q
+No
+Solve mass, momentum, species and energy
+equation with V.q to obtain T
+Converged ?
+t-t+△t
+Yes
+Reached steady
+state ?
+No
+Yes
+EndFigure 3: Variation of Planck mean absorption coefficient of pure H2O and CO2 with different temperature at 1 bar pressure
+geometry by creating a wedge of angle θ = 2.5o of unit cell in θ direction. It contains 38,400 cells and the
+distance of first cell center from the wall is maintained at y+ ≈ 30. The standard k − ϵ model has been
+used to model turbulence. Pressure-implicit split algorithm (PISO) is used to solve the governing flow and
+energy equations. Thermophysical and transport properties for air are taken constant as, Cp = 1005kJ/kgK,
+γ = 1.4, µ = 1.789 × 10−5PaS and Pr = 0.7. The time step used for the present simulation is 10−8 s. The
+simulation has been performed for 7ms. The pressure and Mach number variation along centerline of nozzle
+along with the results reported by Darwish et al. [1], are plotted in Figure. 4 and 5, respectively. The present
+results are in good agreement with the literature results . There are no shocks or sudden discontinuities
+inside the nozzle as the flow is perfectly expanded inside the nozzle. Since, the nozzle is designed with 1D
+isentropic calculations and the present simulations are performed for 2D axisymmetric case, there is deviation
+from 1D isentropic flow. Thus the small expansion and compression waves are formed which create small
+diamond pattern that can be seen in profiles of pressure and Mach number along the axis of geometry.
+14
+
+50
+Present Calculations
+45
+B-
+Chuetal.
+40
+35
+30
+co.
+25
+20
+15
+H.O
+10
+5
+500
+1000
+1500
+2000
+T (K)Figure 4: Variation of pressure along the axis of geometry
+Figure 5: Variation of Mach number along the axis of
+geometry
+3.3. Verification of Rocket base plate heating with assumed plume shape
+The axisymmetric approximation for RTE has been tested for rocket base plate heating problem from fixed
+plume shape. The plume is assumed as connical shape with half cone angle of 15o having non-dimensional
+length Z/R = 50 as shown in Figure. 6. The temperature of the plume Tp is uniform. The environment is
+assumed to be cold and non-participating i.e., κ = 0 and the absorption coefficient of plume is κ = 0.5 m−1.
+Figure 7, shows the radiative heat flux at the base plate from exhaust plume by both axisymmetric
+and three-dimensional calculations. The result obtained from 3D simulations is in good agreement with
+the results published by Baek and Kim [6], whereas axisymmetric simulation result of radiative transfer
+equations is very far from the result published. This requires reformulation of axisymmetric approximation
+of radiative heat transfer in OpenFOAM. Therefore, a three dimensional geometry has been used for the
+further simulations as shown in Figure. 8a.
+4. Results and discussion
+The heating of rocket base plate by thermal radiation from different plumes made of constituents of
+pure H2O plume, CO2 plume and 50%- 50% mixture of H2O and CO2 plume are studied numerically
+with OpenFOAM, an open source CFD package.
+The present simulations are carried out on a full 3D
+geometry with a pressure based compressible flow application sonicRadFoam. It has additional features
+than existing sonicFoam, like work done due to viscous forces in energy equation, species transport equation
+and emission/absorption due to gaseous radiation. The Planck mean radiation heat transfer model with
+15
+
+PresentSimulations
+B-Darwishetal
+6
+5
+Pressure (bar)
+3
+2
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+Nozzle Central Axis (mm)2
+1.8
+1.6
+1.4
+MachNumber
+1.2
+0.8
+0.6
+PresentSimulations
+0.4
+B-- Darwish etal.
+0.2
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+NozzleCentralAxis(mm)Figure 6: Geometry of conical plume
+Figure 7: Variation of non-dimensional radiative heat flux
+by axisymmetric and 3D RTE solution at the base plate
+from assumed plume shape
+(a)
+(b)
+Figure 8: (a) Three dimensional geometry and meshing for simulation of plumes with radiation; (b)Cross sectional view of three
+dimensional geometry
+multidimensional linear interpolation technique for properties is also incorporated to perform radiation heat
+transfer calculations due to validity of optically thin approximation.
+The results of thermal load on the rocket base plate from exhaust plume of three different constituents,
+i.e., pure H2O plume, pure CO2 plume and 50%-50% mixture of H2O and CO2 plume are presented in the
+subsequent sections.
+4.1. Pure H2O plume
+Pure H2O plume is formed by the combustion of pure H2 with liquid oxidizer LOX. The resulting
+product contains mole-fraction of H2O (x = 1) which emanates from the nozzle in the form of the plume.
+Initially the medium is filled with N2, and the H2O expands from 7.11 bar and 2000 K to a quiescent medium
+16
+
+Plumeemission
+Environment
+ExhaustPlume
+R
+15°
+BasePlate
+Z0.8
+3Dcalculation
+0.7
+Axisymmetriccalculation
+BaekandKim
+0.6
+0.5
+-10/b
+0.4
+0.3
+0.2
+ACAAC
+0.1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+5.5
+6
+r/Rof 1 atm and 288 K Temperature.
+The pressure remains constant in the convergent part of the nozzle, however it suddenly decreases at the
+throat and the divergent part of the nozzle as shown in Figure.10a. The exit pressure at nozzle for H2O
+plume is slightly higher than the pressure of quiescent medium, i.e., 1.4 bar, this essentially means that the
+flow is underexpanded [29]. Due to this underexpansion scenario, there forms the series of strong expansion
+and compression waves (oblique shocks) which evolves from the lip of the nozzle, as pressure tries to adjust
+itself against medium pressure. The shock which evolve from the lip of the nozzle is in the shape of barrel so
+it can be called as ”barrel shock” and a Mach disc appears after the shock which is formed due to singular
+reflection. The pressure variation in divergent part of the nozzle enables the temperature reduction as shown
+in Figure. 10b. Similar effect of pressure variation in the plume is seen on the temperature variation as
+well. Thus, the temperature variation in the divergent part of the nozzle and in the plume enables the
+heat transfer mechanism. However, heat transfer mechanism does not occur in the convergent part of the
+nozzle, due to uniform temperature inside the convergent part of the nozzle. The physical quantities such
+as pressure, temperature and velocity or Mach number vary rapidly across the shock. The shock pattern
+is in the form of a diamond also known as diamond flow structure. The pressure varies between 1.4 bar to
+0.58 bar across the shock as in Figure 10a. Similarly, the temperature also varies sharply, i.e., upto 300 K
+in the region from 23 mm to 25 mm as it can be seen from temperature profile across the axis in Fig. 10b.
+The temperature first decreases due to expansion of gases and then it increases due to compression wave
+and this pattern continues till pressure comes in equilibrium with the buffer zone pressure. After 40 mm,
+flow stabilizes, as the pressure of fluid at that point becomes same as that of medium pressure. The trend
+is opposite for Mach number as gas expands, the velocity of the flow increases and the maximum value of
+Mach number achieved in this case is 2.25. The contour of Mach number and its profile along the centerline
+distribution are shown in Fig. 9c and 10c, respectively. In the near field region of plume, after the inviscid
+core central region, there forms a mixing layer where viscosity effects are felt and the primary species (H2O)
+starts getting mixed with the atmospheric species (N2) and forms shear layer. The region just outside the
+nozzle where species starts mixing is called as entrainment region of the plume. Moving downstream in the
+direction of the flow, mixing layer widens for H2O being lighter molecule (molecular weight=18), as in Fig.
+9d. In the far field region, i.e., the region after the shock, species mixes completely till the centerline as it
+can be seen in the H2O and N2 profiles along the centerline. Fig. 10d shows the profiles of H2O and N2
+along the axis and contours of H2O and N2 are represented in the Figs. 9d and 9e, respectively.
+The pressure, temperature and the species concentration of H2O contours constitutes the thermodynamic
+17
+
+state of the H2O vapour, and Planck mean absorption coefficient of H2O has been accessed through lookup
+tables and its contours is shown in Figure.11a. It has very high value in the convergent portion of the nozzle
+due to very high pressure and decreases as pressure decreases in the divergent section of the nozzle and its
+value is further reduced in the plume. The absorption coefficient is zero where only N2 gas is available plume
+being very small thickness, the reabsorption does not occur and the major emission comes from the core of
+the plume, as emission and absorption are almost same in the shear layer as the divergence of radiative heat
+flux is almost zero in the shear layer and the regions of zero absorption coefficient as shown in Figure. 11b.
+One thing to notice that the range of divergence of radiative flux is negative to positive, both the positive
+value of the divergence of radiative flux reveals radiative sink term while negative value tells radiative source
+term, Thus, radiation is heating the gas inside the divergent part of the nozzle while it is cooling the plume.
+Further the energy is transferred by radiation mode of heat transfer to other region without any change.
+The high temperature plume after emanating from the nozzle gets diffused and develop very high flux
+and temperature in a very narrow region around the lip of the nozzle on the base plate. Barring this region,
+the base plate receives the radiation energy emanating from the shear layer of plume. The radiative heat
+flux on the base plate is shown in Fig. 12a, baring some region near to the lip of the nozzle. The maximum
+value of radiative heat flux is 1300 W/m2 and it decreases along the radial direction as the view factor of
+plume decreases. Similarly, the temperature developed due to this radiative flux is shown in Fig. 12b. The
+maximum value which base plate attains due to radiation energy is 308 K and it decreases in the similar
+manner of radiation flux along the radius.
+4.2. Pure CO2 plume
+Although generation of pure CO2 plume is not very much realistic, however, for the theoretical understanding
+the simulation has been performed for pure CO2 plume. The simulations for pure CO2 are performed by
+supplying pure CO2 (x = 1) at the inlet of the nozzle and rest conditions are kept same as that of H2O
+plume. This is also the case of underexpansion, so pressure at the lip of the nozzle varies from 1.4 bar to
+0.5 bar across the shocks. There is a formation of Mach disc at the end of the first shock. The contour of
+pressure and distribution of pressure along the centerline is shown in Fig. 13a and 14a, respectively. The
+temperature drop across the shock in the CO2 plume is less compared to H2O, however there is more drop
+in temperature towards the end of the plume.
+The temperature contour is shown in fig. 13b. The variation of temperature across the first shock is not
+much drastic in comparison to H2O plume, also this plume cools faster than H2O plume i.e., minimum
+temperature of this plume is 1200 K at the ends while it is 1350 K for H2O plume. The Mach number
+18
+
+(a)
+(b)
+(c)
+(d)
+(e)
+Figure 9: The contours of (a) Pressure (b) Temperature (c) Mach number (d) H2O (e) N2 for pure H2O plume
+19
+
+p(N/m2)
+7.11e+05
+6.50e+5
+6.00e+5
+5.50e+5
+5.00e+5
+4.50e+5
+4.00e+5
+3.50e+5
+3.00e+5
+2.50e+5
+2.00e+5
+1.50e+5
+1.00e+5
+5.76e+04T(K)
+2000
+1900
+1800
+1700
+1600
+1500
+1400
+1300
+1200
+1100
+1000
+006
+800
+700
+600
+500
+400
+295Ma
+2.22
+2.00
+1.80
+1.60
+1.40
+1.20
+1.00
+0.80
+0.60
+0.40
+0.20
+0.00H20
+1.00
+0.90
+0.80
+0.70
+0.60
+0.50
+0.40
+0.30
+0.20
+0.10
+0.00N2
+1.00
+0.90
+0.80
+0.70
+0.60
+0.50
+0.40
+0.30
+0.20
+0.10
+0.00(a)
+(b)
+(c)
+(d)
+Figure 10: Profile of (a) Pressure (b) Temperature (c) Mach number (d) Species along the centerline for pure H2O plume
+20
+
+8
+6
+5
+Pressure (bar)
+3
+2
+1
+0
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+Nozzlecentralaxis(mm)2100
+2000
+1900
+1800
+Temperature (K)
+1700
+1600
+1500
+1400
+1300
+1200
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+Nozzle central axis (mm)2.5
+2.25
+2
+1.75
+Machnumber
+1.5
+1.25
+0.75
+0.5
+0.25
+0
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+Nozzlecentralaxis(mm)0.9
+0.8
+0.7
+0.6
+H,O
+0.5
+N.
+0.4
+0.3
+0.2
+0.1
+25
+35
+40
+45
+50
+Nozzlecentral axis (mm)(a)
+(b)
+Figure 11: The contours of (a) Absorption coefficient (b) Divergence of radiative heat flux for pure H2O plume
+(a)
+(b)
+Figure 12: Profile of (a) Radiative heat flux (b) Temperature along the radius of the base plate for pure H2O plume
+21
+
+k(m-1)
+5.60
+5.00
+4.50
+4.00
+3.50
+3.00
+2.50
+2.00
+1.50
+1.00
+0.50
+0.00(gw/m)b- A
+2.63e+06
+2.00e+6
+1.50e+6
+-1.00e+6
+5.00e+5
+0.00
+-5.00e+5
+-1.00e+6
+-1.50e+6
+-2.00e+6
+-2.50e+6
+-3.00e+6
+-3.76e+061400
+1200
+1000
+q,(W/m")
+800
+600
+400
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+Radiusofbaseplate(mm)310
+309
+308
+307
+Temperature (K)
+306
+305
+304
+303
+302
+301
+300
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+Radiusof baseplate (mm)(a)
+(b)
+(c)
+(d)
+(e)
+Figure 13: Contours of (a) Pressure (b) Temperature (c) Mach number (d) CO2 (e) N2 for pure CO2 plume
+22
+
+p(N/m2)
+7.11e+05
+6.50e+5
+6.00e+5
+5.50e+5
+5.00e+5
+4.50e+5
+4.00e+5
+3.50e+5
+3.00e+5
+2.50e+5
+2.00e+5
+1.50e+5
+1.00e+5
+5.84e+04T (K)
+2000
+1900
+1800
+1700
+1600
+1500
+1400
+1300
+1200
+1100
+1000
+900
+800
+700
+600
+500
+400
+292Ma
+2.22
+2.00
+1.80
+1.60
+1.40
+1.20
+1.00
+0.80
+0.60
+0.40
+0.20
+0.00CO2
+1.00
+0.90
+0.80
+0.70
+0.60
+0.50
+0.40
+0.30
+0.20
+0.10
+0.00N2
+1.00
+ 0.90
+ 0.80
+0.70
+ 0.60
+0.50
+0.40
+0.30
+0.20
+0.10
+0.00(a)
+(b)
+(c)
+(d)
+Figure 14: Profile of (a) Pressure (b) Temperature (c) Mach number (d) Species for pure CO2 plume
+23
+
+8
+6
+5
+Pressure (bar)
+4
+3
+2
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+Nozzlecentralaxis(mm)2100
+2000
+1900
+1800
+1700
+Temperature(
+1600
+1500
+1400
+1300
+1200
+1100
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+Nozzlecentralaxis(mm)2.5
+2.25
+2
+1.75
+Machnumber
+1.5
+1.25
+0.75
+0.5
+0.25
+0
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+Nozzle central axis (mm)0.9
+0.8
+0.7
+Species distribution
+0.6
+CO2
+0.5
+N2
+0.4
+0.3
+0.2
+0.1
+10
+15
+25
+30
+35
+40
+45
+50
+Nozzlecentralaxis(mm)(a)
+(b)
+Figure 15: Contours of (a) Absorption coefficient (b) Divergence of radiative heat flux for pure CO2 plume
+(a)
+(b)
+Figure 16: Profile of (a) Radiative heat flux (b) Temperature along the radius of base plate for pure CO2 plume
+24
+
+k(m-1)
+30
+28
+26
+24
+22
+20
+18
+16
+14
+12
+10
+8
+6
+4
+2
+0(gw/m)b- A
+1.79e+07
+1.60e+7
+1.40e+7
+1.20e+7
+1.00e+7
+8.00e+6
+6.00e+6
+4.00e+6
+2.00e+6
+0.00
+-2.00e+6
+-4.00e+6
+-6.00e+6
+-8.00e+6
+-1.00e+7
+-1.20e+7
+-1.42e+074500
+4000
+3500
+3000
+2500
+2000
+1500
+1000
+500
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+Radius ofbaseplate(mm)325
+320
+315
+Temperature (K)
+310
+305
+300
+295
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+Radiusofbaseplate(mm)(a)
+(b)
+Figure 17: Profile of (a) Radiative heat flux (b) Temperature on base plate along the radius of base plate for 50-50% mixture
+of CO2 − H2O plume
+contour and its distribution along the centerline are shown in Fig. 13c and 14c, respectively. The diffusion
+of CO2 in N2 is less in comparison to H2O due to higher molecular weight of CO2 (44) compared to H2O
+(18) as shown in Fig. 14d. The contours of CO2 and N2 mole fraction are shown in Fig. 13d and 13e,
+respectively.
