Sparse Representation and Low Rank Methods for Image Restoration and Classification. Lei Zhang

Transcription

1 Sparse Representation and Low Rank Methods for Image Restoration and Classification Lei Zhang Dept. of Computing The Hong Kong Polytechnic University

2 My recent research focuses Sparse Representation, Dictionary Learning, Low Rank Image restoration Collaborative representation based pattern classification Image Quality Assessment Full reference and no reference IQA models Visual Tracking Fast and robust trackers Image Segmentation Evaluation Biometrics (face, finger-knuckle-print, palmprint)

3 A linear system??.?? A dense solution A sparse solution 3

4 Sparse solutions 4

5 How to solve the sparse coding problem? Greedy Search Approach for L 0 -minimization Orthogonal Matching Pursuit Least angle regression Convex Optimization Method for L 1 -minimization Interior Point Gradient Projection Proximal Gradient Descent (Iterative soft-thresholding) Augmented Lagrangian Methods Alternating Direction Method of Multipliers Non-convex L p -minimization W. Zuo, D. Meng, L. Zhang, X. Feng and D. Zhang, A Generalized Iterated Shrinkage Algorithm for Non-convex Sparse Coding, In ICCV

6 Example applications Denoising 6

7 Example applications Deblurring 7

8 Example applications Superresolution 8

9 Example applications Medical image reconstruction (e.g., CT) TV-based method Our method Qiong Xu, Hengyong Yu, Xuanqin Mou, Lei Zhang, Jiang Hsieh, and Ge Wang, Low-dose X-ray CT Reconstruction via Dictionary Learning, IEEE Transactions on Medical Imaging, vol. 31, pp ,

10 Example applications Inpainting 10

11 Example applications Morphological component analysis (cartoon-texture decomposition) = + J. Bobin, J.-L. Starck J. Fadili, Y. Moudden and D.L Donoho, "Morphological Component Analysis: an adaptive thresholding strategy", IEEE Transactions on Image Processing, Vol 16, No 11, pp ,

12 Why sparse: neuroscience perspective Observations on Primary Visual Cortex The Monkey Experiment by Hubel and Wiesel, 1968 Responses of a simple cell in monkeys right striate cortex. David Hubel and Torsten Wiesel Nobel Prize Winner 12

13 Why sparse: neuroscience perspective Olshausen and Field s Sparse Coding, 1996 The basis function can be updated by gradient descent: Resulted basis functions. Courtesy by Olshausen and Field

14 Why sparse: probabilistic Bayes perspective Signal recovery in a Bayesian viewpoint xˆ arg max P x y arg max P y x P x Represent x as a linear combination of bases (dictionary atoms) x And assume that the representation coefficients are i.i.d. and follow some prior distribution: ~ exp( i i ) l p x Likelihood Prior x 14

15 Why sparse: probabilistic Bayes perspective The maximum a posteriori (MAP) solution: MAP We have: arg max p( y) arg max log p( y ) log p( ) arg m in y If p=0, it is the L 0 -norm sparse coding problem. If p=1, it becomes the convex L 1 -norm sparse coding. If 0<p<1, it will be non-convex L p -norm minimization. 2 2 i i l p 15

16 Why sparse: signal processing perspective x is called K-sparse if it is a linear combination of only K basis vectors. If K<<N, we say x is compressible. Measurement of x K x i 1 i i y x 16

17 Why sparse: signal processing perspective Reconstruction If x is k-sparse, we can reconstruct x from y with M (M<<N) measurements: ˆ arg min, s.t. y But the measurement matrix should satisfy the restricted isometry property (RIP) condition: 0 For any vector v sharing the same K nonzero entries as : 17

18 Image reconstruction: the problem Reconstruct x from its degraded measurement y x y y = Hx + v H: the degradation matrix v: Gaussian white noise 18

19 Image reconstruction by sparse coding: the basic procedures 1. Partition the degraded image into overlapped patches. 2. Denote by the employed dictionary. For each patch, solve the following L 1 -norm sparse coding problem: ˆ arg min y 3. Reconstruct each patch by. ˆx ˆ 4. Put the reconstructed patch back to the original image. For the overlapped pixels between patches, average them. 5. In practice, the above procedures can be iterated for several rounds

20 How sparsity helps? An illustrative example You are looking for a girlfriend/boyfriend. i.e., you are reconstructing the desired signal. Your objective is that she/he is 白 - 富 - 美 / 高 - 富 - 帅. i.e., you want a clean and perfect reconstruction. However, the candidates are limited. i.e., the dictionary is small. Can you find your ideal girlfriend/boyfriend? 20

