# American Institute of Mathematical Sciences

February  2019, 13(1): 117-147. doi: 10.3934/ipi.2019007

## Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters

 1 SK hynix Inc., Icheon, Korea 2 Department of Mathematics, Chungnam National University, Daejeon, Korea 3 Department of Mathematics, Hankuk University of Foreign Studies, Yongin, Korea 4 Department of Mathematical Sciences, Seoul National University, Seoul, Korea

* Corresponding author: Myungjoo Kang

Received  January 2018 Revised  May 2018 Published  December 2018

In this article, we introduce a novel variational model for the restoration of images corrupted by multiplicative Gamma noise. The model incorporates a convex data-fidelity term with a nonconvex version of the total generalized variation (TGV). In addition, we adopt a spatially adaptive regularization parameter (SARP) approach. The nonconvex TGV regularization enables the efficient denoising of smooth regions, without staircasing artifacts that appear on total variation regularization-based models, and edges and details to be conserved. Moreover, the SARP approach further helps preserve fine structures and textures. To deal with the nonconvex regularization, we utilize an iteratively reweighted $\ell_1$ algorithm, and the alternating direction method of multipliers is employed to solve a convex subproblem. This leads to a fast and efficient iterative algorithm for solving the proposed model. Numerical experiments show that the proposed model produces better denoising results than the state-of-the-art models.

Citation: Hanwool Na, Myeongmin Kang, Miyoun Jung, Myungjoo Kang. Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters. Inverse Problems & Imaging, 2019, 13 (1) : 117-147. doi: 10.3934/ipi.2019007
##### References:
 [1] A. Almansa, C. Ballester, V. Caselles and G. Haro, A TV based restoration model with local constraints, Journal of Scientific Computing, 34 (2008), 209-236. doi: 10.1007/s10915-007-9160-x. [2] M. Artina, M. Fornasier and F. Solombrino, Linearly constrained nonsmooth and nonconvex minimization, SIAM Journal on Optimization, 23 (2013), 1904-1937. doi: 10.1137/120869079. [3] H. Attouch, J. Bolte and B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized GaussSeidel methods, Mathematical Programming, 137 (2013), 91-129. doi: 10.1007/s10107-011-0484-9. [4] G. Aubert and J.-F. Aujol, A variational approach to removing multiplicative noise, SIAM Journal on Applied Mathematics, 68 (2008), 925-946. doi: 10.1137/060671814. [5] J. Bolte, A. Daniilidis and A. Lewis, The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM Journal on Optimization, 17 (2007), 1205-1223. doi: 10.1137/050644641. [6] S. Boyd, Alternating direction method of multipliers, in Talk at NIPS Workshop on Optimization and Machine Learning, 2011. [7] K. Bredies, K. Kunisch and T. Pock, Total generalized variation, SIAM Journal on Imaging Sciences, 3 (2010), 492-526. doi: 10.1137/090769521. [8] E. J. Candes, M. B. Wakin and S. P. Boyd, Enhancing sparsity by reweighted $\ell_1$ minimization, Journal of Fourier analysis and applications, 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x. [9] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik, 76 (1997), 167-188. doi: 10.1007/s002110050258. [10] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1. [11] T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM Journal on Scientific Computing, 22 (2000), 503-516. doi: 10.1137/S1064827598344169. [12] D.-Q. Chen and L.-Z. Cheng, Spatially adapted regularization parameter selection based on the local discrepancy function for Poissonian image deblurring, Inverse Problems, 28 (2011), 015004, 24pp. doi: 10.1088/0266-5611/28/1/015004. [13] D.-Q. Chen and L.-Z. Cheng, Spatially adapted total variation model to remove multiplicative noise, IEEE Transactions on Image Processing, 21 (2012), 1650-1662. doi: 10.1109/TIP.2011.2172801. [14] D.-Q. Chen and L.-Z. Cheng, Fast linearized alternating direction minimization algorithm with adaptive parameter selection for multiplicative noise removal, Journal of Computational and Applied Mathematics, 257 (2014), 29-45. doi: 10.1016/j.cam.2013.08.012. [15] Y. Chen, W. Feng, R. Ranftl, H. Qiao and T. Pock, A higher-order MRF based variational model for multiplicative noise reduction, IEEE Signal Processing Letters, 21 (2014), 1370-1374. [16] K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3-D transfor-mdomain collaborative filtering, IEEE Transactions on Image Processing, 16 (2007), 2080-2095. doi: 10.1109/TIP.2007.901238. [17] A. Dauwe, B. Goossens, H. Q. Luong and W. Philips, A fast non-local image denoising algorithm, in Electronic Imaging 2008, International Society for Optics and Photonics, 2008, 681210-681210. [18] Y. Dong, M. Hintermüller