Hybrid non-convex regularizers model for removing multiplicative noise. (English) Zbl 1524.94025
Summary: Obtaining natural and realistic restorations from the noisy images contaminated by multiplicative noise is a challenging task in image processing. To get over this conundrum, by introducing the non-convex potential functions into the total variation and high-order total variation regularizers, we investigate a novel hybrid non-convex optimization model for image restoration. Numerically, to optimize the resulting high-order PDE system, a proximal linearized alternating minimization method, based on the classical iteratively reweighted \(\ell_1\) algorithm and variable splitting technique, is designed in detail. Meanwhile, the convergence of the constructed algorithm is also established on the basis of convex analysis. The provided numerical experiments point out that our new scheme shows superiorities in both visual effects and quantitative comparison, especially in terms of the staircase aspects suppression and edge details preservation, compared with some popular denoising methods.
MSC:
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 68U10 | Computing methodologies for image processing |
| 65K10 | Numerical optimization and variational techniques |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
| 62H35 | Image analysis in multivariate analysis |
Keywords:
multiplicative noise; non-convex regularizer; total variation; high-order derivative; alternating minimization methodReferences:
| [1] | Aubert, G.; Aujol, J.-F., A variational approach to removing multiplicative noise, SIAM J. Appl. Math., 68, 4, 925-946 (2008) · Zbl 1151.68713 |
| [2] | Shi, J.; Osher, S., A nonlinear inverse scale space method for a convex multiplicative noise model, SIAM J. Imaging Sci., 1, 3, 294-321 (2008) · Zbl 1185.94018 |
| [3] | Huang, Y.-M.; Ng, M. K.; Wen, Y.-W., A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci., 2, 1, 20-40 (2009) · Zbl 1187.68655 |
| [4] | Dahl, J.; Hansen, P. C.; Jensen, S. H.; Jensen, T. L., Algorithms and software for total variation image reconstruction via first-order methods, Numer. Algorithms, 53, 1, 67-92 (2010) · Zbl 1181.94009 |
| [5] | Woo, H.; Yun, S., Proximal linearized alternating direction method for multiplicative denoising, SIAM J. Sci. Comput., 35, 2, B336-B358 (2013) · Zbl 1326.94025 |
| [6] | Zhao, X.-L.; Wang, F.; Ng, M. K., A new convex optimization model for multiplicative noise and blur removal, SIAM J. Imaging Sci., 7, 1, 456-475 (2014) · Zbl 1296.65093 |
| [7] | Chan, T.; Marquina, A.; Mulet, P., High-order total variation-based image restoration, SIAM J. Sci. Comput., 22, 2, 503-516 (2000) · Zbl 0968.68175 |
| [8] | Lv, X.-G.; Le, J.; Huang, J.; Jun, L., A fast high-order total variation minimization method for multiplicative noise removal, Math. Probl. Eng., 2013, Article 834035 pp. (2013) |
| [9] | Liu, G.; Huang, T.-Z.; Liu, J., High-order TVL1-based images restoration and spatially adapted regularization parameter selection, Comput. Math. Appl., 67, 10, 2015-2026 (2014) · Zbl 1366.94058 |
| [10] | Lysaker, M.; Lundervold, A.; Tai, X.-C., Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12, 12, 1579-1590 (2003) · Zbl 1286.94020 |
| [11] | Siddig, A.; Guo, Z.; Zhou, Z.; Wu, B., An image denoising model based on a fourth-order nonlinear partial differential equation, Comput. Math. Appl., 76, 5, 1056-1074 (2018) · Zbl 1435.94047 |
| [12] | Bredies, K.; Kunisch, K.; Pock, T., Total generalized variation, SIAM J. Imaging Sci., 3, 3, 492-526 (2010) · Zbl 1195.49025 |
| [13] | Ma, T.-H.; Huang, T.-Z.; Zhao, X.-L., Spatially dependent regularization parameter selection for total generalized variation-based image denoising, Comput. Appl. Math., 37, 1, 277-296 (2018) · Zbl 1423.94009 |