21 How sparsity helps? An illustrative example Candidate A is tall; however, he is not handsome. Candidate B is rich; however, he is too fat. Candidate C is handsome; however, he is poor. If you sparsely select one of them, none is ideal for you i.e., a sparse representation vector such as [0, 1, 0]. How about a dense solution: (A+B+C)/3? i.e., a dense representation vector [1, 1, 1]/3 The reconstructed boyfriend is a compromise of 高 - 富 - 帅, and he is fat (i.e., has some noise) at the same time. 21

22 How sparsity helps? An illustrative example So what s wrong? This is because the dictionary is too small! If you can select your boyfriend/girlfriend from boys/girls all over the world (i.e., a large enough dictionary), there is a very high probability (nearly 1) that you will find him/her! i.e., a very sparse solution such as [0,, 1,, 0] In summary, a sparse solution with an over-complete dictionary often works! Sparsity and redundancy are the two sides of the same coin. 22

23 The dictionary Usually, an over-complete dictionary is required in doing sparse representation. The dictionary can be formed by the off-the-shelf bases such as DCT bases, wavelets, curvelets, etc. Learning dictionaries from natural images has shown very promising results in image reconstruction. Dictionary learning has become a hot topic in image processing and computer vision. M. Aharon, M. Elad, and A.M. Bruckstein, The K-SVD: An algorithm for designing of overcomplete dictionaries for sparse representation, IEEE Trans. Signal Processing, vol. 54, no. 11, pp , Nov

24 Nonlocal self-similarity In natural images, usually we can find many similar patches to a given path, which can be spatially far from it. This is called nonlocal self-similarity. Nonlocal self-similarity has been widely and successfully used in image reconstruction.

25 Representative image restoration methods K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, Image denoising by sparse 3-d transform-domain collaborative filtering," TIP (BM3D) J. Mairal, F. Bach, J. Ponce, G. Sapiro, and A. Zisserman, Non-local sparse models for image restoration," ICCV (LSSC) J. Yang, J. Wright, T. Huang and Y. Ma, Image super-resolution via sparse representation, TIP (ScSR) D. Zoran and Y. Weiss, From learning models of natural image patches to whole image restoration, ICCV (EPLL) W. Dong, L. Zhang, G. Shi and X. Wu, Image deblurring and supper-resolution by adaptive sparse domain selection and adaptive regularization, TIP (ASDS) S. Wang, L. Zhang, Y. Liang and Q. Pan, Semi-Coupled Dictionary Learning with Applications to Image Superresolution and Photo-Sketch Image Synthesis, CVPR (SCDL) W. Dong, L. Zhang, G. Shi, and X. Li, Nonlocally Centralized Sparse Representation for Image Restoration, TIP (NCSR) 25

26 NCSR (ICCV 11, TIP 13) A simple but very effective sparse representation model was proposed. It outperforms many state-of-the-arts in image denoising, deblurring and super-resolution. W. Dong, L. Zhang and G. Shi, Centralized Sparse Representation for Image Restoration, in ICCV W. Dong, L. Zhang, G. Shi and X. Li, Nonlocally Centralized Sparse Representation for Image Restoration, IEEE Trans. on Image Processing, vol. 22, no. 4, pp , April

27 NCSR: The idea For true signal arg min, s.t. x For degraded signal x The sparse coding noise (SCN) 1 2 y arg min, s.t. y H = y x 1 2 To better reconstruct the signal, we need to reduce the SCN because: x = xˆ x Φα Φα Φυ y x α

28 NCSR: The objective function The proposed objective function Key idea: Suppressing the SCN How to estimate x? arg min yh + ˆ y 2 2 x l p The unbiased estimate: ˆ E[ ] x x The zero-mean property of SCN makes ˆ E[ ] E[ ] x x y 28

29 NCSR: The solution The nonlocal estimation of μ i i, j i, j, jc i [ ] E y,, The simplified objective function exp( ˆ ˆ i j xi xi j / h) / W 2 2 N 2 arg min + y y H μ 2 i1 The iterative solution: i i l N ( j) 2 ( j1) arg min + y y H 2 i μi i1 p l p 29

