# American Institute of Mathematical Sciences

October  2019, 13(5): 903-930. doi: 10.3934/ipi.2019041

## Piecewise constant signal and image denoising using a selective averaging method with multiple neighbors

 1 Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China 2 School of Mathematical Sciences, Nankai University, Tianjin 300071, China 3 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

* Corresponding author: Chunlin Wu

Received  March 2018 Revised  March 2019 Published  July 2019

Fund Project: The first author is supported by HDU grant KYS075618129. The second author is supported by NSFC grant 11871035 and 11531013 and Recruitment Program of Global Young Expert. The third author is supported by NSFC grant 11771420.

Piecewise constant signals and images are an important kind of data. Typical examples include bar code signals, logos, cartoons, QR codes (Quick Response codes), and text images, which are widely used in both general commercial and automotive industry use. One previous work called a general selective averaging method (GSAM) was introduced to remove noise from them. It chooses homogeneous neighbors from the two closest pixels (one pixel at each side) to update the current pixel. One limitation is that it suffered from appearing sparse noisy pixels in the denoised result when the noise level is high. In this paper, we try to solve this problem by proposing a selective averaging method with multiple neighbors. To update the intensity value at each pixel, the proposed algorithm averages more homogeneous neighbors selected from a large domain, which is based on the property of the local geometry of signals and images. This greatly reduces sparse noisy pixels left in the final result by GSAM. Similarly, our method adopts the Neumann boundary condition at edges, and thus preserves edges well. In 1D case, some theoretical results are given to guarantee the convergence of our algorithm. In 2D case, except eliminating additive Gaussian noise, this algorithm can be used for restoring noisy images corrupted by speckle noise. Intensive experiments on both gray and color image denoising demonstrate that the proposed method is quite effective for piecewise constant image denoising and achieves superior performance visually and quantitatively.

Citation: Weina Wang, Chunlin Wu, Jiansong Deng. Piecewise constant signal and image denoising using a selective averaging method with multiple neighbors. Inverse Problems and Imaging, 2019, 13 (5) : 903-930. doi: 10.3934/ipi.2019041
##### References:
 [1] P. Blomgren and T. F. Chan, Color TV: total variation methods for restoration of vector-valued images, IEEE Trans. Image Process., 7 (1998), 304-309. [2] J. E. Boyd and J. Meloche, Binary restoration of thin objects in multidimensional imagery, IEEE Trans. Pattern Anal. Mach. Intell., 20 (1998), 647-651.  doi: 10.1109/34.683781. [3] C. Brito-Loeza, K. Chen and V. Uc-Cetina, Image denoising using the Gaussian curvature of the image surface, Numer. Math. Part. D. E., 32 (2016), 1066-1089.  doi: 10.1002/num.22042. [4] A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms with a new one, Multiscale Model. Simul., 4 (2005), 490-530.  doi: 10.1137/040616024. [5] J. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Model. Simul., 8 (2009), 337-369.  doi: 10.1137/090753504. [6] J. Cai, B. Dong and Z. Shen, Image restoration: A wavelet frame based model for piecewise smooth functions and beyond, Appl. Comput. Harmon. Anal., 41 (2015), 94-138.  doi: 10.1016/j.acha.2015.06.009. [7] J. Cai, H. Ji, Z. Shen and G. Ye, Data-driven tight frame construction and image denoising, Appl. Comput. Harmon. Anal., 37 (2014), 89-105.  