and M. M. Rincon-Camacho, Automated regularization parameter selection in multi-scale total variation models for image restoration, Journal of Mathematical Imaging and Vision, 40 (2011), 82-104. doi: 10.1007/s10851-010-0248-9. [19] Y. Dong and T. Zeng, A convex variational model for restoring blurred images with multiplicative noise, SIAM Journal on Imaging Sciences, 6 (2013), 1598-1625. doi: 10.1137/120870621. [20] W. Feng, H. Lei and Y. Gao, Speckle reduction via higher order total variation approach, IEEE Transactions on Image Processing, 23 (2014), 1831-1843. doi: 10.1109/TIP.2014.2308432. [21] P. Getreuer, Total variation deconvolution using split Bregman, Image Processing On Line, 2 (2012), 158-174. [22] P. Getreuer, M. Tong and L. A. Vese, A variational model for the restoration of MR images corrupted by blur and Rician noise, in International Symposium on Visual Computing, Springer, 2011,686-698 [23] G. Gilboa, N. Sochen and Y. Y. Zeevi, Variational denoising of partly textured images by spatially varying constraints, IEEE Transactions on Image Processing, 15 (2006), 2281-2289. [24] T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [25] M. L. Gonçalves, J. G. Melo and R. D. Monteiro, Convergence rate bounds for a proximal ADMM with over-relaxation stepsize parameter for solving nonconvex linearly constrained problems, arXiv preprint, arXiv: 1702.01850. [26] K. Guo, D. Han and T.-T. Wu, Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints, International Journal of Computer Mathematics, 94 (2017), 1653-1669. doi: 10.1080/00207160.2016.1227432. [27] W. Guo, J. Qin and W. Yin, A new detail-preserving regularization scheme, SIAM Journal on Imaging Sciences, 7 (2014), 1309-1334. doi: 10.1137/120904263. [28] M. Kang, M. Kang and M. Jung, Nonconvex higher-order regularization based Rician noise removal with spatially adaptive parameters, Journal of Visual Communication and Image Representation, 32 (2015), 180-193. [29] M. Kang, S. Yun and H. Woo, Two-level convex relaxed variational model for multiplicative denoising, SIAM Journal on Imaging Sciences, 6 (2013), 875-903. doi: 10.1137/11086077X. [30] F. Knoll, K. Bredies, T. Pock and R. Stollberger, Second order total generalized variation (tgv) for MRI, Magnetic Resonance in Medicine, 65 (2011), 480-491. [31] F. Knoll, G. Schultz, K. Bredies, D. Gallichan, M. Zaitsev, J. Hennig and R. Stollberger, Reconstruction of undersampled radial PatLoc imaging using total generalized variation, Magnetic Resonance in Medicine, 70 (2013), 40-52. [32] D. Krishnan and R. Fergus, Fast image deconvolution using hyper-Laplacian priors, in Advances in Neural Information Processing Systems, 2009, 1033-1041. [33] A. Lanza, S. Morigi and F. Sgallari, Convex image denoising via non-convex regularization, in International Conference on Scale Space and Variational Methods in Computer Vision, Springer, 9087 (2015), 666-677. doi: 10.1007/978-3-319-18461-6_53. [34] T. Le, R. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, Journal of Mathematical Imaging and Vision, 27 (2007), 257-263. doi: 10.1007/s10851-007-0652-y. [35] J.-S. Lee, Digital image enhancement and noise filtering by use of local statistics, IEEE transactions on pattern analysis and machine intelligence, 2 (1980), 165-168. [36] F. Li, M. K. Ng and C. Shen, Multiplicative noise removal with spatially varying regularization parameters, SIAM Journal on Imaging Sciences, 3 (2010), 1-20. doi: 10.1137/090748421. [37] F. Li, C. Shen, J. Fan and C. Shen, Image restoration combining a total variational filter and a fourth-order filter, Journal of Visual Communication and Image Representation, 18 (2007), 322-330. [38] G. Liu, T.-Z. Huang and J. Liu, High-order TVL1-based images restoration and spatially adapted regularization parameter selection, Computers & Mathematics with Applications, 67 (2014), 2015-2026. doi: 10.1016/j.camwa.2014.04.008. [39] R. W. Liu, L. Shi, W. Huang, J. Xu, S. C. H. Yu and D. Wang, Generalized total variation-based MRI Rician denoising model with spatially adaptive regularization parameters, Magnetic resonance imaging, 32 (2014), 702-720. [40] S. Łojasiewicz, Sur la géométrie semi-et sous-analytique, Ann. Inst. Fourier, 43 (1993), 1575-1595. doi: 10.5802/aif.1384. [41] J. Lu, L. Shen, C. Xu and Y. Xu, Multiplicative noise removal in imaging: An exp-model and its fixed-point proximity algorithm, Applied and Computational Harmonic Analysis, 41 (2016), 518-539. doi: 10.1016/j.acha.2015.10.003. [42] V. Luminita and T. F. Chan, Reduced non-convex functional approximations for imag restoration & segmentation, UCLA CAM Website 97-56, 1997. [43] M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Transactions on image processing, 12 (2003), 1579-1590. [44] H. Na, M. Kang, M. Jung and M. Kang, An exp model with spatially adaptive regularization parameters for multiplicative noise removal, Journal of Scientific Computing, 75 (2018), 478-509. doi: 10.1007/s10915-017-0550-4. [45] M. Nikolova, M. K. Ng and C.