+The absorption coefficient distribution by considering Planck mean absorption coefficient for CO2 plume
+is shown in Fig.
+15a.
+Its value is almost zero everywhere except in the core of the plume and in the
+shear layer. As the absorption coefficient of CO2 is higher in the shear layer compared to H2O plume, the
+radiative heat flux on the rocket base plate is also higher, i.e., around 4000 W/m2 as shown in Fig. 16a. The
+corresponding temperature distribution on the base plate is shown in Fig. 16b, having a maximum value of
+323 K, barring the diffusion region.
+4.3. Mixture plume (50 % H2O and 50 % CO2)
+The combustion of hydrocarbon fuel with liquid oxidizer (LOX) gives 50-50% mixture of CO2 and H2O.
+Thus, for the present problem, we supply 50-50% mixture of both CO2 and H2O at the inlet of the nozzle
+and other conditions are kept same as previous cases for the simulation of this plume. This is also a case
+of underexpanded plume. The temperature variation along the centerline at the end of the buffer section is
+somewhat the average of both pure CO2 and H2O plume.
+The radiative transfer calculations are performed to determine the heat flux on the base plate from
+25
+
+2400
+2200
+2000
+1800
+1600
+(w/M)"b
+1400
+1200
+1000
+800
+600
+400
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+Radiusofbaseplate(mm)314
+313
+312
+311
+310
+309
+308
+307
+306
+305
+304
+303
+302
+301
+300
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+Radius ofbaseplate(mm)CO2 − H2O plume. The maximum radiative heat flux on the base plate is 2300 W/m2 (Fig. 17a) and it
+decays with the radius of the base plate. The corresponding profiles of the temperature on the base plate
+is shown in Fig. 17b. It is noted that the flux and temperature profiles for CO2 and mixture plume are
+exponential decaying with radius, while it is almost linear for H2O. This is owing to the fact that, high
+diffusion of H2O causes more spreading of H2O and this emission from H2O has high view factor, while this
+is not the case with CO2 and mixture plume.
+5. Conclusions
+The thermal load calculation on the base plate of nozzle from exhaust plume is performed in OpenFOAM.
+The ability of pressure based compressible flow application, ”sonicFoam” is tested to capture the flow fields
+for air expanding in a convergent divergent nozzle. The stagnation pressure and temperature at the inlet
+of the nozzle are 7.11 bar and 288 K due to which flow expands and achieves Mach 2.1 at the exit of the
+nozzle. The resulting pressure and Mach number variation at the centerline matches well with the standard
+published results.
+The same nozzle is then used with elevated stagnation temperature of 2000 K and same pressure at inlet,
+to estimate the heat load on base plate for three different plumes namely, pure H2O plume, pure CO2 plume
+and mixture plume. The ”sonicFoam” application is then modified by incorporating the work done due to
+viscous forces, species transport equation and finally clubbed with the RTE solver fvDOM along with Planck
+mean absorption emission model and named as ”radSonicFOAM”. All three plumes exit from the nozzle at
+underexpanded flow conditions, where exit pressure is higher than the back pressure. The expansion waves
+start from the lip of the nozzle due to which the temperature decreases as flow exit from the nozzle and
+Mach number increases to a maximum value of 2.25.
+The maximum amount of heat load in the present study due to thermal radiation on base plate is from
+pure CO2 plume, i.e., 4000 W/m2 due to the high value of absorption coefficient, barring the diffusion zone.
+This flux heats up the base plate and its temperature rises upto 323 K, followed by mixture plume, which
+receives maximum radiative heat flux of 2300 W/m2 and the corresponding rise in temperature is 312 K.
+For pure H2O plume, the heat flux is least, i.e., 1300 W/m2 with temperature rise of 308 K. For different
+plumes the variation in flux is different and this is mostly due to the difference in the absorption coefficient
+of the gases. Further, their molecular weights are also different, due to which there is difference in the flow
+field of the gases and also the different nature of flux and temperature variations on the nozzle base plate.
+Due to small length scale, the current case falls in optically thin regime, thus, the Planck mean absorption
+26
+
+model provides the satisfactory results, however, Planck mean absorption model may not be useful for other
+cases with big length scale. Therefore, The full spectrum radiative properties models are needed with the
+properties for all thermodynamic states existing in the plume. Furthermore, the solid fuel emanates particles
+which contribute most of the radiative thermal load on the nozzle base plate, therefore the current radiation
+heat transfer feature needs to further enhance by including the scattering model.
+References
+[1] M. Darwish, L. Orazi, D. Angeli, Simulation and analysis of the jet flow patterns from supersonic nozzles
+of laser cutting using openfoam, The International Journal of Advanced Manufacturing Technology
+102 (9) (2019) 3229–3242.
+[2] F. Simmons,
+Rocket exhaust plume phenomenology,
+American
+Institute
+of Aeronautics
+and
+Astronautics, Inc., 2000.
+[3] M. F. Modest, Radiative heat transfer, Academic press, 2013.
+[4] C. Tien, M. Abu-Romia, A method of calculating rocket plume radiation to the base region, Journal of
+Spacecraft and Rockets 1 (4) (1964) 433–435.
+[5] H. Nelson, Backward monte carlo modeling for rocket plume base heating, Journal of Thermophysics
+and Heat Transfer 6 (3) (1992) 556–558.
+[6] S. W. Baek, M. Y. Kim, Analysis of radiative heating of a rocket plume base with the finite-volume
+method, International Journal of Heat and Mass Transfer 40 (7) (1997) 1501–1508.
+[7] H.-P. Tan, Y. Shuai, S.-K. Dong, Analysis of rocket plume base heating by using backward monte-carlo
+method, Journal of thermophysics and heat transfer 19 (1) (2005) 125–127.
+[8] J. Everson, H. Nelson, Rocket plume radiation base heating by reverse monte carlo simulation, Journal
+of thermophysics and heat transfer 7 (4) (1993) 717–723.
+[9] K. R. S. Sunil Kumar, Prediction of radiation from plumes, considering spatial temperature variations,
+Heat Transfer Engineering 21 (1) (2000) 55–73.
+[10] B. Gu, M. Y. Kim, S. W. Baek, Analysis of the ir signature and radiative base heating from a supersonic
+solid rocket exhaust plume, International Journal of Aeronautical and Space Sciences 20 (2) (2019) 423–
+432.
+27
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+[11] L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown,
+A. Campargue, J.-P. Champion, et al., The hitran 2008 molecular spectroscopic database, Journal of
+Quantitative Spectroscopy and Radiative Transfer 110 (9-10) (2009) 533–572.
+[12] S. Tashkun, V. Perevalov, Cdsd-4000: High-resolution, high-temperature carbon dioxide spectroscopic
+databank, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (9) (2011) 1403–1410.
+[13] L. Rothman, I. Gordon, R. Barber, H. Dothe, R. Gamache, A. Goldman, V. Perevalov, S. Tashkun,
+J. Tennyson, Hitemp, the high-temperature molecular spectroscopic database, Journal of Quantitative
+Spectroscopy and Radiative Transfer 111 (15) (2010) 2139–2150.
+[14] M. F. Modest, H. Zhang, The full-spectrum correlated-k distribution for thermal radiation from
+molecular gas-particulate mixtures, Journal of heat transfer 124 (1) (2002) 30–38.
+[15] C. Wang, W. Ge, M. F. Modest, B. He, A full-spectrum k-distribution look-up table for radiative transfer
+in nonhomogeneous gaseous media, Journal of Quantitative Spectroscopy and Radiative Transfer 168
+(2016) 46–56.
+[16] V. P. Solovjov, B. W. Webb, Slw modeling of radiative transfer in multicomponent gas mixtures, Journal
+of Quantitative Spectroscopy and Radiative Transfer 65 (4) (2000) 655–672.
+[17] S. Parvatikar, K. Khemani, P. Kumar, Benchmark test cases for non-gray radiative heat transfer
+calculation using fsk look-up table, in: Journal of Physics: Conference Series, Vol. 2116, IOP Publishing,
+2021, p. 012066.
+[18] K. Khemani, S. Parvatikar, P. Kumar, Radiative heat transfer calculations using full spectrum k-
+distribution method for benchmark test cases, S¯adhan¯a 48 (1) (2023) 1–18.
+[19] K. Khemani, P. Kumar, Radiative heat transfer calculation for mixture of gases using full spectrum
+k-distribution method, in: Journal of Physics: Conference Series, Vol. 2116, IOP Publishing, 2021, p.
+012065.
+[20] OpenCFD, OpenFOAM - The Open Source CFD Toolbox - User’s Guide, OpenCFD Ltd. (11 Apr.
+2007).
+[21] D. C. Wilcox, et al., Turbulence modeling for CFD, Vol. 2, DCW industries La Canada, CA, 1998.
+[22] T. F. Edgar, R. M. Felder, J. McKenna, R. W. Rousseau, S. I. Sandier, R. C. Seagrave, Bird, stewart
+and lightfoot: Transport phenomena.
+28
+
+[23] G. Chanakya, P. Kumar, Investigation of thermal adiabatic boundary condition on semitransparent
+wall in combined radiation and natural convection, International Journal for Computational Methods
+in Engineering Science and Mechanics 23 (4) (2022) 349–366.
+[24] P. Kumar, Radiative heat transfer in a participating gray medium and its interaction with fluid flow,
+Ph.D. thesis, Indian Institute of Technology Kanpur (January 2009).
+[25] N. Bartwal, G. Chanakya, P. Kumar, Calculation of non-gray radiation transmissivity, absorptivity and
+absorption coefficient of water vapour from hitemp-2010 database at high temperature, in: 6th Asian
+Symposium on Computational Heat Transfer and Fluid Flow, IITM, Chennai, India, 2017.
+[26] N. Bartwal, P. Kumar, Calculation of non-gray radiation transmissivity, absorptivity of carbon-dioxide
+from hitemp2010 database at high temperature, in: 24th National & 2nd International ISHMT-ASTFE
+Heat and Mass Transfer Conference, BITS-Pilani, Hyderabad, India, 2017.
+[27] N. Bartwal, P. Kumar, Calculation of non-gray radiation absorptivity and absorption coefficient of
+mixture of gases from hitemp-2010 database, in: International Heat Transfer Conference Digital Library,
+Begel House Inc., 2018.
+[28] H. Chu, M. Gu, H. Zhou, F. Liu, Calculations of narrow-band transimissities and the planck mean
+absorption coefficients of real gases using line-by-line and statistical narrow-band models, Frontiers in
+Energy 8 (1) (2014) 41–48.
+[29] E. Franquet, V. Perrier, S. Gibout, P. Bruel, Free underexpanded jets in a quiescent medium: A review,
+Progress in Aerospace Sciences 77 (2015) 25–53.
+29
+

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+Charge collection and efficiency measurements of the
+TJ-Monopix2 DMAPS in 180 nm CMOS technology
+Christian Bespin,𝑎,∗ Ivan Caicedo,𝑎 Jochen Dingfelder,𝑎 Tomasz Hemperek,𝑎,𝑒 Toko
+Hirono,𝑎,𝑏 Fabian Hügging,𝑎 Hans Krüger,𝑎 Konstantinos Moustakas,𝑎,𝑐 Heinz
+Pernegger,𝑑 Petra Riedler,𝑑 Lars Schall,𝑎 Walter Snoeys𝑑 and Norbert Wermes𝑎
+𝑎Physikalisches Institut, Universität Bonn,
+Nußallee 12, Bonn, Germany
+𝑏Deutsches Elektronen-Synchrotron (DESY)
+Notkestaße. 85, Hamburg, Germany
+𝑐Paul Scherrer Institut,
+Forschungsstrasse 111, Villingen, Switzerland
+𝑑CERN
+Espl. des Particules 1, Meyrin, Switzerland
+𝑒DECTRIS AG
+Täfernweg 1, Baden-Dättwil, Switzerland
+E-mail: [email protected]
+Monolithic CMOS pixel detectors have emerged as competitive contenders in the field of high-
+energy particle physics detectors. The use of commercial processes offers high-volume production
+of such detectors. A series of prototypes has been designed in a 180 nm Tower CMOS process with
+depletion of the sensor material and a column-drain readout architecture. The latest iteration, TJ-
+Monopix2, features a large 2 × 2 cm2 matrix consisting of 512 × 512 pixels with 33.04 µm pitch. A
+small collection electrode design aims at low power consumption and low noise while the radiation
+tolerance for high-energy particle detector applications needs extra attention. With a goal to reach
+radiation tolerance to levels of NIEL damage of 1 × 1015 1 MeV neq/cm2, a modification of the
+standard process has been implemented by adding a low-dosed n-type silicon implant across the
+pixel in order to allow for homogeneous depletion of the sensor volume. Recent lab measurements
+and beam tests were conducted for unirradiated modules to study electrical characteristics and hit
+detection efficiency.
+10th International Workshop on Semiconductor Pixel Detectors for Particles and Imaging (Pixel2022)
+12-16 December 2022
+Santa Fe, New Mexico, USA
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.13638v1  [physics.ins-det]  31 Jan 2023
+
+TJ-Monopix2: DMAPS in 180 nm CMOS technology
+Christian Bespin
+1.
+Introduction
+In recent years, advances in CMOS technologies have fueled the development of a new gener-
+ation of monolithic active pixel sensors (MAPS) with fast readout and high radiation tolerance by
+depleting the charge sensitive volume [1]. These depleted MAPS (DMAPS) devices are therefore an
+interesting candidate for high-energy particle physics experiments with high radiation environments
+and high particle rate. Depletion is achieved by either using high-voltage add-ons in the CMOS
+technology and/or high resistivity substrates. The increasing availability of these features in com-
+mercial CMOS processes could combine the features of the detector concept with possibly faster
+and cheaper production than common hybrid pixel detectors for the mentioned purposes. The idea
+behind and measurements results from one of multiple DMAPS prototypes, TJ-Monopix2 [2, 3],
+will be presented in the following.
+2.
+Design of TJ-Monopix2
+TJ-Monopix2 is the latest DMAPS prototype from the TJ-Monopix development line which is
+based on the ALPIDE pixel detector developed for the ALICE ITS upgrade [4]. It is fabricated in
+the same 180 nm commercial CMOS process provided by Tower Semiconductor1. A modification
+of the process used for ALPIDE has been implemented to increase the radiation tolerance to
+levels ≥ 1 × 1015 neq cm−2 by adding a low dose n-type implant for homogeneous growth of the
+depletion zone with applied bias voltage. In measurements on first prototypes with this modification,
+a drop in hit detection efficiency was observed after irradiation [5, 6]. This could be improved
+significantly by adding a gap in the n-type blanket or a deep p-type implant in the pixel corners
+to shape the electrical field towards the collection electrode [7].
+The cross-sections of these
+two sensor designs is shown in fig. 1. Additionally, chips have been produced on Czochralski
+P+ SUBSTRATE
+P- EPITAXIAL LAYER
+COLLECTION N-WELL
+LOW DOSE N-TYPE IMPLANT
+DEEP PWELL
+PWELL
+PWELL
+NWELL
+DEEP PWELL
+PWELL
+PWELL
+NWELL
+(a) Modification with gap in low dose n-type implant be-
+low pixel electronics.
+P+ SUBSTRATE
+P- EPITAXIAL LAYER
+COLLECTION N-WELL
+LOW DOSE N-TYPE IMPLANT
+DEEP PWELL
+PWELL
+PWELL
+NWELL
+DEEP PWELL
+PWELL
+PWELL
+NWELL
+EXTRA DEEP PWELL
+EXTRA DEEP PWELL
+(b) Modification with continuous n-type implant and deep
+p-type implant below pixel electronics.
+Figure 1: Cross-section variants of modified sensor process for TJ-Monopix2.
+silicon to increase the available depletable volume compared to the thickness of the epitaxial layer
+(O(10 µm)). Measurements on Czochralski silicon chips in TJ-Monopix1 showed a further increase
+in hit detection efficiency after irradiation [8].
+TJ-Monopix2 follows a small collection electrode approach with a pixel capacitance of
+about 3 fF. The pixels of size 33 × 33 µm2 are read out using an established synchronous column-
+drain technique from the FE-I3 readout chip [9]. Further changes from the predecessor TJ-Monopix1
+1https://towersemi.com
+2
+
+TJ-Monopix2: DMAPS in 180 nm CMOS technology
+Christian Bespin
+include an improved front-end design, a new pixel masking scheme and a 3-bit DAC for local
+threshold tuning. With these changes the threshold is expected to be reduced by a factor of 3 while
+improving the threshold dispersion and noise behavior.
+The digital periphery contains logic for register configuration, data handling and LVDS output
+drivers. Slow control is done via a command protocol and decoder that was taken from the RD53B
+readout chip [10]. Both pixel and register data is 8b10b encoded in a frame-based data stream
+which allows operating the chip with four differential data lines.