doi: 10.1016/j.acha.2013.10.001. [8] R. H. Chan, T. F. Chan, L. Shen and Z. Shen, Wavelet algorithms for high-resolution image reconstruction, SIAM J. Sci. Comput., 24 (2003), 1408-1432.  doi: 10.1137/S1064827500383123. [9] T. F. Chan, S. Esedoglu and M. Nikolova, Finding the global minimum for binary image restoration, in IEEE International Conference on Image Processing 2005, (2005), 75-89. doi: 10.1109/ICIP.2005.1529702. [10] C. S. Cho and S. Lee, Effective Five Directional Partial Derivatives-based Image Smoothing and a Parallel Structure Design, IEEE Trans. Image Process., 25 (2016), 1617-1625.  doi: 10.1109/TIP.2016.2526785. [11] R. Choksi, Y. van Gennip and A. Oberman, Anisotropic total variation regularized L1 approximation and denoising/deblurring of 2D bar codes, Inverse Probl. Imaging, 5 (2011), 591-617.  doi: 10.3934/ipi.2011.5.591. [12] N. Chumchob, K. Chen and C. Brito-Loeza, A new variational model for removal of combined additive and multiplicative noise and a fast algorithm for its numerical approximation, Int. J. Comput. Math., 90 (2013), 140-161.  doi: 10.1080/00207160.2012.709625. [13] P. Coupé, P. Hellier, C. Kervrann and C. Barillot, Nonlocal means-based speckle filtering for ultrasound images, IEEE Trans. Image Process., 18 (2009), 2221-2229.  doi: 10.1109/TIP.2009.2024064. [14] K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238. [15] C. A. Deledalle, V. Duval and J. Salmon, Anisotropic non-local means with spatially adaptive patch shapes, SSVM: Scale Space and Variational Methods in Computer Vision, 6667 (2011), 231-242.  doi: 10.1007/978-3-642-24785-9_20. [16] C. A. Deledalle, A. Charles, V. Duval and J. Salmon, Non-local Methods with Shape-Adaptive Patches (NLM-SAP), J. Math. Imaging Vis., 43 (2012), 103-120.  doi: 10.1007/s10851-011-0294-y. [17] F. Dong and Y. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Probl. Imag., 10 (2016), 1066-1089.  doi: 10.3934/ipi.2016.10.27. [18] D. L. Donoho and J. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81 (1994), 425-455.  doi: 10.1093/biomet/81.3.425. [19] D. L. Donoho, De-noising by soft-thresholding, IEEE Trans. Inf. Theory, 41 (1995), 613-627.  doi: 10.1109/18.382009. [20] M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15 (2006), 3736-3745.  doi: 10.1109/TIP.2006.881969. [21] M. Figueiredo and R. Nowak, An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process., 12 (2003), 906-916.  doi: 10.1109/TIP.2003.814255. [22] R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2 edition, Prentice Hall, 2002. [23] X. Gu, H. Wang and D. Yu, Binary Image Restoration Using Pulse Coupled Neural Network, in ICNIP, (2001), 922-927. [24] W. Guo, J. Qin and V. A. Luminita, A geometry guided image denoising scheme, Inverse Probl. Imag., 7 (2013), 1066-1089. [25] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1990. [26] Y. Huang, M. K. Ng and Y. Wen, A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci., 2 (2009), 20-40.  doi: 10.1137/080712593. [27] H. Ji, Y. Luo and Z. Shen, Image recovery via geometrically structured approximation, Appl. Comput. Harmon. Anal., 41 (2016), 75-93.  doi: 10.1016/j.acha.2015.08.012. [28] Z. Jin and X. Yang, A variational model to remove the multiplicative noise in ultrasound images, J. Math. Imaging Vis., 39 (2011), 62-74.  doi: 10.1007/s10851-010-0225-3. [29] M. Karaman, M. A. Kutay and G. Bozdagi, An adaptive speckle suppression filter for medical ultrasonic imaging, IEEE Trans. Med. Imag., 14 (1995), 283-292.  doi: 10.1109/42.387710. [30] D. T. Kuan, A. Sawchuk, C. Timothy and P. Chavel, Adaptive noise smoothing filter for images with signal-dependent noise, IEEE Trans. Pattern Anal. Mach. Intell., 7 (1985), 165-177.  