iccv iccv2013 iccv2013-357 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Deyu Meng, Fernando De_La_Torre
Abstract: Many problems in computer vision can be posed as recovering a low-dimensional subspace from highdimensional visual data. Factorization approaches to lowrank subspace estimation minimize a loss function between an observed measurement matrix and a bilinear factorization. Most popular loss functions include the L2 and L1 losses. L2 is optimal for Gaussian noise, while L1 is for Laplacian distributed noise. However, real data is often corrupted by an unknown noise distribution, which is unlikely to be purely Gaussian or Laplacian. To address this problem, this paper proposes a low-rank matrix factorization problem with a Mixture of Gaussians (MoG) noise model. The MoG model is a universal approximator for any continuous distribution, and hence is able to model a wider range of noise distributions. The parameters of the MoG model can be estimated with a maximum likelihood method, while the subspace is computed with standard approaches. We illustrate the benefits of our approach in extensive syn- thetic and real-world experiments including structure from motion, face modeling and background subtraction.
Reference: text
sentIndex sentText sentNum sentScore
1 x Abstract Many problems in computer vision can be posed as recovering a low-dimensional subspace from highdimensional visual data. [sent-4, score-0.07]
2 Factorization approaches to lowrank subspace estimation minimize a loss function between an observed measurement matrix and a bilinear factorization. [sent-5, score-0.141]
3 However, real data is often corrupted by an unknown noise distribution, which is unlikely to be purely Gaussian or Laplacian. [sent-8, score-0.196]
4 To address this problem, this paper proposes a low-rank matrix factorization problem with a Mixture of Gaussians (MoG) noise model. [sent-9, score-0.364]
5 The MoG model is a universal approximator for any continuous distribution, and hence is able to model a wider range of noise distributions. [sent-10, score-0.236]
6 The parameters of the MoG model can be estimated with a maximum likelihood method, while the subspace is computed with standard approaches. [sent-11, score-0.095]
7 We illustrate the benefits of our approach in extensive syn- thetic and real-world experiments including structure from motion, face modeling and background subtraction. [sent-12, score-0.081]
8 These linear models have been widely used in computer vision to solve problems such as structure from motion [39], face recognition [43], photometric stereo [19], object recognition [40], motion segmentation [41] and plane-based pose estimation [36]. [sent-15, score-0.127]
9 Standard approaches to subspace learning optimize 1Bold uppercase letters denote matrices, bold lowercase letters denote vectors, and non-bold letters represent scalar variables. [sent-18, score-0.168]
10 di and di represent the ith column and row vectors of the matrix D, respectively, and dij denotes the scalar in the ith row and jth column of D. [sent-19, score-0.07]
11 A closed-form solution for U and V can be computed with the Singular Value Decomposition (SVD) when all data is available (no missing data). [sent-55, score-0.102]
12 However, the L2 norm is only optimal for Gaussian noise and provides biased estimates in presence of outliers and non-Gaussian noise distributions. [sent-56, score-0.375]
13 In order to introduce robustness to outliers, the L2 can be replaced by robust functions [11] or the L1 norm [13, 14, 18, 20, 22, 39, 43]. [sent-57, score-0.079]
14 1337 While L2 or L1 norms are only optimal if the noise follows a Gaussian or Laplacian distribution, this is not the case in most real problems. [sent-59, score-0.144]
15 For instance, consider the case of face images of one subject taken under varying illumination conditions (e. [sent-60, score-0.051]
16 Under the assumption that the face is a Lambertian surface, the faces under different point light sources can be recovered by a three dimensional subspace (e. [sent-64, score-0.152]
17 If the diffusion or background illumination is considered in the model the subspace will be of dimension four [21]. [sent-67, score-0.1]
18 However, in real images there are different types of noise sources. [sent-68, score-0.17]
