iccv iccv2013 iccv2013-45 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Yu-Tseh Chi, Mohsen Ali, Muhammad Rushdi, Jeffrey Ho
Abstract: This paper proposes a novel approach for sparse coding that further improves upon the sparse representation-based classification (SRC) framework. The proposed framework, Affine-Constrained Group Sparse Coding (ACGSC), extends the current SRC framework to classification problems with multiple input samples. Geometrically, the affineconstrained group sparse coding essentially searches for the vector in the convex hull spanned by the input vectors that can best be sparse coded using the given dictionary. The resulting objectivefunction is still convex and can be efficiently optimized using iterative block-coordinate descent scheme that is guaranteed to converge. Furthermore, we provide a form of sparse recovery result that guarantees, at least theoretically, that the classification performance of the constrained group sparse coding should be at least as good as the group sparse coding. We have evaluated the proposed approach using three different recognition experiments that involve illumination variation of faces and textures, and face recognition under occlusions. Prelimi- nary experiments have demonstrated the effectiveness of the proposed approach, and in particular, the results from the recognition/occlusion experiment are surprisingly accurate and robust.
Reference: text
sentIndex sentText sentNum sentScore
1 edu @ Abstract This paper proposes a novel approach for sparse coding that further improves upon the sparse representation-based classification (SRC) framework. [sent-5, score-0.703]
2 The proposed framework, Affine-Constrained Group Sparse Coding (ACGSC), extends the current SRC framework to classification problems with multiple input samples. [sent-6, score-0.172]
3 Geometrically, the affineconstrained group sparse coding essentially searches for the vector in the convex hull spanned by the input vectors that can best be sparse coded using the given dictionary. [sent-7, score-1.353]
4 Furthermore, we provide a form of sparse recovery result that guarantees, at least theoretically, that the classification performance of the constrained group sparse coding should be at least as good as the group sparse coding. [sent-9, score-1.409]
5 The dictionary is obtained virtually without cost by simply stacking the training samples into a matrix D. [sent-17, score-0.216]
6 edu s ing testing, a test image x is sparse coded with respect to the dictionary by optimizing a (convex) objective function E(c; D) of the sparse coefficient c that is usually a sum of an c? [sent-24, score-0.701]
7 The proposed affineconstrained group sparse coding (ACGSC) model aims to further improve the effectiveness of SRC by addressing two of these shortcomings: its generalizability and its extension for multiple input images. [sent-35, score-0.712]
8 Therefore, there is a need to properly generalize SRC for classification problems that require decision on a group of test data x1, . [sent-38, score-0.42]
9 Right: Illustration of the selected atoms from D and the sparse coefficients c of x∗ shown on the right. [sent-54, score-0.398]
10 (Figure best viewed in color) ent setting for SRC, and a straightforward application of SRC would be to apply SRC individually to each xi and generate the final classification decision by pooling or voting among the individual classification results. [sent-56, score-0.391]
11 Group sparse coding [1] (GSC) offers a solution that takes into account the potential relation among [x1, . [sent-58, score-0.392]
12 , xk] by sparse coding all data simultaneously using the objective function mCin E(C;X,D) = ? [sent-61, score-0.392]
13 The matrix C of sparse coefficients can be used as in SRC to generate classification decision by applying voting or pooling across its rows. [sent-66, score-0.482]
14 For classification problems in computer vision, this pa- per argues that a generalization of the group sparse coding, affine-constrained group sparse coding, using the following objective function, offers a more principled and flexible approach: E(a, c; X, D) = ? [sent-68, score-0.916]
15 , is a k-dimensional vector with nonnegative components satisfying the affine constraint a1 + . [sent-74, score-0.225]
16 (2), the ACGSC enlarges its feasible domain by including a kdimensional vector a. [sent-79, score-0.171]
17 However, the feature vector c used in classification is in fact a vector not a matrix, and comparing with GSC, the classification decision based on Eq. [sent-80, score-0.252]