+3.
+Injection-based threshold and noise measurements
+Initial tests have been performed in a laboratory setup to measure the threshold and noise
+performance of TJ-Monopix2. All of these values are extracted from injecting different amounts of
+charge into the pixel a given number of times and recording the amount of registered hits 𝑛hits. The
+response function is a smeared step function of the form
+𝑛hits(𝑞) = 1
+2 · 𝑛injections ·
+�
+erf
+�𝑞 − 𝑞thr
+𝜎
+√
+2
+�
++ 1
+�
+(1)
+with 𝑞 the injected charge amount, 𝑛injections the number of consecutive injections of 𝑞 and 𝑞thr the
+charge at the threshold. 𝜎 denotes the gaussian smearing of the step function which is defined
+as the electronic noise of the pixel. The threshold is defined as the charge at which 50 % of the
+injected hits are recorded from the pixel. Histogramming the individual thresholds from each pixel
+leads to the distribution shown in fig. 2a. The distribution has a mean value of 164 e− and a width
+50
+100
+150
+200
+250
+300
+Threshold / e
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+# of pixels
+Fit results:
+= 164.3 e
+= 13.0 e
+Threshold distribution for enabled pixels
+1
+2
+3
+4
+5
+6
+7
+TDAC
+(a) Threshold distribution before in-pixel threshold tuning.
+50
+100
+150
+200
+250
+300
+Threshold / e
+0
+2000
+4000
+6000
+8000
+# of pixels
+Fit results:
+= 163.2 e
+= 2.7 e
+Threshold distribution for enabled pixels
+1
+2
+3
+4
+5
+6
+7
+TDAC
+(b) Threshold distribution after in-pixel threshold tuning.
+Figure 2: Threshold distribution of TJ-Monopix2 before and after adjusting the in-pixel threshold DAC to
+lower the dispersion. Color-coded is the value of the DAC setting for every pixel.
+of 13 e− which is defined as the threshold dispersion. By adjusting the threshold DAC in each pixel
+in order to even out the deviations from the target threshold, the threshold dispersion can be reduced
+significantly. The resulting distribution after this so-called tuning process is shown in fig. 2b. While
+the mean threshold stays basically the same, the dispersion could be reduced by a factor of almost 5.
+Both the mean threshold and threshold dispersion are significantly lower than in TJ-Monopix1,
+where losses in hit detection efficiency could be observed due to too large thresholds [8].
+3
+
+TJ-Monopix2: DMAPS in 180 nm CMOS technology
+Christian Bespin
+0
+5
+10
+15
+20
+25
+30
+35
+ENC / e
+0
+100
+200
+300
+400
+500
+600
+700
+800
+# of pixels
+Fit results:
+= 11.8 e
+= 1.5 e
+Noise distribution for enabled pixels
+(a) Noise distribution of TJ-Monopix1 with noticeable tail
+towards larger values.
+2
+4
+6
+8
+10
+ENC / e
+0
+200
+400
+600
+800
+1000
+1200
+1400
+# of pixels
+Fit results:
+= 5.6 e
+= 0.6 e
+Noise distribution for enabled pixels
+(b) Noise distribution of TJ-Monopix2. There is no observ-
+able tail and lower noise overall.
+Figure 3: Noise distribution of TJ-Monopix1 (left) and TJ-Monopix2 (right) for comparison.
+The corresponding histogram of the electronic noise is depicted in fig. 3. As a comparison,
+the distribution from the predecessor TJ-Monopix1 is included, where a large tail towards higher
+values was observed that led to a high operational threshold in order to limit the amount of noisy
+pixels. It can be seen, that this tail is largely removed with slight changes to the analog front-end,
+which in turn lowers the threshold for a regular operation of the chip.
+4.
+Hit detection efficiency measurements
+Two different pixel variations were investigated regarding their hit detection efficiency, that
+will be presented in the following – a DC-coupled, more standard design which makes up most part
+of the matrix and an AC-coupled investigative design realized in only a few columns of the matrix.
+While the former was measured in more detail, some first results of the latter are included as well.
+4.1 Standard DC-coupled pixel flavor
+First measurements to determine the hit detection efficiency have been performed in a 5 GeV
+electron beam at the DESY II testbeam facility at DESY, Hamburg [11]. Three unirradiated modules
+were tested with different sensor geometries: two chips with 30 µm thick epitaxial silicon and the
+two geometries depicted in fig. 1 as well as one chip built on 300 µm Czochralski silicon with a gap
+in the low dose n-type implant (see fig. 1a). It should be noted that the different substrate materials
+offer different sensor thicknesses and therefore charge-sensitive volume depending on the depletion.
+The measurements are not targeting a comparison between different types of silicon.
+Figures 4a and 4b show the recorded cluster charge for a chip with epitaxial layer and with
+Czochralski substrate. It can be observed that the collected charge is about 25 % larger in the Cz
+sample, because the depletion depth is only limited by the thickness of the sensor (300 µm) which
+is by far not fully depleted, but more depleted than the 30 µm thick epitaxial layer in the other chip.
+The average cluster size is significantly larger in the Cz sample as well which results in a high
+spatial resolution due to charge-weighted clustering. The cluster size distributions for the same
+samples as above are depicted in figs. 4c and 4d. While cluster size 1 is predominant in the epitaxial
+4
+
+TJ-Monopix2: DMAPS in 180 nm CMOS technology
+Christian Bespin
+2000
+4000
+6000
+8000
+10000
+12000
+Cluster charge / e
+0
+1000
+2000
+3000
+4000
+5000
+#
+MPV charge: 2579 e
+Data
+(a) Cluster charge distribution for an epitaxial silicon chip.
+2000
+4000
+6000
+8000
+10000
+12000
+Cluster charge / e
+0
+500
+1000
+1500
+2000
+2500
+#
+MPV charge: 3235 e
+Data
+(b) Cluster charge for a Czochralski silicon chip.
+1
+2
+3
+4
+5
+6
+Cluster size
+0
+20000
+40000
+60000
+80000
+100000
+#
+Mean cluster size: 1.51
+(c) Cluster size distribution for an epitaxial silicon chip.
+1
+2
+3
+4
+5
+6
+Cluster size
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+#
+Mean cluster size: 1.95
+(d) Cluster size distribution for a Czochralski silicon chip.
+Figure 4: Cluster charge and size distributions for a chip with 30 µm epitaxial silicon (left) and 300 µm
+Czochralski silicon (right) at −6 V bias voltage. The latter can be depleted further than 30 µm resulting in a
+larger cluster charge and size. Both chips were operated at a threshold of 200 e−.
+sample, the Cz sample has mainly clusters of size 2. The corresponding average cluster size is 1.55
+and 1.95, respectively.
+Taking the pointing resolution of the beam telescope into account, an
+intrinsic spatial resolution of 8.6 µm could be achieved in a Czochralski silicon sample.
+The hit detection efficiency was measured with a beam telescope with six Mimosa26 planes
+and a FE-I4 time reference plane which are all connected to a trigger logic unit to synchronize
+individual detector hits time-wise. The efficiency for all three modules is shown in fig. 5 where
+the result for every pixel was mapped onto a two by two pixel cell to increase the statistics to see
+possible effects or efficiency losses within a single pixel. All samples were running at a threshold of
+about 200 e− and achieve a hit detection efficiency around 99.80 % with slight deviations within the
+error (estimated < 0.1 %). There are no losses observable in the pixel corners or between pixels.
+4.2 AC-coupled pixel flavor
+Another pixel variation with different analog front-end was tested as well to determine its
+performance in a particle beam. In this design the (positive) bias voltage is applied via a diode on
+the top side of the chip and connected to the charge collection n-well. To avoid breakdown of the
+front-end electronics due to the high voltage (≤ 50 V) on that well, the input signal is AC coupled to
+5
+
+TJ-Monopix2: DMAPS in 180 nm CMOS technology
+Christian Bespin
+0
+10
+20
+30
+40
+50
+60
+column [ m]
+0
+10
+20
+30
+40
+50
+60
+row [ m]
+Region 1 (Center): In-pixel efficiency
+for DUT
+90.00
+91.25
+92.50
+93.75
+95.00
+96.25
+97.50
+98.75
+100.00
+(a) (99.80 ± 0.10) % efficiency for a chip built on epitaxial
+silicon with gap in n-layer modification from fig. 1a.
+0
+10
+20
+30
+40
+50
+60
+column [ m]
+0
+10
+20
+30
+40
+50
+60
+row [ m]
+Region 1 (Center): In-pixel efficiency
+for DUT
+90.00
+91.25
+92.50
+93.75
+95.00
+96.25
+97.50
+98.75
+100.00
+(b) (99.79 ± 0.10) % efficiency for a chip built on Cz sili-
+con with gap in n-layer modification from fig. 1a.
+0
+10
+20
+30
+40
+50
+60
+column [ m]
+0
+10
+20
+30
+40
+50
+60
+row [ m]
+Region 1 (Center): In-pixel efficiency
+for DUT
+90.00
+91.25
+92.50
+93.75
+95.00
+96.25
+97.50
+98.75
+100.00
+(c) (99.85 ± 0.10) % efficiency for a chip built on epitaxial
+silicon with additional p-well modification from fig. 1b.
+Figure 5: Hit detection efficiencies for different substrate materials with different thicknesses and sensor
+geometries. Results were mapped onto a 2 x 2 pixel array for higher statistics and in-pixel resolution. The
+chips were operated with −6 V bias voltage and at a 200 e− threshold.
+the amplifier. This approach can potentially deplete the substrate further due to the higher voltage
+than what can be applied in the standard pixel design. The hit detection efficiency was measured
+for different bias voltages and is shown in fig. 6. At 5 V the efficiency is already above 99 % and
+reaches the same value as for the DC coupled pixel flavor of 99.85 % at or before 25 V bias voltage.
+This is, taking the slightly higher threshold into account, in agreement with the expectation that
+there should be no noticeable difference in hit detection efficiency before irradiation between the
+two pixel flavors. The larger applicable bias voltage could prove superior after irradiation to achieve
+more depletion and therefore higher charge signal.
+5.
+Conclusion
+In summary, the performance of TJ-Monopix2 shows a significant improvement in threshold
+value and dispersion compared to TJ-Monopix1, although the former is higher than its design value
+6
+
+TJ-Monopix2: DMAPS in 180 nm CMOS technology
+Christian Bespin
+0
+10
+20
+30
+40
+50
+60
+column [ m]
+0
+10
+20
+30
+40
+50
+60
+row [ m]
+Region 1 (HV CASC): In-pixel efficiency
+for DUT
+90.00
+91.25
+92.50
+93.75
+95.00
+96.25
+97.50
+98.75
+100.00
+(a) (99.21 ± 0.10) % efficiency of an AC-coupled pixel
+flavor at 5 V bias voltage and 250 e− threshold.
+0
+10
+20
+30
+40
+50
+60
+column [ m]
+0
+10
+20
+30
+40
+50
+60
+row [ m]
+Region 1 (HV CASC): In-pixel efficiency
+for DUT
+90.00
+91.25
+92.50
+93.75
+95.00
+96.25
+97.50
+98.75
+100.00
+(b) (99.85 ± 0.10) % efficiency of an AC-coupled pixel
+flavor at 25 V bias voltage and 200 e− threshold.
+Figure 6: Hit detection efficiency of an AC-coupled pixel flavor at (6a) 5 V and (6b) 25 V bias voltage.
+(120 e−). With the measured amount of charge in the sensor the threshold is still small enough
+to detect a majority of hits even from large clusters before irradiation. The large signal compared
+to the small pixel leads to a large cluster size and therefore high spatial resolution, where chips
+on Czochralski substrate perform slightly better due to the larger sensor volume. For the tested
+sensor materials with different thicknesses and sensor geometries, the hit detection efficiency is
+around 99.8 % or better in all cases. The modified front-end version with bias applied on the charge
+collection node achieves similar values for the hit detection efficiency while providing a larger
+headroom in bias voltage to achieve efficient performance after radiation damage. The results for
+irradiated modules will be presented in a forthcoming publication.
+Acknowledgments
+This project has received funding from the Deutsche Forschungsgemeinschaft DFG (grant WE
+976/4-1), the German Federal Ministry of Education and Research BMBF (grant 05H15PDCA9),
+and the European Union’s Horizon 2020 research and innovation program under grant agreements
+no. 675587 (Maria Sklodowska-Curie ITN STREAM), 654168 (AIDA-2020), and 101004761
+(AIDAinnova). The measurements leading to these results have been performed at the Test Beam
+Facility at DESY Hamburg (Germany), a member of the Helmholtz Association (HGF).
+References
+[1] I. Perić, A novel monolithic pixelated particle detector implemented in high-voltage CMOS
+technology, Nuclear Instruments and Methods in Physics Research Section A: Accelerators,
+Spectrometers, Detectors and Associated Equipment 582 (2007) 876.
+[2] K. Moustakas, M. Barbero, I. Berdalovic, C. Bespin, P. Breugnon, I. Caicedo et al., CMOS
+monolithic pixel sensors based on the column-drain architecture for the HL-LHC upgrade,
+7
+
+TJ-Monopix2: DMAPS in 180 nm CMOS technology
+Christian Bespin
+Nuclear Instruments and Methods in Physics Research Section A: Accelerators,
+Spectrometers, Detectors and Associated Equipment 936 (2019) 604.
+[3] Konstantinos Moustakas, Design and Development of Depleted Monolithic Active Pixel
+Sensors with Small Collection Electrode for High-Radiation Applications, Ph.D. thesis,
+Rheinische Friedrich-Wilhelms-Universität Bonn, Sept., 2021.
+[4] M. Mager, ALPIDE, the Monolithic Active Pixel Sensor for the ALICE ITS upgrade, Nuclear
+Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers,
+Detectors and Associated Equipment 824 (2016) 434 .
+[5] I. Caicedo, M. Barbero, P. Barrillon, I. Berdalovic, S. Bhat, C. Bespin et al., The Monopix
+chips: depleted monolithic active pixel sensors with a column-drain read-out architecture for
+the ATLAS Inner Tracker upgrade, Journal of Instrumentation 14 (2019) C06006.
+[6] C. Bespin, M. Barbero, P. Barrillon, I. Berdalovic, S. Bhat, P. Breugnon et al., DMAPS
+Monopix developments in large and small electrode designs, Nuclear Instruments and
+Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and
+Associated Equipment 978 (2020) 164460.
+[7] M. Dyndal, V. Dao, P. Allport, I.A. Tortajada, M. Barbero, S. Bhat et al., Mini-MALTA:
+radiation hard pixel designs for small-electrode monolithic CMOS sensors for the High
+Luminosity LHC, Journal of Instrumentation 15 (2020) P02005.
+[8] C. Bespin, I. Berdalovic, I. Caicedo, R. Cardella, J. Dingfelder, L. Flores et al., Development
+and characterization of a DMAPS chip in TowerJazz 180 nm technology for high radiation
+environments, Nuclear Instruments and Methods in Physics Research Section A:
+Accelerators, Spectrometers, Detectors and Associated Equipment 1040 (2022) 167189.
+[9] I. Perić, L. Blanquart, G. Comes, P. Denes, K. Einsweiler, P. Fischer et al., The FEI3 readout
+chip for the ATLAS pixel detector, Nuclear Instruments and Methods in Physics Research
+Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 565 (2006)
+178.
+[10] RD53 collaboration, RD53B Manual, Tech. Rep. CERN-RD53-PUB-19-002, CERN, Geneva
+(Mar, 2019).
+[11] R. Diener, J. Dreyling-Eschweiler, H. Ehrlichmann, I. Gregor, U. Kötz, U. Krämer et al., The
+DESY II test beam facility, Nuclear Instruments and Methods in Physics Research Section A:
+Accelerators, Spectrometers, Detectors and Associated Equipment 922 (2019) 265.