19 Third, the camera noise (“read noise”) is amplified in the dark areas [30] (see Fig. [sent-71, score-0.204]
20 These different types of noise can have different distributions, and minimizing either the L2 or L1 loss is unlikely to produce a good model to factorize the illumination component (see Fig. [sent-73, score-0.203]
21 The key idea is to model the noise as a Mixture of Gaussians (MoG) [26], which is an universal approximator to any continuous density function [25]. [sent-77, score-0.236]
22 1(d) illustrates how the proposed MoG noise model can better account for the different types of noise and provide a better estimate of the underlying face. [sent-80, score-0.314]
23 The effectiveness of our MoG method is shown in synthetic, Structure From Motion (SFM), face modeling and background subtraction experiments. [sent-82, score-0.13]
24 Previous work The L2-norm LRMF with missing data has been studied in the statistical literature since the early 80’s. [sent-84, score-0.102]
25 Gabriel and Zamir [15] proposed a weighted SVD technique that used alternated minimization (or criss-cross regression) to find the principal subspace of the data. [sent-85, score-0.181]
26 De la Torre and Black [11] proposed Robust PCA by changing the L2 norm to a robust function to deal with outliers. [sent-86, score-0.109]
27 This approach can handle missing data by setting weights to zero in the IRLS algorithm, but it is prone to local minima. [sent-88, score-0.102]
28 [1] introduced a globally optimal solution to L2 LRMF with missing data under the assumption that the missing data has a special Young diagram structure. [sent-95, score-0.204]
29 [28] showed that the matrix factorization problem can be formulated as a lowrank semidefinite program and proposed an augmented La- grangian method. [sent-98, score-0.22]
30 In order to introduce robustness to outliers, Ke and Kanade [20] suggested replacing the L2 loss with the L1 norm, minimized by alternated linear or quadratic programming (ALP/AQP). [sent-100, score-0.065]
31 Similarly to the L2 Wiberg approach [3 1], Eriksson and Hengel [14] experimentally showed that the alternated convex programming approach frequently does not converge to the desired point, and introduced the L1Wiberg approach to address this. [sent-103, score-0.065]
32 [45] extended [ 14] by adding a nuclear norm regularizer on V and the orthogonality constraints in U, which resulted in improvements on the structure from motion problem. [sent-105, score-0.071]
33 A major advantage of this approach lies in its convex formulation even in the case of sparse outliers and missing data. [sent-108, score-0.142]
34 Beyond these deterministic LRMF methods, there has been several probabilistic extensions of matrix factorizations. [sent-110, score-0.088]
35 Factor analysis (FA) [4] is a probabilistic extension of PCA that assumes normally distributed coefficients (U) and a diagonal Gaussian noise model. [sent-111, score-0.221]
36 Unlike FA, PPCA assumes an isotropic Gaussian noise models. [sent-113, score-0.144]
37 Other probabilistic extensions include the mixture of PPCA [37] that extends PPCA by considering a mixture model in which the components are probabilistic PCA models (Mixture PCA). [sent-114, score-0.244]
38 Recently, some probabilistic frameworks for robust matrix factorization [42, 23, 3] have been further proposed and model the noise with a Laplacian or student-t distributions. [sent-115, score-0.413]
39 Unlike 1338 previous work, we model our noise as a MoG and not a particular unimodal distribution. [sent-116, score-0.144]
40 LRMF with MoG noise This section proposes a new LRMF method with a MoG noise model, a new matrix factorization method that accounts for multi-modal noise distributions. [sent-118, score-0.652]
41 The subspace-MoG model In LRMF, each element xij (i = 1, 2, · · · , d, j = 1, 2, · · · , n) of the input matrix X can = be m 1o,2d,e·le·d· as xij = (ui)Tvj + εij, (2) where ui and vi are the ith row vectors of U and V, respectively, and εij denotes the noise in xij . [sent-121, score-0.912]
42 Since it is an universal approximator to any continuous distributions [25]. [sent-124, score-0.092]
43 Then, th≥e probability oxifn egac phr oeploemrtieonnt xij of X? [sent-135, score-0.208]
