iccv iccv2013 iccv2013-14 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Wangmeng Zuo, Deyu Meng, Lei Zhang, Xiangchu Feng, David Zhang
Abstract: In many sparse coding based image restoration and image classification problems, using non-convex ?p-norm minimization (0 ≤ p < 1) can often obtain better results than timhei convex 0?1 -norm m 1)ini camniza otfiteonn. Ab naiunm bbeetrt of algorithms, e.g., iteratively reweighted least squares (IRLS), iteratively thresholding method (ITM-?p), and look-up table (LUT), have been proposed for non-convex ?p-norm sparse coding, while some analytic solutions have been suggested for some specific values of p. In this paper, by extending the popular soft-thresholding operator, we propose a generalized iterated shrinkage algorithm (GISA) for ?p-norm non-convex sparse coding. Unlike the analytic solutions, the proposed GISA algorithm is easy to implement, and can be adopted for solving non-convex sparse coding problems with arbitrary p values. Compared with LUT, GISA is more general and does not need to compute and store the look-up tables. Compared with IRLS and ITM-?p, GISA is theoretically more solid and can achieve more accurate solutions. Experiments on image restoration and sparse coding based face recognition are conducted to validate the performance of GISA. ××
Reference: text
sentIndex sentText sentNum sentScore
1 cn Abstract In many sparse coding based image restoration and image classification problems, using non-convex ? [sent-11, score-0.21]
2 , iteratively reweighted least squares (IRLS), iteratively thresholding method (ITM-? [sent-16, score-0.256]
3 p-norm sparse coding, while some analytic solutions have been suggested for some specific values of p. [sent-18, score-0.134]
4 In this paper, by extending the popular soft-thresholding operator, we propose a generalized iterated shrinkage algorithm (GISA) for ? [sent-19, score-0.196]
5 Unlike the analytic solutions, the proposed GISA algorithm is easy to implement, and can be adopted for solving non-convex sparse coding problems with arbitrary p values. [sent-21, score-0.251]
6 Experiments on image restoration and sparse coding based face recognition are conducted to validate the performance of GISA. [sent-25, score-0.291]
7 Introduction Sparse coding [7, 18, 3 1] is an effective tool in a myriad of applications such as compressed sensing [11], image restoration [24, 25], face recognition [38], etc. [sent-27, score-0.271]
8 Originally, it aims to solve the following minimization problem: mxin21 ? [sent-28, score-0.06]
9 p-norm non-convex sparse coding problems, and they have been applied to various vision and learning tasks, e. [sent-71, score-0.151]
10 , compressed sensing [10], image restoration [25], face recognition [29], and variable selection [33]. [sent-73, score-0.186]
11 Several typical algorithms include iteratively reweighted least squares (IRLS) [12, 14, 23, 24, 28], iteratively reweighted ? [sent-74, score-0.138]
12 LUT uses look-up tables to store the solutions w. [sent-81, score-0.053]
13 Other algorithms, such as the analytic solutions in [25, 39], can only be used for some specific values of p. [sent-88, score-0.068]
14 217 Inspired by the great success of soft thresholding [16] and iterative shrinkage/thresholding (IST) [15] methods, in this paper, we propose a generalized iterated shrinkage algorithm (GISA) for ? [sent-89, score-0.391]
15 p-norm sparse coding problems with arbitrary p, λ and y values. [sent-92, score-0.174]
16 It is easy to implement and can be readily used to solve the many ? [sent-95, score-0.052]
17 (4), IRLS and IRL1 sometimes cannot converge to the desired solutions. [sent-156, score-0.092]
18 3, by initializing x(0) = y, IRLS and IRL1 would converge to the same local minimum. [sent-160, score-0.092]
19 (4) is for 1D optimization, one can define a proper thresholding function [33] or construct look-up tables (LUTs) [25] in advance. [sent-162, score-0.162]