+8
+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf,len=275
+page_content='Charge collection and efficiency measurements of the  TJ-Monopix2 DMAPS in 180 nm CMOS technology  Christian Bespin,𝑎,∗ Ivan Caicedo,𝑎 Jochen Dingfelder,𝑎 Tomasz Hemperek,𝑎,𝑒 Toko  Hirono,𝑎,𝑏 Fabian Hügging,𝑎 Hans Krüger,𝑎 Konstantinos Moustakas,𝑎,𝑐 Heinz  Pernegger,𝑑 Petra Riedler,𝑑 Lars Schall,𝑎 Walter Snoeys𝑑 and Norbert Wermes𝑎  𝑎Physikalisches Institut, Universität Bonn,  Nußallee 12, Bonn, Germany  𝑏Deutsches Elektronen-Synchrotron (DESY)  Notkestaße.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 85, Hamburg, Germany  𝑐Paul Scherrer Institut,  Forschungsstrasse 111, Villingen, Switzerland  𝑑CERN  Espl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' des Particules 1, Meyrin, Switzerland  𝑒DECTRIS AG  Täfernweg 1, Baden-Dättwil, Switzerland  E-mail: bespin@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='uni-bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='de  Monolithic CMOS pixel detectors have emerged as competitive contenders in the field of high-  energy particle physics detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The use of commercial processes offers high-volume production  of such detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' A series of prototypes has been designed in a 180 nm Tower CMOS process with  depletion of the sensor material and a column-drain readout architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The latest iteration, TJ-  Monopix2, features a large 2 × 2 cm2 matrix consisting of 512 × 512 pixels with 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='04 µm pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' A  small collection electrode design aims at low power consumption and low noise while the radiation  tolerance for high-energy particle detector applications needs extra attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' With a goal to reach  radiation tolerance to levels of NIEL damage of 1 × 1015 1 MeV neq/cm2, a modification of the  standard process has been implemented by adding a low-dosed n-type silicon implant across the  pixel in order to allow for homogeneous depletion of the sensor volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' Recent lab measurements  and beam tests were conducted for unirradiated modules to study electrical characteristics and hit  detection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  10th International Workshop on Semiconductor Pixel Detectors for Particles and Imaging (Pixel2022)  12-16 December 2022  Santa Fe, New Mexico, USA  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='13638v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='ins-det]  31 Jan 2023  TJ-Monopix2: DMAPS in 180 nm CMOS technology  Christian Bespin  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Introduction  In recent years, advances in CMOS technologies have fueled the development of a new gener-  ation of monolithic active pixel sensors (MAPS) with fast readout and high radiation tolerance by  depleting the charge sensitive volume [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' These depleted MAPS (DMAPS) devices are therefore an  interesting candidate for high-energy particle physics experiments with high radiation environments  and high particle rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' Depletion is achieved by either using high-voltage add-ons in the CMOS  technology and/or high resistivity substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The increasing availability of these features in com-  mercial CMOS processes could combine the features of the detector concept with possibly faster  and cheaper production than common hybrid pixel detectors for the mentioned purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The idea  behind and measurements results from one of multiple DMAPS prototypes, TJ-Monopix2 [2, 3],  will be presented in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Design of TJ-Monopix2  TJ-Monopix2 is the latest DMAPS prototype from the TJ-Monopix development line which is  based on the ALPIDE pixel detector developed for the ALICE ITS upgrade [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' It is fabricated in  the same 180 nm commercial CMOS process provided by Tower Semiconductor1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' A modification  of the process used for ALPIDE has been implemented to increase the radiation tolerance to  levels ≥ 1 × 1015 neq cm−2 by adding a low dose n-type implant for homogeneous growth of the  depletion zone with applied bias voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' In measurements on first prototypes with this modification,  a drop in hit detection efficiency was observed after irradiation [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' This could be improved  significantly by adding a gap in the n-type blanket or a deep p-type implant in the pixel corners  to shape the electrical field towards the collection electrode [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  The cross-sections of these  two sensor designs is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' Additionally, chips have been produced on Czochralski  P+ SUBSTRATE  P- EPITAXIAL LAYER  COLLECTION N-WELL  LOW DOSE N-TYPE IMPLANT  DEEP PWELL  PWELL  PWELL  NWELL  DEEP PWELL  PWELL  PWELL  NWELL  (a) Modification with gap in low dose n-type implant be-  low pixel electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  P+ SUBSTRATE  P- EPITAXIAL LAYER  COLLECTION N-WELL  LOW DOSE N-TYPE IMPLANT  DEEP PWELL  PWELL  PWELL  NWELL  DEEP PWELL  PWELL  PWELL  NWELL  EXTRA DEEP PWELL  EXTRA DEEP PWELL  (b) Modification with continuous n-type implant and deep  p-type implant below pixel electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Figure 1: Cross-section variants of modified sensor process for TJ-Monopix2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  silicon to increase the available depletable volume compared to the thickness of the epitaxial layer  (O(10 µm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' Measurements on Czochralski silicon chips in TJ-Monopix1 showed a further increase  in hit detection efficiency after irradiation [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  TJ-Monopix2 follows a small collection electrode approach with a pixel capacitance of  about 3 fF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The pixels of size 33 × 33 µm2 are read out using an established synchronous column-  drain technique from the FE-I3 readout chip [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' Further changes from the predecessor TJ-Monopix1  1https://towersemi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='com  2  TJ-Monopix2: DMAPS in 180 nm CMOS technology  Christian Bespin  include an improved front-end design, a new pixel masking scheme and a 3-bit DAC for local  threshold tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' With these changes the threshold is expected to be reduced by a factor of 3 while  improving the threshold dispersion and noise behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  The digital periphery contains logic for register configuration, data handling and LVDS output  drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' Slow control is done via a command protocol and decoder that was taken from the RD53B  readout chip [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' Both pixel and register data is 8b10b encoded in a frame-based data stream  which allows operating the chip with four differential data lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Injection-based threshold and noise measurements  Initial tests have been performed in a laboratory setup to measure the threshold and noise  performance of TJ-Monopix2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' All of these values are extracted from injecting different amounts of  charge into the pixel a given number of times and recording the amount of registered hits 𝑛hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The  response function is a smeared step function of the form  𝑛hits(𝑞) = 1  2 · 𝑛injections ·  �  erf  �𝑞 − 𝑞thr  𝜎  √  2  �  + 1  �  (1)  with 𝑞 the injected charge amount, 𝑛injections the number of consecutive injections of 𝑞 and 𝑞thr the  charge at the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 𝜎 denotes the gaussian smearing of the step function which is defined  as the electronic noise of the pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The threshold is defined as the charge at which 50 % of the  injected hits are recorded from the pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' Histogramming the individual thresholds from each pixel  leads to the distribution shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The distribution has a mean value of 164 e− and a width  50  100  150  200  250  300  Threshold / e  0  250  500  750  1000  1250  1500  1750  # of pixels  Fit results:  = 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='3 e  = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='0 e  Threshold distribution for enabled pixels  1  2  3  4  5  6  7  TDAC  (a) Threshold distribution before in-pixel threshold tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  50  100  150  200  250  300  Threshold / e  0  2000  4000  6000  8000  # of pixels  Fit results:  = 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='2 e  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='7 e  Threshold distribution for enabled pixels  1  2  3  4  5  6  7  TDAC  (b) Threshold distribution after in-pixel threshold tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Figure 2: Threshold distribution of TJ-Monopix2 before and after adjusting the in-pixel threshold DAC to  lower the dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' Color-coded is the value of the DAC setting for every pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  of 13 e− which is defined as the threshold dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' By adjusting the threshold DAC in each pixel  in order to even out the deviations from the target threshold, the threshold dispersion can be reduced  significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The resulting distribution after this so-called tuning process is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' While  the mean threshold stays basically the same, the dispersion could be reduced by a factor of almost 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Both the mean threshold and threshold dispersion are significantly lower than in TJ-Monopix1,  where losses in hit detection efficiency could be observed due to too large thresholds [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  3  TJ-Monopix2: DMAPS in 180 nm CMOS technology  Christian Bespin  0  5  10  15  20  25  30  35  ENC / e  0  100  200  300  400  500  600  700  800  # of pixels  Fit results:  = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='8 e  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='5 e  Noise distribution for enabled pixels  (a) Noise distribution of TJ-Monopix1 with noticeable tail  towards larger values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  2  4  6  8  10  ENC / e  0  200  400  600  800  1000  1200  1400  # of pixels  Fit results:  = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='6 e  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='6 e  Noise distribution for enabled pixels  (b) Noise distribution of TJ-Monopix2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' There is no observ-  able tail and lower noise overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Figure 3: Noise distribution of TJ-Monopix1 (left) and TJ-Monopix2 (right) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  The corresponding histogram of the electronic noise is depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' As a comparison,  the distribution from the predecessor TJ-Monopix1 is included, where a large tail towards higher  values was observed that led to a high operational threshold in order to limit the amount of noisy  pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' It can be seen, that this tail is largely removed with slight changes to the analog front-end,  which in turn lowers the threshold for a regular operation of the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Hit detection efficiency measurements  Two different pixel variations were investigated regarding their hit detection efficiency, that  will be presented in the following – a DC-coupled, more standard design which makes up most part  of the matrix and an AC-coupled investigative design realized in only a few columns of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  While the former was measured in more detail, some first results of the latter are included as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='1 Standard DC-coupled pixel flavor  First measurements to determine the hit detection efficiency have been performed in a 5 GeV  electron beam at the DESY II testbeam facility at DESY, Hamburg [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' Three unirradiated modules  were tested with different sensor geometries: two chips with 30 µm thick epitaxial silicon and the  two geometries depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 1 as well as one chip built on 300 µm Czochralski silicon with a gap  in the low dose n-type implant (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' It should be noted that the different substrate materials  offer different sensor thicknesses and therefore charge-sensitive volume depending on the depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  The measurements are not targeting a comparison between different types of silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Figures 4a and 4b show the recorded cluster charge for a chip with epitaxial layer and with  Czochralski substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' It can be observed that the collected charge is about 25 % larger in the Cz  sample, because the depletion depth is only limited by the thickness of the sensor (300 µm) which  is by far not fully depleted, but more depleted than the 30 µm thick epitaxial layer in the other chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  The average cluster size is significantly larger in the Cz sample as well which results in a high  spatial resolution due to charge-weighted clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The cluster size distributions for the same  samples as above are depicted in figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 4c and 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' While cluster size 1 is predominant in the epitaxial  4  TJ-Monopix2: DMAPS in 180 nm CMOS technology  Christian Bespin  2000  4000  6000  8000  10000  12000  Cluster charge / e  0  1000  2000  3000  4000  5000  #  MPV charge: 2579 e  Data  (a) Cluster charge distribution for an epitaxial silicon chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  2000  4000  6000  8000  10000  12000  Cluster charge / e  0  500  1000  1500  2000  2500  #  MPV charge: 3235 e  Data  (b) Cluster charge for a Czochralski silicon chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  1  2  3  4  5  6  Cluster size  0  20000  40000  60000  80000  100000  #  Mean cluster size: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='51  (c) Cluster size distribution for an epitaxial silicon chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  1  2  3  4  5  6  Cluster size  0  5000  10000  15000  20000  25000  30000  35000  #  Mean cluster size: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='95  (d) Cluster size distribution for a Czochralski silicon chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Figure 4: Cluster charge and size distributions for a chip with 30 µm epitaxial silicon (left) and 300 µm  Czochralski silicon (right) at −6 V bias voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The latter can be depleted further than 30 µm resulting in a  larger cluster charge and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' Both chips were operated at a threshold of 200 e−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  sample, the Cz sample has mainly clusters of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The corresponding average cluster size is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='55  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='95, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Taking the pointing resolution of the beam telescope into account, an  intrinsic spatial resolution of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='6 µm could be achieved in a Czochralski silicon sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  The hit detection efficiency was measured with a beam telescope with six Mimosa26 planes  and a FE-I4 time reference plane which are all connected to a trigger logic unit to synchronize  individual detector hits time-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The efficiency for all three modules is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 5 where  the result for every pixel was mapped onto a two by two pixel cell to increase the statistics to see  possible effects or efficiency losses within a single pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' All samples were running at a threshold of  about 200 e− and achieve a hit detection efficiency around 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='80 % with slight deviations within the  error (estimated < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='1 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' There are no losses observable in the pixel corners or between pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='2 AC-coupled pixel flavor  Another pixel variation with different analog front-end was tested as well to determine its  performance in a particle beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' In this design the (positive) bias voltage is applied via a diode on  the top side of the chip and connected to the charge collection n-well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' To avoid breakdown of the  front-end electronics due to the high voltage (≤ 50 V) on that well, the input signal is AC coupled to  5  TJ-Monopix2: DMAPS in 180 nm CMOS technology  Christian Bespin  0  10  20  30  40  50  60  column [ m]  0  10  20  30  40  50  60  row [ m]  Region 1 (Center): In-pixel efficiency  for DUT  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='25  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='50  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='75  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='25  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='50  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='75  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  (a) (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='10) % efficiency for a chip built on epitaxial  silicon with gap in n-layer modification from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  0  10  20  30  40  50  60  column [ m]  0  10  20  30  40  50  60  row [ m]  Region 1 (Center): In-pixel efficiency  for DUT  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='25  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='50  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='75  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='25  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='50  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='75  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  (b) (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='10) % efficiency for a chip built on Cz sili-  con with gap in n-layer modification from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  0  10  20  30  40  50  60  column [ m]  0  10  20  30  40  50  60  row [ m]  Region 1 (Center): In-pixel efficiency  for DUT  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='25  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='50  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='75  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='25  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='50  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='75  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  (c) (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='10) % efficiency for a chip built on epitaxial  silicon with additional p-well modification from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Figure 5: Hit detection efficiencies for different substrate materials with different thicknesses and sensor  geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' Results were mapped onto a 2 x 2 pixel array for higher statistics and in-pixel resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The  chips were operated with −6 V bias voltage and at a 200 e− threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  the amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' This approach can potentially deplete the substrate further due to the higher voltage  than what can be applied in the standard pixel design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The hit detection efficiency was measured  for different bias voltages and is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' At 5 V the efficiency is already above 99 % and  reaches the same value as for the DC coupled pixel flavor of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='85 % at or before 25 V bias voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  This is, taking the slightly higher threshold into account, in agreement with the expectation that  there should be no noticeable difference in hit detection efficiency before irradiation between the  two pixel flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The larger applicable bias voltage could prove superior after irradiation to achieve  more depletion and therefore higher charge signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Conclusion  In summary, the performance of TJ-Monopix2 shows a significant improvement in threshold  value and dispersion compared to TJ-Monopix1, although the former is higher than its design value  6  TJ-Monopix2: DMAPS in 180 nm CMOS technology  Christian Bespin  0  10  20  30  40  50  60  column [ m]  0  10  20  30  40  50  60  row [ m]  Region 1 (HV CASC): In-pixel efficiency  for DUT  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='25  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='50  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='75  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='25  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='50  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='75  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  (a) (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='10) % efficiency of an AC-coupled pixel  flavor at 5 V bias voltage and 250 e− threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  0  10  20  30  40  50  60  column [ m]  0  10  20  30  40  50  60  row [ m]  Region 1 (HV CASC): In-pixel efficiency  for DUT  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='25  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='50  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='75  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='25  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='50  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='75  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='00  (b) (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='10) % efficiency of an AC-coupled pixel  flavor at 25 V bias voltage and 200 e− threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Figure 6: Hit detection efficiency of an AC-coupled pixel flavor at (6a) 5 V and (6b) 25 V bias voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  (120 e−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' With the measured amount of charge in the sensor the threshold is still small enough  to detect a majority of hits even from large clusters before irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The large signal compared  to the small pixel leads to a large cluster size and therefore high spatial resolution, where chips  on Czochralski substrate perform slightly better due to the larger sensor volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' For the tested  sensor materials with different thicknesses and sensor geometries, the hit detection efficiency is  around 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='8 % or better in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The modified front-end version with bias applied on the charge  collection node achieves similar values for the hit detection efficiency while providing a larger  headroom in bias voltage to achieve efficient performance after radiation damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The results for  irradiated modules will be presented in a forthcoming publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content='  Acknowledgments  This project has received funding from the Deutsche Forschungsgemeinschaft DFG (grant WE  976/4-1), the German Federal Ministry of Education and Research BMBF (grant 05H15PDCA9),  and the European Union’s Horizon 2020 research and innovation program under grant agreements  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' 675587 (Maria Sklodowska-Curie ITN STREAM), 654168 (AIDA-2020), and 101004761  (AIDAinnova).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
+page_content=' The measurements leading to these results have been performed at the Test Beam  Facility at DESY Hamburg (Germany), a member of the Helmholtz Association (HGF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFRT4oBgHgl3EQfwTji/content/2301.13638v1.pdf'}
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DNE0T4oBgHgl3EQfygIs/content/tmp_files/2301.02659v1.pdf.txt

@@ -0,0 +1,1969 @@
+Bayesian Modelling
+of
+Visual Discrimination Learning in Mice
+Pouya Baniasadi, PhD
+Department of Physiology, Development and Neuroscience
+UNIVERSITY OF CAMBRIDGE
+August 2020
+This project report is written in partial fulfilment of the requirement for the
+Master of Philosophy in Basic and Translational Neuroscience
+Supervised by
+Dr. Jasper Poort
+Prof. Máté Lengyel
+Selective Vision Laboratory
+Computational and Biological
+Learning Laboratory
+Department of Psychology
+Department of Engineering
+arXiv:2301.02659v1  [q-bio.NC]  15 Nov 2022
+
+*
+入Dedication
+For my parents Mahin and Ghasem, who taught me about pursuing dreams,
+for their endless love, support and sacrifices
+i
+
+Declaration
+This report describes work carried out at Cambridge University from Jan 2020 to Jul 2020
+under the supervision of Dr Jasper Poort (Selective Vision Laboratory at the Department
+of Psychology) and Prof. Máté Lengyel (Computational and Biological Learning Lab at
+the Department of Engineering) as a part of the MPhil program in Basic and Translational
+Neuroscience. I confirm that the material in this report is not copied from any published
+material, nor is it a paraphrase or abstract of any published material unless it is identified
+as such and a full source reference is given. I confirm that, other than where indicated
+above, this document is my own work.