44 E Step: Assume a latent variable zijk in the model, with zijk ∈ {0, 1} and zijk = 1, indicating the assignment ∈of { th0e, n1o}i saen εij? [sent-163, score-0.225]
45 nsibility of mixture k (= 1, 2, · · · , K) fTohre generating ethspeo nnsoibisiel oyf xij x(it r=e 1, 2, ·1 1·, ,·2 , ·d,· j K=) Π, Π, ? [sent-167, score-0.317]
46 , K = 6) that is large enough to fit the noise distribution in all our experiments. [sent-214, score-0.144]
47 In our subspace-MoG model, however, an outlier will belong to a mixture with large variance, and wij will have a small value based on Eq. [sent-225, score-0.123]
48 Experiments To evaluate the performance of the proposed subspaceMoG method, we conducted extensive synthetic, Structure From Motion (SFM), face modeling and background subtraction experiments. [sent-229, score-0.13]
49 In the synthetic and SFM experiments we analyzed the performance of our algorithm in situations when the ground truth is known and we added different types of noise to it. [sent-230, score-0.214]
50 The experiments in face modeling and background subtraction illustrate how the the MoG noise model is a realistic assumption for visual data. [sent-231, score-0.274]
51 To properly measure the capability of various nonconvex LRMF optimization models, all competing methods (except SVD and nuclear-norm based Robust PCA [43]) are run with 10 random initializations, and the best result is selected. [sent-234, score-0.058]
52 The NN-Robust method provides the rank of the matrix as a function of the regularization param2http://www. [sent-239, score-0.085]
53 Synthetic experiments Four sets of synthetic experiments were designed to evaluate the performance of our method against other LRMF methods with different types of noise. [sent-266, score-0.07]
54 In each experiment, we randomly specified 2u0ti%on no Nf missing enn etraicehs in Xgt and further added different types of noise as follows: (1) No noise added. [sent-273, score-0.416]
55 (4) Mixture nuoniisfeor: 2l0y% d ostfr tbhuet ednt nrioeiss were corrupted 5w]. [sent-281, score-0.052]
56 ith ( uniformly distributed noise over [−5, 5], 20% are contaminated with dGiasutrsisbiuatne dno nioseis e N o(v0e, r0 [. [sent-282, score-0.202]
57 2−)5, ,a5n]d, 2t0he% remaining m40in%a are corrupted aGnau nossisiaen N Nno(0is,e0 . [sent-283, score-0.052]
58 It can be observed from Table 1 that although the L1 and L2 LRMF methods generally perform better in terms of E1 and E2, respectively, the proposed subspace-MoG method performs best or the second best in all experiments in estimating a better subspace from noisy data (measurements E3-E6). [sent-356, score-0.07]
59 Particularly, in the fourth set of experiments(when the noise is a mixture) our method always performs best in all errors. [sent-357, score-0.144]
60 The matorvixe rc 3on6ta firanms aers,ou lenadd i7n7g% t missing ×da 7ta2 d muaet rtiox . [sent-362, score-0.102]
61 We added four types of noise to the matrix: (1) No noise added. [sent-364, score-0.314]
62 h (e2 non-missing eoilseemNe nt(s0 were corrupted by uniformly distributed noise Mcoirxrtuuprete dn boyise u: f1o0r%m oyf d tihster non-missing corrupted by uniformly distributed noise remaining b9y0% un were yco dnitsatrmiibnuatetedd woiisthe N(0, 10). [sent-367, score-0.58]
63 Similar to our synthetic experiments, the proposed subspace-MoG method does not perform best among all competing methods in terms of E1 and E2. [sent-426, score-0.102]
64 Especially, it performs the best in terms of both E3 and E4 in the mixture noise experiments. [sent-428, score-0.223]
65 The data matrix is generated by extracting the first subset of the Extended Yale B database [16, 5], containing 64 faces of one subject with size 192 [11668, 5u]n,d ceor ndtiafifneirnegnt 6 i4ll fuamceisna otifo onns. [sent-432, score-0.076]
66 Assuming perfect conditions, these face images would lie on a 4-D subspace [5]. [sent-438, score-0.121]
67 we thus set the rank to 4 for all competing methods except nuclear-norm based Robust PCA, which automatically selects the rank. [sent-439, score-0.098]
68 2 compares some reconstructed images obtained by these competing methods. [sent-441, score-0.058]
69 2, it is easy to observe that the proposed method, as well as the other competing methods, are capable of removing the cast shadows and saturations in the faces, as shown in the first row of Fig. [sent-443, score-0.109]