18 Geometrically, ACGSC is easy to explain as it simply searches for the vector in the convex hull S generated by x1, . [sent-82, score-0.321]
19 xk that can best be sparse coded using the dictionary D. [sent-85, score-0.431]
20 From classification viewpoint, ACGSC benefits from multiple inputs by using a linear generative model to “align” the test images with the dictionary elements. [sent-86, score-0.319]
21 For example, if the test (face) images were taken under illumination conditions that are very different from the ones for the dictionary elements, sparse coding test images individually or in group can be expected to provide poor classification results. [sent-87, score-1.086]
22 Heuristically, this can be explained as a kind of generalized misalignment (in terms of illumination effects) between the training and test images. [sent-88, score-0.218]
23 For ACGSC, the coefficient a is used to “align” the test images with the dictionary el- ements, and it is precisely this online “alignment” of the input images xi that provides ACGSC with an edge over other SRC methods. [sent-89, score-0.339]
24 Second, in most applications, the linear generative model given by Xa is valid only for restricted a, and the nonnegative affine constraint proposed here is sufficiently general to provide a bounded and tractable domain for efficient optimization. [sent-93, score-0.275]
25 We remark that for image-based applications, the test images xi have nonnegative intensity values, and therefore, their convex hull would never contain the zero vector (i. [sent-94, score-0.576]
26 Likewise, for a typical group of data xi, their convex hull will not contain the zero vector. [sent-97, score-0.486]
27 The argument in favor of affine-constrained group sparse coding relies mainly on a form of sparse recovery guarantee presented in Theorem 1below. [sent-100, score-0.895]
28 We propose a novel sparse representation-based classification framework based on affine-constrained group sparse coding, and it provides a principled extension of the current SRC framework to classification problems with multiple input samples. [sent-104, score-0.946]
29 We provide theoretical analysis of the proposed frame668822 work in the form of a sparse recovery result. [sent-107, score-0.271]
30 Theory and Method Let x1, · · · , xk denote a group of input test data, and D is the given dictionary. [sent-113, score-0.359]
31 Our proposed affine-constrained group sparse coding sxeeks to minimize the objective function: ECGSC(a, c; X, D) = ? [sent-115, score-0.594]
32 2 + λ Ψ(c) (4) subject to the nonnegative affine constraint on the group coefficient a ai = 1, and a1, a2 , . [sent-117, score-0.566]
33 arse coding [1] (GSC), there are no group eco theaftficients a an? [sent-122, score-0.393]
34 d the sparse coefficients c are given as a matrix. [sent-123, score-0.291]
35 A schematic illustration of the difference between the group sparse coding and our constrained version is shown in Fig. [sent-124, score-0.69]
36 The GSC-based classification scheme sparse codes the input feature vectors xi simultaneously. [sent-126, score-0.399]
37 While some group sparsity can be claimed for this approach based on the appropriate regularizer Ψ, it is generally difficult to provide any guarantee on the behavior of the sparse coefficient matrix C. [sent-127, score-0.523]
38 On the other hand, for our constrained version, the discrete set of the input vectors has been completed to form a convex set S, and our approach is designed to capture any vector in this convex set that can best be sparse coded by the dictionary D. [sent-128, score-0.74]
39 The situation here shares some similarity with the LP-relaxation of integer programming [24] or the convexification of a non- convex program [18], in which one enlarges the feasible domain in order to achieve convexity and thereby, efficiently compute approximate solution. [sent-129, score-0.302]
40 It is clear that the optimization problem is indeed convex and it is completely tractable as the feasible domain and objective function are both convex. [sent-132, score-0.256]
41 The only complication is the projection onto the simplex defined by the group co- (? [sent-134, score-0.202]
42 Figure 2: Illustration of the difference between group sparse coding and constrained group sparse coding, and its effect on classification results. [sent-139, score-1.138]
43 The cone represents the subspace spanned by a block of the dictionary D. [sent-140, score-0.327]
44 Shaded areas represent the convex hull spanned columns in X. [sent-141, score-0.451]