-P. Tam, Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction, IEEE Transactions on Image Processing, 19 (2010), 3073-3088. doi: 10.1109/TIP.2010.2052275. [46] P. Ochs, A. Dosovitskiy, T. Brox and T. Pock, An iterated $\ell_1$ algorithm for non-smooth nonconvex optimization in computer vision, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2013, 1759-1766. [47] P. Ochs, A. Dosovitskiy, T. Brox and T. Pock, On iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision, SIAM Journal on Imaging Sciences, 8 (2015), 331-372. doi: 10.1137/140971518. [48] S. Oh, H. Woo, S. Yun and M. Kang, Non-convex hybrid total variation for image denoising, Journal of Visual Communication and Image Representation, 24 (2013), 332-344. [49] A. Parekh and I. W. Selesnick, Convex denoising using non-convex tight frame regularization, IEEE Signal Processing Letters, 22 (2015), 1786-1790. [50] S. Parrilli, M. Poderico, C. V. Angelino and L. Verdoliva, A nonlocal SAR image denoising algorithm based on LLMMSE wavelet shrinkage, IEEE Transactions on Geoscience and Remote Sensing, 50 (2012), 606-616. [51] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. [52] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [53] S. Setzer, Operator splittings, Bregman methods and frame shrinkage in image processing, International Journal of Computer Vision, 92 (2011), 265-280. doi: 10.1007/s11263-010-0357-3. [54] M.-G. Shama, T.-Z. Huang, J. Liu and S. Wang, A convex total generalized variation regularized model for multiplicative noise and blur removal, Applied Mathematics and Computation, 276 (2016), 109-121. [55] J. Shi and S. Osher, A nonlinear inverse scale space method for a convex multiplicative noise model, SIAM Journal on Imaging Sciences, 1 (2008), 294-321. doi: 10.1137/070689954. [56] S. Sra, Scalable nonconvex inexact proximal splitting, in Advances in Neural Information Processing Systems, 2012,530-538. [57] C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in Computer Vision, 1998. Sixth International Conference on, IEEE, 1998,839-846. [58] L. van den Dries, Tame topology and $o$-minimal structures, Bull. of the AMS, 37 (2000), 351-357. [59] Y. Wang, W. Yin and J. Zeng, Global convergence of ADMM in nonconvex nonsmooth optimization, in Journal of Scientific Computing, Springer, 2018, 1-35, URL https://link.springer.com/article/10.1007/s10915-018-0757-z. [60] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE transactions on image processing, 13 (2004), 600-612. [61] A. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, Journal of the American Mathematical Society, 9 (1996), 1051-1094. doi: 10.1090/S0894-0347-96-00216-0. [62] H. Woo and S. Yun, Proximal linearized alternating direction method for multiplicative denoising, SIAM Journal on Scientific Computing, 35 (2013), B336-B358. doi: 10.1137/11083811X. [63] J. Yang and Y. Zhang, Alternating direction algorithms for $\ell_1$-problems in compressive sensing, SIAM Journal on Scientific Computing, 33 (2011), 250-278. doi: 10.1137/090777761.

show all references

##### References:
 [1] A. Almansa, C. Ballester, V. Caselles and G. Haro, A TV based restoration model with local constraints, Journal of Scientific Computing, 34 (2008), 209-236. doi: 10.1007/s10915-007-9160-x. [2] M. Artina, M. Fornasier and F. Solombrino, Linearly constrained nonsmooth and nonconvex minimization, SIAM Journal on Optimization, 23 (2013), 1904-1937. doi: 10.1137/120869079. [3] H. Attouch, J. Bolte and B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized GaussSeidel methods, Mathematical Programming, 137 (2013), 91-129. doi: 10.1007/s10107-011-0484-9. [4] G. Aubert and J.-F. Aujol, A variational approach to removing multiplicative noise, SIAM Journal on Applied Mathematics, 68 (2008), 925-946. doi: 10.1137/060671814. [5] J. Bolte, A. Daniilidis and A. Lewis, The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM Journal on Optimization, 17 (2007), 1205-1223. doi: 10.1137/050644641. [6] S. Boyd, Alternating direction method of multipliers, in Talk at NIPS Workshop on Optimization and Machine Learning, 2011. [7] K. Bredies, K. Kunisch and T. Pock, Total generalized variation, SIAM Journal on Imaging Sciences, 3 (2010), 492-526. doi: 10.1137/090769521. [8] E. J. Candes, M. B. Wakin and S. P. Boyd, Enhancing sparsity by reweighted $\ell_1$ minimization, Journal of Fourier analysis and applications, 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x. [9] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik, 76 (1997), 167-188. doi: 10.1007/s002110050258. [10] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1. [11] T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM Journal on Scientific Computing, 22 (2000), 503-516. doi: 10.1137/S1064827598344169. [12] D.