30 NSCR: The parameters and dictionaries The L p -norm is set to L 1 -norm since SCN is generally Laplacian distributed. The regularization parameter is adaptively determined based on the MAP estimation principle. Local PCA dictionaries are used, and they are adaptively learned from the image. We cluster the image patches, and for each cluster, a PCA dictionary is learned and used to code the patches within this cluster. 30

31 Denoising results From left to right and top to bottom: original image, noisy image (=100), denoised images by SAPCA-BM3D (PSNR=25.20 db; FSIM=0.8065), LSSC (PSNR=25.63 db; FSIM=0.8017), EPLL (PSNR=25.44 db; FSIM= ), and NCSR (PSNR=25.65 db; FSIM=0.8068). 31

32 Deblurring results Blurred FISTA (27.75 db) BM3D (28.61 db) NCSR (30.30 db) Blurred Fergus, et al [SIGGRAPH06] NCSR Close up View 32

33 Super-resolution results Low resolution TV (31.24 db) ScSR (32.87 db) NCSR (33.68 db) Low resolution TV (31.34 db) ScSR (31.55 db) NCSR (34.00 db) 33

34 GHP (CVPR 13, TIP 14) Like noise, textures are fine scale structures in images, and most of the denoising algorithms will remove the textures while removing noise. Is it possible to preserve the texture structures, to some extent, in denoising? We made a good attempt in: W. Zuo, L. Zhang, C. Song, and D. Zhang, Texture Enhanced Image Denoising via Gradient Histogram Preservation, in CVPR W. Zuo, L. Zhang, C. Song, D. Zhang, and H. Gao, Gradient Histogram Estimation and Preservation for Texture Enhanced Image Denoising, in TIP

35 GHP The key is to estimate and preserve the gradient histogram of true images and preserve it in the denoised image: xˆ y x α 2 i i βi 1 arg minx, F 2 F( x) x s.t. xd α, h h F r An iterative histogram specification algorithm is developed for the efficient solution of the GHP model. Similar PSNR/SSIM measures to BM3D, LSSC and NCSR, but more natural and visually pleasant denoising results. 35

36 GHP results: CVPR 13 logo Original Noisy BM3D GHP 36

37 Noisy image BM3D LSSC NCSR GHP Ground truth 37

38 Group sparsity?????? ?????? A sparse solution A group sparse solution 38

39 From 1D to 2D: rank minimization 39

40 Nuclear norm 40

41 Nuclear Norm Minimization(NNM) 41

42 NNM: pros and cons Pros Tightest convex envelope of rank minimization. Closed form solution. Cons Treat equally all the singular values. Ignore the different significances of matrix singular values. 42

43 Weighted nuclear norm minimization (WNNM) 43

44 Optimization of WNNM Q. Xie, D. Meng, S. Gu, L. Zhang, W. Zuo, X. Feng, and Z. Xu, On the optimization of weighted nuclear norm minimization, Technical Report, to be online soon. 44

45 An important corollary Q. Xie, D. Meng, S. Gu, L. Zhang, W. Zuo, X. Feng, and Z. Xu, On the optimization of weighted nuclear norm minimization, Technical Report, to be online soon. 45

46 Application of WNNM to image denoising 1. For each noisy patch, search in the image for its nonlocal similar patches to form matrix Y. 2. Solve the WNNM problem to estimate the clean patches X from Y. 3. Put the clean patch back to the image. 4. Repeat the above procedures several times to obtain the denoised image. 46

47 WNNM based image denoising WNNM S. Gu, L. Zhang, W. Zuo and X. Feng, Weighted Nuclear Norm Minimization with Application to Image Denoising, CVPR

48 The weights 48

49 Experimental Results (a) Ground truth (b) Noisy image (PSNR: 14.16dB) (c) BM3D (PSNR: 26.78dB) (d) EPLL (PSNR: 26.65dB) (e) LSSC (PSNR: 26.77dB) (f) NCSR (PSNR: 26.66dB) (g) SAIST (PSNR: 26.63dB) (h) WNNM (PSNR: 26.98dB) Denoising results on image Boats by different method (noise level sigma=50). 49

50 Experimental Results (a) Ground truth (b) Noisy image (PSNR:dB) (c) BM3D (PSNR: 24.22dB) (d) EPLL (PSNR: 22.46dB) (e) LSSC (PSNR: 24.04dB) (f) NCSR (PSNR: 23.76dB) (g) SAIST (PSNR: 24.26dB) (h) WNNM (PSNR: 24.68dB) Denoising results on image Fence by different method (noise level sigma=75). 50