20 p converge to the same local minimum, but GISA can converge to a better solution. [sent-170, score-0.184]
21 , 1/2 or 2/3, the analytic solutions can be derived [25, 39]. [sent-173, score-0.068]
22 (11) cannot always guarantee to converge to the global solution. [sent-180, score-0.092]
23 p-norm non-convex sparse coding problems where the values of x, λ and p are unconstrained, LUT will not be an effective and efficient solution. [sent-185, score-0.174]
24 (13) Generally, if |y| ≤ λ, the soft-thresholding operator uses the thresholding r|yu|le ≤ t oλ assign T1(y; λ) otol d0in; goth oeprewraistoer, uses tthhee shrinkage rule to assign T1(y; λ) to sgn(y)(|y| − λ). [sent-202, score-0.349]
25 Generalization of soft-thresholding Inspired by soft-thresholding, we proposed a generalized shrankage/thresholding operator to solve the ? [sent-205, score-0.108]
26 (4) by modifying the thresholding and the shrinkage rules. [sent-207, score-0.275]
27 Thus, to generalize soft thresholding for solving the problem in Eq. [sent-219, score-0.203]
28 (19) In ITM, She [33] extended the soft-thresholding with the thresholding function in Eq. [sent-235, score-0.162]
29 (11) actually is not a good generalization of the soft-thresholding operator for ? [sent-242, score-0.061]
30 Thus, to generalize softthresholding, we should solve the following nonlinear equation system to determine a correct thresholding value τpGST(λ) and its corresponding x∗p: (x0(λ,p), 21? [sent-246, score-0.207]
31 e range of (x(0λ,p), +∞) can be obtained as xp∗ = (2λ(1 − p))2−1p, (24) and the thresholding value τpGST(λ) is τpGST(λ) = (2λ(1 − p))2−1p + λp(2λ(1 − p))p2−−1p. [sent-270, score-0.162]
32 Theorem 1 For any y ∈ (τpGST(λ), +∞), f(x) has one unique minimum SpGST(y; λ) in the range of (xp∗, +∞), which can be obtained by solving the following equation: SGpST(y;λ) − y+ λp? [sent-272, score-0.066]
33 In Algorithm 1, the output would converge to the correct solution when J → ∞. [sent-298, score-0.113]
34 s Finally, we propose a generalized soft-thresholding (GST) function for solving the ? [sent-301, score-0.083]
35 (28) Like the soft-thresholding function, the GST function also involves a thresholding rule TpGST(y; λ) = 0 when |y| ≤ τpGST(λ) and a shrinkage rule TpGST(y; λ) = sgn(y)SpGST(y; λ) when |y| > τpGST(λ). [sent-305, score-0.339]
36 Compared with the thresholding functwiohne nin | [|3 >3], τ in GST we adopt a different thresholding val- ue τpGST(λ), and propose an algorithm, i. [sent-306, score-0.324]
37 When p = 1, GST will converge after one iteration. [sent-320, score-0.092]
38 Since pl→im1τpGST(λ) = λp l→im1(1 − p)p−1 = λ, (29) the thresholding value of GST will become λ, and the GST function becomes T1GST(y;λ) =? [sent-321, score-0.162]
39 When p = 0, GST will also converge after one iteration. [sent-324, score-0.092]
40 Generalized iterated shrinkage algorithm With the proposed GST in Eq. [sent-330, score-0.154]
41 (28), we can readily have a generalized iterated shrinkage algorithm (GISA) for solving the ? [sent-331, score-0.237]
42 GISA The proposed GISA is an iterative algorithm, and in each iteration it involves a gradient descent step based on A or y, followed by a generalized shrinkage/thresholding step: x(k+1) = TGpST(x(k) − ? [sent-336, score-0.097]
43 Output: x Actually, GISA is a generalization of the iterative shrinkage/thresholding (IST) method [15], and an example of the iterative thresholding method (ITM) [33]. [sent-360, score-0.247]
44 In [33], She proved that, for any thresholding function Θ (y; λ) defined for −∞ < y < +∞ and 0 ≤ λ < +∞, if Θ (y; λ) satisfies the following properties: i) Θ(−y; λ) = −Θ(y; λ), ii) Θ(y; λ) ≤ −Θ(y? [sent-361, score-0.162]
45 , iii) limy→∞Θ(y; λ) =; ∞, iv) 0 ≤ Θ(y; λ) ≤λ y =fo ∞r 0, ≤ y < ∞, the ITM0 ≤m eΘth(yo;dλ )w ≤ou yld f converge t o∞ a, stationary point. [sent-363, score-0.092]