-Q. Chen and L.-Z. Cheng, Spatially adapted regularization parameter selection based on the local discrepancy function for Poissonian image deblurring, Inverse Problems, 28 (2011), 015004, 24pp. doi: 10.1088/0266-5611/28/1/015004. [13] D.-Q. Chen and L.-Z. Cheng, Spatially adapted total variation model to remove multiplicative noise, IEEE Transactions on Image Processing, 21 (2012), 1650-1662. doi: 10.1109/TIP.2011.2172801. [14] D.-Q. Chen and L.-Z. Cheng, Fast linearized alternating direction minimization algorithm with adaptive parameter selection for multiplicative noise removal, Journal of Computational and Applied Mathematics, 257 (2014), 29-45. doi: 10.1016/j.cam.2013.08.012. [15] Y. Chen, W. Feng, R. Ranftl, H. Qiao and T. Pock, A higher-order MRF based variational model for multiplicative noise reduction, IEEE Signal Processing Letters, 21 (2014), 1370-1374. [16] K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3-D transfor-mdomain collaborative filtering, IEEE Transactions on Image Processing, 16 (2007), 2080-2095. doi: 10.1109/TIP.2007.901238. [17] A. Dauwe, B. Goossens, H. Q. Luong and W. Philips, A fast non-local image denoising algorithm, in Electronic Imaging 2008, International Society for Optics and Photonics, 2008, 681210-681210. [18] Y. Dong, M. Hintermüller and M. M. Rincon-Camacho, Automated regularization parameter selection in multi-scale total variation models for image restoration, Journal of Mathematical Imaging and Vision, 40 (2011), 82-104. doi: 10.1007/s10851-010-0248-9. [19] Y. Dong and T. Zeng, A convex variational model for restoring blurred images with multiplicative noise, SIAM Journal on Imaging Sciences, 6 (2013), 1598-1625. doi: 10.1137/120870621. [20] W. Feng, H. Lei and Y. Gao, Speckle reduction via higher order total variation approach, IEEE Transactions on Image Processing, 23 (2014), 1831-1843. doi: 10.1109/TIP.2014.2308432. [21] P. Getreuer, Total variation deconvolution using split Bregman, Image Processing On Line, 2 (2012), 158-174. [22] P. Getreuer, M. Tong and L. A. Vese, A variational model for the restoration of MR images corrupted by blur and Rician noise, in International Symposium on Visual Computing, Springer, 2011,686-698 [23] G. Gilboa, N. Sochen and Y. Y. Zeevi, Variational denoising of partly textured images by spatially varying constraints, IEEE Transactions on Image Processing, 15 (2006), 2281-2289. [24] T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [25] M. L. Gonçalves, J. G. Melo and R. D. Monteiro, Convergence rate bounds for a proximal ADMM with over-relaxation stepsize parameter for solving nonconvex linearly constrained problems, arXiv preprint, arXiv: 1702.01850. [26] K. Guo, D. Han and T.-T. Wu, Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints, International Journal of Computer Mathematics, 94 (2017), 1653-1669. doi: 10.1080/00207160.2016.1227432. [27] W. Guo, J. Qin and W. Yin, A new detail-preserving regularization scheme, SIAM Journal on Imaging Sciences, 7 (2014), 1309-1334. doi: 10.1137/120904263. [28] M. Kang, M. Kang and M. Jung, Nonconvex higher-order regularization based Rician noise removal with spatially adaptive parameters, Journal of Visual Communication and Image Representation, 32 (2015), 180-193. [29] M. Kang, S. Yun and H. Woo, Two-level convex relaxed variational model for multiplicative denoising, SIAM Journal on Imaging Sciences, 6 (2013), 875-903. doi: 10.1137/11086077X. [30] F. Knoll, K. Bredies, T. Pock and R. Stollberger, Second order total generalized variation (tgv) for MRI, Magnetic Resonance in Medicine, 65 (2011), 480-491. [31] F. Knoll, G. Schultz, K. Bredies, D. Gallichan, M. Zaitsev, J. Hennig and R. Stollberger, Reconstruction of undersampled radial PatLoc imaging using total generalized variation, Magnetic Resonance in Medicine, 70 (2013), 40-52. [32] D. Krishnan and R. Fergus, Fast image deconvolution using hyper-Laplacian priors, in Advances in Neural Information Processing Systems, 2009, 1033-1041. [33] A. Lanza, S. Morigi and F. Sgallari, Convex image denoising via non-convex regularization, in International Conference on Scale Space and Variational Methods in Computer Vision, Springer, 9087 (2015), 666-677. doi: 10.1007/978-3-319-18461-6_53. [34] T. Le, R. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, Journal of Mathematical Imaging and Vision, 27 (2007), 257-263. doi: 10.1007/s10851-007-0652-y. [35] J.