nips nips2012 nips2012-103 knowledge-graph by maker-knowledge-mining

103 nips-2012-Distributed Probabilistic Learning for Camera Networks with Missing Data

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Author: Sejong Yoon, Vladimir Pavlovic

Abstract: Probabilistic approaches to computer vision typically assume a centralized setting, with the algorithm granted access to all observed data points. However, many problems in wide-area surveillance can beneﬁt from distributed modeling, either because of physical or computational constraints. Most distributed models to date use algebraic approaches (such as distributed SVD) and as a result cannot explicitly deal with missing data. In this work we present an approach to estimation and learning of generative probabilistic models in a distributed context where certain sensor data can be missing. In particular, we show how traditional centralized models, such as probabilistic PCA and missing-data PPCA, can be learned when the data is distributed across a network of sensors. We demonstrate the utility of this approach on the problem of distributed afﬁne structure from motion. Our experiments suggest that the accuracy of the learned probabilistic structure and motion models rivals that of traditional centralized factorization methods while being able to handle challenging situations such as missing or noisy observations. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract Probabilistic approaches to computer vision typically assume a centralized setting, with the algorithm granted access to all observed data points. [sent-5, score-0.401]

2 However, many problems in wide-area surveillance can beneﬁt from distributed modeling, either because of physical or computational constraints. [sent-6, score-0.16]

3 Most distributed models to date use algebraic approaches (such as distributed SVD) and as a result cannot explicitly deal with missing data. [sent-7, score-0.497]

4 In this work we present an approach to estimation and learning of generative probabilistic models in a distributed context where certain sensor data can be missing. [sent-8, score-0.297]

5 In particular, we show how traditional centralized models, such as probabilistic PCA and missing-data PPCA, can be learned when the data is distributed across a network of sensors. [sent-9, score-0.661]

6 We demonstrate the utility of this approach on the problem of distributed afﬁne structure from motion. [sent-10, score-0.204]

7 Our experiments suggest that the accuracy of the learned probabilistic structure and motion models rivals that of traditional centralized factorization methods while being able to handle challenging situations such as missing or noisy observations. [sent-11, score-0.783]

8 1 Introduction Traditional computer vision algorithms, particularly those that exploit various probabilistic and learning-based approaches, are often formulated in centralized settings. [sent-12, score-0.464]

9 A scene or an object is observed by a single camera with all acquired information centrally processed and stored in a single knowledge base (e. [sent-13, score-0.332]

10 Even if the problem setting relies on multiple cameras, as may be the case in multi-view or structure from motion (SfM) tasks, all collected information is still processed and organized in a centralized fashion. [sent-16, score-0.491]

11 Nevertheless, the overall goal of such distributed device (camera) networks may still be to exchange information and form a consensus interpretation of the visual scene. [sent-18, score-0.304]

12 For instance, even if a camera observes a limited set of object views, one would like its local computational model to reﬂect a general 3D appearance of the object visible by other cameras in the network. [sent-19, score-0.548]

13 A number of distributed algorithms have been proposed to address the problems such as calibration, pose estimation, tracking, object and activity recognition in large camera networks [1–3]. [sent-20, score-0.497]

14 In order to deal with high dimensionality of vision problems, distributed latent space search such as decentralized variants of PCA have been studied in [4, 5]. [sent-21, score-0.282]

15 A more general framework using distributed least squares [6] based on distributed averaging of sensor fusions [7] was introduced for PCA, triangulation, pose estimation and SfM. [sent-22, score-0.394]

16 Similar approaches have been extended to settings such as the distributed object tracking and activity interpretation [8,9]. [sent-23, score-0.248]

17 Even though the methods such as PCA or Kalman ﬁltering have their well-known probabilistic counterparts, the aforementioned approaches do not use probabilistic formulation when dealing with the distributed setting. [sent-24, score-0.308]

18 One critical challenge in distributed data analysis includes dealing with missing data. [sent-25, score-0.332]

19 In camera networks, different nodes will only have access to a partial set of data features because of varying camera views or object movement. [sent-26, score-0.647]

20 For instance, object points used for SfM may be visible only 1 in some cameras and only in particular object poses. [sent-27, score-0.318]

21 As a consequence, different nodes will be frequently exposed to missing data. [sent-28, score-0.18]

22 However, most current distributed data analysis methods are algebraic in nature and cannot seamlessly handle such missing data. [sent-29, score-0.31]