46 2-norm, GISA can al- so converge to the optimal solution. [sent-369, score-0.092]
47 Moreover, if p = 1, GISA would degenerate to IST, and would converge to the global minimum. [sent-370, score-0.092]
48 Sparse GST gradient based deconvolution using One important application of sparse coding is image restoration. [sent-378, score-0.342]
49 n A typical image deconvolution model usually includes a fidelity term and a regularization term, where the fidelity term is modeled based on the degradation process, and the regularization term is modeled based on image priors. [sent-382, score-0.245]
50 Recent studies on natural image statistics have shown that the marginal distributions of filtering responses can be modeled as hyper-Laplacian with 0 < p < 1 [25, 28, 35], which had been adopted in many low level vision problems [13, 36]. [sent-383, score-0.062]
51 By using the sparse gradient based image prior, the image deconvolution model can be formulated as mxin21 ? [sent-384, score-0.257]
52 pp, (37) where λ is the regularization parameter, D = [Dh, Dv] denotes the gradient operator, and Dh and Dv are the horizontal and vertical gradient operators, respectively. [sent-388, score-0.063]
53 We adopt an alternating minimization strategy to solve the problem in Eq. [sent-399, score-0.06]
54 In each iteration, given a fixed d, x can be obtained by solving the following subproblem η2λ mxin21 ? [sent-401, score-0.068]
55 Given a fixed x, let dref = Dx, and d can be obtained by solving the following subproblem: mdin 2η ? [sent-413, score-0.108]
56 (42) Finally, we summarize the GST based image deconvolution algorithm in Algorithm 3. [sent-429, score-0.169]
57 Algorithm 3 is similar to the algorithms in [25, 37], but Wang and Yin [37] only studied the Laplacian prior (p = 1), and Krishnan and Fergus [25] used look-up table (LUT) to solve the subproblem in Eq. [sent-430, score-0.051]
58 Here we empirically choose J = 1, making our algorithm very efficient for sparse gradient based image deconvolution. [sent-432, score-0.088]
59 Experimental results In this section, we evaluate the proposed GISA on two representative vision applications: image deconvolution and face recognition. [sent-434, score-0.23]
60 In image deconvolution experiments, we compare GISA with four state-of-the-art algorithms of ? [sent-435, score-0.169]
61 In face recognition, we use GISA to solve the sparse representation-based classification (SRC) model [38], and show that the performance of SRC can be improved by using GISA with p < 1. [sent-456, score-0.151]
62 Image deconvolution In image deconvolution, we followed the experiment setting in [25]. [sent-465, score-0.169]
63 5 shows the deconvolution results of GISA on a test image by using p = 1and p = 0. [sent-485, score-0.169]
64 7 is much better than that with p = 1in terms of suppressing noise and ring effects and preserving edge details, which indicates that non-convex image deconvolution can much improve the deconvolution performance. [sent-488, score-0.338]
65 Image deconvolution with GISA: (a) original image, (b) blurry image, (c) deconvolution result (PSNR: 27. [sent-509, score-0.369]
66 17) of GISA with p = 1, and (d) deconvolution result (PSNR: 28. [sent-510, score-0.169]
67 Face recognition via sparse coding Given a test sample y and the training data matrix X = [X1,X2, ,XK], where Xk, k = 1, 2, ,K, is the sample matrix of class k, Wright et al. [sent-536, score-0.171]
68 [38] proposed a sparse representation based classification (SRC) method for face recognition (FR). [sent-537, score-0.147]
69 SRC first seeks the solution of the following sparse coding problem: × αˆ = argmαin? [sent-538, score-0.172]
70 Then, by simply replacing the soft-thresholding operator in ALM by the proposed GST operator, we can embed the proposed GISA algorithm into the ALM method for solving the SRC model with arbitrary values of p and q. [sent-553, score-0.1]