-S. Lee, Digital image enhancement and noise filtering by use of local statistics, IEEE transactions on pattern analysis and machine intelligence, 2 (1980), 165-168. [36] F. Li, M. K. Ng and C. Shen, Multiplicative noise removal with spatially varying regularization parameters, SIAM Journal on Imaging Sciences, 3 (2010), 1-20. doi: 10.1137/090748421. [37] F. Li, C. Shen, J. Fan and C. Shen, Image restoration combining a total variational filter and a fourth-order filter, Journal of Visual Communication and Image Representation, 18 (2007), 322-330. [38] G. Liu, T.-Z. Huang and J. Liu, High-order TVL1-based images restoration and spatially adapted regularization parameter selection, Computers & Mathematics with Applications, 67 (2014), 2015-2026. doi: 10.1016/j.camwa.2014.04.008. [39] R. W. Liu, L. Shi, W. Huang, J. Xu, S. C. H. Yu and D. Wang, Generalized total variation-based MRI Rician denoising model with spatially adaptive regularization parameters, Magnetic resonance imaging, 32 (2014), 702-720. [40] S. Łojasiewicz, Sur la géométrie semi-et sous-analytique, Ann. Inst. Fourier, 43 (1993), 1575-1595. doi: 10.5802/aif.1384. [41] J. Lu, L. Shen, C. Xu and Y. Xu, Multiplicative noise removal in imaging: An exp-model and its fixed-point proximity algorithm, Applied and Computational Harmonic Analysis, 41 (2016), 518-539. doi: 10.1016/j.acha.2015.10.003. [42] V. Luminita and T. F. Chan, Reduced non-convex functional approximations for imag restoration & segmentation, UCLA CAM Website 97-56, 1997. [43] M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Transactions on image processing, 12 (2003), 1579-1590. [44] H. Na, M. Kang, M. Jung and M. Kang, An exp model with spatially adaptive regularization parameters for multiplicative noise removal, Journal of Scientific Computing, 75 (2018), 478-509. doi: 10.1007/s10915-017-0550-4. [45] M. Nikolova, M. K. Ng and C.-P. Tam, Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction, IEEE Transactions on Image Processing, 19 (2010), 3073-3088. doi: 10.1109/TIP.2010.2052275. [46] P. Ochs, A. Dosovitskiy, T. Brox and T. Pock, An iterated $\ell_1$ algorithm for non-smooth nonconvex optimization in computer vision, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2013, 1759-1766. [47] P. Ochs, A. Dosovitskiy, T. Brox and T. Pock, On iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision, SIAM Journal on Imaging Sciences, 8 (2015), 331-372. doi: 10.1137/140971518. [48] S. Oh, H. Woo, S. Yun and M. Kang, Non-convex hybrid total variation for image denoising, Journal of Visual Communication and Image Representation, 24 (2013), 332-344. [49] A. Parekh and I. W. Selesnick, Convex denoising using non-convex tight frame regularization, IEEE Signal Processing Letters, 22 (2015), 1786-1790. [50] S. Parrilli, M. Poderico, C. V. Angelino and L. Verdoliva, A nonlocal SAR image denoising algorithm based on LLMMSE wavelet shrinkage, IEEE Transactions on Geoscience and Remote Sensing, 50 (2012), 606-616. [51] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. [52] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [53] S. Setzer, Operator splittings, Bregman methods and frame shrinkage in image processing, International Journal of Computer Vision, 92 (2011), 265-280. doi: 10.1007/s11263-010-0357-3. [54] M.-G. Shama, T.-Z. Huang, J. Liu and S. Wang, A convex total generalized variation regularized model for multiplicative noise and blur removal, Applied Mathematics and Computation, 276 (2016), 109-121. [55] J. Shi and S. Osher, A nonlinear inverse scale space method for a convex multiplicative noise model, SIAM Journal on Imaging Sciences, 1 (2008), 294-321. doi: 10.1137/070689954. [56] S. Sra, Scalable nonconvex inexact proximal splitting, in Advances in Neural Information Processing Systems, 2012,530-538. [57] C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in Computer Vision, 1998. Sixth International Conference on, IEEE, 1998,839-846. [58] L. van den Dries, Tame topology and $o$-minimal structures, Bull. of the AMS, 37 (2000), 351-357. [59] Y. Wang, W. Yin and J. Zeng, Global convergence of ADMM in nonconvex nonsmooth optimization, in Journal of Scientific Computing, Springer, 2018, 1-35, URL https://link.springer.com/article/10.1007/s10915-018-0757-z. [60] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE transactions on image processing, 13 (2004), 600-612. [61] A. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, Journal of the American Mathematical Society, 9 (1996), 1051-1094. doi: 10.1090/S0894-0347-96-00216-0. [62] H. Woo and S. Yun, Proximal linearized alternating direction method for multiplicative denoising, SIAM Journal on Scientific Computing, 35 (2013), B336-B358. doi: 10.1137/11083811X. [63] J. Yang and Y. Zhang, Alternating direction algorithms for $\ell_1$-problems in compressive sensing, SIAM Journal on Scientific Computing, 33 (2011), 250-278. doi: 10.1137/090777761.