+Pouya Baniasadi
+August 2020
+ii
+
+Abstract
+The brain constantly turns large flows of sensory information into selective representations
+of the environment. It, therefore, needs to learn to process those sensory inputs that
+are most relevant for behaviour. It is not well understood how learning changes neural
+circuits in visual and decision-making brain areas to adjust and improve its visually guided
+decision-making. To address this question, head-fixed mice were trained to move through
+virtual reality environments and learn visual discrimination while neural activity was
+recorded with two-photon calcium imaging. Previously, descriptive models of neuronal
+activity were fitted to the data, which was used to compare the activity of excitatory and
+different inhibitory cell types. However, the previous models did not take the internal
+representations and learning dynamics into account. Here, I present a framework to infer
+a model of internal representations that are used to generate the behaviour during the
+task. We model the learning process from untrained mice to trained mice within the
+normative framework of the ideal Bayesian observer and provide a Markov model for
+generating the movement and licking. The framework provides a space of models where
+a range of hypotheses about the internal representations could be compared for a given
+data set.
+iii
+
+Contents
+Contents
+Declaration
+ii
+Abstract
+iii
+1
+Introduction
+1
+1.1
+Mathematical preliminaries
+. . . . . . . . . . . . . . . . . . . . . . . . .
+2
+2
+The experiment
+6
+2.1
+Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.2
+Behavioral data and observations . . . . . . . . . . . . . . . . . . . . . .
+8
+3
+Behavioral model part 1: internal representations
+13
+3.1
+Structure of spatial states
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+3.2
+Space of models for spatial states . . . . . . . . . . . . . . . . . . . . . .
+16
+3.3
+Bayesian learning model . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+3.3.1
+Learning reward probability within a state . . . . . . . . . . . . .
+19
+3.3.2
+Learning state transitions
+. . . . . . . . . . . . . . . . . . . . . .
+21
+4
+Behavioral model part 2: the generative model
+25
+4.1
+Spatial state parameter ˜λk: licking rate . . . . . . . . . . . . . . . . . . .
+26
+4.2
+Parameter ˜νk: target speed within the current spatial state
+. . . . . . .
+29
+4.3
+Generative model of licking and speed
+. . . . . . . . . . . . . . . . . . .
+30
+4.4
+Estimation of model parameters . . . . . . . . . . . . . . . . . . . . . . .
+33
+5
+Discussion
+34
+5.1
+Limitations
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+34
+5.2
+Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+35
+Bibliography
+37
+iv
+
+Chapter 1
+Introduction
+Learning modifies neural representations of behaviourally relevant information. While
+changes in response selectivity to behaviourally relevant stimuli have been observed in
+many studies across different species (Yang & Maunsell 2004, Yan et al. 2014, Poort
+et al. 2015). There has been growing evidence that different cell types, classified using
+molecular and cellular properties (Kepecs & Fishell 2014), have specific roles in learning
+(Khan et al. 2018, Fishell & Kepecs 2019). However, the nature of these changes and
+how they relate to sensory coding is not well understood (Yap & Greenberg 2018).
+Probabilistic models of behavioural learning are an important approach to link the
+changes in neural representations to internal representation of the environment and
+decision-making (Fiser et al. 2010, Berkes et al. 2011, Heeger 2017). Given the non-
+deterministic nature of events in the real world, human and animal learning must involve
+at least some internal representations of the uncertainties in the environment (Barlow
+et al. 1961). There has been an extensive body of research on how the nervous system
+represents uncertainly about the environment (Pouget et al. 2003, Beck et al. 2008, Fiser
+et al. 2010, Kriegeskorte & Douglas 2018).
+Bayesian learning theory provides a normative framework of learning that represents
+uncertainty in probabilistic outcomes(Bishop 2006).
+In particular, the ideal observer
+analysis uses the Bayesian learning theory for achieving optimal learning performance in
+a given task (Geisler 2003, 2011). Learning can be conceptualised as the incorporation
+of sensory information to update and improve performance on a given task. the ideal
+observer performs at the theoretical limits of information processing to update their
+1
+
+1.1.
+Mathematical preliminaries
+beliefs. However, it is important to note that optimality in this context refers to the
+optimal incorporation of information, which is not equivalent to achieving the optimal
+solution in all trials. While the nervous system may or may not have representations
+similar to an ideal observer, the ideal observer analysis provides a systematic framework to
+formulate hypotheses about the internal representations and learning dynamics (Maloney
+& Mamassian 2009, Orbán et al. 2008).
+In this thesis, we describe a Bayesian learning model using the framework of ideal observer
+learning. Our goal is to develop a model of internal representations of reward and space
+that are used for learning and adjusting behaviour in the visual discrimination task. This
+model will allow us in future work to relate the neuronal activity measurements (Poort
+et al. 2015, Khan et al. 2018) to the internal representations that guide behaviour. We
+continue this chapter with a brief overview of the basic mathematical ideas used to develop
+the model. Then, in Chapter 2, we explain the experimental setup and describe the
+behavioural data. A space of models (for the structure of Markov models) is introduced
+in Chapter 3 which defines the internal representations of reward and state transitions.
+Then, a Bayesian model of learning reward probabilities and state transitions is described
+that uses the ideal observer framework. In Chapter 4, we introduce a generative Markov
+model that uses internal representations to generate behaviour. We also discuss the use
+of maximum likelihood estimation to estimate the model parameters. Finally, in Chapter
+5, we discuss the potential applications and limitations of the model and set out a path
+for the continuation of the research.
+1.1
+Mathematical preliminaries
+In this section, I briefly introduce the concepts that provide the mathematical foundation
+of the Behavioral model.
+Markov chain model
+A system has the Markov property if the predictions about future events only require the
+knowledge of the system’s present state. In other words, given the present state of the
+system, future events are conditionally independent of past events. A Markov chain is a
+stochastic model of a sequence of events with the Markov property.
+Let S = s1, s2, ..., sr be a set of states for a Markov chain. The process starts in one of
+these states and moves sequentially from one state to another. Each move is called a step.
+Let Xn be the current step. We denote by pij = P(Xn+1 = sj|Xn = si), the transition
+2
+
+1.1.
+Mathematical preliminaries
+probability of visiting state sj after visiting si. Note that by Markov property, given Xn,
+Xn+1 is conditionally independent of the past states. A transition from si to sj can be
+represented as a directed edge (si, sj) with a corresponding transition probability pij. The
+sum of transition probabilities of the outgoing edges from a state should add up to 1.
+Figure 1.1 illustrates a Markov chain with 4 states and transition probabilities.
+Figure 1.1: Movement in a corridor simulated in the VR environment.
+Let T = [pij] be the transition probability matrix for the Markov chain and let u be
+the probability vector which represents the starting distribution (i.e., Xk ∼ u). Then
+the probability that the chain is in state si after m steps is the i-th entry in the vector
+u(m) := u T m. That is,
+P(Xk+m|Xk) = u(i)(m),
+where u(m) = uT m.
+(1.1)
+Bayesian learning
+The probability of an event A is denoted by P(A). Consider another event B and its
+corresponding probability P(B). The conditional probability P(A|B) is the conditional
+probability of A given B. Bayes Theorem states that
+P(A|B) = P(B|A) P(A)
+P(B)
+Consider a system that generates data and a space of possible models for describing
+the behaviour of the system.
+The probability distribution over the space of models
+P(Model) represents the prior knowledge about the system. Suppose that a set of data
+D is observed from the system. Then P(D | Model) is called the likelihood and P(Data)
+3
+
+0.3
+0.7
+S1
+S2
+0.4
+0.6
+1.0
+S4
+S3
+0.5
+0.51.1.
+Mathematical preliminaries
+is called the model evidence or marginal likelihood. The posterior distribution over the
+models P(Model | D) represents our beliefs about the system after observing the data
+D. Bayes rule provides a principled way of updating our beliefs about the system after
+observing data. Formally,
+P(Model | Data) = P(Data | Model) P(Model)
+P(Data)
+.
+(1.2)
+Dirichlet distribution learning of categorical probability values
+Consider a random variable which can take on K possible categories. The categorical
+distribution is a discrete probability distribution for the random variable, where the
+probability of each category is separately specified.
+The categorical distribution is a
+generalisation of the Bernoulli distribution for a discrete variable with more than two
+outcomes, such as the probability of outcomes for a 6-sided die. It is also a special case
+of the multinomial distribution where the number of trials in one.
+If the probabilities of each outcome for a categorical distribution are unknown, using
+Bayesian learning, we can update prior probability distributions of probability values.
+The Dirichlet distribution is a conjugate before the multinomial (and categorical) distri-
+bution, meaning starting with a Dirichlet prior and multinomial likelihood, the resulting
+posterior is also a Dirichlet distribution. The probability mass function for the Dirichlet
+distribution DirK(α) with K categories is
+f(p|α) =
+1
+B(α)
+K
+�
+i=1
+pα(i)−1
+i
+,
+(1.3)
+where α = (α(1), . . . , α(K)) is the vector of parameters. Furthermore,
+B(α) =
+K
+�
+i=1
+Γ(α(i))
+Γ
+� �K
+i=1 α(i)�
+where for positive real number n,
+Γ(n) =
+� ∞
+0
+xn−1e−xdx.
+For integer values, Γ(n) = n! .
+To learn probabilities for a categorical distribution, given a prior distribution DirK(α)
+4
+
+1.1.
+Mathematical preliminaries
+over the probability vector p = (p1, . . . , pK), and data c = {c1, . . . , ck} representing the
+number of observation Dirichlet category, the posterior distribution is
+P(p|x) = (p|x + α)
+p|x ∼ DirK(x + α)
+(1.4)
+Finally, the Beta distribution is a special case of the Dirichlet distribution where the
+outcomes are binary (true or false). To distinguish this special case, we may use the
+notation Beta(β(1), β(2)) ≡ Dir2(α) where α = {β(1), β(2)}.
+5
+
+Chapter 2
+The experiment
+In this chapter, I describe the experimental setup in Khan et al. (2018) and Poort et al.
+(2015) for which we have developed a behavioural model in the later chapters. A summary
+of previous findings and a description of the behavioural data accompanied by figures are
+also included.
+2.1
+Experimental setup
+The experimental setup involves the placement of the mouse on a cylindrical treadmill
+where its head is fixed to enable imaging of neural activity. The mouse can move forward
+(and backward). In front of the mouse, a screen is shown to the animal where visual
+feedback connected to the movement can simulate the movement of the subject in an
+environment. By controlling the setup of the space and visual stimulus while allowing
+imaging, the VR setup has been extensively used for studying the visual cortex and
+hippocampus in mice in recent years (Harvey et al. 2009, Dombeck et al. 2010, Khan
+et al. 2018, Poort et al. 2015, Saleem et al. 2018). Figure 2.1 illustrates the VR setup.
+Figure 2.1: Movement in a corridor simulated in the VR environment.
+6
+
+2.1.
+Experimental setup
+Specifics of the corridor space and reward administration
+We specifically consider the experimental setup described in Khan et al. (2018), Poort
+et al. (2015). In these two studies, the activity of populations of neurons in V1 was mea-
+sured with two-photon calcium imaging Chen et al. (2013) during a visual discrimination
+task in a virtual reality (VR) environment. Head-fixed mice ran through a simulated
+corridor where different types of visual stimuli were displayed on the walls. Three types
+of wall patterns characterise the different corridors. In the grey corridor a short stretch
+of circle patterns followed by grey walls for a random distance, before the pattern on
+the walls abruptly changes to one of the grating corridors. The grating corridors either
+displayed vertical gratings (illustrated in Figure 2.1) or angled gratings for a fixed length
+(60 VR length units), before the grey corridor. An illustration of the corridor space is
+displayed in Figure 2.2.
+Figure 2.2: Illustration of the corridor space.
+A milk dispenser was placed in front of the mouse to administer rewards. Mice received a
+reward for licking the dispenser in a reward zone starting halfway in the vertical grating
+corridor and halfway for around 10 VR-length units. If the mouse licked the dispenser in
+the reward zone, it would trigger the opening of the reward valve and a drop of soy milk
+would appear at the dispenser. No punishment was given for licking in the corridors with
+grey and angled grating walls. All mice learnt to discriminate the two stimuli, starting at
+the chance performance (behavioural d′ close to zero) and reaching the threshold criterion
+of d′ > 2.0 within 5-9 days.
+7
+
+reward zone2.2.
+Behavioral data and observations
+summary of previous findings
+The motivation behind developing a behavioural model is to take advantage of the be-
+havioural data for the future analysis of experiments similar to Khan et al. (2018). A
+summary of results in Khan et al. (2018) is as follows. After learning the visual discrim-
+ination task, neurons showed increased stimulus selectivity for the angled and vertical
+gratings. Interestingly, this effect depended on the cell types. In particular, stimulus se-
+lectivity for populations of pyramidal cells (PYR) along with parvalbumin (PV), somato-
+statin (SOM), and vasoactive intestinal peptide-expressing (VIP) inhibitory interneurons
+in layer 2/3 (L2/3) of the primary visual cortex (V1) were compared. Selectivity was
+increased for PYR and PV cells. PV neurons became as selective as the PYR cells, and
+showed changes in functional interactions, particularly with PYR cells. On the other
+hand, SOM neurons became decorrelated from the network and PYR–SOM coupling
+before learning predicted selectivity increases in individual PYR cells. While SOM inhi-
+bition seemed to gate changes in selectivity, PV cells provided strong stimulus selective
+inhibition after learning. A multivariate autoregressive linear model (MVAR model) fit-
+ted the activity of the neurons, and further supported the statistical analysis results.
+However, the MVAR model arguably neglects potentially important information in the
+behavioural data. Even though speed is taken into account, its contribution to the be-
+haviour of the MVAR model is negligible. Accordingly, one of the primary motivations of
+the behavioural model proposed in this report is potential improvements in the (MVAR
+model). This is discussed in more detail in Chapter 5.
+2.2
+Behavioral data and observations
+Behavioural data were collected during the experiment. The distance travelled from the
+onset of the current corridor and the corridor type (determined by the wall patterns) is
+continuously recorded. The observed variables of spatial location and visual stimuli at
+each time are marked by a pair (x, y) ∈ (xLoc × Cor), where x is the distance travelled
+from the onset of the current corridor pattern, and y is the corridor pattern. Set xLoc =
+[0, max(x)] is an interval from 0 to the maximal length of an interval max(x) and set
+Cor = {grey, vertical, angled} is the set of corridor types. The speed of the subject at
+each time is also recorded. A list of licking times and valve opening times (indicating
+reward administration) is also given by the data.
+For the generative behavioural model in Chapter 4, we discretize the data into time
+intervals of ∆τ seconds, each identified by an index t ∈ {1, 2, . . . , N}.
+The value of
+∆τ determines the time resolution of the behavioural data. Since the imaging data of
+8
+
+2.2.
+Behavioral data and observations
+Khan et al. (2018) is taken in 1
+8 second intervals, time resolutions lower than 1
+8 seconds
+are not useful.
+Higher time resolutions may be desirable because they will decrease
+the computational cost of the analysis, but the cost of losing time resolution must be
+discussed. However, unless explicitly discussed, we can assume ∆τ =
+1
+8 for the data
+analysis. Table 2.1 describes the notation used to describe the data. Note that some
+of the records are behavioural, while others specify the values that are observed by the
+subject.
+Table 2.1: Behavioral and observational records for t ∈ {1, 2 . . . , N}.
+Data
+Type
+Description
+xt
+Observation
+xt is the true value of distance from the onset of the current
+corridor at time step t.
+yt
+Observation
+yt ∈ Cor = {grey, vertical, angled} is the true value of the
+corridor type, which determines the visual stimuli at time
+step t.
+ot
+Observation
+ot is a binary value for whether the reward valve has opened
+during the time step.