51 Experimental Results (a) Ground truth (b) Noisy image ( PSNR: 8.10dB) (c) BM3D (PSNR: 22.52dB) (d) EPLL (PSNR: 22.23dB) (e) SSC (PSNR: 22.24dB) (f) NCSR (PSNR: 22.11dB) (g) SAIST (PSNR: 22.61dB) (h) WNNM (PSNR: 22.91dB) Denoising results on image Monarch by different method (noise level sigma=100). 51

52 Experimental Results (a) Ground truth (b) Noisy image (PSNR:8.10dB) (c) BM3D (PSNR: 33.05dB) (d) EPLL (PSNR: 32.61dB) (e) LSSC (PSNR: 32.88dB) (f) NCSR (PSNR: 32.95dB) (g) SAIST (PSNR: 33.08dB) (h) WNNM (PSNR: 33.12dB) Denoising results on image House by different method (noise level sigma=100). 52

53 Experimental Results Sigma=20 BM3D EPLL LSSC NCSR SAIST WNNM C-Man House Peppers Montage Leaves Starfish Mornar Airplane Paint JellyBean Fence Parrot Lena Barbara Boat Hill F.print Man Couple Straw AVE

54 Experimental Results Sigma=40 BM3D EPLL LSSC NCSR SAIST WNNM C-Man House Peppers Montage Leaves Starfish Mornar Airplane Paint JellyBean Fence Parrot Lena Barbara Boat Hill F.print Man Couple Straw AVE

55 Experimental Results Sigma=50 BM3D EPLL LSSC NCSR SAIST WNNM C-Man House Peppers Montage Leaves Starfish Mornar Airplane Paint JellyBean Fence Parrot Lena Barbara Boat Hill F.print Man Couple Straw AVE

56 Experimental Results Sigma=75 BM3D EPLL LSSC NCSR SAIST WNNM C-Man House Peppers Montage Leaves Starfish Mornar Airplane Paint JellyBean Fence Parrot Lena Barbara Boat Hill F.print Man Couple Straw AVE

57 Experimental Results Sigma=100 BM3D EPLL LSSC NCSR SAIST WNNM C-Man House Peppers Montage Leaves Starfish Mornar Airplane Paint JellyBean Fence Parrot Lena Barbara Boat Hill F.print Man Couple Straw AVE

58 Low Rank Sparse representation Nonlocal Dictionary Learning Image Patches With patch based image modeling, nonlocal, sparse representation, low rank and dictionary leaning can be used individually or jointly for image processing.

59 What s next? Actually I don t know Probably Sparse/Low-rank + Big Data? Theoretical analysis? Algorithms and implementation? W.r.t. image restoration, one interesting topic (at least I think) is perceptual quality oriented image restoration. 59

60 Sparse representation: data perspective Curse of dimensionality For real-world high-dimensional data, the available samples are usually insufficient. Fortunately, real data often lie on low-dimensional, sparse, or degenerate structures in the high-dimensional space. Subspace methods: PCA, LLE, ISOMAP, ICA, Coding methods: Bag-of-words, Mixture-models, 60

61 Sparse representation based classification (SRC) y X = X1, X2,..., X K = ; ;...; K 1 2 Test image Training dictionary coefficients Representation: min s.t. y X 1 is sparse: ideally, only supported on images of the same subject Classification: r label ( ) arg min k k r yx ˆ y k k k 2 J. Wright, A. Yang, A. Ganesh, S. S. Sastry, and Y. Ma. Robust Face Recognition via Sparse Representation, PAMI

62 How to process a corrupted face? y X = X1, X2,..., X K = 1; 2;...; K e Test image Training dictionary coefficients Representation: min s.t. y X, I 1 Equivalent to min e s.t. y X e 1 1 Classification: r label ( ) arg min k k r y X ˆ eˆ y k k k 2 62

63 Regularized robust coding Can we have a more principled way to deal with various types of outliers in face images? Our solution: Meng Yang Lei Zhang, Jian Yang and David Zhang. Robust sparse coding for face recognition. In CVPR Meng Yang, Lei Zhang, Jian Yang, and David Zhang, Regularized robust coding for face recognition. IEEE Trans. Image Processing,

64 One big question! Is it true that the sparse representation helps face recognition? L. Zhang, M. Yang, and X. Feng. Sparse Representation or Collaborative Representation: Which Helps Face Recognition? In ICCV L. Zhang, M. Yang, X. Feng, Y. Ma and D. Zhang, Collaborative Representation based Classification for Face Recognition, arxiv:

65 Within-class or across-class representation Analyze the working mechanism of SRC. Training samples Yes Within-class representation Enough? No Across-class representation Regularized nearest subspace Collaborative representation 65

66 Regularized nearest subspace Training samples Enough? Yes Regularized nearest subspace (RNS) The query sample is represented on the training samples from the same class with regularized representation coefficient: min yx st.. 2 i 2 l p 66

67 Why regularization? Assume that we have enough training samples for each class so that all the images of class i can be faithfully represented by X i. All the face images are somewhat similar, while some subjects may have very similar face images. Let X j = X i +. If is small (meet some condition), there is e j e y 2 i ( Xi) min 1, m n e j and e j are the representation error of X j and X i to represent y without any constraint on the representation coefficient. 67

68 Regularized nearest subspace with L p -norm Representation residual min yx st i 2 l p Representation residual Correct class Wrong class L 0 norm of representation 0.46 coefficients Correct class Wrong class L 1 norm of representation coefficients L 2 norm of representation coefficients Regularization makes classification more stable. L 2 -norm regularization can play the similar role to L 0 and L 1 norm in this classification task. Representation residual Correct class Wrong class 68

69 Why collaborative representation? Training samples Enough? No Collaborative Representation FR is a typical small-sample-size problem, and X i is under-complete in general. Face images of different classes share similarities. Samples from other classes can be used to collaboratively represent the sample in one class. 2 min y X X [ X, X,, X ] 2 l p 1 2 Dilute the small-size-sample problem Consider the competition between different classes K 69

70 Why collaborative representation? Without considering the l p -norm regularization in coding, the associated representation is actually the perpendicular projection of y onto the space spanned by X. 2 α 2 i αˆ arg min y Xα yˆ X ˆ iαi Only e yˆ X αˆ * i i i 2 2 works for classification The double checking in e* makes the classification more effective and robust. 70

71 L 1 vs. L 2 in regularization min yx Coefficients of l 1 -regularized minimization 0.2 Coefficients of l 2 -regularized minimization l p Sparse! Non-sparse! Though L 1 leads to sparser coefficients, the classification rates are similar. Recognition rate l 1 -regularized minimization l 2 -regularized minimization 0 0 1e-6 5e-5 5e-4 5e-3 1e

72 Collaborative representation model min y X p, q 1or 2 l l q q=2,p=1, Sparse Representation based Classification (S-SRC) q=2,p=2, Collaborative Representation based Classification with regularized least square (CRC_RLS) q=1,p=1, Robust Sparse Representation based Classification (R-SRC) q=1,p=2, Robust Collaborative Representation based Classification (R-CRC) CRC_RLS has a closed-form solution; others have iterative solutions. p 72

73 Gender classification 700 Male Samples 700 Female Samples Training Set Male? Female? Feature (AR) RNS_L1 RNS_L2 CRC-RLS S-SRC SVM LRC NN 300-d Eigenface 94.9% 94.9% 93.7% 92.3% 92.4% 27.3% 90.7% Big benefit (67% improvements) brought by regularization on coding vector! 73

74 Face recognition without occlusion Training samples per subject are limited. Training Set Identity? 95 AR database Recognition rate NN LRC SVM S-SRC CRC_RLS Highest accuracy when feature dimension is not too low Eigenface dimension 74

75 Recognition rate Face recognition with pixel corruption Original image 70% random corruption Identity? ~ Corruption percent (%) on EYB R-SRC R-CRC 75

76 Face recognition with real disguise Identity? Disguise(AR) Sunglass (test 1) Scarf (test 1) Sunglass (test 2) Scarf (test 2) R-SRC 87.0% 59.5% 69.8% 40.8% CRC-RLS 68.5% 90.5% 57.2% 71.8% R-CRC 87.0% 86.0% 65.8% 73.2% Significant improvement in the case of scarf 76

77 Running time No occlusion (MPIE) L1_ls Homotopy FISTA ALM CRC-RLS Recognition rate 92.6% 92.0% 79.6% 92.0% 92.2% Running time (s) Speed-up times! Corruption (MPIE) L1_ls Homotopy SpaRSA FISTA ALM R-CRC Average running time Speed-up times! 77