doi: 10.1109/TPAMI.1985.4767641. [31] J. Lee, Digital image enhancement and noise filtering by use of local statistics, IEEE Trans. Pattern Anal. Mach. Intell., 2 (1980), 165-168.  doi: 10.1109/TPAMI.1980.4766994. [32] F. Li, M. K. Ng and C. Shen, Multiplicative noise removal with spatially varying regularization parameters, SIAM J. Imaging Sci., 3 (2010), 1-20.  doi: 10.1137/090748421. [33] J. E. Odegard, H. Guo, M. Lang, C. Burrus, R. Wells, L. Novak and M. Margarita, Wavelet based SAR speckle reduction and image compression, in SPIE, (1995), 17-21. [34] S. Ono, K. Morinaga and S. Nakayama, Two-dimensional barcode decoration based on real-coded genetic algorithm, in IEEE CEC, (2008), 1068-1073. [35] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.  doi: 10.1109/34.56205. [36] S. V. Richard, Matrix Iterative Analysis, 2nd edition, Springer-Verlag, New York, 2009. [37] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F. [38] Y. Shen, B. Han and E. Braverman, Adaptive frame-based color image denoising, Appl. Comput. Harmon. Anal., 41 (2015), 54-74.  doi: 10.1016/j.acha.2015.04.001. [39] Y. Shen, E. Y. Lam and N. Wong, A signomial programming approach for binary image restoration by penalized least squares, IEEE Trans. Circuits Sys. Ⅱ: Exp. Briefs, 55 (2008), 41-45.  doi: 10.1109/TCSII.2007.907751. [40] C. Sheng, Y. Xin, L. Yao and S. Kun, Total variation-based speckle reduction using multi-grid algorithm for ultrasound images, in ICIAP, (2005), 245-252. doi: 10.1007/11553595_30. [41] M. E. Taylor, Partial Differential Equations Ⅰ: Basic Theory, 2nd edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-7055-8. [42] C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in ICCV, (1998), 839-846. doi: 10.1109/ICCV.1998.710815. [43] S. Wang, T. Huang, X. Zhao, J. Mei and J. Huang, Speckle noise removal in ultrasound images by first- and second-order total variation, Numer. Algor., 78 (2018), 513-533.  doi: 10.1007/s11075-017-0386-x. [44] W. Wang, C. Wu and J. Deng, A general selective averaging method for piecewise constant signal and image processing, J. Sci. Comput., 76 (2018), 1078-1104.  doi: 10.1007/s10915-018-0650-9. [45] W. Wang, S. Wen, C. Wu and J. Deng, Denoising piecewise constant images with selective averaging and outlier removal, Numer. Math. Theor. Meth. Appl., 12 (2019), 467-491.  doi: 10.4208/nmtma.OA-2017-0130. [46] C. Wu and X. Tai, Augmented Lagrangian Method, Dual Methods, and Split Bregman Iteration for ROF, Vectorial TV, and High Order Models, SIAM J. Imaging Sci., 3 (2010), 300-339.  doi: 10.1137/090767558. [47] L. Xu, C. Lu, Y. Xu and J. Jia, Image smoothing via L0 gradient minimization, ACM Trans. Graph., 30 (2011), 174: 1-174: 12. [48] J. Yu, J. Tan and Y. Wang, Ultrasound speckle reduction by a SUSAN-controlled anisotropic diffusion method, Pattern Recogn., 43 (2010), 3083-3092.  doi: 10.1016/j.patcog.2010.04.006. [49] J. Zhang and K. Chen, A total fractional-order variation model for image restoration with non-homogeneous boundary conditions and its numerical solution, SIAM J. Imaging Sci., 8 (2015), 2487-2518.  doi: 10.1137/14097121X. [50] X. Zhao, F. Wang and M. K. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM J. Imaging Sci., 7 (2014), 456-475. arXiv: 1809.09783v2 arXiv: 1809.03948v1 doi: 10.1137/13092472X.

show all references

