iccv iccv2013 iccv2013-257 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Peihua Li, Qilong Wang, Wangmeng Zuo, Lei Zhang
Abstract: The symmetric positive de?nite (SPD) matrices have been widely used in image and vision problems. Recently there are growing interests in studying sparse representation (SR) of SPD matrices, motivated by the great success of SR for vector data. Though the space of SPD matrices is well-known to form a Lie group that is a Riemannian manifold, existing work fails to take full advantage of its geometric structure. This paper attempts to tackle this problem by proposing a kernel based method for SR and dictionary learning (DL) of SPD matrices. We disclose that the space of SPD matrices, with the operations of logarithmic multiplication and scalar logarithmic multiplication de?ned in the Log-Euclidean framework, is a complete inner product space. We can thus develop a broad family of kernels that satis?es Mercer’s condition. These kernels characterize the geodesic distance and can be computed ef?ciently. We also consider the geometric structure in the DL process by updating atom matrices in the Riemannian space instead of in the Euclidean space. The proposed method is evaluated with various vision problems and shows notable per- formance gains over state-of-the-arts.
Reference: text
sentIndex sentText sentNum sentScore
1 nite (SPD) matrices have been widely used in image and vision problems. [sent-6, score-0.284]
2 Though the space of SPD matrices is well-known to form a Lie group that is a Riemannian manifold, existing work fails to take full advantage of its geometric structure. [sent-8, score-0.213]
3 This paper attempts to tackle this problem by proposing a kernel based method for SR and dictionary learning (DL) of SPD matrices. [sent-9, score-0.19]
4 We disclose that the space of SPD matrices, with the operations of logarithmic multiplication and scalar logarithmic multiplication de? [sent-10, score-0.397]
5 ned in the Log-Euclidean framework, is a complete inner product space. [sent-11, score-0.155]
6 We can thus develop a broad family of kernels that satis? [sent-12, score-0.149]
7 These kernels characterize the geodesic distance and can be computed ef? [sent-14, score-0.159]
8 We also consider the geometric structure in the DL process by updating atom matrices in the Riemannian space instead of in the Euclidean space. [sent-16, score-0.392]
9 nite (SPD) matrices can be introduced in the imaging, pre-processing, feature extraction and representation processes, and have been widely adopted in many computer vision applications [4, 29, 27]. [sent-20, score-0.305]
10 As to feature extraction, the covariance matrices that model the second-order statistics of image features also result in SPD matrices, which have been successfully applied to detection [27], recognition [18], and classi? [sent-29, score-0.21]
11 Due to their wide applications, the investigations on learning methods for SPD matrices have recently received considerable research interests. [sent-31, score-0.149]
12 It is known that the space of n n SPD matrices, denoted by Sknn+o,w isn n thoat a hlien esapra space n1 ×but n fo SPrmDs a aLtrieic group nthotaetd dis b a RSiemannian manifold [1]. [sent-33, score-0.141]
13 Hence, the mathematical modeling in this space is different from what is commonly done in the Euclidean space [6, 3 1], and new operators for SPD matrices should be introduced for SPD matrix-based learning. [sent-34, score-0.248]
14 In this work, we take sparse representation (SR) and dictionary learning (DL) [8] as examples, and focus on extending SR and DL to SPD matrix-based learning. [sent-35, score-0.164]
15 Since conventional SR and DL methods are proposed for vector data in the Euclidean space rather than the SPD matrices in the Riemannian manifold, in order to use SR and DL for SPD matrix-based learning, we should consider the following issues in developing new operators in Sn+ . [sent-36, score-0.209]
16 (1) In the Euicnlgide isasnu space tehvee l ionpeianrg cnoemwb oipneatriaotno rosfi anto Sm vectors can be naturally obtained using the conventional matrix operators; but it would be challenging to represent an SPD matrix as a linear combination of atom matrices since Sn+ is not a linear space. [sent-37, score-0.462]
