SUPPLEMENTARY MATERIAL

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1 SUPPLEMENTARY MATERIAL Zhiyuan Zha 1,3, Xin Liu 2, Ziheng Zhou 2, Xiaohua Huang 2, Jingang Shi 2, Zhenhong Shang 3, Lan Tang 1, Yechao Bai 1, Qiong Wang 1, Xinggan Zhang 1 1 School of Electronic Science and Engineering, Nanjing University, Nanjing , China. 2 The Center for Machine Vision and Signal Analysis, University of Oulu, 90014, Finland. 3 School of Information Engineering and Automation, Kunming University of Science and Technology, Kunming , China. 1. MODELING OF GROUP SPARSITY RESIDUAL CONSTRAINT The flowchart of the proposed method is illustrated in Fig DISCUSSION This section will provide the detailed discussion about the main difference among the traditional first-pass estimation, NCSR method [1] and patch selection via k nearest neighbors (knn) method. Instead of using the traditional first-pass estimation denoising methods [2, 3, 4, 5], in our method, the noise image Y is only operated by BM3D, because BM3D result is regarded as a good approximation of the original image X, without continuing denoising operator to BM3D result. We are motivated by the fact that natural images often possess similar repetitive patterns, i.e., a large number of nonlocal redundancies [6]. By searching many nonlocal similar patches to given reference patch, NCSR [1] first obtains good estimates of the sparse coding coefficients of the original image by the principle of NLM, and then centralizes the sparse coding coefficients of the observed image to those estimates to improve the performance of denoising. However, since the fact that NLM depends on the weighted graph [7], it is unavoidable that the weighted manner leads to disturbance and inaccuracy [8]. It is indeed fortunate that the proposed GSRC model does not involve the weighted graph. In addition, NCSR is actually a patchbased sparse representation method. knn method has been widely used to similar patch s- election. Given a noisy reference patch and a target This work was supported by the NSFC ( , , ) and nd the open research fund of National Mobile Commune. Research Lab., Southeast University (No.2015D08). dataset. The goal is to find the k most similar patches. However, since the given reference patch is noisy, knn has a drawback that some of the k selected patches may not be truly similar to given reference patch. For example, the noisy similar patches via knn and the clean patches matched by these noisy similar patches are shown in Fig. 2(a) and Fig. 2(b), respectively. It is clear that the 7-th patch is deviating from given reference patch. Because the BM3D operation result is regarded as a good estimation of the original image, in this paper we adopt BM3D result as the target image to fetch the k most similar patches. Fig. 2(c) shows the BM3D similar patches via knn and Fig. 2(d) shows the clean patches matched by BM3D similar patches. It can be seen that BM3D image can achieve more similar patches index than a noisy image. Moreover, in order to obtain an effective similar patches index by knn, we empirically define SSIM (Z, ˆX t+1 )-SSIM (Z, ˆX t )< τ, then ˆX t+1 is regarded as target image to fetch the k similar patches indices, where SSIM represents structural similarity, ˆX t represents the t-th iteration denosing result and τ is a small constant. 3. ADAPTIVE GROUP SPARSITY REGULARIZATION PARAMETER SETTING The parameter λ that balances the fidelity term and the regularization term should be adaptively determined for better denoising performance. In this subsection, we propose a more robust method for computing λ i of each group by formulating the group sparse estimation as a Maximum a Posterior (MAP) estimation problem. For a given B i, the MAP estimation of R i can be calculated as ˆR i = arg max R i {log P(R i Y i )} = arg min R i { log P(Y i R i ) log P(R i )} Since Y i is contaminated with some additive Gaussian white noises of standard deviation σ n, the likelihood term is often (1)

2 Patch yi Group Y i Group Sparsity i Group Sparsity Residual i i Noisy Image PSNR=18.56dB Extracting Matching PCA Operator A D Y 1 i i i i i - Recoverd Image PSNR=32.66dB Initilization by BM3D Patch z i Group Zi Dictionary Di Group Sparsity i -2-6 i i Distribution of ˆ i i i X D Aggregating Recovered Group Xˆ i Extracting Matching B D Z 1 i i i -10 BM3D Result PSNR=32.09dB Distribution of A i -B i Fig. 1. Flowchart of image denoising by group sparsity residual constraint prior model. (a) Noisy image patches (b) Original image patches matched by noisy image patches (c) Image patches by BM3D (d) Original image patches matched by BM3D patches Fig. 2. Patch selection between noisy image and BM3D result via knn method (where green box represents the reference patch).