##### References:
 [1] P. Blomgren and T. F. Chan, Color TV: total variation methods for restoration of vector-valued images, IEEE Trans. Image Process., 7 (1998), 304-309. [2] J. E. Boyd and J. Meloche, Binary restoration of thin objects in multidimensional imagery, IEEE Trans. Pattern Anal. Mach. Intell., 20 (1998), 647-651.  doi: 10.1109/34.683781. [3] C. Brito-Loeza, K. Chen and V. Uc-Cetina, Image denoising using the Gaussian curvature of the image surface, Numer. Math. Part. D. E., 32 (2016), 1066-1089.  doi: 10.1002/num.22042. [4] A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms with a new one, Multiscale Model. Simul., 4 (2005), 490-530.  doi: 10.1137/040616024. [5] J. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Model. Simul., 8 (2009), 337-369.  doi: 10.1137/090753504. [6] J. Cai, B. Dong and Z. Shen, Image restoration: A wavelet frame based model for piecewise smooth functions and beyond, Appl. Comput. Harmon. Anal., 41 (2015), 94-138.  doi: 10.1016/j.acha.2015.06.009. [7] J. Cai, H. Ji, Z. Shen and G. Ye, Data-driven tight frame construction and image denoising, Appl. Comput. Harmon. Anal., 37 (2014), 89-105.  doi: 10.1016/j.acha.2013.10.001. [8] R. H. Chan, T. F. Chan, L. Shen and Z. Shen, Wavelet algorithms for high-resolution image reconstruction, SIAM J. Sci. Comput., 24 (2003), 1408-1432.  doi: 10.1137/S1064827500383123. [9] T. F. Chan, S. Esedoglu and M. Nikolova, Finding the global minimum for binary image restoration, in IEEE International Conference on Image Processing 2005, (2005), 75-89. doi: 10.1109/ICIP.2005.1529702. [10] C. S. Cho and S. Lee, Effective Five Directional Partial Derivatives-based Image Smoothing and a Parallel Structure Design, IEEE Trans. Image Process., 25 (2016), 1617-1625.  doi: 10.1109/TIP.2016.2526785. [11] R. Choksi, Y. van Gennip and A. Oberman, Anisotropic total variation regularized L1 approximation and denoising/deblurring of 2D bar codes, Inverse Probl. Imaging, 5 (2011), 591-617.  doi: 10.3934/ipi.2011.5.591. [12] N. Chumchob, K. Chen and C. Brito-Loeza, A new variational model for removal of combined additive and multiplicative noise and a fast algorithm for its numerical approximation, Int. J. Comput. Math., 90 (2013), 140-161.  doi: 10.1080/00207160.2012.709625. [13] P. Coupé, P. Hellier, C. Kervrann and C. Barillot, Nonlocal means-based speckle filtering for ultrasound images, IEEE Trans. Image Process., 18 (2009), 2221-2229.  doi: 10.1109/TIP.2009.2024064. [14] K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238. [15] C. A. Deledalle, V. Duval and J. Salmon, Anisotropic non-local means with spatially adaptive patch shapes, SSVM: Scale Space and Variational Methods in Computer Vision, 6667 (2011), 231-242.  doi: 10.1007/978-3-642-24785-9_20. [16] C. A. Deledalle, A. Charles, V. Duval and J. Salmon, Non-local Methods with Shape-Adaptive Patches (NLM-SAP), J. Math. Imaging Vis., 43 (2012), 103-120.  doi: 10.1007/s10851-011-0294-y. [17] F. Dong and Y. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Probl. Imag., 10 (2016), 1066-1089.  doi: 10.3934/ipi.2016.10.27. [18] D. L. Donoho and J. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81 (1994), 425-455.  doi: 10.1093/biomet/81.3.425. [19] D. L. Donoho, De-noising by soft-thresholding, IEEE Trans. Inf. Theory, 41 (1995), 613-627.  doi: 10.1109/18.382009. [20] M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15 (2006), 3736-3745.  doi: 10.1109/TIP.2006.881969. [21] M. Figueiredo and R. Nowak, An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process., 12 (2003), 906-916.  doi: 10.1109/TIP.2003.814255. [22] R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2 edition, Prentice Hall, 2002. [23] X. Gu, H. Wang and D. Yu, Binary Image Restoration Using Pulse Coupled Neural Network, in ICNIP, (2001), 922-927. [24] W. Guo, J. Qin and V. A. Luminita, A geometry guided image denoising scheme, Inverse Probl. Imag., 7 (2013), 1066-1089. [25] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1990. [26] Y. Huang, M. K. Ng and Y. Wen, A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci., 2 (2009), 20-40.  doi: 10.1137/080712593. [27] H. Ji, Y. Luo and Z. Shen, Image recovery via geometrically structured approximation, Appl. Comput. Harmon. Anal., 41 (2016), 75-93.  