45 None of the xi lie within the subspace; however some of the points on the convex hull do and the proposed algorithm is designed to capture these points. [sent-142, score-0.34]
46 Theoretical Guarantee Given a dictionary D, a vector x has sparsity s if it can be written exactly as a linear combination of s columns of D. [sent-145, score-0.247]
47 An important result that underlies all SRC frameworks is the guarantee provided by the sparse recovery result that for a feature vector x with sparsity bounded from above by a constant depending on D [4, 6], x can be recovered by minimizing the ? [sent-146, score-0.388]
48 We remark that the two programs, while related, are in fact different, with most sparse recovery results given by minimizing (P1). [sent-156, score-0.315]
49 i gA typical SRC method will determine its classification based on its sparse coefficients obtained by minimizing the program (Pλ1). [sent-158, score-0.401]
50 Compared to them, our proposed framework enlarges (thPe optimization domain by introducing the group coefficients a, and it is possible that with larger domain, spurious and incorrect solutions could arise. [sent-159, score-0.447]
51 The following theorem rules out this possibility, at least when the sparse vector x can be exactly recovered by a typical SRC method and classified correctly: (Pλ1) Theorem 1. [sent-160, score-0.269]
52 gF Purthermore, we assume that the global minimum of ECGSC, given a group of input adta ttha eX g, lios unique. [sent-163, score-0.266]
53 The proofis straightforward and it consists ofchecking the sparse vector x also corresponds to the global minimum of ECGSC (a, c). [sent-168, score-0.233]
54 Since x is a sparse vector that can be recovered exactly by minimizing (Pλ1) in Eq. [sent-170, score-0.24]
55 (h6a)t, we bleet c cboev eirtse sparse yco ebfyfi mcieinntism, aznindg we have x = Dc. [sent-171, score-0.201]
56 This shows that (a, c) is the global minimum of the convex function ECGSC (a, c), regardless wmhineitmheurm x oisf on eth ceo boundary toiof nth Ee convex hull S. [sent-178, score-0.447]
57 However, the behavior of GSC in this case is difficult to predict because other (noisy) input feature vectors will affect the sparse coding of the (noiseless) input vector, and the result of the subsequence pooling based on the matrix C can be uncertain. [sent-185, score-0.505]
58 Second, if there is a sparse vector x lying inside the convex hull S spanned by x1, . [sent-186, score-0.586]
59 In the standard ACGSC, the online reconstructed image X a is simply the convex combination of the input images as the columns of X. [sent-194, score-0.281]
60 This model can be augmented using a known image domain partition and define the reconstructed image as a composite image that uses a different convex combination in each region of the partition. [sent-195, score-0.233]
61 (a): ai are the group coefficients corresponding to the sample xi. [sent-207, score-0.359]
62 The nonnegative affine constraint here is a1 + a2 = 1. [sent-208, score-0.225]
63 The vector ap can then be optimized individually under the nonnegative affine constraints given in Eq. [sent-271, score-0.329]
64 Sparse representations have been successfully applied to many vision problems such as face recognition [12], [17], [22], image classification [23], [5], [7], denoising and inpainting [14], and other areas [25]. [sent-279, score-0.228]
65 Interestingly, one of the original motivations for using sparse representations in solving vision problems is the realization that sparse representations of visual signals are produced in the first stage of visual processing by human brain [15]. [sent-280, score-0.46]
66 In many cases, simply replacing the original features with their sparse representations leads to surprisingly better recognition/classification results [25]. [sent-281, score-0.261]
67 Group sparse coding was first proposed in [1] and its extension to block-structured dictionaries has been studied in [2, 19]. [sent-282, score-0.422]
68 In these methods, group sparsity was promoted using a matrix norm that encourages features in the same group to share the same atoms in the dictionary. [sent-283, score-0.559]
69 From classification viewpoint, [1] has two undesirable features: First, the classification is based on matrices (multiple vectors) and this increase in dimension complicates the process. [sent-284, score-0.22]
70 Second, while promoting group sparsity, [1] does not go beyond the test data xi themselves to search for potentially more useful features that are better-aligned with the dictionary. [sent-285, score-0.334]