78 One even bigger question! SRC/CRC represents the query face by gallery faces from all classes. However, it uses the representation residual by each class for classification. So what kind of classifier SRC/CRC is? Why SRC/CRC works? L. Zhang, W. Zuo, X. Feng, and Y. Ma, A Probabilistic Formulation of Collaborative Representation based Classifier, Preprint, to be online soon. 78

79 Probabilistic subspace of X k Samples of class k: X k = [x 1, x 2,..., x n ]. S : the subspace spanned by X k. Each data point x in S can be written as: x= X k. We assume that the probability that x belongs to class k is determined by : P( label( x) k) exp c α 2 2 It can be shown that such a probability depends on the distribution of the samples of X k. The red point will have much higher probability than the green one. 79

80 Representation of query sample y The query sample y usually lies outside the subspace {X k }. The probability that y belongs to class k is determined by two factors: Given x= X k, how likely y has the same class label as x? What is the probability that x belong to y class k? By maximizing the product of the two probabilities, we have X k 2 2 p max log P( label( y) k) min y X α α k α k

81 Two classes X 1 = [x 1,1, x 1,2,..., x 1,n ]; X 2 = [x 2,1, x 2,2,..., x 2,n ] S : the subspace spanned by [X 1 X 2 ] Each data point x in S can be written as: x= X X 2 2. x belongs to X 1 or X 2 with certain probability: P( label( x) 1) 2 2 exp x X α α x P( label( x) 2) 2 2 exp x X α α

82 Collaborative representation of y y lies outside the subspace {X X 2 2 }. The probability that y belongs to class 1 or 2 depends on how likely y has the same class label as x=x 1 1 +X 2 2 and the probability that x belongs to class 1 or 2: y p 1 { α, α } 1 2 maxlog P( label( y) 1) min y ( X α X α ) Xα X α α x p 2 { α, α } 1 2 maxlog P( label( y) 2) min y ( X α X α ) Xα X α α

83 General case The probability that query sample y belongs to class k can be computed as: p K 2 K 2 min y X α X α X α α k { } 1 i i 2 1 i i k k k α i i i 2 The classification rule: label( y) arg max { p } k k 2 2 Problem: For each class k, we need to solve the optimization once. This can be costly. 83

84 Joint probability For a data point x in the subspace spanned by all classes X, we define the following joint probability: For the query sample y outside the subspace of X, we have We use the marginal probability for classification: We only need to solve the optimization once. i1 2 2 P( label( x) 1,..., label( x) K) exp x X α α K i i 2 i 2 2 max log( P K K ( )) min y X α Xα X α α α 2 2 i1 i i 1 i i 2 i 2 2 i p P( label( y) k) exp y Xαˆ Xαˆ X αˆ αˆ k 2 k k 2 k 2 label( y) arg max { p } k k 84

85 Variants ProCRC-l 2 (closed form solution) min α ProCRC-l 1 min α Robust ProCRC (ProCRC-r ) K 2 K 2 2 y X α Xα X α α i1 i i 1 i i 2 i 2 2 i K 2 K y X α Xα X α α i1 i i 1 i i i i 2 min α y K X α K i1 i i 1 i1 i i 2 i 1 x X α 2 α 85

86 Face recognition: AR Dim SVM NSC CRC SRC CROC ProCRC-l ProCRC-l

87 Face recognition: Extended Yale B Dim SVM NSC CRC SRC CROC ProCRC-l ProCRC-l

88 Robust face recognition Random corruption (YaleB) Corruption ratio 10% 20% 40% 60% SRC-r ProCRC-r Block occlusion (YaleB) Occlusion ratio 10% 20% 30% 40% SRC-r ProCRC-r Disguise (AR) Disguise Sunglasses Scarf SRC-r ProCRC-r

89 Handwritten digit recognition: MNIST Number of training samples per class SVM NSC CRC SRC CROC ProCRC-l ProCRC-l

90 Handwritten digit recognition: USPS Number of training samples per class SVM NSC CRC SRC CROC ProCRC-l ProCRC-l

91 Running time Intel Core (TM) i7-2720qm 2.20 GHz CPU with 8 GB RAM Running time (second) of different methods on the Extended Yale B dataset:

92 Remarks ProCRC provides a good probabilistic interpretation of collaborative representation based classifiers (NSC, SRC and CRC). ProCRC achieves higher classification accuracy than the competing classifiers in most experiments. ProCRC has small performance variation under different number of training samples and feature dimension. It is robust to training sample size and feature dimension.

93 Take Research as Fun! Thank you!