Original images. First row: Boat $(256 \times 256)$, Elaine $(256 \times 256)$, Face $(255 \times 255)$, Girl $(336 \times 254)$, and Mountain $(400 \times 200)$. Second row: Peppers $(256 \times 256)$, Remote1 $(350 \times 228)$, Remote2 $(350 \times 253)$, Remote3 $(308 \times 236)$, and Remote4 $(275 \times 275)$
Evolution of the function $\lambda$ (top) and denoised image $\tilde{u}$ (bottom). Top: (a) initial $\lambda^1$, (b) $\lambda^2$, (c) $\lambda^3$. Bottom: denoised images (a) $\tilde{u}^1$ (PSNR: 22.68), (b) $\tilde{u}^2$ (PSNR: 24.79), (c) $\tilde{u}^3$ (PSNR: 24.83).
Denoised images (top) and final $\lambda$ (bottom) with different number of window sizes. (a) $(r_1, r_2) = (7, 21)$ (PSNR : 25.31 dB), (b) $(r_1, r_2) = (7,256)$ (PSNR: 25.63 dB), (c) $(r_1, r_2, r_3) = (7, 21,256)$ (PSNR: 25.65 dB)
Denoised images of (a) exp-SARP, (b) our model with a fixed $\lambda$ (without SARP), (c) our NTGV-SARP model. 1st row: $M = 10$, 2nd row: $M = 5$, 3th row: $M = 3$. PSNR: (1st row, left to right) 24.62, 24.77, 24.88; (2nd row) 25.53, 26.42, 26.50; (3rd row) 22.47, 22.49, 22.57.
Denoising results of our model when $M = 10$, and comparisons with other models. (a) data $f$ with $M = 10$. Denoised images: (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
Denoising results of our model when $M = 10$, and comparisons with other models. (a) data $f$ with $M = 10$. Denoised images: (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
Denoising results of our model when (a) $M = 10$ (b) $M = 5$ (c) $M = 3$, and comparisons with other models. Top to bottom rows: TwL-4V [29], exp-SARP [44], SO-TGV [20], DZ-TGV [54], and our model
Denoising results of our model when (a) $M = 10$ (b) $M = 5$ (c) $M = 3$, and comparisons with other models. Top to bottom rows: TwL-4V [29], exp-SARP [44], SO-TGV [20], DZ-TGV [54], and our model
Denoising results of our model when $M = 3$, and comparisons with other models. (a) data $f$ with $M = 3$. Denoised images. (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
Denoising results of our model when $M = 3$, and comparisons with other models. (a) data $f$ with $M = 3$. Denoised images: (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
Denoising results of our model when $M = 10$, and comparisons with other models. (a) data $f$ with $M = 10$. Denoised images: (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
Denoising results of our model when $M = 5$, and comparisons with other models. (a) data $f$ with $M = 5$. Denoised images: (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
Denoising results of our model when $M = 3$, and comparisons with other models. (a) data $f$ with $M = 3$. Denoised images: (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
Denoising results of our model when (a) $M = 10$ (b) $M = 5$ (c) $M = 3$, and comparisons with other models. Top to bottom rows: TwL-4V [29], exp-SARP [44], SO-TGV [20], DZ-TGV [54], our model
Comparison of denoised images of our model and SAR-BM3D model [50] when (a) $M = 10$ (b) $M = 5$ (c) $M = 3$. 1st, 3rd rows: our model. 2nd, 4th rows: SAR-BM3D model. PSNR: (1st row, left to right) 27.91, 26.25, 24.80; (2nd row) 28.23, 26.47, 24.99; (3rd row) 30.57, 29.00, 27.50; (4th row) 30.04, 28.35, 26.95