3 characterized by the Gaussian distribution, P(Y i R i ) = P(Y i A i, B i ) = exp( 1 2σn 2 Y i D i A i 2 2) (2) where R i and B i are assumed to be independent. Since the signal R i can be well characterized by the Laplacian distribution from Fig. 4. Thus, the prior distribution P(R i ) is characterized by an i.i.d Laplacian distribution, P(R i ) = i c 2 exp( R i (j) ) (3) 2σi, j σ i, j j where R i (j) is the j-th element of R i, and σ i, j is the standard deviation of R i (j). Then we substitute (2) and (3) into (1), and we can readily derive the desired regularization parameter λ i, j, i.e., λ i, j = c 2 2σ n 2 σ i, j (4) where σ i, j denotes the estimated variance of R i, and c is a small constant. 4. EXPERIMENTAL RESULTS 4.1. Synthetic image results We first evaluate the competing methods on 12 test images, whose scenes are shown in Fig. 2. Gaussian white noise with standard deviation σ = 30, 40, 50, 100 is added to those test images. The visual comparison of competing denoising methods are shown in Figs Real image results Next, we evaluate the competing method on 4 real images, the visual comparison of competing denoising methods are shown in Figs It can be seen that the proposed GSRC is able to preserve the sharp edges and small fine details more effectively than the other competing methods Image dataset results We also comprehensively evaluate the proposed method on randomly 300 test images from ASDS [9] 1. Table 1 shows qualitative comparisons of the competing denosing methods on five noise levels (σ = 30, 40, 50, 75, 100). It can be seen that the proposed GSRC can outperform the other competing methods in terms of PSNR. Table 1. Denoising PSNR (db) results by different denoising methods on 300 test images with additive Gaussian white noise. σ BM3D NCSR WNNM AST MSEPLL GSRC REFERENCES [1] W. Dong, L. Zhang, G. Shi, and X. Li, Nonlocally centralized sparse representation for image restoration, IEEE Transactions on Image Processing, vol. 22, no. 4, pp , [2] H. Talebi and P. Milanfar, Global image denoising, IEEE Transactions on Image Processing, vol. 23, no. 2, pp , [3] Y. Romano and M. Elad, Boosting of image denoising algorithms, SIAM Journal on Imaging Sciences, vol. 8, no. 2, pp , [4] E. Luo, S. H. Chan, and T. Q. Nguyen, Adaptive image denoising by targeted databases, IEEE Transactions on Image Processing, vol. 24, no. 7, pp , [5] V. Papyan and M. Elad, Multi-scale patch-based image restoration, IEEE Transactions on Image Processing, vol. 25, no. 1, pp , [6] A. Buades, B. Coll, and J.-M. Morel, A non-local algorithm for image denoising, in 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 05), vol. 2. IEEE, 2005, pp [7] G. Peyré, Image processing with nonlocal spectral bases, Multiscale Modeling & Simulation, vol. 7, no. 2, pp , [8] X. Zhang, M. Burger, X. Bresson, and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM Journal on Imaging Sciences, vol. 3, no. 3, pp , [9] W. Dong, L. Zhang, G. Shi, and X. Wu, Image deblurring and super-resolution by adaptive sparse domain selection and adaptive regularization, IEEE Transactions on Image Processing, vol. 20, no. 7, pp , Download in xliu/research/gsrc/gsrc.html.

4 Fig. 3. Denoising images of lin by different methods (σ = 30). (a) Original image; (b) Noisy image; (c) BM3D (P- SNR=31.07dB); (d) NCSR (PSNR=30.84dB); (e) WNNM (PSNR=30.96dB); (f) AST-NLS (PSNR=30.84dB); (g) MSEPLL (PSNR=30.96dB); (h) GSRC (PSNR=31.21dB). Fig. 4. Denoising images of P arrot by different methods (σ = 30). (a) Original image; (b) Noisy image; (c) BM3D (P- SNR=30.33dB); (d) NCSR (PSNR=30.38dB); (e) WNNM (PSNR=30.66dB); (f) AST-NLS (PSNR=30.52dB); (g) MSEPLL (PSNR=30.29dB); (h) GSRC (PSNR=30.79dB).

5 Fig. 5. Denoising images of Couple by different methods (σ = 30). (a) Original image; (b) Noisy image; (c) BM3D (P- SNR=30.58dB); (d) NCSR (PSNR=30.22dB); (e) WNNM (PSNR=30.56dB); (f) AST-NLS (PSNR=30.30dB); (g) MSEPLL (PSNR=30.42dB); (h) GSRC (PSNR=30.67dB). Fig. 6. Denoising images of Barbara by different methods (σ = 40). (a) Original image; (b) Noisy image; (c) BM3D (PSNR=27.33dB); (d) NCSR (PSNR=27.36dB); (e) WNNM (PSNR=27.84dB); (f) AST-NLS (PSNR=27.51dB); (g) MSEPLL (PSNR=26.05dB); (h) GSRC (PSNR=28.00dB).