70 However, our method performs better over the faces with large dark regions, as shown in the 2on − 4th rows of Fig. [sent-445, score-0.056]
71 Unlike existing methods, our approach is able to model a mixture of Gaussians noise. [sent-449, score-0.079]
72 One mixture, with large variance models shadows, saturated regions and outliers in the face (see the blue and yellow areas in the second noise image of Fig. [sent-450, score-0.235]
73 The other mixture, with smaller variance, accounts for the camera noise 1341 MoGIRLS[11]WLRA[35]DN[6]ALP[20]L1Wiberg[14]RegL1ALM[45]CWM[27]NN-Robust PCA[7] No Noise (log values) E1−18. [sent-452, score-0.144]
74 Six measures of error for the synthetic example with different noise models. [sent-718, score-0.188]
75 Performance comparison of the competing methods in the SFM experiments. [sent-849, score-0.058]
76 From left to right: RegL1ALM PCAL1 CWM NN-Robust PCA original face images, faces re- × constructed by MoG, Robust PCA (IRLS), SVD, RegL1ALM, PCAL1, CWM and nuclear-norm based Robust PCA, respectively. [sent-852, score-0.082]
77 which are specially amplified in the dark area of the face (see the first noise image of Fig. [sent-853, score-0.255]
78 Background subtraction experiments This experiment compares our approach in the problem of background subtraction [11]. [sent-857, score-0.128]
79 The dimension of the subspace was set to 6 to the videos from [24] and 15 for the video in [ 1 1]. [sent-872, score-0.07]
80 The foreground extracted by the other− competing methods is more coarse because it merges the object and its shadow (see Frame 402 for easy visualization). [sent-878, score-0.085]
81 Conclusions This paper proposes a new low-rank factorization method to estimate subspaces with an unknown noise distribution. [sent-883, score-0.319]
82 The noise is modeled as a MoG, and the parameters of the subspace-MoG model are learned from data automatically. [sent-884, score-0.144]
83 Compared to existing L2 and L1LRMF methods that are optimal for Gaussian or Laplacian noises, our method performs better (on average) in a wide variety of synthetic and real noise experiments. [sent-885, score-0.188]
84 Our method has proven useful in modeling different types of noise in faces under different illuminations and background subtraction. [sent-886, score-0.231]
85 Finally, adding robustness to different types of noise can be similarly applied to other analysis methods (e. [sent-889, score-0.17]
86 Damped Newton algorithms for matrix factorization with missing data. [sent-932, score-0.296]
87 Optimization algorithms on subspaces: Revisiting missing data problem in low-rank matrix. [sent-949, score-0.102]
88 Maximum likelihood from incomplete data via the em algorithm. [sent-968, score-0.065]
89 R1-PCA: Rotational invariant l1norm principal component analysis for robust subspace factorization. [sent-977, score-0.181]
90 Efficient computation of robust low-rank matrix approximations in the presence of missing data using the l1 norm. [sent-982, score-0.179]
91 Lower rank approximation of matrices by least squares with any choice of weights. [sent-988, score-0.065]
92 Robust l1norm factorization in the presence of outliers and missing data by alternative convex programming. [sent-1027, score-0.291]
93 A cyclic weighted median method for l1low-rank matrix factorization with missing entries. [sent-1074, score-0.296]
94 Large-scale matrix factorization with missing data under additional constraints. [sent-1080, score-0.296]
95 On the wiberg algorithm for matrix factorization in the presence of missing components. [sent-1099, score-0.348]
96 Efficient algorithm for low-rank matrix factorization with missing components and performance comparison of latest algorithms. [sent-1105, score-0.296]
97 Shape and motion from image streams under orthography: a factorization method. [sent-1141, score-0.173]
98 Multiframe motion segmentation with missing data using power factorization and GPCA. [sent-1152, score-0.275]
99 Robust principal component analysis: Exact recovery of corrupted low-rank matrices by convex optimization. [sent-1172, score-0.156]
100 Successively alternate least square for low-rank matrix factorization with bounded missing data. [sent-1178, score-0.296]
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