17 (3) The updating of dictionary atoms involves solving a constrained optimization problem in Sn+ and it is more appro1S+n is not a linear space with the operations of conventional matrix additSion and scalar-matrix multiplication. [sent-40, score-0.2]
18 However, with the operations of logarithmic multiplication and scalar logarithmic multiplication Sn+ is not loonglyar a hlimneiacr m space [ic1]a tbiount a alnsod a ccaolmarp l oegtae i nthnmeri cpr moduultcitp space as Sshown in section 2. [sent-41, score-0.406]
19 In Tensor Spare Coding (TSC) [22], an SPD matrix is linearly decomposed as a set of of atom matrices and LogDet (or Bregman matrix) divergence [15] was adopted to measure the reconstruction error. [sent-51, score-0.433]
20 Based on this framework, dictionary learning methods are further proposed to learn atom matrices [23]. [sent-52, score-0.449]
21 In the generalized dictionary learning (GDL) algorithm [25], each SPD matrix is represented as a linear combination of rank-1 atom matrices; the error between one SPD matrix and its linear combination is evaluated by matrix Frobenius norm. [sent-53, score-0.405]
22 Second, we can explicitly or implicitly map SPD matrices to some Reproducing Kernel Hilbert Space (RKHS), and use the kernel SR or DL framework for SPD matrix- based learning. [sent-55, score-0.243]
23 They adopted Stein kernel tmo map dth feo rS SPDR manadtri DcLes tno higher dimensional RKHS. [sent-59, score-0.094]
24 This method is in contrast with those methods which directly embed SPD matrices into Euclidean space (LogE-SR) [11, 34] and achieves state-of-the-art performance compared to its counterparts. [sent-60, score-0.188]
25 The main difference is that we develop a novel family of kernel functions based on the Log-Euclidean framework [1]. [sent-72, score-0.125]
26 The proposed kernels characterize the geodesic distance and thus can accurately × measure the reconstruction error; they also satisfy the Mercer’s condition under broad conditions. [sent-73, score-0.193]
27 These are in contrast to the Stein kernel which is only an approximation of the geodesic distance and satis? [sent-74, score-0.147]
28 Our work differs from them in that we disclose the inner product structure of Sn+, by which we can develop a broad variety of kernel foufn Sctions and the Gaussian kernel is a special case of ours. [sent-78, score-0.378]
29 Log-Euclidean Kernel This section starts with a brief introduction of LogEuclidean framework [1]; subsequently, we show that Sn+ Efourcmlisd an ni fnrnaemr product space; ebqauseedn on twheis ,s we design a family of kernel functions. [sent-80, score-0.178]
30 In the Log-Euclidean framework, an operation of logarithmic multiplication ? [sent-87, score-0.171]
31 ee identity matrix and with the inverse operation the regular matrix inverse. [sent-92, score-0.12]
32 cTohme geodesics equipped wSith a bi-invariant metric are the left translates of the geodesics through the identity element, given by one-parameter subgroup exp(tV), where t ∈ R and V ∈ Sn. [sent-99, score-0.136]
33 nally the geodesic distance between two SPD matrices S and T as follows: ρgeo(S, T) = ? [sent-101, score-0.202]
34 Sn+ as a Complete Inner Product Space It is known that Sn+ is not a linear space with the operationIst iosf k tnhoew wconn tvheant Stional matrix addition and scalar-matrix multiplication but forms a Riemannian manifold [1, 3 1]. [sent-111, score-0.181]
35 However, as shown in [1], in the Log-Euclidean framework it is endowed with a linear space structure with the logarithmic multiplication (1) and the following scalar logarithmic multiplication [1]: × λ ⊗ S = exp(λ log(S)) = Sλ (3) where λ is a real number. [sent-112, score-0.337]
36 and ⊗ satisfy the conditions of a linear space, ewraithti tnhse identity m saattrisixfy being otnhed identity eale limneenart and regular matrix inversion operation as inverse mapping. [sent-114, score-0.113]
37 Indeed, not only a linear space, Sn+ is also an inner productI space as dte osnclryib aed li by rth sep following corollary which is not disclosed previously: Corollary 1 With two operations ? [sent-115, score-0.247]
38 log = tr(log(S) log(T)) (4) is an inner product, where tr denotes the matrix trace, and Sn+ is a complete inner product space (Hilbert space). [sent-120, score-0.423]
39 log = ≥0 0if, aanndd only bifv iSo iss thhaet identity matrix I. [sent-163, score-0.238]