71 We use the principal component analysis (PCA) to reduce the dimensionality of face images, and test the algorithms in both the original image space (1, 024 dimensions) and the PCA subspace with feature dimension 500, 300, and 100, respectively. [sent-559, score-0.061]
72 6 we show the recognition rates of SRC versus different p values. [sent-562, score-0.048]
73 Table 2 lists the recognition rates of SRC and SRC-p (p = 0. [sent-568, score-0.068]
74 One can see that, GISA based SRC-p can always achieve higher recognition rates than the original SRC with p = 1. [sent-570, score-0.048]
75 This is because when the face feature dimension is small, the matrix X tends to be redundant. [sent-572, score-0.061]
76 Thus, the coding solution should be sparser, and GISA is more probable to obtain the correct solution. [sent-573, score-0.106]
77 By choosing q = p = 1, the SRC method would become robust to face corruption/occlusion [38]. [sent-574, score-0.061]
78 The face recognition rate of SRC by varying the value of p with GISA. [sent-590, score-0.081]
79 09 275 0 < q = p < 1, and embed the proposed GISA into ALM to implement SRC-p, q for robust face recognition. [sent-600, score-0.106]
80 Two types of face image corruption are considered: random pixel corruption and random block occlusion. [sent-601, score-0.181]
81 Table 3 lists the recognition rates of SRC and SRC-p, q under different ratios of random corruption. [sent-605, score-0.068]
82 One can see that SRC-p, q can always outperform SRC for recognizing face images with random corruption. [sent-606, score-0.061]
83 Table 4 lists the recognition rates of SRC and SRC-p, q under different ratios of block occlusion. [sent-608, score-0.096]
84 Again, SRC-p, q can obtain better recognition rates than SRC for face recognition with random block occlusion. [sent-609, score-0.157]
85 Recognition rate (%) on face images with random corruption. [sent-611, score-0.061]
86 Recognition rate (%) on face images with block occlusion. [sent-623, score-0.089]
87 We proposed a generalized shrinkage/thresholding (GST) function and the associated gen- eralized iterated shrinkage algorithm (GISA) for ? [sent-629, score-0.196]
88 Compared with the state-of-theart methods, GISA is theoretically more solid, easier to understand and more efficient to implement, and it can converge to a more accurate solution. [sent-631, score-0.114]
89 Our experimental results on image deconvolution verify the effectiveness and efficiency of GISA, and our experiments on sparse coding based face recognition showed that ? [sent-632, score-0.401]
90 A fast iterative shrinkage / thresholding algorithm for linear inverse problems. [sent-658, score-0.308]
91 [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] thresholding algorithms for image restoration. [sent-668, score-0.162]
92 An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. [sent-742, score-0.218]
93 For most large underdetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution. [sent-753, score-0.074]
94 Recovering sparse signals with a certain family of nonconvex penalties and DC programming. [sent-774, score-0.108]
95 From few to many: Illumination cone models for face recognition under variable lighting and pose. [sent-780, score-0.081]
96 q minimization with 0 < q < 1 for sparse solution of under-determined linear systems. [sent-808, score-0.123]
97 Acquiring linear subspaces for face recognition under variable lighting. [sent-814, score-0.081]
98 An iterative algorithm for fitting nonconvex penalized generalized linear models with grouped predictors. [sent-857, score-0.117]
99 1/2 regularization: A thresholding rep- resentation theory and a fast solver. [sent-897, score-0.162]
100 A generalized accelerated proximal gradient approach for total-variation-based image restoration. [sent-912, score-0.064]
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