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The regularized monotonicity method: Detecting irregular indefinite inclusions
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 |
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.
References:
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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. Google Scholar |
[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. |
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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. Google Scholar |
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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. |
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Automated regularization parameter selection in multi-scale total variation models for image restoration, Journal of Mathematical Imaging and Vision, 40 (2011), 82-104.
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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. |
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P. Getreuer, Total variation deconvolution using split Bregman, Image Processing On Line, 2 (2012), 158-174. Google Scholar |
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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 Google Scholar |
[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. Google Scholar |
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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. |
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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. Google Scholar |
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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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[32] |
D. Krishnan and R. Fergus, Fast image deconvolution using hyper-Laplacian priors, in Advances in Neural Information Processing Systems, 2009, 1033-1041. Google Scholar |
[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. Google Scholar |
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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. |
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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. Google Scholar |
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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. |
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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. Google Scholar |
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Sur la géométrie semi-et sous-analytique, Ann. Inst. Fourier, 43 (1993), 1575-1595.
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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. |
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V. Luminita and T. F. Chan, Reduced non-convex functional approximations for imag restoration & segmentation, UCLA CAM Website 97-56, 1997. Google Scholar |
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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. Google Scholar |
[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. |
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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. Google Scholar |
[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. Google Scholar |
[49] |
A. Parekh and I. W. Selesnick, Convex denoising using non-convex tight frame regularization, IEEE Signal Processing Letters, 22 (2015), 1786-1790. Google Scholar |
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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. Google Scholar |
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Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.
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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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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 Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[32] |
D. Krishnan and R. Fergus, Fast image deconvolution using hyper-Laplacian priors, in Advances in Neural Information Processing Systems, 2009, 1033-1041. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[49] |
A. Parekh and I. W. Selesnick, Convex denoising using non-convex tight frame regularization, IEEE Signal Processing Letters, 22 (2015), 1786-1790. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |
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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 |
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 |
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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