doi: 10.1016/j.acha.2015.08.012. [28] Z. Jin and X. Yang, A variational model to remove the multiplicative noise in ultrasound images, J. Math. Imaging Vis., 39 (2011), 62-74.  doi: 10.1007/s10851-010-0225-3. [29] M. Karaman, M. A. Kutay and G. Bozdagi, An adaptive speckle suppression filter for medical ultrasonic imaging, IEEE Trans. Med. Imag., 14 (1995), 283-292.  doi: 10.1109/42.387710. [30] D. T. Kuan, A. Sawchuk, C. Timothy and P. Chavel, Adaptive noise smoothing filter for images with signal-dependent noise, IEEE Trans. Pattern Anal. Mach. Intell., 7 (1985), 165-177.  doi: 10.1109/TPAMI.1985.4767641. [31] J. Lee, Digital image enhancement and noise filtering by use of local statistics, IEEE Trans. Pattern Anal. Mach. Intell., 2 (1980), 165-168.  doi: 10.1109/TPAMI.1980.4766994. [32] F. Li, M. K. Ng and C. Shen, Multiplicative noise removal with spatially varying regularization parameters, SIAM J. Imaging Sci., 3 (2010), 1-20.  doi: 10.1137/090748421. [33] J. E. Odegard, H. Guo, M. Lang, C. Burrus, R. Wells, L. Novak and M. Margarita, Wavelet based SAR speckle reduction and image compression, in SPIE, (1995), 17-21. [34] S. Ono, K. Morinaga and S. Nakayama, Two-dimensional barcode decoration based on real-coded genetic algorithm, in IEEE CEC, (2008), 1068-1073. [35] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.  doi: 10.1109/34.56205. [36] S. V. Richard, Matrix Iterative Analysis, 2nd edition, Springer-Verlag, New York, 2009. [37] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F. [38] Y. Shen, B. Han and E. Braverman, Adaptive frame-based color image denoising, Appl. Comput. Harmon. Anal., 41 (2015), 54-74.  doi: 10.1016/j.acha.2015.04.001. [39] Y. Shen, E. Y. Lam and N. Wong, A signomial programming approach for binary image restoration by penalized least squares, IEEE Trans. Circuits Sys. Ⅱ: Exp. Briefs, 55 (2008), 41-45.  doi: 10.1109/TCSII.2007.907751. [40] C. Sheng, Y. Xin, L. Yao and S. Kun, Total variation-based speckle reduction using multi-grid algorithm for ultrasound images, in ICIAP, (2005), 245-252. doi: 10.1007/11553595_30. [41] M. E. Taylor, Partial Differential Equations Ⅰ: Basic Theory, 2nd edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-7055-8. [42] C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in ICCV, (1998), 839-846. doi: 10.1109/ICCV.1998.710815. [43] S. Wang, T. Huang, X. Zhao, J. Mei and J. Huang, Speckle noise removal in ultrasound images by first- and second-order total variation, Numer. Algor., 78 (2018), 513-533.  doi: 10.1007/s11075-017-0386-x. [44] W. Wang, C. Wu and J. Deng, A general selective averaging method for piecewise constant signal and image processing, J. Sci. Comput., 76 (2018), 1078-1104.  doi: 10.1007/s10915-018-0650-9. [45] W. Wang, S. Wen, C. Wu and J. Deng, Denoising piecewise constant images with selective averaging and outlier removal, Numer. Math. Theor. Meth. Appl., 12 (2019), 467-491.  doi: 10.4208/nmtma.OA-2017-0130. [46] C. Wu and X. Tai, Augmented Lagrangian Method, Dual Methods, and Split Bregman Iteration for ROF, Vectorial TV, and High Order Models, SIAM J. Imaging Sci., 3 (2010), 300-339.  doi: 10.1137/090767558. [47] L. Xu, C. Lu, Y. Xu and J. Jia, Image smoothing via L0 gradient minimization, ACM Trans. Graph., 30 (2011), 174: 1-174: 12. [48] J. Yu, J. Tan and Y. Wang, Ultrasound speckle reduction by a SUSAN-controlled anisotropic diffusion method, Pattern Recogn., 43 (2010), 3083-3092.  doi: 10.1016/j.patcog.2010.04.006. [49] J. Zhang and K. Chen, A total fractional-order variation model for image restoration with non-homogeneous boundary conditions and its numerical solution, SIAM J. Imaging Sci., 8 (2015), 2487-2518.  doi: 10.1137/14097121X. [50] X. Zhao, F. Wang and M. K. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM J. Imaging Sci., 7 (2014), 456-475. arXiv: 1809.09783v2 arXiv: 1809.03948v1 doi: 10.1137/13092472X.