+vt
+Behavior
+Speed (average) at time step t
+lt
+Behavior
+Number of licks at time step t
+Instance of data visualisations
+The Figures below are instances of behavioural data visualizations from the experimental
+data. Figures 2.4 and 2.3 (Poort et al. 2015) illustrate the licking behaviour at different
+positions in different corridors, and Figures 2.5 (Poort et al. 2015) and 2.6 (Khan et al.
+2018) give a colour map of speed at the different positions in the different corridors. For
+all Figures, the horizontal axis represents the position concerning the onset of the grating
+corridor 1, and the vertical axis is the trial index. Higher trial numbers are later. The
+black or red labels are data labels the ls associated with the experimental sessions.
+The following observations about the licking behaviour have influenced parameter defini-
+tions and assumptions about prior beliefs of the animal in Chapter 3. These observations
+are consistent among all subjects.
+reward association prior: The mice do not know reward associations before the
+reward. However, the mice know that moving forward and licking the dispenser may
+lead to a reward. Initially, the licking behaviour is frequent to explore the space
+and discover reward associations. A uniformly random prior for reward probability
+may be appropriate.
+1note that for the grey corridor this is obtained by shifting xt by the length of the grey corridor
+9
+
+2.2.
+Behavioral data and observations
+Change of visual discrimination: The behaviour of the mice in the grating
+area and the grey area starts to diverge immediately, and the behaviour of the
+mouse in angled and vertical grating corridors seems to be similar at first; the
+differences of licking behaviour seem to be only after the reward is present in the
+vertical grating corridor. The dissociation of the award from the angled grating is
+realised substantially later than the dissociation of the reward from the grey area.
+It seems that at different points in the trial, the set of visually discriminated stimuli
+is different.
+Location is also taken into account As the learning progresses, the licking
+concentrates close to the reward zone. It seems that the mice associate a spatial
+region, characterised by both visual stimuli and spatial positioning, with the reward
+area.
+The following observations about the speed have influenced our generative model of speed
+in Chapter 4.3. These observations are consistent among all subjects.
+Reward association influences speed: the graphs suggest that the dissociation
+of reward in upcoming regions is associated with higher speed while anticipation of
+reward in upcoming regions is associated with reduction of speed.
+Evidence for change in the internal model: while speed behaviour in the
+grey corridor diverges from the grating corridor quickly, the divergence of speed
+behaviour for angled grating and angled grating happen at a later point.
+This
+suggests that the mice initially correlate the grating areas with the reward, and
+then learn to differentiate between the grating areas to dissociate the angled grating
+with the reward.
+Change of visual discrimination:: Similar to the licking behaviour, initially
+speed behaviour seems to discriminate between the angled and vertical gratings
+only after the reward is present in the vertical grating corridor.
+This suggests
+that the mice initially correlate the grating areas with reward, and then learn to
+discriminate between the vertical and angled grating areas.
+10
+
+2.2.
+Behavioral data and observations
+Figure 2.3: Lick locations for M27. See the figure descriptions below.
+Figure 2.4: Lick locations for M31 in all trials. The horizontal axis represents the location in
+a corridor, with 0 being set at the onset of a grating corridor. Negative values are in the grey
+corridors and positive values are in the grating corridors. The licking locations are marked by
+coloured points. Red dots represent licking within 1 length unit before a valve opening, and
+yellow indicates the licking after the opening of the reward valve, in a grating corridor. All
+other lick locations are marked in black. The trial number on the vertical axis shows the
+sequential order of the trials in each plot. The right plot shows all trials, where each trial is
+passing through one grey corridor followed by a grating corridor. The middle and the left plots
+show a closer look at the vertical and angled grating corridors. The red labels are labels for the
+experimental sessions.
+11
+
+grey walls
+vertical gratings
+ angled gratings
+Unrewarded Lick
+Lick after valve opens Lick within 1 unit before valve opens
+M27-date:20130515b-B2
+M27-date:20130515b-B2
+M27-date:20 130515b-B2
+2000
+900
+M27-date:20130514b-B3
+M27-date:20130514b-B3-
+M27-date:20 130514b-B3
+1000
+M27-date:20130513b-B3-..
+M27-date:20130513b-B3
+800
+M27-date:20130513-B1
+M27-date:20130513-B1
+M27-date:20130513-B
+1600
+M27-date:20130511-B2
+M27-date:20130511-B2---
+800  M27-date:20130510-B
+1400
+M27-date:201305
+M27-date:20130510-B1
+ 600
+No.
+M27-date:20130509-B
+1200
+M27-date:20130509-B
+Trial
+2
+M27-date:20130509-
+600
+.
+Angled-g 
+1000
+rtical
+M27-date:20130508-B
+M27-date:20130508-B
+400上
+ M27-date:20130508-B
+800
+400
+300
+600
+M27-date:20130507-B
+M27-date:20130507-B
+M27-date:20130507-B
+200
+400
+200
+date:20130506
+M27-date:201305
+B
+100
+M27-date:20 130
+200
+0
+0
+.
+0
+-250
+-200
+-150
+-100
+-50
+0
+50
+0
+20
+40
+60
+0
+20
+40
+60
+x (VR length unit)
+x (VR length unit)
+x (VR length unit)grey walls
+vertical gratings
+ angled gratings
+Unrewarded Lick
+Lick after valve opensLick within 1 unit before valve opens
+M31-date:20130612b-B2
+M31-date:20130612b-B2
+800
+1600 M31-date:20130612-B1
+M31-date:20130612-B
+M31-date:20130612-B1
+M31-date:20130611b-B8
+800
+M31-date:20130611b-B8
+M31-date:20130611b-B8 -
+700
+1400 M31-date:20130611-B2
+M31-date:20130611-B2
+M31-date:20130611-B2
+700
+M31-date:20130610b-B3
+600
+M31-date:20130610b-B3
+M31-date:20130610b-B3
+1200
+600
+M31-date:20130610-B1
+No.
+M31-date:20130610-B1
+诚
+500
+ M31-date:20130610-B1
+....
+1000
+Trial
+500
+Trial
+M31-date:20130609b-B2
+M31-date:20130609b-B2
+9
+M31-date:20130609b-B2
+M31-date:20130609-B1
+M31-date:20130609-B1
+.
+800
+M31-date:20130609-B1
+Ver
+M31-date:20130608-B2
+M31-date:20130608-B2
+M31-date:20130608-B2
+600
+...
+300
+300
+M31-date:20130607-B3--
+M31-date:20130607-B3
+M31-date:20130607-B3
+M31-date:20130606-B2
+400
+200
+ M31-date:20130606-B2
+200  M31-date:20130605-B4
+M31- date:20130605-B4
+M31-date:20130605-B4
+M31-date:20130604:B3 :
+M31-date:20130604-B3
+200
+100
+100
+M31-date:20130604-B3-
+M31-date:201306031
+M31-date:20 130603b-B3
+M31-date:20130603b-B3
+0
+M31-date:20130603
+M31-date:20130603-B2
+-300
+-200
+-100
+0
+0
+20
+40
+60
+20
+40
+60
+x (VR length unit)
+x (VR length unit)
+x (VR length unit)2.2.
+Behavioral data and observations
+Figure 2.5: Speed vs location for M31. See the figure descriptions below.
+Figure 2.6: Speed and licks vs location for M70. The horizontal axis represents the location in
+the corridor, with 0 being set at the onset of a grating corridor. Negative values are in the grey
+corridors and positive values are in the grating corridors. The trial number on the vertical axis
+shows the sequential order of the trials in each plot. The right plot shows all trials, where each
+trial is passing through one grey corridor followed by a grating corridor. The middle and the
+left plots show a closer look at the vertical and angled grating corridors. The colour for each
+location of each trial represents the speed of the animal at that point according to the colour
+scale; warmer colours represent higher speeds and cooler colours represent lower speeds. Note
+that for Figure 2.5, the speed is averaged over 5 unit intervals due to virtual memory limits.
+The white points show the lock locations for M70, and the small black star indicates a valve
+opening location during a trial. The black labels are data labels associated with experimental
+sessions.
+12
+
+speed (units per second)
+0
+10
+20
+40
+50
+M31-date:20130612b-B2
+L M31-date:20130612b-B2
+900
+M31-date:20130612b-B2
+M
+800
+1600 M31-date:20130612-B1
+M31-date:20130612-B1
+M31-date:20130612-B1
+M31-date:20130611b-B8
+800
+M31-date:20130611b-B8
+M31-date:20130611b B8
+700
+1400 |M31-date:20130611-B2
+M31-date:20 130611-B2
+M31-date:20130611-B2
+700
+M31-date:20 1306 10b-B
+600
+M31-date:20130610b-B3
+600
+No.
+M31-date:201306 10-B1
+M31-date:20 1306 10-B1
+500
+FM31-date:20130610-B1
+1000
+Trial
+500
+Trial
+M31-date:20130609b-B2
+M31-date:20130609b-B2
+9
+M31-date:20130609b-B2
+M31-date:20 130609-B1
+M31-date:20130609-B1
+800
+M31-date:20130609-B1
+400
+Ver
+M31-date:20130608-B2
+M31-date:20130608-B2
+M31-date:20130608-B2
+600
+300
+M31-date 20 130607- B3
+M31-da
+M31-da
+qe:20130607-B3
+200M31-da
+400
+te:20130606-B2
+M31-date:20130605 B4
+M31-date-20130604 B3
+M31-date:20 130604-B3
+200
+100
+100
+M31-date:20 130604 B3
+M31-date: 20130603b- B
+M31-date:20 130603b-B3
+M31-date:20130603b-B3
+0
+M31-date.20130603-B2
+-300
+-200
+-100
+0
+0
+20
+40
+60
+U
+20
+40
+60
+x (VR length unit)
+x (VR length unit)
+x (VR length unit)speed (units per second)
+0
+10
+20
+30
+40
+50
+M70-date:20141028-B1
+M70-date:20141028 B7
+M70-date:20141028-B1
+200
+160
+350
+180
+140
+300
+160
+120
+250
+140
+I No.
+No.
+Angled-g Trial l
+100
+Trial No.
+Trial
+120
+200
+Vertical-g
+100
+80
+150
+80
+M70-date:20141022-B1
+60
+M70-date:20141022-B1
+60
+M70-date:20141022-B1
+100
+40
+40
+50
+20
+20
+0
+0
+-250
+-200
+-150
+-100
+-50
+0
+50
+0
+20
+40
+60
+0
+20
+40
+60
+x (VR length unit)
+x (VR length unit)
+x (VR length unit)Chapter 3
+Behavioral model part 1: internal
+representations
+The behavioural model presented here provides a framework for inferring an internal
+model that can predict the animal’s behaviour at a given time. Before getting into the
+specifics, consider a broad perspective on inferring a model that generates the current be-
+haviour by incorporating past experiences. Figure 3.1 is a graphical model of big-picture
+relation between the history of animal’s observations H, the internal model M that in-
+corporates experience into internal representations, and the observed behaviour B. This
+chapter discusses the relationship between the history of observations and behaviorally
+relevant representations in the internal model (H → M in the graphical model of Figure
+3.1). I introduce a space of models where a range of hypotheses about the internal model
+can be systematically examined. The internal representations about reward and space
+are then used in the next chapter to construct a generative model of behaviour (M → B
+in the graphical model of Figure 3.1). Then using a systematic approach, an internal
+model is inferred that best describes the data (H and B).
+Figure 3.1: Relation between history of experimental observations H, internal model M, and
+behavior B. H and B are observed in the experimental data, but the internal model M is
+unobserved.
+13
+
+H
+M
+B3.1.
+Structure of spatial states
+By exploring and experiencing the environment, the brain uses experience to update its
+beliefs (i.e., learning) about the environment using its internal representations. In this
+learning model, the normative framework of Bayesian ideal observer analysis (Geisler
+2003, 2011) is used to learn behaviorally relevant internal representations. These include
+learning about the probability of reward in different regions of the VR corridor, and
+expectations about upcoming spatial regions when moving forward1.
+Model of spatial states in Section 3.1 describes how the space (VR corridor) is divided into
+states corresponding to spatial segments, where the representation of reward probability
+within a state only depends on the information (history of reward outcomes) obtained at
+that state. The structure of these states is a Markov chain. The space of models in Section
+3.2 prescribes a range of Markov chain structures of spatial states within which a model is
+selected. For given states of a model, the dynamics for learning reward associations and
+state transitions are considered within the normative framework of the Bayesian ideal
+observer model in Section 3.3.
+3.1
+Structure of spatial states
+Animals’ observation of visual stimuli and spatial positioning is an observation of the
+current (x, y) ∈ {xLoc, Cor}.
+Observations about reward association at the current
+location (x, y) may be relevant to reward association at some other locations.
+It is
+therefore necessary to define spatial regions where reward observations are relevant to
+the entire region but explicitly irrelevant to other regions. To formalise this concept,
+the objective of this section is to associate the segments of space with states where the
+information about reward association is relevant to the current state and no other state.
+A reasonable way to define such states is to group areas that are spatially close by, visually
+similar, or both.
+Defining states associated with spatial segments
+Taking into account both spatial proximity and visual similarity, consider sectioning xLoc
+into a finite set of mutually exclusive spatial segments each identified by a fixed y, and an
+interval Ix for x values. We illustrate an example of spatial segmentation in Figure 3.2.
+Denote by S a set of states and associate each segment with only one state (note that
+multiple segments may be associated with the same state). Then we say that the mouse
+is in state s if its position (x, y) is inside a segment that is associated with s. We associate
+all positions in all corridors with only one state with the function f : (xLoc × Cor) → S.
+1The subject can only move forward due to the experimental setup.
+14
+
+3.1.
+Structure of spatial states
+The mouse may map locations onto states in multiple ways. By considering various ways
+to map between locations and states, we can infer the mapping that best matches the
+behavioural data (see 4.4).
+Spatial state transition event and structural properties
+Let Xk be the random variable describing the k-th visited spatial state, where a spatial
+state transition event (i.e., transition to the next spatial step) happens when the subject
+crosses the initial point of a segment associated with a state2. Given the current position,
+the future positions do not depend on the history of visited positions, so given Xk, state
+Xk+1 is conditionally independent of Xn for n < k. It follows that the state structure as
+defined above satisfies the Markov property.
+We assume that the spatial states are fully observable. In other words, given a state
+structure, we assume that the subject always knows which state is the current state.
+Observations of the animal may be noisy and inaccurate, so assuming fully observable
+states is a simplification that may be contended with in a more sophisticated future
+model. However, states are associated with intervals of space rather than precise points
+in space, and they already incorporate some approximation about the spatial awareness
+of the subject.
+We assume that the mouse learns two things from the visual stimuli and licking in state
+s. First, it learns the reward association in that state. Second, it learns the transition
+from that state to other states. Let r(s) be the probability that licking in state s leads to
+reward in state s. Also, denote by p(s,s′) = P(Xk+1 = s′|Xk = s) the transition probability
+Figure 3.2: An example of dividing the corridor space into mutually exclusive spatial
+segments. Each segment is then associated with exactly one state.
+2Note that the time spent in each state is not fixed in this Markov model.
+15
+
+3.2.
+Space of models for spatial states
+of visiting any state s′ after s. These parameters are initially unknown to the mouse and
+should be learned. In Section 3.3, I discuss a semi-normative model of learning for these
+parameters using the ideal observer framework.
+It is worth noting that the state transitions of the Markov chain are sparse. To understand
+the sparsity of state transitions, first note that x is a positive real value, which ranges
+from 0 to the maximal length of a corridor with the same patterns, and y is a discrete
+value with three possible entries. From the onset of a corridor, until the onset of the next
+corridor, the spatial location is a continuous function of time. Within the period between
+two consecutive onsets, if a state transition happens, it can only be to the state associated
+with the next interval of x, with the same y. Moreover, when passing the onset of the
+next corridor, there is a discrete change in the value of y, and x = 0 at the onset of the
+new corridor. This event can only be a state transition to the start of a new corridor (a
+state that starts at x = 0) so there are at most three such possible transitions. It follows
+that the structure of states is a sparse Markov chain.
+3.2
+Space of models for spatial states
+To define a space of models M , we use two parameters for identifying a model in the
+model space; one for the set of discriminated patterns (V), and one for the length of
+segments (d).
+Spatial model parameter V: set of discriminated visual stimuli
+Let V be the set of visual stimuli that are discriminated in the spatial state model. The
+set of possible choices for V is {V1, V2, V3} which are described below.
+• V1 = {u := undifferentiated}, where the grey and grating are not discriminated.
+• V2 = {g := grey, va := angled or vertical grating}, where the grey corridor is dis-
+criminated from the grating corridors, but where angled and vertical grating corri-
+dors are not discriminated.
+• V3 = {g := grey, v := vertical, a := angled}, where the grey corridor, the angled and
+vertical grating corridor are discriminated.
+While set Cor contains the types of visual stimuli on the corridors, set V refers to subjec-
+tive visual discrimination (or classification) between corridors by the mouse. Also note
+that the choices for set V implicitly contain a mapping from Cor to V.