40 nite dimension and therefore it is a complete inner product space (Hilbert space) [26]. [sent-165, score-0.3]
41 The norm induced by the inner product is expressed as S? [sent-166, score-0.126]
42 However, a linear space or a normed liner space is not necessarily an inner product space unless a function that satis? [sent-187, score-0.273]
43 r oacneds sicngala dri logarithmic multiplication ⊗, unlike [1] twiohnich ? [sent-194, score-0.149]
44 i annvodlv sceasl mapping hSmPiDc mmuatlrtiipcleics atotio logarithmic d [1o-] main, performing data processing therein and then mapping back to Sn+ again. [sent-195, score-0.1]
45 kee:r n(1el); T T(h2e) The tensor product φ1 ∗ φ? [sent-268, score-0.125]
46 kernels, such as polynomial, exponential, radial basis, B-Spline kernels, or Fourier kernel etc. [sent-284, score-0.094]
47 T Here we compare the proposed kernels with Stein kernel = >− [24]. [sent-335, score-0.178]
48 ne-Riemannian distance between the two matrices is dA (S, T) = ? [sent-342, score-0.149]
49 1 shows the histogram computed from the SPD matrices used in texture classi? [sent-368, score-0.178]
50 kernel under restricted co =nd eitxipo(n−, tβhdat is, β = 21, . [sent-375, score-0.094]
51 The logarithm of SPD matrices can be computed through the eigen-decomposition. [sent-385, score-0.206]
52 The logarithms of the involved SPD matrices can generally be computed beforehand because oftheir “decoupling” property either in the inner product or distance; in these cases, the complexity of the proposed kernels becomes O(n3) and is the same as that of the Stein kernel. [sent-394, score-0.359]
53 While this method outperforms state-of-the-arts, the symmetric Stein divergence only approximates the Riemannian metric and the Stein kernel only satis? [sent-400, score-0.206]
54 Let φ be th∈e fSunction that maps SPD matrices to RKHS, SR of Y can be formulated as the following kernelised LASSO problem [12]: xm∈RinN? [sent-409, score-0.149]
55 1 Minimaization of the above equation is similar to regular sparse coding in Euclidean space [10], and we use the method introduced in [12] for its solution. [sent-459, score-0.109]
56 , M, the atom matrices can be obtained by learning method so that they have more powerful representation capability. [sent-465, score-0.374]
57 First, suppose that the atom matrices Si ∈ Sn+ , i= 1, . [sent-498, score-0.353]
58 , M, we compute its sparse vector xj as described in the previ- ous section; then, let xj be ? [sent-505, score-0.097]
59 xed, we update dictionary atom matrices Si, i= 1, . [sent-506, score-0.449]
60 In the following, we illustrate the atom matrices update scheme using Gaussian kernel κg. [sent-510, score-0.447]
61 Re-writing (7) in kernel function κ, we have the partial derivative of f(·) w. [sent-513, score-0.094]
62 (8) One may update log Sr instead of Sr, which is equivalent to transforming by logarithm the SPD matrices to Euclidean space in which atoms are updated. [sent-520, score-0.395]
63 We thus instead update the atom matrices in the Lie group as follows: Sr = exp ? [sent-523, score-0.425]
64 rst evaluate the performance of the × proposed family of kernels on sparse representation without dictionary learning. [sent-532, score-0.321]
65 As in [30], the training samples are adopted as atom matrices and the reconstruction errors are used for classi? [sent-533, score-0.353]
66 Then we learn the atom matrices from the training data and the sparse codes obtained from the learned atom matrices are used for classi? [sent-535, score-0.753]
67 cation × mbye taho 43d ×in 4[330 c]o, vaanrdia tnhcee preprocessing ompetth thoed cilna s[1si2? [sent-552, score-0.296]
68 2 shows the recognition accuracy of the proposed kernels with the regularization parameter λ = 10−3, where the recognition rates of RSR using Stein kernel [12] are also shown as baseline (red dash-dotted). [sent-555, score-0.232]
69 ,Irte can ibneg seen thaat ta btoheu proposed kernels are clearly better than the Stein kernel on all datasets. [sent-560, score-0.178]
70 cation rates of RSR that uses the Stein kernel [12] are shown as baseline (red dash-dotted line). [sent-619, score-0.444]
71 TSC has unsatisfactory performance and we owe it to the linear representation of SPD matrices in the Euclidean space without use of the Riemannian metric. [sent-621, score-0.209]
72 By using the Riemannian metric, LogE-SR has improved recognition rates but the sparse decomposition is performed in the logarithm domain rather than in the original Riemannian manifold. [sent-622, score-0.158]