An illustration of the updating order from $i = 1$ to $i = m$
The first two rows show the results by GSAM [44] ($T = 0.44, p = 0.49$), Algorithm 1 ($T = 0.44, k = 2, p = 0.25$), TV[37] ($\alpha = 1.3, r_p = 10$) and $L_{0}$[47] ($\lambda = 0.03, \kappa = 1.01$), when $\sigma = 0.12$. The last two rows show the results by GSAM [44] ($T = 0.43, p = 0.49$), Algorithm 1 ($T = 0.43, k = 3, p = 0.11$), TV[37] ($\alpha = 1.4, r_p = 10$) and $L_{0}$[47] ($\lambda = 0.05, \kappa = 1.01$), when $\sigma = 0.16$. The corresponding SNR values are listed in brackets
Test images. (a) Logo. (b) Cartoon1. (c) QRcode1. (d) Text. (e) QArtcode. (f) Blobs. (g) QRcode2. (h) QRcode3. (i) Cartoon2
From top to down and left to right: the noisy Logo, the results by TV[37], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the top-left image are displayed for further comparison. The corresponding SNR values are listed in brackets
From top to down and left to right: the noisy Cartoon1, the results by TV[37], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The zoom-in views of the region indicited by the little line in each image are displayed for further comparison. The corresponding SNR values are listed in brackets
From top to down and left to right: the noisy QRcode1, the results by TV[37], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The corresponding SNR values are listed in brackets
From top to down and left to right: the noisy Text, the results by TV[37], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The corresponding SNR values are listed in brackets
From top to down and left to right: the noisy QArtcode, the results by TV[37], $L_{0}$[47], NLM-SAP[16], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the top-left image are displayed for further comparison. The corresponding SNR values are listed in brackets
From left to right: the noisy QRcode2, the results by TV[1], $L_{0}$[47], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the first image are displayed for further comparison. The corresponding SNR values are listed in brackets
From up to down: Denoising results by different methods for two noisy QRcode3 and Cartoon2 images. From left to right: the noisy images, the results by TV[1], $L_{0}$[47], AGSAM[44] and our method. The zoom-in views of the region indicited by the little red box in the second row-left image are displayed for further comparison. The corresponding SNR values are listed in brackets
From top to down and left to right: the clean and noisy "Blobs", the results by Wang et.al [43], $L_{0}$ [47], AGSAM [44] and our method. The zoom-in views of the region indicited by the little line in the each image are displayed for further comparison. The corresponding SNR values are listed in brackets
From up to down: Denoising results by different methods for Cartoon1 corrupted by two levels of noise. From left to right: the noisy images, the results by Wang et.al [43], $L_{0}$ [47], AGSAM [44] and our method. The corresponding SNR values are listed in brackets
Different types to compute $u^{(n+1)}_{i}$ based on five cases of $K^{l}_{i-1}$ and $K^{r}_{i-1}$ from (2) to (6). (a1-a2), (b1-b2), (c1-c2), (d) and (e) correspond to Case 1, Case 2, Case 3, Case 4 and Case 5, respectively
Iterative number and time comparison(in seconds) by GSAM [44] and Algorithm 1 in Fig. 3
 Signal $\sigma$ GSAM [44] Proposed iter t iter t Fig. 3 $0.12$ 6231 1.61 3695 1.07 $0.16$ 3322 0.98 2181 0.79
 Signal $\sigma$ GSAM [44] Proposed iter t iter t Fig. 3 $0.12$ 6231 1.61 3695 1.07 $0.16$ 3322 0.98 2181 0.79