23 In this work we propose a distributed consensus learning approach for parametric probabilistic models with latent variables that can effectively deal with missing data. [sent-30, score-0.566]

24 Furthermore, we assume that some of the data features may be missing across different nodes. [sent-34, score-0.15]

25 The goal of the network of sensors is to learn a single consensus probabilistic model (e. [sent-35, score-0.214]

26 , 3D object structure) without ever resorting to a centralized data pooling and centralized computation. [sent-37, score-0.787]

27 We will demonstrate that this task can be accomplished in a principled manner by local probabilistic models and in-network information sharing, implemented as recursive distributed probabilistic learning. [sent-38, score-0.337]

28 In particular, we focus on probabilistic PCA (PPCA) as a prototypical example and derive its distributed version, the D-PPCA. [sent-39, score-0.223]

29 We then suggest how missing data can be handled in this setting using a missing-data PPCA and apply this model to solve the distributed SfM task in a camera network. [sent-40, score-0.563]

30 Our model is inspired by the consensus-based distributed Expectation-Maximization (EM) algorithm for Gaussian mixtures [10], which we extend to deal with generalized linear Gaussian models [11]. [sent-41, score-0.187]

31 These assumptions are reasonably applicable to many real world camera network settings. [sent-44, score-0.285]

32 In Section 2, we ﬁrst explain the general distributed probabilistic model. [sent-45, score-0.223]

33 Section 3 shows how DPPCA can be formulated as a special case of the probabilistic framework and propose the means for handling missing data. [sent-46, score-0.213]

34 2 Distributed Probabilistic Model We start our discussion by ﬁrst considering a general parametric probabilistic model in a centralized setting and then we show how to derive its distributed form. [sent-50, score-0.608]

35 Our model is a joint density deﬁned on (xn , zn ) with a global parameter θ (xn , zn ) ∼ p(xn , zn |θ), with p(X, Z|θ) = n p(xn , zn |θ), as depicted in Fig. [sent-56, score-0.25]

36 It is important to point out that each posterior density estimate at point n depends solely on the corresponding measurement xn and does not depend on any other xk , k = n. [sent-60, score-0.105]

37 , Ni }, where xin ∈ RD is n-th measurement vector and Ni is the number of samples collected in i-th node. [sent-70, score-0.207]

38 2 (a) Centralized (b) Distributed (c) Augmented Figure 1: Centralized, distributed and augmented models for probabilistic PCA. [sent-75, score-0.264]

39 Still, the centralized model can be equivalently deﬁned using the set of local parameters, with an additional constraint on their consensus, θ1 = θ2 = · · · = θ|V | . [sent-78, score-0.384]

40 The simple consensus tying can be more conveniently deﬁned using a set of auxiliary variables ρij , one for each edge eij (Fig. [sent-81, score-0.159]

41 This now leads to the ﬁnal distributed consensus learning formulation, similar to [10]: ˆ θ = arg min − log p(X|θ, G) s. [sent-83, score-0.279]

42 The last term (modulated by η) is not strictly necessary for consensus but introduces additional regularization. [sent-91, score-0.119]

43 This classic (ﬁrst introduced in 1970s) meta decompose algorithm can be used to devise a distributed counterpart for any centralized problem that attempts to maximize a global log likehood function over a connected network. [sent-94, score-0.555]

44 3 Distributed Probabilistic PCA (D-PPCA) We now apply the general distributed probabilistic learning explained above to the speciﬁc case of distributed PPCA. [sent-95, score-0.383]

45 Traditional centralized formulation of probabilistic PCA (PPCA) [17] assumes that latent variable zin ∼ N (zin |0, I), with a generative relation xin = Wi zin + µi + where i i, (3) ∼ N ( i |0, a−1 I) and ai is the noise precision. [sent-96, score-1.345]

46 i T where Li = Wi Wi + We can ﬁnd optimal parameters Wi , µi , ai by ﬁnding the maximum likelihood estimates of the marginal data likelihood or by applying the EM algorithm on expected complete data log likelihood with respect to the posterior density p(Zi |Xi ). [sent-98, score-0.143]

47 1 Distributed Formulation The distributed algorithm developed in Section 2 can be directly applied to this PPCA model. [sent-100, score-0.16]

