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. AlmansaC. BallesterV. 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. ArtinaM. Fornasier and F. Solombrino, Linearly constrained nonsmooth and nonconvex minimization, SIAM Journal on Optimization, 23 (2013), 1904-1937. doi: 10.1137/120869079.

[3]

H. AttouchJ. 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. BolteA. 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. BrediesK. Kunisch and T. Pock, Total generalized variation, SIAM Journal on Imaging Sciences, 3 (2010), 492-526. doi: 10.1137/090769521.

[8]

E. J. CandesM. 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. ChanA. 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. ChenW. FengR. RanftlH. 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. DabovA. FoiV. 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. DongM. 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. FengH. 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. GilboaN. 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. GuoD. 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. GuoJ. 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. KangM. 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. KangS. 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. KnollK. BrediesT. Pock and R. Stollberger, Second order total generalized variation (tgv) for MRI, Magnetic Resonance in Medicine, 65 (2011), 480-491.

[31]

F. KnollG. SchultzK. BrediesD. GallichanM. ZaitsevJ. 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. LeR. 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. LiM. 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. LiC. ShenJ. 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. LiuT.-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. LiuL. ShiW. HuangJ. XuS. 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. LuL. ShenC. 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. LysakerA. 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. NaM. KangM. 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. NikolovaM. 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. OchsA. DosovitskiyT. 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. OhH. WooS. 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. ParrilliM. PodericoC. 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. RudinS. 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. ShamaT.-Z. HuangJ. 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. WangA. C. BovikH. 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. AlmansaC. BallesterV. 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. ArtinaM. Fornasier and F. Solombrino, Linearly constrained nonsmooth and nonconvex minimization, SIAM Journal on Optimization, 23 (2013), 1904-1937. doi: 10.1137/120869079.

[3]

H. AttouchJ. 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. BolteA. 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. BrediesK. Kunisch and T. Pock, Total generalized variation, SIAM Journal on Imaging Sciences, 3 (2010), 492-526. doi: 10.1137/090769521.

[8]

E. J. CandesM. 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. ChanA. 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. ChenW. FengR. RanftlH. 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. DabovA. FoiV. 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. DongM. 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. FengH. 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. GilboaN. 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. GuoD. 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. GuoJ. 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. KangM. 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. KangS. 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. KnollK. BrediesT. Pock and R. Stollberger, Second order total generalized variation (tgv) for MRI, Magnetic Resonance in Medicine, 65 (2011), 480-491.

[31]

F. KnollG. SchultzK. BrediesD. GallichanM. ZaitsevJ. 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. LeR. 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. LiM. 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. LiC. ShenJ. 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. LiuT.-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. LiuL. ShiW. HuangJ. XuS. 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. LuL. ShenC. 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. LysakerA. 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. NaM. KangM. 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. NikolovaM. 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. OchsA. DosovitskiyT. 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. OhH. WooS. 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. ParrilliM. PodericoC. 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. RudinS. 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. ShamaT.-Z. HuangJ. 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. WangA. C. BovikH. 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.