The parameter settings of different methods for gray images
 Image $\sigma$ TV[37] $L_{0}$[47] NLM-SAP[16] AGSAM[44] Proposed $\alpha$ $\lambda$ $h$/$T$ $T$ $T$/$p$ Logo $10$ 30 0.01 1.5/0.6 5 5/0.13 $15$ 20 0.02 0.9/1.12 4 4/0.13 $20$ 14 0.03 0.7/2 2.4 3.3/0.05 Cartoon1 $15$ 20 0.02 1.6/1.35 5 5/0.13 $20$ 15 0.03 1.1/2.8 4.5 4.5/0.13 $25$ 12 0.04 0.8/5 4 4/0.06 QRcode1 $15$ 20 0.02 1.8/1.12 5 5/0.13 $25$ 12 0.04 1.1/3.12 4.5 4.5/0.13 $35$ 8 0.1 0.9/11.03 3.2 3.2/0.07 Text $15$ 24 0.02 1.7/1.8 4.5 4.5/0.13 $25$ 14 0.05 1.1/3.75 4.5 4.5/0.13 $35$ 10 0.06 1/9.8 3.2 3.5/0.13 QArtcode $15$ 23 0.02 1.7/1.8 5 5/0.13 $25$ 14 0.04 0.9/4.38 4.5 4.5/0.13 $35$ 10 0.07 0.6/9.8 3.4 3.4/0.07
 Image $\sigma$ TV[37] $L_{0}$[47] NLM-SAP[16] AGSAM[44] Proposed $\alpha$ $\lambda$ $h$/$T$ $T$ $T$/$p$ Logo $10$ 30 0.01 1.5/0.6 5 5/0.13 $15$ 20 0.02 0.9/1.12 4 4/0.13 $20$ 14 0.03 0.7/2 2.4 3.3/0.05 Cartoon1 $15$ 20 0.02 1.6/1.35 5 5/0.13 $20$ 15 0.03 1.1/2.8 4.5 4.5/0.13 $25$ 12 0.04 0.8/5 4 4/0.06 QRcode1 $15$ 20 0.02 1.8/1.12 5 5/0.13 $25$ 12 0.04 1.1/3.12 4.5 4.5/0.13 $35$ 8 0.1 0.9/11.03 3.2 3.2/0.07 Text $15$ 24 0.02 1.7/1.8 4.5 4.5/0.13 $25$ 14 0.05 1.1/3.75 4.5 4.5/0.13 $35$ 10 0.06 1/9.8 3.2 3.5/0.13 QArtcode $15$ 23 0.02 1.7/1.8 5 5/0.13 $25$ 14 0.04 0.9/4.38 4.5 4.5/0.13 $35$ 10 0.07 0.6/9.8 3.4 3.4/0.07
The parameter settings of different methods for color images
 Image $\sigma$ TV[1] $L_{0}$[47] AGSAM[44] Proposed $\alpha$ $\lambda$ $T$ $T$/$p$ QRcode2 $10$ 16 0.01 5 5/0.13 $20$ 8 0.04 4 4/0.13 $30$ 5 0.08 3.5 3.5/0.13 $40$ 4 1.6 2.3 2.6/0.1 QRcode3 $10$ 17 0.02 5 5/0.13 $20$ 9 0.04 4 4/0.13 $30$ 6 0.12 2.5 2.6/0.11 $40$ 4 0.5 2 2/0.06 Cartoon2 $10$ 19 0.01 5 5/0.13 $20$ 10 0.05 4 4.2/0.13 $30$ 6 0.09 2.5 2.8/0.13 $40$ 5 1.16 1.9 2.3/0.1
 Image $\sigma$ TV[1] $L_{0}$[47] AGSAM[44] Proposed $\alpha$ $\lambda$ $T$ $T$/$p$ QRcode2 $10$ 16 0.01 5 5/0.13 $20$ 8 0.04 4 4/0.13 $30$ 5 0.08 3.5 3.5/0.13 $40$ 4 1.6 2.3 2.6/0.1 QRcode3 $10$ 17 0.02 5 5/0.13 $20$ 9 0.04 4 4/0.13 $30$ 6 0.12 2.5 2.6/0.11 $40$ 4 0.5 2 2/0.06 Cartoon2 $10$ 19 0.01 5 5/0.13 $20$ 10 0.05 4 4.2/0.13 $30$ 6 0.09 2.5 2.8/0.13 $40$ 5 1.16 1.9 2.3/0.1
The SNR values by different methods for gray images
 Image $\sigma$ TV[37] $L_{0}$[47] NLM-SAP[16] AGSAM[44] Proposed Logo $10$ 30.07 46.80 46.41 48.05 $\bf{49.26}$ $20$ 24.41 $\bf{39.15}$ 34.96 29.05 34.77 Cartoon1 $15$ 26.86 44.60 43.22 44.95 $\bf{46.87}$ $25$ 22.64 40.17 36.01 41.20 $\bf{42.35}$ QRcode1 $15$ 27.20 41.98 42.00 43.78 $\bf{44.97}$ $25$ 22.93 37.54 35.77 39.85 $\bf{40.78}$ Text $15$ 23.55 39.19 39.75 42.57 $\bf{43.06}$ $35$ 16.48 $\bf{30.46}$ 25.00 29.79 29.97 QArtcode $15$ 24.45 36.14 $\bf{40.01}$ 39.37 39.51 $25$ 20.16 31.70 35.08 35.20 $\bf{35.25}$
 Image $\sigma$ TV[37] $L_{0}$[47] NLM-SAP[16] AGSAM[44] Proposed Logo $10$ 30.07 46.80 46.41 48.05 $\bf{49.26}$ $20$ 24.41 $\bf{39.15}$ 34.96 29.05 34.77 Cartoon1 $15$ 26.86 44.60 43.22 44.95 $\bf{46.87}$ $25$ 22.64 40.17 36.01 41.20 $\bf{42.35}$ QRcode1 $15$ 27.20 41.98 42.00 43.78 $\bf{44.97}$ $25$ 22.93 37.54 35.77 39.85 $\bf{40.78}$ Text $15$ 23.55 39.19 39.75 42.57 $\bf{43.06}$ $35$ 16.48 $\bf{30.46}$ 25.00 29.79 29.97 QArtcode $15$ 24.45 36.14 $\bf{40.01}$ 39.37 39.51 $25$ 20.16 31.70 35.08 35.20 $\bf{35.25}$
The iterative number and time comparison(in seconds) by different methods for gray images