48 The local parameter i estimates are then computed using the consensus updates that combine local summary data statistics with the information about the model conveyed through neighboring network nodes. [sent-103, score-0.191]

49 Let Θi = {Wi , µi , ai } be the set of parameters for each node i. [sent-105, score-0.159]

50 The global constrained consensus optimization now becomes Wi = ρij , ρij = Wj , i ∈ V, j ∈ Bi , µi = φij , φij = µj , i ∈ V, j ∈ Bi , ai = ψij , ψij = aj , i ∈ V, j ∈ Bi min{Wi ,µi ,ai :i∈V } −F (Θi ) s. [sent-106, score-0.26]

51 Exploiting the posterior density in (4), we compute the expected mean and variance of latent variables in each node as T E[zin ] = L−1 Wi (xin − µi ), i E[zin zT ] = a−1 L−1 + E[zin ]E[zin ]T . [sent-114, score-0.152]

52 (12) j∈Bi For ai , we solve the quadratic equation 0=− + Ni D (t+1) 2 (t+1) + 2η|Bi |ai + ai · 2 1 2 Ni (t) (t) 2βi − η (t) ai + aj T E[zin ]T Wi (xin − µi ) − n=1 j∈Bi Ni T ||xin − µi ||2 + tr E[zin zT ]Wi Wi in . [sent-116, score-0.339]

53 (13) n=1 The overall distributed EM algorithm for D-PPCA is summarized in Algorithm 1. [sent-117, score-0.16]

54 4 , Algorithm 1 Distributed Probabilistic PCA (D-PPCA) (0) (0) (0) (0) Require: For every node i initialize Wi , µi , ai randomly and set λi for t = 0, 1, 2, . [sent-119, score-0.159]

55 Hence, we adopt D-PPCA as a method to deal with missing data in a distributed consensus setting. [sent-132, score-0.456]

56 Generalization to missing data D-PPCA from D-PPCA is straightforward and follows [18]. [sent-133, score-0.15]

57 4 D-PPCA for Structure from Motion (SfM) In this section, we consider a speciﬁc formulation of the modiﬁed distributed probabilistic PCA for application in afﬁne SfM. [sent-141, score-0.223]

58 In SfM, our goal is to estimate the 3D location of N points on a rigid object based on corresponding 2-D points observed from multiple cameras (or views). [sent-142, score-0.278]

59 The dimension D of our measurement matrix is thus twice the number of frames each camera observed. [sent-143, score-0.334]

60 Given a 2D (image coordinate) measurement matrix X, of size 2 · #f rames × #points, the matrix is factorized into a 2 · #f rames × 3 motion matrix M and the 3 × #points 3D structure matrix S. [sent-145, score-0.337]

61 In the centralized setting this can be easily computed using SVD on X. [sent-146, score-0.364]

62 Equivalently, the estimates of M and S can be found using inference and learning in a centralized PPCA, where M is treated as the PPCA parameter and S is the latent structure. [sent-147, score-0.414]

63 However, the above deﬁned (2 · #f rames × #points) data structure of X is not amenable to distribution of different views (cameras, nodes), as considered in Section 3 of D-PPCA. [sent-149, score-0.167]

64 The latent D-PPCA variables will model the unknown and uncertain motion of each camera (and/or object in its view). [sent-158, score-0.441]

65 One should note that we have implicitly assumed, in a standard D-PPCA manner, that each column of Zi is iid and distributed as N (0, I). [sent-160, score-0.186]

66 However, each pair of subsequent Zi columns represents one 3 × 2 afﬁne motion matrix. [sent-161, score-0.103]

67 The reason is that occlusions, the main source of missing data, cannot be treated as a random process. [sent-168, score-0.174]

68 However, as we demonstrate in experiments this assumption does not adversely affect SfM when the number of missing points is within a reasonable range. [sent-171, score-0.221]

69 1 Empirical Convergence Analysis Using synthetic data generated from Gaussian distribution, we observed that D-PPCA works well regardless of the number of network nodes, topology, choice of the parameter η or even with missing values in both MAR and MNAR cases. [sent-175, score-0.229]

70 2 Afﬁne Structure from Motion We now show that the modiﬁed D-PPCA can be used as an effective framework for distributed afﬁne SfM. [sent-178, score-0.16]

71 We assume that correspondences across frames and cameras are known. [sent-180, score-0.186]

72 For the missing values of MNAR case, we either used the actual occlusions to induce missing points or simulated consistently missing points over several frames. [sent-181, score-0.57]