Figure 1.  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 $(275 \times 275)$
Figure 2.  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).
Figure 3.  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)
Figure 4.  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.
Figure 5.  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
Figure 6.  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
Figure 7.  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
Figure 8.  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
Figure 9.  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
Figure 10.  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
Figure 11.  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
Figure 12.  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
Figure 13.  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
Figure 14.  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
Figure 15.  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
Figure 16.  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
Table 1.  Comparisons of denoising results when $M = 10$ (PSNR/SSIM)
TwL-4V [29]exp-SARP [44]SO-TGV [20]DZ-TGV [54]Our model
Boat25.30 / 0.685125.54 / 0.698524.93 / 0.676225.22 / 0.685325.65 / 0.7085
Elaine27.05 / 0.764627.06 / 0.768827.54 / 0.797727.21 / 0.779227.91 / 0.8104
Face26.70 / 0.815827.08 / 0.850827.99 / 0.877926.84 / 0.845528.28 / 0.8851
Girl28.99 / 0.819928.79 / 0.821830.39 / 0.872429.06 / 0.865530.57 / 0.8788
Mountain24.73 / 0.649124.70 / 0.653924.42 / 0.648624.16 / 0.634824.82 / 0.6666
Peppers27.10 / 0.806427.22 / 0.816127.10 / 0.814327.12 / 0.812427.51 / 0.8303
Remote124.97 / 0.709125.07 / 0.707224.96 / 0.709524.23 / 0.688525.20 / 0.7098
Remote224.70 / 0.691224.74 / 0.701024.66 / 0.677523.89 / 0.665024.83 / 0.7026
Remote330.33 / 0.769030.36 / 0.770130.60 / 0.781630.01 / 0.758330.82 / 0.7846
Remote424.62 / 0.628724.64 / 0.661124.72 / 0.644224.33 / 0.631124.89 / 0.6756
average26.45 / 0.733826.52 / 0.744926.73 / 0.7526.21 / 0.736527.05 / 0.7652
TwL-4V [29]exp-SARP [44]SO-TGV [20]DZ-TGV [54]Our model
Boat25.30 / 0.685125.54 / 0.698524.93 / 0.676225.22 / 0.685325.65 / 0.7085
Elaine27.05 / 0.764627.06 / 0.768827.54 / 0.797727.21 / 0.779227.91 / 0.8104
Face26.70 / 0.815827.08 / 0.850827.99 / 0.877926.84 / 0.845528.28 / 0.8851
Girl28.99 / 0.819928.79 / 0.821830.39 / 0.872429.06 / 0.865530.57 / 0.8788
Mountain24.73 / 0.649124.70 / 0.653924.42 / 0.648624.16 / 0.634824.82 / 0.6666
Peppers27.10 / 0.806427.22 / 0.816127.10 / 0.814327.12 / 0.812427.51 / 0.8303
Remote124.97 / 0.709125.07 / 0.707224.96 / 0.709524.23 / 0.688525.20 / 0.7098
Remote224.70 / 0.691224.74 / 0.701024.66 / 0.677523.89 / 0.665024.83 / 0.7026
Remote330.33 / 0.769030.36 / 0.770130.60 / 0.781630.01 / 0.758330.82 / 0.7846
Remote424.62 / 0.628724.64 / 0.661124.72 / 0.644224.33 / 0.631124.89 / 0.6756
average26.45 / 0.733826.52 / 0.744926.73 / 0.7526.21 / 0.736527.05 / 0.7652
Table 2.  Comparisons of denoising results when $M = 5$ (PSNR/SSIM)
TwL-4V [29]exp-SARP [44]SO-TGV [20]DZ-TGV [54]Our model
Boat23.84 / 0.620723.89 / 0.630923.62 / 0.616523.41 / 0.608124.17 / 0.6515
Elaine25.54 / 0.712225.41 / 0.714026.01 / 0.755225.17 / 0.717326.25 / 0.7638
Face25.02 / 0.770725.53 / 0.806526.12 / 0.840524.51 / 0.797426.50 / 0.8493
Girl27.63 / 0.780627.29 / 0.772828.87 / 0.839426.90 / 0.813129.00 / 0.8446