 Image $\sigma$ TV[37] $L_{0}$[47] NLM-SAP[16] AGSAM[44] Proposed iter/t iter/t iter/t iter/t iter/t Logo $10$ 90/0.7 1551/5.5 1/141.4 139/0.5 $\bf{66}$/$\bf{0.2}$ $20$ $\bf{91}$/0.6 1481/5.0 1/145.7 182/0.6 138/$\bf{0.4}$ Cartoon1 $15$ $\bf{96}$/2.5 1481/19.6 1/698.3 170/3.5 102/$\bf{2.2}$ $25$ $\bf{98}$/$\bf{2.6}$ 1355/18.2 1/720.5 235/5.0 177/3.8 QRcode1 $15$ $\bf{83}$/$\bf{2.2}$ 1479/18.0 1/994.1 177/3.6 103/2.3 $25$ $\bf{84}$/$\bf{2.1}$ 1411/17.2 1/998.0 254/5.2 150/3.5 Text $15$ $\bf{112}$/$\bf{0.7}$ 1482/3.9 1/165.9 295/1.6 147/0.8 $35$ $\bf{109}$/$\bf{0.6}$ 1355/3.5 1/166.4 502/2.7 324/1.8 QArtcode $15$ $\bf{67}$/0.3 1478/2.9 1/77.6 175/0.3 109/$\bf{0.2}$ $25$ $\bf{69}$/0.3 1389/2.3 1/78.3 254/0.4 155/$\bf{0.2}$
 Image $\sigma$ TV[37] $L_{0}$[47] NLM-SAP[16] AGSAM[44] Proposed iter/t iter/t iter/t iter/t iter/t Logo $10$ 90/0.7 1551/5.5 1/141.4 139/0.5 $\bf{66}$/$\bf{0.2}$ $20$ $\bf{91}$/0.6 1481/5.0 1/145.7 182/0.6 138/$\bf{0.4}$ Cartoon1 $15$ $\bf{96}$/2.5 1481/19.6 1/698.3 170/3.5 102/$\bf{2.2}$ $25$ $\bf{98}$/$\bf{2.6}$ 1355/18.2 1/720.5 235/5.0 177/3.8 QRcode1 $15$ $\bf{83}$/$\bf{2.2}$ 1479/18.0 1/994.1 177/3.6 103/2.3 $25$ $\bf{84}$/$\bf{2.1}$ 1411/17.2 1/998.0 254/5.2 150/3.5 Text $15$ $\bf{112}$/$\bf{0.7}$ 1482/3.9 1/165.9 295/1.6 147/0.8 $35$ $\bf{109}$/$\bf{0.6}$ 1355/3.5 1/166.4 502/2.7 324/1.8 QArtcode $15$ $\bf{67}$/0.3 1478/2.9 1/77.6 175/0.3 109/$\bf{0.2}$ $25$ $\bf{69}$/0.3 1389/2.3 1/78.3 254/0.4 155/$\bf{0.2}$
The SNR values by different methods for color images
 Image $\sigma$ TV[1] $L_{0}$[47] AGSAM[44] Proposed QRcode2 $10$ 31.29 46.49 46.02 $\bf{47.62}$ $30$ 22.23 36.37 37.53 $\bf{37.81}$ $40$ 20.01 29.14 33.39 $\bf{33.85}$ QRcode3 $10$ 29.68 44.10 44.44 $\bf{46.11}$ $20$ 23.97 38.08 38.91 $\bf{40.77}$ $40$ 18.48 $\bf{25.57}$ 21.10 25.30 Cartoon2 $10$ 29.39 40.31 40.93 $\bf{41.05}$ $20$ 23.65 38.36 38.46 $\bf{39.53}$ $30$ 20.37 36.09 35.80 $\bf{36.99}$
 Image $\sigma$ TV[1] $L_{0}$[47] AGSAM[44] Proposed QRcode2 $10$ 31.29 46.49 46.02 $\bf{47.62}$ $30$ 22.23 36.37 37.53 $\bf{37.81}$ $40$ 20.01 29.14 33.39 $\bf{33.85}$ QRcode3 $10$ 29.68 44.10 44.44 $\bf{46.11}$ $20$ 23.97 38.08 38.91 $\bf{40.77}$ $40$ 18.48 $\bf{25.57}$ 21.10 25.30 Cartoon2 $10$ 29.39 40.31 40.93 $\bf{41.05}$ $20$ 23.65 38.36 38.46 $\bf{39.53}$ $30$ 20.37 36.09 35.80 $\bf{36.99}$
The iterative number and time comparison(in seconds) by different methods for color images
 Image $\sigma$ TV[1] $L_{0}$[47] AGSAM[44] Proposed iter/t iter/t iter/t iter/t QRcode2 $10$ 76/3.0 1551/33.3 144/2.7 $\bf{68}$/$\bf{1.4}$ $30$ $\bf{82}$/3.3 1342/29.6 253/3.4 172/$\bf{2.9}$ $40$ $\bf{75}$/$\bf{3.0}$ 1041/23.9 311/5.8 215/4.4 QRcode3 $10$ $\bf{66}$/2.6 1411/31.3 139/2.7 68/$\bf{1.5}$ $20$ $\bf{63}$/2.8 1340/32.3 183/3.5 118/$\bf{2.7}$ $40$ $\bf{72}$/$\bf{2.9}$ 1158/25.9 308/6.0 283/6.1 Cartoon2 $10$ $\bf{70}$/3.4 1551/46.1 146/3.4 79/$\bf{1.9}$ $20$ $\bf{54}$/3.2 1355/44.1 189/4.2 116/$\bf{2.8}$ $30$ $\bf{78}$/3.8 1240/39.5 245/5.4 152/$\bf{3.7}$
 Image $\sigma$ TV[1] $L_{0}$[47] AGSAM[44] Proposed iter/t iter/t iter/t iter/t QRcode2 $10$ 76/3.0 1551/33.3 144/2.7 $\bf{68}$/$\bf{1.4}$ $30$ $\bf{82}$/3.3 1342/29.6 253/3.4 172/$\bf{2.9}$ $40$ $\bf{75}$/$\bf{3.0}$ 1041/23.9 311/5.8 215/4.4 QRcode3 $10$ $\bf{66}$/2.6 1411/31.3 139/2.7 68/$\bf{1.5}$ $20$ $\bf{63}$/2.8 1340/32.3 183/3.5 118/$\bf{2.7}$ $40$ $\bf{72}$/$\bf{2.9}$ 1158/25.9 308/6.0 283/6.1 Cartoon2 $10$ $\bf{70}$/3.4 1551/46.1 146/3.4 79/$\bf{1.9}$ $20$ $\bf{54}$/3.2 1355/44.1 189/4.2 116/$\bf{2.8}$ $30$ $\bf{78}$/3.8 1240/39.5 245/5.4 152/$\bf{3.7}$
The parameter settings of different methods for gray images with speckle noise
 Image $\sigma$ Wang et.al [43] $L_{0}$[47] AGSAM[44] Proposed $\lambda$/$\mu$ $\lambda$ $T$ $T$/$p$/$k$ Blobs $2$ 1.2/11 0.04 60 60/0.04/12 Cartoon1 $2$ 0.8/2 0.05 58 58/0.13/4 $2.5$ 0.6/2 0.06 45 45/0.03/5
 Image $\sigma$ Wang et.al [43] $L_{0}$[47] AGSAM[44] Proposed $\lambda$/$\mu$ $\lambda$ $T$ $T$/$p$/$k$ Blobs $2$ 1.2/11 0.04 60 60/0.04/12 Cartoon1 $2$ 0.8/2 0.05 58 58/0.13/4 $2.5$ 0.6/2 0.06 45 45/0.03/5
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