6 Fig. 7. Denoising images of House by different methods (σ = 40). (a) Original image; (b) Noisy image; (c) BM3D (P- SNR=30.67dB); (d) NCSR (PSNR=30.82dB); (e) WNNM (PSNR=31.35dB); (f) AST-NLS (PSNR=31.16dB); (g) MSEPLL (PSNR=30.47dB); (h) GSRC (PSNR=31.64dB). Fig. 8. Denoising images of lin by different methods (σ = 40). (a) Original image; (b) Noisy image; (c) BM3D (P- SNR=29.53dB); (d) NCSR (PSNR=29.45dB); (e) WNNM (PSNR=29.80dB); (f) AST-NLS (PSNR=29.43dB); (g) MSEPLL (PSNR=29.68dB); (h) GSRC (PSNR=29.98dB).

7 Fig. 9. Denoising images of C.man by different methods (σ = 50). (a) Original image; (b) Noisy image; (c) BM3D (P- SNR=26.22dB); (d) NCSR (PSNR=26.17dB); (e) WNNM (PSNR=26.45dB); (f) AST-NLS (PSNR=26.34dB); (g) MSEPLL (PSNR=26.12dB); (h) GSRC (PSNR=26.71dB). Fig. 10. Denoising images of M onarch by different methods (σ = 50). (a) Original image; (b) Noisy image; (c) BM3D (PSNR=25.89dB); (d) NCSR (PSNR=25.78dB); (e) WNNM (PSNR=26.42dB); (f) AST-NLS (PSNR=26.10dB); (g) MSEPLL (PSNR=25.93dB); (h) GSRC (PSNR=26.64dB).

8 (e)(a) (f)(b) (g) (c) (h)(d) Fig. 11. Denoising images of Lena by different methods (σ = 50). (a) Original image; (b) Noisy image; (c) BM3D (P- SNR=27.03dB); (d) NCSR (PSNR=27.02dB); (e) WNNM (PSNR=27.16dB); (f) AST-NLS (PSNR=27.08dB); (g) MSEPLL (PSNR=26.97dB); (h) GSRC (PSNR=27.28dB). Fig. 12. Denoising images of House by different methods (σ = 100). (a) Original image; (b) Noisy image; (c) BM3D (PSNR=25.87dB); (d) NCSR (PSNR=25.57dB); (e) WNNM (PSNR=26.72dB); (f) AST-NLS (PSNR=26.92dB); (g) MSEPLL (PSNR=25.99dB); (h) GSRC (PSNR=27.24dB).

9 Fig. 13. Denoising images of P arrot by different methods (σ = 100). (a) Original image; (b) Noisy image; (c) BM3D (PSNR=24.62dB); (d) NCSR (PSNR=24.46dB); (e) WNNM (PSNR=24.94dB); (f) AST-NLS (PSNR=24.88dB); (g) MSEPLL (PSNR=24.37dB); (h) GSRC (PSNR=25.23dB). Fig. 14. Denoising images of f oreman by different methods (σ = 100). (a) Original image; (b) Noisy image; (c) BM3D (PSNR=26.31dB); (d) NCSR (PSNR=26.55dB); (e) WNNM (PSNR=27.40dB); (f) AST-NLS (PSNR=27.06dB); (g) MSEPLL (PSNR=26.84dB); (h) GSRC (PSNR=27.55dB).

10 Fig. 15. Denoising images of C.M an by different methods (σ = 100). (a) Original image; (b) Noisy image; (c) BM3D (PSNR=23.09dB); (d) NCSR (PSNR=22.94dB); (e) WNNM (PSNR=23.40dB); (f) AST-NLS (PSNR=23.34dB); (g) MSEPLL (PSNR=23.01dB); (h) GSRC (PSNR=23.60dB). (a) (b) (c) (d) (e) (f) (g) Fig. 16. Denoising images of Old Image 1 by different methods. (a) Original image; (b) BM3D; (c) NCSR; (d) WNNM; (e) AST-NLS; (f) MSEPLL ; (g) GSRC.

11 (a) (b) (c) (d) (e) (f) (g) Fig. 17. Denoising images of Old Image 2 by different methods. (a) Original image; (b) BM3D; (c) NCSR; (d) WNNM; (e) AST-NLS; (f) MSEPLL ; (g) GSRC. (a) (b) (c) (d) (e) (f) (g) Fig. 18. Denoising images of Old Image 3 by different methods. (a) Original image; (b) BM3D; (c) NCSR; (d) WNNM; (e) AST-NLS; (f) MSEPLL ; (g) GSRC.

12 (a) (b) (c) (d) (e) (f) (g) Fig. 19. Denoising images of Old Image 2 by different methods. (a) Original image; (b) BM3D; (c) NCSR; (d) WNNM; (e) AST-NLS; (f) MSEPLL ; (g) GSRC. (h)