Mountain23.52 / 0.586923.40 / 0.584323.16 / 0.579122.66 / 0.554423.52 / 0.6013
Peppers25.68 / 0.760125.82 / 0.776325.60 / 0.773225.17 / 0.762126.16 / 0.7959
Remote123.53 / 0.632923.60 / 0.622023.54 / 0.642222.55 / 0.600523.70 / 0.6461
Remote223.38 / 0.611923.39 / 0.617123.34 / 0.583522.22 / 0.575523.58 / 0.6249
Remote329.32 / 0.728829.36 / 0.728629.45 / 0.735128.20 / 0.697729.76 / 0.7434
Remote423.53 / 0.558823.42 / 0.578723.63 / 0.575022.91 / 0.550023.75 / 0.6075
average25.10 / 0.676325.11 / 0.683125.33 / 0.693924.37 / 0.667625.64 / 0.7128
TwL-4V [29]exp-SARP [44]SO-TGV [20]DZ-TGV [54]Our model
Boat23.84 / 0.620723.89 / 0.630923.62 / 0.616523.41 / 0.608124.17 / 0.6515
Elaine25.54 / 0.712225.41 / 0.714026.01 / 0.755225.17 / 0.717326.25 / 0.7638
Face25.02 / 0.770725.53 / 0.806526.12 / 0.840524.51 / 0.797426.50 / 0.8493
Girl27.63 / 0.780627.29 / 0.772828.87 / 0.839426.90 / 0.813129.00 / 0.8446
Mountain23.52 / 0.586923.40 / 0.584323.16 / 0.579122.66 / 0.554423.52 / 0.6013
Peppers25.68 / 0.760125.82 / 0.776325.60 / 0.773225.17 / 0.762126.16 / 0.7959
Remote123.53 / 0.632923.60 / 0.622023.54 / 0.642222.55 / 0.600523.70 / 0.6461
Remote223.38 / 0.611923.39 / 0.617123.34 / 0.583522.22 / 0.575523.58 / 0.6249
Remote329.32 / 0.728829.36 / 0.728629.45 / 0.735128.20 / 0.697729.76 / 0.7434
Remote423.53 / 0.558823.42 / 0.578723.63 / 0.575022.91 / 0.550023.75 / 0.6075
average25.10 / 0.676325.11 / 0.683125.33 / 0.693924.37 / 0.667625.64 / 0.7128
Table 3.  Comparisons of denoising results when $M = 3$ (PSNR/SSIM)
TwL-4V [29]exp-SARP [44]SO-TGV [20]DZ-TGV [54]Our model
Boat22.93 / 0.578623.01 / 0.583722.71 / 0.570421.92 / 0.535523.11 / 0.6005
Elaine24.27 / 0.670623.95 / 0.668924.50 / 0.711723.06 / 0.643624.80 / 0.7240
Face23.50 / 0.734724.12 / 0.762424.66 / 0.805822.45 / 0.738925.00 / 0.8181
Girl26.32 / 0.746526.25 / 0.733227.32 / 0.803825.30 / 0.744727.50 / 0.8125
Mountain22.58 / 0.542022.57 / 0.538422.30 / 0.533021.12 / 0.488522.63 / 0.5600
Peppers24.16 / 0.732724.42 / 0.737124.27 / 0.736823.21 / 0.707424.78 / 0.7534
Remote122.61 / 0.576022.65 / 0.572322.64 / 0.573120.77 / 0.514222.75 / 0.5754
Remote222.43 / 0.552522.45 / 0.552622.46 / 0.532720.32 / 0.497222.57 / 0.5555
Remote328.37 / 0.690828.41 / 0.693128.56 / 0.704327.08 / 0.651428.75 / 0.7138
Remote422.81 / 0.511622.71 / 0.52122.92 / 0.530921.95 / 0.484923.02 / 0.5513
average24.00 / 0.633624.05 / 0.636224.23 / 0.650222.72 / 0.600624.49 / 0.6665
TwL-4V [29]exp-SARP [44]SO-TGV [20]DZ-TGV [54]Our model
Boat22.93 / 0.578623.01 / 0.583722.71 / 0.570421.92 / 0.535523.11 / 0.6005
Elaine24.27 / 0.670623.95 / 0.668924.50 / 0.711723.06 / 0.643624.80 / 0.7240
Face23.50 / 0.734724.12 / 0.762424.66 / 0.805822.45 / 0.738925.00 / 0.8181
Girl26.32 / 0.746526.25 / 0.733227.32 / 0.803825.30 / 0.744727.50 / 0.8125
Mountain22.58 / 0.542022.57 / 0.538422.30 / 0.533021.12 / 0.488522.63 / 0.5600
Peppers24.16 / 0.732724.42 / 0.737124.27 / 0.736823.21 / 0.707424.78 / 0.7534
Remote122.61 / 0.576022.65 / 0.572322.64 / 0.573120.77 / 0.514222.75 / 0.5754
Remote222.43 / 0.552522.45 / 0.552622.46 / 0.532720.32 / 0.497222.57 / 0.5555
Remote328.37 / 0.690828.41 / 0.693128.56 / 0.704327.08 / 0.651428.75 / 0.7138
Remote422.81 / 0.511622.71 / 0.52122.92 / 0.530921.95 / 0.484923.02 / 0.5513
average24.00 / 0.633624.05 / 0.636224.23 / 0.650222.72 / 0.600624.49 / 0.6665
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