73 1 Synthetic Data (Cube) We ﬁrst generated synthetic data with a rotating unit cube and 5 cameras facing the cube in a 3D space, similar to synthetic experiments in [6]. [sent-184, score-0.436]

74 We extracted 8 cube points projected on each camera view every 6◦ , i. [sent-186, score-0.37]

75 For all synthetic and real SfM experiments, we picked η = 10 and initialized Wi matrix with feature point coordinates of the ﬁrst frame visible in the i-th camera with some small noise. [sent-191, score-0.355]

76 To measure the performance, we computed maximum subspace angle between the ground truth 3D coordinates and our estimated 3D structure matrix. [sent-193, score-0.264]

77 Red circles are camera locations and blue arrows indicate each camera’s facing direction. [sent-212, score-0.28]

78 Green and red crosses in the right plot are outliers for centralized SVD-based SfM and D-PPCA for SfM, respectively. [sent-213, score-0.364]

79 The mean subspace angle tends to be slightly larger than that estimated by the centralized SVD SfM, however both reside within the overlapping conﬁdence intervals. [sent-215, score-0.526]

80 66◦ for 20% missing points averaged over 10 different missing point samples. [sent-217, score-0.343]

81 Intuitively, this is because the missing points in the scene are naturally not random. [sent-219, score-0.213]

82 However, we argue that D-PPCA can still handle missing points given the evidence below. [sent-220, score-0.193]

83 The dataset provides various objects rotating on a turntable under different lighting conditions. [sent-224, score-0.124]

84 The views of most objects were taken every 5◦ which make it challenging to extract feature points with correspondence across frames. [sent-225, score-0.13]

85 Due to the lack of the ground truth 3D coordinates, we compared the subspace angles between the structure inferred using the traditional centralized SVD-based SfM and the D-PPCA-based SfM. [sent-235, score-0.549]

86 Experimenal results indicate existance of differences between the reconstructions obtained by centralized factorization approach and that of D-PPCA. [sent-238, score-0.401]

87 Moreover, re-projecting back to the camera coordinate space resulted in close matching with the tracked feature points, as shown in videos provided in supplementary materials. [sent-241, score-0.277]

88 We collected 135 single-object sequences containing image coordinates of points and we simulated multi-camera setting by partitioning the frames sequentially and almost equally for 5 nodes and the network was connected using ring topology. [sent-244, score-0.22]

89 Again, we computed maximum subspace angle between centralized SVD-based SfM and distributed D-PPCA for SfM. [sent-245, score-0.686]

90 MAR results provide variances over both various initializations and missing value settings. [sent-254, score-0.15]

91 Object BallSander BoxStuff Rooster Standing StorageBin # Points # Frames 62 67 189 310 102 30 30 30 30 30 Subspace angle b/w centralized SVD SfM and D-PPCA (degree) Mean 1. [sent-255, score-0.454]

92 2002 Subspace angle b/w fully observable centralized PPCA SfM and D-PPCA with MAR (degree) Mean 6. [sent-265, score-0.454]

93 0444 Subspace angle b/w fully observable centralized PPCA SfM and D-PPCA with MNAR (degree) Mean 3. [sent-282, score-0.454]

94 Moreover, more than 53% of all objects yielded the subspace angle below 1◦ , 77% of them below 5◦ and more than 94% were less than 15◦ . [sent-296, score-0.226]

95 6 Discussion and Future Work In this work we introduced a general approach for learning parameters of traditional centralized probabilistic models, such as PPCA, in a distributed setting. [sent-301, score-0.629]

96 Our synthetic data experiments showed that the proposed algorithm is robust to choices of initial parameters and, more importantly, is not adversely affected by variations in network size, topology or missing values. [sent-302, score-0.282]

97 In the SfM problems, the algorithm can be effectively used to distribute computation of 3D structure and motion in camera networks, while retaining the probabilistic nature of the original model. [sent-303, score-0.464]

98 In particular, we assume the independence of the afﬁne motion matrix parameters in (15). [sent-305, score-0.103]

99 The assumption is clearly inconsistent with the modeling of motion on the SE(3) manifold. [sent-306, score-0.103]

100 Shape and motion from image streams under orthography: a factorization method. [sent-423, score-0.14]

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