73 cation We employ the Brodatz dataset and follow the experimental setting in [22, 23, 12] for fair comparison. [sent-639, score-0.296]
74 cation algorithm [17] but to testify the proposed method with closely related work. [sent-641, score-0.326]
75 In the Brodatz dataset each class contains only one image and we use the mosaics of 5-texture (‘5c’ , ‘5m’, ‘5v’, ‘5v2’, ‘5v3’), 10texture (‘ 10’,‘ 10v’), and 16-texture (‘ 16c’, ‘ 16v’). [sent-642, score-0.09]
76 Among the 64 covariance matrices per class, 5 are randomly selected for training and the remaining ones are for testing. [sent-646, score-0.21]
77 Dictionary Learning To testify the effectiveness of the proposed dictionary learning method, we compare three methods: random sampling, K-Means clustering and dictionary learning. [sent-673, score-0.222]
78 The K-Means clustering is performed in the LogEuclidean framework [1]: the covariance matrices are ? [sent-675, score-0.21]
79 rst mapped to the linear space Sn by matrix logarithm, in which mthea clustering ilisn performed and the results are then mapped ×× back to Sn+ . [sent-676, score-0.116]
80 cation We use the Brodatz dataset and follow the experimental setting in [12]. [sent-678, score-0.296]
81 T phiexe 5ls-d,i frmoemns wiohnicahl afe 5at×u5re voevcatroirasn ctoe compute the covariance matrix comprise grayscale intensity, and the 1st and 2nd partial derivatives with respect to spatial coordinates. [sent-681, score-0.096]
82 We thus have 2200 covariance matrices in total for dictionary learning. [sent-683, score-0.306]
83 It can be seen that the dictionary learning method is consistently superior to random dictionary and Log-E K-Means, particularly when the number of atom matrices are small. [sent-690, score-0.545]
84 It is interesting to notice that the random dictionary is better than the learned dictionary via Log-E K-Means if the number of atom matrices are less than 80. [sent-691, score-0.545]
85 We also observe that the performance of both random sampling and Log-E K-Means improves with the increase of atom matrix number. [sent-693, score-0.239]
86 cation accuracy on the Brodatz dataset Scene Categorization We use the popular benchmark database Scene15 [16] for classi? [sent-696, score-0.296]
87 In each image, we extract 8 8 covariance matrices at dense grids with a stride of t8r a pcitxe 8l×s. [sent-699, score-0.21]
88 First, among covariance matrices ofall images, 50,000 ones are randomly chosen which are used to obtain atom matrices. [sent-705, score-0.414]
89 cation rates of the proposed method are over 18 percent higher than the random dictionary. [sent-712, score-0.383]
90 We can also observe that the proposed method has over 8 percent, 4 percent, and 2 percent advantages over Log-E K-Means Clustering for 32, 64, and 128 atom matrices, respectively. [sent-713, score-0.237]
91 From both of the above experiments, we observe that as atom matrix number grows, the performance gains of the dictionary learning over the other two methods gets smaller. [sent-714, score-0.357]
92 As the current dictionary is generative without discriminative information, more powerful representational capability does not necessarily mean better discriminability. [sent-715, score-0.096]
93 ndings and we think that the performance difference between the three methods will get smaller or even negligible as the atom matrix number becomes much larger. [sent-717, score-0.239]
94 Conclusion This paper presented a novel Riemannian metric based kernel method for SR and DL in Sn+. [sent-728, score-0.114]
95 We disclosed that the space of SPD matrices is a complete inner product space, and developed a broad 11660077 family of p. [sent-734, score-0.404]
96 Action recognition using sparse representation on covariance manifolds of optical ? [sent-828, score-0.129]
97 Sparse coding and dictionary learning for symmetric positive de? [sent-838, score-0.166]
98 Kernel methods on the riemannian manifold of symmetric positive de? [sent-854, score-0.236]
99 Gabor feature based sparse representation for face recognition with gabor occlusion dictionary. [sent-990, score-0.096]
100 Action recognition using sparse representation on covariance manifolds of optical ? [sent-998, score-0.129]
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