+16
+
+3.2.
+Space of models for spatial states
+Spatial model parameter d: length of states
+Denote by d a value in the interval (0, max(x)] for the length of spatial segments. Value d
+uniquely defines a sequence of intervals of x values. For example, the associated sequence
+of intervals to d = 30 is {[0, 30), [30, 60), . . .}. Then state sij is associated with the j-th
+interval of x, which is [(j −1)d , jd), and i ∈ C identifies the visual stimuli. For example,
+for V = {g, p} and d = 30, the state sp,2 refers to intervals of x ∈ [30, 60) for both the
+vertical and angled grating corridors.
+Model space
+Now it is possible to introduce a Markov model MV,d ∈ M with the set of states S that
+are associated with the spatial intervals induced by V and d. Since the length of the a
+Figure 3.3: Nine instances of Markov chain models MV,d for choices of V selected instances
+of d. For d = xmax, there is only one state per and self transition event only occurs when the
+corridor type changes. The length of the angled and vertically grating corridors is exactly 60
+(VR length units) in the experiment. So for d = 60 and d = 20, there are exactly 1 and 3 states
+associated with the relevant element in V. Note that the figure illustrates only selected instances
+of the model space M .
+17
+
+Mv,d
+V = V1
+V = V2
+V = V3
+[u]
+(g ,va)
+(g ,V,a)
+d = max(x)
+09 = p
+d = 203.3.
+Bayesian learning model
+Table 3.1: Parameters for the model of spatial states
+.
+Parameter
+Type
+Description
+V
+Spatial model
+parameter
+Set of discriminated visual stimuli on the corridors in the
+model MV,d; Possible options are V1 = {u}, V2 = {g, p}
+and V3 = {g, v, a}.
+d
+Spatial model
+parameter
+A constant length in (0, max(x)] for the length of the
+spatial for model MV,d.
+corridor is bounded by max(x), model MV,d is a finite state Markov model. For example,
+MV1,max(x) and MV3,max(x) have exactly one and three states, respectively.
+Figure 3.3
+illustrates the states of Markov chain models MV,D for example cases of V and d.
+Parameters V and d are free parameters that will be set during the model selection, which
+will be further discussed in Section 4.4. The fit for parameter V, selected from V1, V2
+or V3, is determined by which stimuli the animal discriminates. The true value for d is
+the length of spatial segments where information about reward associations and state
+transitions in the current segment is reasonably independent of segments associated with
+other states. For the sake of simplicity, it is assumed that d is a fixed value, and it is
+the same across different visual stimuli. However, relaxing this assumption is possible by
+having more free parameters, for example, by introducing a free parameter of distance
+for each element of V. For example, suppose V = V3. Then instead of a free parameter d,
+we could use three parameters in D = {dg, da, dv} which contains one free parameter of
+distance for every element of V. In the initial implementation of the model, one parameter
+d is considered.
+In summary, parameters V and d for a model MV,d determine the structure of the states
+in the Markov chain, where for each state the learning dynamics about reward association
+and state transitions is only dependent on the observations in that state. The learning
+dynamics are discussed in the next section.
+3.3
+Bayesian learning model
+As first noted in Section 3.1, in any state s, the subject uses sensory information to
+learn r(s), the probability that licking in s leads to the administration of reward in s, or
+reward probability of s for short. Furthermore, state transition probability p(s,s′), which
+is the probability of visiting state s′ after visiting s, is also unknown to the subject and
+it is learned.
+Here, we use the ideal observer framework (Geisler 2003) to develop a
+semi-normative model for learning both reward associations and state transitions. In this
+18
+
+3.3.
+Bayesian learning model
+section, the learning dynamics are discussed for a given model M ∈ M . Therefore, states
+S and their associated spatial intervals are unambiguous.
+3.3.1
+Learning reward probability within a state
+Recall that reward is given to the subject immediately after the subject licks the dispenser
+in the reward zone (see Section 2.1 for details of the experimental setup). The reward is
+a fixed amount of milk administered via the dispenser. We noticed that even in trained
+animals, licking started before the reward zone (see example mice in Figures 2.3 and 2.4).
+This suggests that the mouse associates an extended region with the reward delivery
+which starts before the reward zone set by the experimenters.
+Reward outcome Rk of current spatial step k
+If the mouse licks the dispenser in state s, it collects some information about the unknown
+parameter r(s). If the subject does not lick the dispenser, it obtains no information about
+r(s). Let the random variable Rk = (R(T)
+k , R(F)
+k ) be the reward outcome of spatial step
+k, where R(T)
+k
+counts the number of positive outcomes, and R(F)
+k
+counts the number of
+negative outcomes in spatial step k. As a consequence of the experimental setup, the
+amount of reward and the frequency of licking in the experiment does not provide any
+additional information about a reward region. Furthermore, spatial states are defined to
+be regions where licking at different points within the region does not provide additional
+information about the reward. Therefore, each visit to a state provides only three possible
+reward outcomes:
+• Rk = (1, 0) for subject licking the dispenser in spatial step k followed by reward
+becoming available in spatial step k,
+• Rk = (0, 1) for subject licking the dispenser in spatial step k followed by no reward
+in spatial step k, and
+• Rk = (0, 0) for subject not licking the dispenser in spatial step k.
+Normative model for updating internal reward representations (Bayesian)
+Let us first discuss how an ideal observer updates its prior beliefs about r(s) after visiting
+state s in spatial step k. The ideal observer provides a theoretical upper limit of perfor-
+mance, given the collected data. It is therefore a normative framework for updating the
+beliefs about reward association. Let prior beliefs about r(s) right before visiting spatial
+19
+
+3.3.
+Bayesian learning model
+step k be a Beta distribution
+Beta(β(1)
+k (s), β(2)
+k (s))
+over the interval [0, 1]. The reward outcome Rk = (R(T)
+k , R(F)
+k ) is the data that is newly
+collected about the reward. By Equation 1.4, the posterior is
+r(s)|Rk ∼ Beta(R(T)
+k
++ β(1)
+k (s), R(F)
+k
++ β(2)
+k (s)).
+Reward learning rate ηr
+The above is a theoretical bound on learning from observations in state s, assuming a
+prior Beta distribution over [0, 1] for the reward probability r(s). Some mice learn faster
+than others, and all of them will perform no better than the ideal observer model above.
+To allow for individual differences, and different learning rates, we introduce a model
+parameter ηr ∈ [0, 1], which dials the amount of data required for the same amount of
+learning as an ideal observer. The update rule (i.e., posterior) is
+r(s)|Rk ∼ Beta(ηrR(T)
+k
++ β(1)
+k (s), ηrR(F)
+k
++ β(2)
+k (s)).
+To keep track of learning parameters, let Bk =
+�
+βk(s) :=
+�
+β(1)
+k (s), β(2)
+k (s)
+�
+: s ∈ S
+�
+be
+the beta parameters for beliefs about reward probabilities of all states in spatial step k.
+Note that after visiting state s in spatial step k,
+βk+1(s) = ηrRk + βk(s)
+for s = Xk, and
+(3.1)
+βk+1(s′) = βk(s)
+for s′ ̸= Xk.
+Note that ηr is defined to have the same value across all states. If ηr = 1, the mice
+performs as well as the normative ideal observer, and if ηr = 0, the mouse never learns
+reward associations. For the values in between 0 and 1, the mouse requires extra data
+points for updating its beliefs to the same extent as an ideal observer model. The model
+parameter ηr can be interpreted as the data efficiency of learning. It could be used to
+compare individual learning differences among subjects. Furthermore, it is interesting
+to assess whether differences of ηr in individuals is predictive of comparative learning
+rates on other learning tasks. It also provides a qualitative way to assess the model. For
+example, if the value is unreasonably high, it may indicate a flaw in the state structure
+20
+
+3.3.
+Bayesian learning model
+Table 3.2: Guide for variables (Var) and parameters (Par) relevant to internal reward
+representations.
+.
+Var/Par
+Type
+Description
+Rk
+observed
+A binary pair representing the reward outcome of step k,
+(1, 0) lick and reward within step k
+(0, 1) lick but no reward within step k
+(0, 0) no lick within step k
+B(k)
+inferred
+List of
+�
+β(1)
+k (s), β(2)
+k (s)
+�
+, for all s ∈ S, where
+Beta(
+�
+β(1)
+k (s), β(2)
+k (s)
+�
+) represents the beliefs about
+r(s) at spatial step k.
+ηr
+model parameter
+A constant in the [0, 1] interval for learning rate of
+reward association.
+or an incorrect choice of prior.
+Implementation notes
+To simplify model implementation, we can derive the posterior distribution at step k by
+merely keeping a list record of the total count of positive and negative reward outcomes
+in state s. In particular, at step k, for state s, let ck(s) =
+�
+c(T)
+k (s), c(F)
+k (s)
+�
+be the total
+count of positive and negative outcomes in state s, from step 1 up to the start of step k.
+That is,
+ck(s) =
+k
+�
+n=1
+Xn=s
+Rk.
+For current spatial state k, a list of numbers can store values of ck(s).
+Assuming a
+uniform prior at the start of the experiment, or β(1)
+1 (s) = β(2)
+1 (s) = 1, the prior probability
+distribution of r(s) at step k is
+r(s) ∼ Beta(ηrc(T)
+k (s) + 1, ηrc(F)
+k (s) + 1),
+for which,
+βk(s) = ηrck(s) + 1.
+(3.2)
+3.3.2
+Learning state transitions
+Learning dynamics for state transitions p(s,s′) is defined similarly to the reward associ-
+ations. Let E be the set of transition edges (directed edges), and let Adj(s) = {s′ :
+21
+
+3.3.
+Bayesian learning model
+(s, s′) ∈ E} be the set of states which for Xk = s, outcome of Xk is in Adj(s). There-
+fore, transition probabilities from s, P(Xk+1|Xk = s) is a distribution of outcomes over
+Adj(s). Assuming fixed probability transitions, P(Xk+1|Xk = s) can be represented by a
+list of probabilities p(s) :=
+�
+p(s,s′) : s′ ∈ Adj(s)
+�
+. Note that if the subject is not familiar
+with the space, the true distribution is unknown, and the subject learns about these
+probabilities through experience.
+Normative model for updating internal transition representations
+(Bayesian)
+Every time the subject leaves state s and the next step is observed, one observation is
+made about the outcome of Xk+1 given Xk = s. Because the outcome is a multinomial
+random variable, where possible outcomes are states in Adj(s), we use a Dirichlet prior
+distribution to represent uncertainties about p(s). Specifically, at spatial step k,
+p(s) ∼ Dir
+�
+αk(s)
+�
+where the list of parameters αk(s) contains an element corresponding to each possible
+outcome. In particular,
+αk(s) =
+�
+αk(s, s′) : s′ ∈ Adj(s)
+�
+.
+Suppose Xk = s and consider an ideal observer whose prior beliefs about p(s) at spatial
+step k is described by Dir(αk(s)). Also suppose, the ideal observer visits the next state
+and makes the observation Xk+1 = ˘s. Then by Equation 1.4, the posterior distribution
+is
+p(s)|(Xk+1 = ˘s, Xk = s) ∼ Dir
+�
+αk+1(s)
+�
+where any element αk(s, s′) of αk+1(s) is updated as follows:
+αk+1(s, s′) = 1 + αk(s, s′)
+for s′ = ˘s, and
+αk+1(s, s′) = αk(s, s′)
+for s′ ̸= ˘s.
+Furthermore, for any other state s′′ ̸= s, it is obvious that the beliefs are not updated,
+i.e., αk+1(s′′ ̸= s) = αk(s′′ ̸= s).
+22
+
+3.3.
+Bayesian learning model
+Table 3.3: Parameter guide for learning transition probabilities
+.
+Parameter(s)
+Type
+Description
+(Xk+1|Xk)
+observed
+Transition outcome from a given state Xk
+Ak
+inferred
+List of αk(s), for all s ∈ S, where Dir(αk(s))
+represents beliefs about p(s) at step k (list of state
+transition probabilities from s to adjacent states)
+ηp
+free parameter
+A constant in the [0, 1] interval for learning rate of
+transition probabilities
+Reward learning rate ηp
+Similar to introducing a learning rate for learning reward association, we introduce a
+ηp ∈ [0, 1] to account for data inefficiency compared to the ideal observer. Denote by Ak,
+the list of all learning parameters of state transition probabilities Ak =
+�
+αk(s) : s ∈ S
+�
+.
+Now, the update rule (posterior distribution) is
+p(s)|(Xk+1, Xk) ∼ Dir
+�
+αk+1(s)
+�
+where any element αk(s, s′) of a list of parameters in Ak is updated as follows:
+αk+1(s, s′) = ηp + αk(s, s′)
+for s = Xk and s′ = Xk+1
+(3.3)
+αk+1(s, s′) = αk(s, s′)
+otherwise.
+For an ideal observer, ηp = 1.
+The lower the value of ηp is, the slower the learning
+becomes, because the subject would require more data for similar updates in beliefs. If
+ηp = 0, the subject never learns from observing consecutive states. Note that the same
+parameter ηp is used for learning all transition probabilities.
+Implementation notes
+For prior beliefs about state transitions, a uniform prior would ensure that the prior does
+not privilege any probability value over another probability value. Then, for any entry
+α1(s, s′) of α1(s, s′), we assume that α1(s, s′) = 1
+So, at spatial step k, for entry αs′(s, k) of α(s, k),
+αk(s, s′) = ηpc(s,s′)(k) + 1.
+(3.4)
+23
+
+3.3.
+Bayesian learning model
+where c(s,s′)(k) is the total number of observed transitions from s to s′ from step 1 to step
+k. By keeping track of c(s,s′)(k) in a matrix, any parameter in A(k) can be calculated on
+demand using Equation 3.4 for the current state.
+24
+
+Chapter 4
+Behavioral model part 2: the
+generative model
+In the previous chapter, I discussed the internal representations of spatial regions and
+reward probabilities within those regions. This chapter describes a model that utilizes
+internal representations to generate behaviour. The learning model for updating beliefs
+about reward probabilities and state transitions utilized a normative model of Bayesian
+learning. In contrast, we present a descriptive model of behaviour that does not explic-
+itly enforce any optimal decision-making criteria. Before making normative assumptions
+about behaviour, it is important to have a descriptive framework for systematically as-
+sessing assumptions about behaviour.
+Recall that location, visual stimulus, licking and speed of the mouse are recorded in the
+experimental data (see Chapter 2.2). To improve readability, Table 4.1 includes notation
+used to represent the behavioural data.
+A spatial state transition event triggers updating internal representations of reward prob-
+ability and spatial transitions. During the period between two transition events, the pa-
+rameters associated with internal representations (specified by elements of Bk and Ak)
+are unchanged. Assuming that the internal representations are guiding the behaviour,
+we define behavioural parameters for speed and licking rate derived from internal rep-
+resentations’ parameters. Figure 4.1 describes the conditional dependence structure of
+parameters associated with a spatial state. In this model, the internal representations
+are used to derive two parameters that guide the licking and speed behaviour. These
+25
+
+4.1.
+Spatial state parameter ˜λk: licking rate
+Table 4.1: Behavioral and observational records for t ∈ {1, 2 . . . , N}.
+Data
+Type
+Description
+xt
+Observation
+xt is the true value of the distance from the onset of the current
+corridor at time step t.
+yt
+Observation
+yt ∈ Cor = {grey, vertical, angled} is the true value of the
+corridor type, which determines the visual stimuli at time
+step t.
+ot
+Observation
+ot is a binary value for whether the reward valve has opened
+during the time step.
+vt
+Behavior
+Speed (average) at time step t
+lt
+Behavior
+Number of licks at time step t
+parameters are target speed ˜νk, and licking rate ˜λk, and they are discussed in detail in
+the Section 4.2 and Section 4.1 respectively.
+Table 4.2: Description of updating internal representations of a given step using the graphical
+model of 4.1. Variables ( Var.) and their parents (Par(.)) are included in the first and second
+columns respectively. The third column (Type) indicates whether the outcome of the variable
+given its parents is stochastic ( Stoch.) or deterministic ( Deter.) given its parents. The
+conditional dependence of the variable on its parents is described in the last column.
+Var.
+Par(.)
+Type
+Update description
+Xk+1
+Xk
+Stoch.
+Stochastic outcome of the state immediately
+following Xk.
+Bk+1
+Bk, Rk, Xk
+Deter.
+Updating reward probability distribution of the
+previous state using Equation 3.1.
+Ak+1
+Ak, Xk+1, Xk
+Deter.
+Updating the transition probability distribution
+for the last transition using Equation 3.3.
+distributions for reward, to Bk+1
+rk
+Bk, Xk
+Deter.