71 Our proposed framework addresses both shortcomings by introducing the group coefficients a in Eq. [sent-286, score-0.36]
72 However, none of the cited work has investigated face recognition under more realistic scenario when there are large differences between the illumination conditions of the training and test images. [sent-292, score-0.261]
73 The training images were used to simulate the well-lit images such as passport photos, and the test images were used to simulate images (e. [sent-297, score-0.177]
74 We used the training images directly as atoms of the dictionary D [5, 26]. [sent-305, score-0.273]
75 We have also compared with two group-regularized sparse coding algorithms proposed by [1] [2]. [sent-329, score-0.392]
76 Their algorithm promotes the data in a group (columns of X) to share same dictionary atoms. [sent-340, score-0.369]
77 In [2], block structure was added to D together with the group structure on the data X. [sent-341, score-0.295]
78 We applied these two methods to every test sample since they do not utilize group structure on data and the results are also shown in Fig. [sent-357, score-0.278]
79 2, is that our method goes beyond the input test images and searches for a more “aligned” image on a larger domain (the convex hull spanned by X) while the group-regularized methods can only rely on the input images that are poorly poorly “aligned” with the dictionary. [sent-361, score-0.612]
80 This is because, in their experiments, they randomly chose half of the dataset as atoms of D and the other half as test samples. [sent-374, score-0.213]
81 Therefore their dictionaries are 50% larger than ours and the variability between training and test samples are much more limited due to the random selection of training images. [sent-375, score-0.192]
82 4) than the mean images if the convex hull spanned by columns of X lies within the subspaces spanned by the blocks in D. [sent-377, score-0.581]
83 Randomly selected ng test samples (X) from the occluded images of person p. [sent-433, score-0.201]
84 Lastly, we compared with the results from directly applying regular sparse coding (Wright et. [sent-464, score-0.392]
85 8(d) shows the part-based group coefficients (ap) × ap of the test samples after the optimization. [sent-475, score-0.488]
86 8(e) shows the part-based affine combination of the test images. [sent-483, score-0.185]
87 Due to the occlusion of the scarf, the test samples were incorrectly matched to training images from a male subject with beard and mustache. [sent-489, score-0.189]
88 2Fo =r e5 6ac1h2 t iemxtaugrees category, we randomly chose 20 images as the training samples and the rest as test samples. [sent-498, score-0.189]
89 (d) Weights of the part-wise group coefficients overlaid on the corresponding test samples. [sent-505, score-0.368]
90 (e) The part-wise affine combination of the test samples after optimization. [sent-507, score-0.235]
91 Use the training samples as columns of the dictionary D. [sent-521, score-0.282]
92 Since there is no group structure defined in their framework, we computed the sparse coefficients for all the test images individually. [sent-534, score-0.569]
93 We have also presented a form of sparse recovery result and based on this result, we have argued that, at least in theory, the classification performance using the proposed method should be as good as if not better than the one using existing SRC-based methods. [sent-561, score-0.381]
94 We have evaluated the proposed approach using three experiments that involves face recognition, texture classification and face recognition under occlusions. [sent-562, score-0.331]
95 We believe that it is possible to obtain a stronger form of the sparse recovery result under noisy assumption, providing a better un- derstanding of the power and limitation of the proposed algorithm. [sent-565, score-0.271]
96 Furthermore, we will also investigate useful and effective prior for the group coefficients a and the resulting (usually non-convex) optimization problem. [sent-566, score-0.292]
97 Block and group regularized sparse modeling for dictionary learning. [sent-581, score-0.536]
98 Multi-layer group sparse coding for concurrent image classification and annotation. [sent-608, score-0.704]
99 Learning multiscale sparse representations for image and video restoration (preprint). [sent-656, score-0.23]
100 Efficient learning of sparse representations with an energy-based model. [sent-662, score-0.23]
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