Comparison of denoised images of our model and SAR-BM3D model [50] when $M = 3$. (a) our model; (b) SAR-BM3D model. PSNR (top to bottom rows): (a) 22.75, 22.57, 28.75, 23.03; (b) 23.38, 23.03, 28.43, 22.97
Comparisons of denoising results when $M = 10$ (PSNR/SSIM)
 TwL-4V [29] exp-SARP [44] SO-TGV [20] DZ-TGV [54] Our model Boat 25.30 / 0.6851 25.54 / 0.6985 24.93 / 0.6762 25.22 / 0.6853 25.65 / 0.7085 Elaine 27.05 / 0.7646 27.06 / 0.7688 27.54 / 0.7977 27.21 / 0.7792 27.91 / 0.8104 Face 26.70 / 0.8158 27.08 / 0.8508 27.99 / 0.8779 26.84 / 0.8455 28.28 / 0.8851 Girl 28.99 / 0.8199 28.79 / 0.8218 30.39 / 0.8724 29.06 / 0.8655 30.57 / 0.8788 Mountain 24.73 / 0.6491 24.70 / 0.6539 24.42 / 0.6486 24.16 / 0.6348 24.82 / 0.6666 Peppers 27.10 / 0.8064 27.22 / 0.8161 27.10 / 0.8143 27.12 / 0.8124 27.51 / 0.8303 Remote1 24.97 / 0.7091 25.07 / 0.7072 24.96 / 0.7095 24.23 / 0.6885 25.20 / 0.7098 Remote2 24.70 / 0.6912 24.74 / 0.7010 24.66 / 0.6775 23.89 / 0.6650 24.83 / 0.7026 Remote3 30.33 / 0.7690 30.36 / 0.7701 30.60 / 0.7816 30.01 / 0.7583 30.82 / 0.7846 Remote4 24.62 / 0.6287 24.64 / 0.6611 24.72 / 0.6442 24.33 / 0.6311 24.89 / 0.6756 average 26.45 / 0.7338 26.52 / 0.7449 26.73 / 0.75 26.21 / 0.7365 27.05 / 0.7652
 TwL-4V [29] exp-SARP [44] SO-TGV [20] DZ-TGV [54] Our model Boat 25.30 / 0.6851 25.54 / 0.6985 24.93 / 0.6762 25.22 / 0.6853 25.65 / 0.7085 Elaine 27.05 / 0.7646 27.06 / 0.7688 27.54 / 0.7977 27.21 / 0.7792 27.91 / 0.8104 Face 26.70 / 0.8158 27.08 / 0.8508 27.99 / 0.8779 26.84 / 0.8455 28.28 / 0.8851 Girl 28.99 / 0.8199 28.79 / 0.8218 30.39 / 0.8724 29.06 / 0.8655 30.57 / 0.8788 Mountain 24.73 / 0.6491 24.70 / 0.6539 24.42 / 0.6486 24.16 / 0.6348 24.82 / 0.6666 Peppers 27.10 / 0.8064 27.22 / 0.8161 27.10 / 0.8143 27.12 / 0.8124 27.51 / 0.8303 Remote1 24.97 / 0.7091 25.07 / 0.7072 24.96 / 0.7095 24.23 / 0.6885 25.20 / 0.7098 Remote2 24.70 / 0.6912 24.74 / 0.7010 24.66 / 0.6775 23.89 / 0.6650 24.83 / 0.7026 Remote3 30.33 / 0.7690 30.36 / 0.7701 30.60 / 0.7816 30.01 / 0.7583 30.82 / 0.7846 Remote4 24.62 / 0.6287 24.64 / 0.6611 24.72 / 0.6442 24.33 / 0.6311 24.89 / 0.6756 average 26.45 / 0.7338 26.52 / 0.7449 26.73 / 0.75 26.21 / 0.7365 27.05 / 0.7652
Comparisons of denoising results when $M = 5$ (PSNR/SSIM)
 TwL-4V [29] exp-SARP [44] SO-TGV [20] DZ-TGV [54] Our model Boat 23.84 / 0.6207 23.89 / 0.6309 23.62 / 0.6165 23.41 / 0.6081 24.17 / 0.6515 Elaine 25.54 / 0.7122 25.41 / 0.7140 26.01 / 0.7552 25.17 / 0.7173 26.25 / 0.7638 Face 25.02 / 0.7707 25.53 / 0.8065 26.12 / 0.8405 24.51 / 0.7974 26.50 / 0.8493 Girl 27.63 / 0.7806 27.29 / 0.7728 28.87 / 0.8394 26.90 / 0.8131 29.00 / 0.8446 Mountain 23.52 / 0.5869 23.40 / 0.5843 23.16 / 0.5791 22.66 / 0.5544 23.52 / 0.6013 Peppers 25.68 / 0.7601 25.82 / 0.7763 25.60 / 0.7732 25.17 / 0.7621 26.16 / 0.7959 Remote1 23.53 / 0.6329 23.60 / 0.6220 23.54 / 0.6422 22.55 / 0.6005 23.70 / 0.6461 Remote2 23.38 / 0.6119 23.39 / 0.6171 23.34 / 0.5835 22.22 / 0.5755 23.58 / 0.6249 Remote3 29.32 / 0.7288 29.36 / 0.7286 29.45 / 0.7351 28.20 / 0.6977 29.76 / 0.7434 Remote4 23.53 / 0.5588 23.42 / 0.5787 23.63 / 0.5750 22.91 / 0.5500 23.75 / 0.6075 average 25.10 / 0.6763 25.11 / 0.6831 25.33 / 0.6939 24.37 / 0.6676 25.64 / 0.7128