+Reward distribution of the current state
+γk(ρ)
+Bk, Ak, Xk
+Deter.
+Discounted reward probability of present and
+future states given by Equation 4.6, with the
+discount factor ρ.
+˜νk
+γk(ρ)
+Deter.
+Value of target speed in spatial step k adjusted by
+value of γk(ρ).
+˜λk
+rk
+Deter.
+Licking rate in step k given by Equation 4.3.
+Rk
+˜λk
+Stoch.
+Reward outcome of spatial state k
+4.1
+Spatial state parameter ˜λk: licking rate
+Consider the relevance of the reward probability distribution for rk to the licking be-
+haviour.
+First, it is reasonable to consider the mouse regulating its licking rate us-
+26
+
+4.1.
+Spatial state parameter ˜λk: licking rate
+ing its perception of expected reward probability in the current state.
+The expected
+value of the reward probability in the current state (in step k) is the expected value of
+Beta(β(1)
+k (s), β(2)
+k (s)), which is
+µ(rk) =
+β(1)
+k
+β(1)
+k β(2)
+k
+.
+(4.1)
+Figure 4.1: Graphical model of updating internal representations at a given spatial step, the
+associated learning parameters (green), and the associated behavioural parameters (blue). The
+dotted squares indicate internal representations that are not observed in the data. Variables
+inside circles have stochastic outcomes given their parents, and variables inside squares have
+deterministic outcomes given their parents. State transitions trigger updating these variables
+for the new step k + 1. Note that the model satisfies the Markov property. A description of the
+conditional dependencies is included in Table 4.2.
+27
+
+Xk
+Xk+1
+μ(rk)
+μ(rk+1)
+Yk(p)
+o(rk)
+Yk+1(p)
+o(rk+1)
+K+
+k+1
+Rk
+Rk+14.1.
+Spatial state parameter ˜λk: licking rate
+Second, independently from the expectation of reward, the degree of uncertainty about
+the true probability of reward may also be relevant to behaviour (Zhao & Warren 2015),
+and in particular, the rate of licking in the current state. More variance in the reward
+probability may mean that the current state should be further explored by licking, to
+decrease the uncertainty about reward values. The variance reward probability beliefs
+can also be calculated from the Beta() distribution.
+σ2(rk) =
+β(1)
+k β(2)
+k
+(β(1)
+k
++ β(2)
+k )2 (β(1)
+k
++ β(2)
+k
++ 1)
+(4.2)
+Let Lt be a random variable for the number of licks at time step t. We assume that the
+licking rate is generated by a Poisson distribution
+Lt ∼ Pois( ˜λk)
+where for model parameters ω1, ω2 and ω3,
+˜λk = ω1µ(rk) + ω2σ(rk) + ω3,
+(4.3)
+is the licking rate at a time step spent within the current spatial step. The probability
+that Lt = lt, for a number of licks lt is given by
+P(Lt = lt) = λlt
+k e−λk
+lt!
+(4.4)
+Table 4.3: Parameters relevant to the licking behaviour.
+.
+Parameter
+Type
+Description
+˜λk
+Spatial state
+parameter
+Rate of the Poisson distribution generating the
+licking behavior within a time step spent in spatial
+step k
+ω1
+Model parameter
+Weight of the expected reward probability of the
+current reward distribution for calculating the
+spatial state parameter ˜λk
+ω2
+Model parameter
+Weight of the standard deviation of the current
+reward distribution for calculating the
+spatial state parameter ˜λk
+ω3
+Model parameter
+base licking rate for calculating ˜λk
+28
+
+4.2.
+Parameter ˜νk: target speed within the current spatial state
+4.2
+Parameter ˜νk: target speed within the current
+spatial state
+We noticed that the mouse tends to speed up if it does not expect a reward in upcoming
+states (for example, see Figures 2.5 and 2.6). We model this behavior using a discounted
+measure of future rewards.
+Discounted future reward
+Expected average reward probability m steps after the current state s can be formulated
+as follows
+�
+s′∈S
+E(r[s′])P(Xk+m = s′|Xk = s)
+(4.5)
+Value of P(Xk+m|Xk) can be estimated by the transition probability matrix obtained
+by the expected value of transition probabilities and standard Markov chain transition
+properties (Equation 1.1) (Häggström et al. 2002). To estimate the values of the transition
+probability matrix, we use the expected value of transition probability for p(s,s′), using
+parameters of Dirichlet distributions for transition probabilities in Ak;
+E[p(s,s′)] =
+αk(s, s′)
+�
+s′′∈Adj(s)
+αk(s, s′′)
+is the estimated probability value for p(s,s′) entry of the transition probability matrix.
+To conclude the discussion for the calculation of expression 4.5, note that E(r[s′]) =
+β(1)
+k (s′)/
+�
+β(1)
+k (s′)β(2)
+k (s′)
+�
+.
+Now, let us define the discounted future reward γk(ρ) for a fixed value of ρ in the current
+step k to be
+γk(ρ) :=
+∞
+�
+m=0
+ρm�
+s′∈S
+�
+E[r(s′)]P(Xk+m|Xk)
+�
+�∞
+m=0 ρm
+(4.6)
+Note that γk(ρ) is a normalised sum of discounted present and future expected re-
+ward probability values. Similar to the value function in reinforcement learning (Sut-
+ton & Barto 2018), or the concept of discounted cash flow in financial asset valuation
+29
+
+4.3.
+Generative model of licking and speed
+(Damodaran 2012), it incorporates all future reward values by iteratively giving less
+weight to future rewards that are further away.
+When transitioning from one state to another, lower discounted future reward γk(ρ) is
+likely to indicate that the next reward is further away. In this case, the mouse may choose
+to adjust its behavior (Kleinfeld et al. 2006), by speeding up to pass the unrewarded
+regions more quickly. Since the discounted value of future reward does not change as
+long as the mouse is in the same spatial state, the desired speed at the current spatial
+step can be modeled as a spatial state parameter. Let the target speed ˜νk for the current
+state be
+˜νk := vmax
+�
+1 − γk(ρ)
+�
+(4.7)
+where vmax is a model parameter that puts an upper bound on the target speed. A simple
+model of speed for time step t is the following
+vt ∼ N( ˜νk, σ2
+ν).
+(4.8)
+However, physical constraints on the movement does not permit an instant jump in speed
+when the spatial state changes. The alternative model of speed that takes the physical
+constraints into considerations (by adding more parameters), is
+vt+1 ∼ N(E[vt+1], V ar[vt+1]),
+(4.9)
+where,
+(E[vt+1], V ar[vt+1]) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(vt + δ+
+v, σ2
+v)
+for vt < ˜νk − ϵ,
+(vt + δ-
+v, σ2
+v)
+for vt > ˜νk + ϵ,
+(vt, σ2
+v)
+otherwise; i.e., for vt ∈ [ ˜νk − ϵ, ˜νk + ϵ].
+(4.10)
+where the model parameters δ+
+v and δ-
+v are constant values for acceleration and deceler-
+ation, σ2
+v is the variance of speed outcome in the next time-step. Furthermore, model
+parameter ϵ determines the range where non-random acceleration or deceleration is not
+enforced.
+4.3
+Generative model of licking and speed
+For given spatial states structure (by fixing parameters V and d), there exists a function
+fV,d : (xLoc × Cor) → S that associates each position to states. Then it is possible to
+30
+
+4.3.
+Generative model of licking and speed
+Table 4.4: Parameters relevant to the speed behavior.
+.
+Parameter
+Type
+Description
+ρ
+Model parameter
+Discount rate of future reward (Expression 4.6)
+˜νk
+Spatial state
+parameter
+Target speed (Expression 4.7)
+σ2
+˜ν
+Model parameter
+Variance of speed in the first model (Expression 4.8)
+σ2
+v
+Model parameter
+Variance of speed change
+Expression 4.9 (second model)
+δ+
+v, δ-
+v
+Model parameter
+Acceleration and deceleration rate (second model)
+ϵ
+Model parameter
+Range of random only of speed change (second model)
+determine time steps associated with state transitions. In Chapter 3.1, we assumed that
+the states are fully observable to the subject. Therefore, the subject knows the value of
+fV,d at any current time step.
+Binary variable Kt: indicator of spatial state transition event
+For the current time step t, let Kt be a binary variable such that
+Kt+1 =
+�
+�
+�
+0,
+for fV,d(xt, yt) = fV,d(xt+1, yt+1)
+1,
+for fV,d(xt, yt) ̸= fV,d(xt+1, yt+1).
+(4.11)
+That is to say, Kt = 1 if (xt, yt) and (xt+1, yt+1) are not in the same state, ans so a state
+transition has occurred. Note that a spatial state transition triggers an update in the
+beliefs about the environment (reward probability within states and state transitions).
+Then the internal representations in the graphical model of Figure 4.1 are updated to the
+next spatial step, and the behavioral parameters λkt+1 and ˜
+nukt+1 correspond to the new
+spatial step. For Kt = 0, the behavioral parameters ˜λkt+1 and ˜
+nukt+1 remain unchanged
+from the previous time-step.
+Figure 4.2 is the graphical model for the generative model of behavior within time steps.
+The model assumes that the spatial state associated with (xt, yt) is unambiguously de-
+termined by the subject (fully observable spatial states). Therefore, the value of Kt+1,
+which indicates a state transition, is also observed by the subject. Furthermore, Kt+1 can
+be deterministically inferred from the experimental data using the Equation 4.11. Hence,
+it is also observed in the behavioral data. If Kt+1 = 1, then the graphical model of up-
+dating internal representations is used to find the new behavioral parameters (indicated
+by green arrows). If Kt+1 = 0, the behavioral parameters remain unchanged from the
+previous step. A description of the relationships is included in Table 4.5.
+31
+
+4.3.
+Generative model of licking and speed
+Table 4.5: Description of relationships in the generative model of behavior in the graphical
+model of 4.2. Variables ( Var.) and their parents (Par(.)) are included in the first and second
+column respectively. Third column (Type) indicates whether the outcome of the variable given
+its parents is stochastic ( Stoch.) or deterministic ( Deter.) given its parents. The conditional
+dependence of the variable on its parents is described in the last column.
+Var.
+Par(.)
+Type
+Update description
+Kt
+(xt, yt)
+(xt+1, yt+1)
+Stoch.
+Transition event indicator (Expression 4.11).
+˜νkt+1
+˜νkt, Kt
+Deter.
+For Kt = 0, ˜νkt+1 = ˜νkt. Otherwise, spatial state changes,
+and graphical model 4.1 updates the value.
+˜λkt+1
+˜λkt, Kt
+Deter.
+For Kt = 0, ˜λkt+1 = ˜λkt. Otherwise, spatial state changes,
+and graphical model 4.1 updates the value.).
+lt
+˜λk
+Stoch.
+Poisson distributed value with rate ˜λk (Expression 4.3)
+vt
+˜νk
+Stoch.
+Speed at time step t by first model (Expression 4.8 ),
+or second model (Expression 4.9.
+Figure 4.2: Graphical model of the generative model of behavior. Note that the variables and
+relationships drawn in yellow and brown are not part of the internal model, and they describe
+the conditional dependence of the observed values to the model variables. See table 4.5 for
+description of the relationships.
+32
+
+(Xt,yt)
+(Xt+1,Yt+1)
+Kt+1
+K
+lt+1
+Vt+14.4.
+Estimation of model parameters
+4.4
+Estimation of model parameters
+Below, the general framework for estimating the model parameters is discussed. For a
+fixed spatial model of space MV,d, let θ be the list of model parameters
+θ := (V, d, ηr, ηp, ω1, ω2, ω3, σ2
+˜ν),
+(using the second speed model), or
+θ := (V, d, ηr, ηp, ω1, ω2, ω3, σ2
+˜v, δ
++
+v, δ-
+v, ϵ)
+(using the first speed model).
+Given the model parameters, and given observational data, parents of vt and lt are deter-
+ministically set at each time point (see graphical model 4.2). Therefore, speed and licking
+are independent. So model likelihood of the generative model of behaviour at time step
+t is
+L
+�
+θ|(vt, lt)
+�
+= P(vt, lt|θ) = P(vt|θ) P(lt|θ)
+∼ f
+�
+µt(θ), σt(θ)
+�
+g
+�
+lt; λt(θ)
+�
+where f are g are probability mass functions for Gaussian and Poisson distributions
+respectively. Note that their distribution parameters are deterministically fixed at each
+time point given the model parameters (see Equations 4.3, 4.8 and 4.9). Then model
+evidence for the generative model for up to time step N is
+L
+�
+θ
+���{(vt, lt) : t = 1 . . . N}
+�
+∝
+N
+�
+t=1
+f
+�
+vt; µt(θ), σt(θ)
+�
+g
+�
+lt; λt(θ)
+�
+(4.12)
+And we can then use the maximum likelihood estimation (MLE) to estimate the fitted
+model parameters
+θ∗ = argmax
+θ
+N
+�
+t=1
+ln
+�
+f
+�
+vt; µt(θ), σt(θ)
+�
+g
+�
+lt; λt(θ)
+��
+(4.13)
+Note that for each spatial step, the graphical model is used for calculating the parameters
+µt(θ), σt(θ) and λt(θ).
+33
+
+Chapter 5
+Discussion
+The next step in the project is to first complete the model validation on synthetic data.
+Before applying the model to real data, it is important to scrutinize the behaviour of the
+generative model. We plan to do so by pre-determining values for a model parameter
+and generating synthetic behavioural data. The generated behaviour is then used as a
+given data set. If the model is well-behaved, the model parameters should be recoverable
+from the synthetic data. As different spatial state structures radically alter the learning
+dynamics, we will conduct the parameter recovery for spatial model parameters more
+diligently. By considering various alternative hypotheses (different values for d and V),
+the model evidence (equation 4.12) of alternative hypotheses will be compared. For a
+well-behaved model, the model evidence for the parameters used to generate data is
+expected to be the best.
+5.1
+Limitations
+While our model assumes fully observable Markov states, noisy observations of the loca-
+tion and visual stimuli introduce uncertainty about the true current state of the system.
+Indeed, observations of the environment are often noisy and some behavioural models
+take this into account (Kang et al. n.d., Kersten & Mamassian 2009). While the learning
+rates of reward probability and transition probability capture some aspects of noisy obser-
+vations, they are not based on normative assumptions. Alternatives should be considered
+for future research (Laquitaine & Gardner 2018). Fortunately, there is an extensive body
+of research on partially observable Markov decision processes (Monahan 1982, Kaelbling
+34
+
+5.2.
+Implications
+et al. 1996) that would provide a clear path for improving the current model.
+An alternative to estimating the model parameters using MLE in Chapter 4.4 is to use
+the maximum a posteriori estimation (MAP) (Murphy 2012, Griffiths & Yuille 2008).
+In contrast to MLE, which gives one estimated value for each parameter, MAP gives
+a distribution for each parameter, characterising the level of uncertainty about each
+parameter. Since some of the model parameters are qualitatively interpretable, MAP
+may be particularly relevant. In particular, a distribution over possible options for V,
+the set of discriminated visual stimuli, is highly relevant to the imaged activity of the
+visual cortex. The potential challenge of MAP is that the computational difficulty of
+the calculation may introduce implementation challenges that are difficult to resolve.
+Nonetheless, its estimation of model parameters are potentially more meaningful for
+studying visual perception.
+5.2
+Implications
+During the experiments, two-photon calcium imaging and optogenetics were performed
+to determine changes in inputs and activity of individual excitatory and inhibitory cells
+within the primary visual cortex. Previously, a multivariate auto-regressive linear model
+(MVAR) was fitted to the neuronal data (Khan et al. 2018):
+qt+1 = qt + A × qt + ut + ξvt
+where qt is the vector of response levels at time step t for all n imaged neurons, A is
+an n × n matrix that includes the fitted interaction parameters, ut is a fitted vector for
+the stimulus-related input, and ξ is a fitted parameter for the contribution of current
+speed vt. The MVAR model was used to compare the activity of populations of different
+inhibitory and excitatory cell types. The only behavioural term that was included was
+speed vt, which did not make a significant contribution. An immediate application of
+the current behavioural model presented in this report is to potentially improve the
+MVAR model by including parameters related to internal representations, In particular,
+learned parameters that are likely to be relevant to behaviour, namely expected reward
+probability µ(rk), variance σ2(rk), and discounted future reward γk(ρ) could potentially
+improve the predictive power of the MVAR model.
+If the internal representation terms from the behavioural model improve the predictive
+power of the MVAR model, it will give new insights into the information encoded in
+neurons located in the primary visual cortex. Future experiments can then be designed to
+35
+
+5.2.
+Implications
+systematically manipulate these internal terms to understand the precise representations
+(Heilbron et al. 2020). This will help us understand how the structure of the environment
+changes learning dynamics and internal representations.
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