 TwL-4V [29] exp-SARP [44] SO-TGV [20] DZ-TGV [54] Our model Boat 23.84 / 0.6207 23.89 / 0.6309 23.62 / 0.6165 23.41 / 0.6081 24.17 / 0.6515 Elaine 25.54 / 0.7122 25.41 / 0.7140 26.01 / 0.7552 25.17 / 0.7173 26.25 / 0.7638 Face 25.02 / 0.7707 25.53 / 0.8065 26.12 / 0.8405 24.51 / 0.7974 26.50 / 0.8493 Girl 27.63 / 0.7806 27.29 / 0.7728 28.87 / 0.8394 26.90 / 0.8131 29.00 / 0.8446 Mountain 23.52 / 0.5869 23.40 / 0.5843 23.16 / 0.5791 22.66 / 0.5544 23.52 / 0.6013 Peppers 25.68 / 0.7601 25.82 / 0.7763 25.60 / 0.7732 25.17 / 0.7621 26.16 / 0.7959 Remote1 23.53 / 0.6329 23.60 / 0.6220 23.54 / 0.6422 22.55 / 0.6005 23.70 / 0.6461 Remote2 23.38 / 0.6119 23.39 / 0.6171 23.34 / 0.5835 22.22 / 0.5755 23.58 / 0.6249 Remote3 29.32 / 0.7288 29.36 / 0.7286 29.45 / 0.7351 28.20 / 0.6977 29.76 / 0.7434 Remote4 23.53 / 0.5588 23.42 / 0.5787 23.63 / 0.5750 22.91 / 0.5500 23.75 / 0.6075 average 25.10 / 0.6763 25.11 / 0.6831 25.33 / 0.6939 24.37 / 0.6676 25.64 / 0.7128
Comparisons of denoising results when $M = 3$ (PSNR/SSIM)
 TwL-4V [29] exp-SARP [44] SO-TGV [20] DZ-TGV [54] Our model Boat 22.93 / 0.5786 23.01 / 0.5837 22.71 / 0.5704 21.92 / 0.5355 23.11 / 0.6005 Elaine 24.27 / 0.6706 23.95 / 0.6689 24.50 / 0.7117 23.06 / 0.6436 24.80 / 0.7240 Face 23.50 / 0.7347 24.12 / 0.7624 24.66 / 0.8058 22.45 / 0.7389 25.00 / 0.8181 Girl 26.32 / 0.7465 26.25 / 0.7332 27.32 / 0.8038 25.30 / 0.7447 27.50 / 0.8125 Mountain 22.58 / 0.5420 22.57 / 0.5384 22.30 / 0.5330 21.12 / 0.4885 22.63 / 0.5600 Peppers 24.16 / 0.7327 24.42 / 0.7371 24.27 / 0.7368 23.21 / 0.7074 24.78 / 0.7534 Remote1 22.61 / 0.5760 22.65 / 0.5723 22.64 / 0.5731 20.77 / 0.5142 22.75 / 0.5754 Remote2 22.43 / 0.5525 22.45 / 0.5526 22.46 / 0.5327 20.32 / 0.4972 22.57 / 0.5555 Remote3 28.37 / 0.6908 28.41 / 0.6931 28.56 / 0.7043 27.08 / 0.6514 28.75 / 0.7138 Remote4 22.81 / 0.5116 22.71 / 0.521 22.92 / 0.5309 21.95 / 0.4849 23.02 / 0.5513 average 24.00 / 0.6336 24.05 / 0.6362 24.23 / 0.6502 22.72 / 0.6006 24.49 / 0.6665
 TwL-4V [29] exp-SARP [44] SO-TGV [20] DZ-TGV [54] Our model Boat 22.93 / 0.5786 23.01 / 0.5837 22.71 / 0.5704 21.92 / 0.5355 23.11 / 0.6005 Elaine 24.27 / 0.6706 23.95 / 0.6689 24.50 / 0.7117 23.06 / 0.6436 24.80 / 0.7240 Face 23.50 / 0.7347 24.12 / 0.7624 24.66 / 0.8058 22.45 / 0.7389 25.00 / 0.8181 Girl 26.32 / 0.7465 26.25 / 0.7332 27.32 / 0.8038 25.30 / 0.7447 27.50 / 0.8125 Mountain 22.58 / 0.5420 22.57 / 0.5384 22.30 / 0.5330 21.12 / 0.4885 22.63 / 0.5600 Peppers 24.16 / 0.7327 24.42 / 0.7371 24.27 / 0.7368 23.21 / 0.7074 24.78 / 0.7534 Remote1 22.61 / 0.5760 22.65 / 0.5723 22.64 / 0.5731 20.77 / 0.5142 22.75 / 0.5754 Remote2 22.43 / 0.5525 22.45 / 0.5526 22.46 / 0.5327 20.32 / 0.4972 22.57 / 0.5555 Remote3 28.37 / 0.6908 28.41 / 0.6931 28.56 / 0.7043 27.08 / 0.6514 28.75 / 0.7138 Remote4 22.81 / 0.5116 22.71 / 0.521 22.92 / 0.5309 21.95 / 0.4849 23.02 / 0.5513 average 24.00 / 0.6336 24.05 / 0.6362 24.23 / 0.6502 22.72 / 0.6006 24.49 / 0.6665
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