| [14] | Mei, J.-J.; Huang, T.-Z.; Wang, S.; Zhao, X.-L., Second order total generalized variation for speckle reduction in ultrasound images, J. Franklin Inst., 355, 1, 574-595 (2018) · Zbl 1380.94029 |
| [15] | Bai, L., A new nonconvex approach for image restoration with Gamma noise, Comput. Math. Appl., 77, 10, 2627-2639 (2019) · Zbl 1442.94003 |
| [16] | Li, F.; Shen, C.; Fan, J.; Shen, C., Image restoration combining a total variational filter and a fourth-order filter, J. Vis. Commun. Image Represent., 18, 4, 322-330 (2007) |
| [17] | Papafitsoros, K.; Schönlieb, C. B., A combined first and second order variational approach for image reconstruction, J. Math. Imaging Vis., 48, 2, 308-338 (2014) · Zbl 1362.94009 |
| [18] | Liu, X., Efficient algorithms for hybrid regularizers based image denoising and deblurring, Comput. Math. Appl., 69, 7, 675-687 (2015) · Zbl 1443.94023 |
| [19] | Wang, S.; Huang, T.-Z.; Zhao, X.-L.; Mei, J.-J.; Huang, J., Speckle noise removal in ultrasound images by first- and second-order total variation, Numer. Algorithms, 78, 2, 513-533 (2018) · Zbl 1391.94105 |
| [20] | Nikolova, M.; Ng, M. K.; Tam, C.-P., Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction, IEEE Trans. Image Process., 19, 12, 3073-3088 (2010) · Zbl 1371.94277 |
| [21] | Han, Y.; Feng, X.-C.; Baciu, G.; Wang, W.-W., Nonconvex sparse regularizer based speckle noise removal, Pattern Recognit., 46, 3, 989-1001 (2013) |
| [22] | Selesnick, I.; Lanza, A.; Morigi, S.; Sgallari, F., Non-convex total variation regularization for convex denoising of signals, J. Math. Imaging Vis., 62, 6-7, 825-841 (2020) · Zbl 1485.94026 |
| [23] | Oh, S.; Woo, H.; Yun, S.; Kang, M., Non-convex hybrid total variation for image denoising, J. Vis. Commun. Image Represent., 24, 3, 332-344 (2013) |
| [24] | Na, H.; Kang, M.; Jung, M.; Kang, M., Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters, Inverse Probl. Imaging, 13, 1, 117-147 (2019) · Zbl 1491.94009 |
| [25] | Liu, X., Adaptive regularization parameter for nonconvex TGV based image restoration, Signal Process., 188, Article 108247 pp. (2021) |
| [26] | Liu, X., Non-convex variational model for image restoration under impulse noise, Signal Image Video Process., 16, 6, 1549-1557 (2022) |
| [27] | Chen, Y. M.; Wunderli, T., Adaptive total variation for image restoration in BV space, J. Math. Anal. Appl., 272, 1, 117-137 (2002) · Zbl 1020.68104 |
| [28] | Lu, C.; Tang, J.; Yan, S.; Lin, Z., Nonconvex nonsmooth low rank minimization via iteratively reweighted nuclear norm, IEEE Trans. Image Process., 25, 2, 829-839 (2016) · Zbl 1408.94866 |
| [29] | Candès, E. J.; Wakin, M. B.; Boyd, S. P., Enhancing sparsity by reweighted \(\ell_1\) minimization, J. Fourier Anal. Appl., 14, 5, 877-905 (2008) · Zbl 1176.94014 |
| [30] | Ochs, P.; Dosovitskiy, A.; Brox, T.; Pock, T., On iteratively reweighted algorithms for nonsmooth nonconvex optimization, SIAM J. Imaging Sci., 8, 1, 331-372 (2015) · Zbl 1326.65078 |
| [31] | Goldstein, T.; Osher, S., The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2, 2, 323-343 (2009) · Zbl 1177.65088 |
| [32] | Cai, J.-F.; Osher, S.; Shen, Z., Split Bregman methods and frame based image restoration, Multiscale Model. Simul., 8, 2, 337-369 (2010) · Zbl 1189.94014 |
| [33] | Fazel, M.; Pong, T. K.; Sun, D.; Tseng, P., Hankel matrix rank minimization with applications to system identification and realization, SIAM J. Matrix Anal. Appl., 34, 3, 946-977 (2013) · Zbl 1302.90127 |
| [34] | Rockafellar, R. T.; Wets, R. J.-B., Variational Analysis (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0888.49001 |
| [35] | Borwein, J. M.; Lewis, A. S., Convex Analysis and Nonlinear Optimization (2006), Springer: Springer New York · Zbl 1116.90001 |
| [36] | Hiriart-Urruty, J.-B.; Lemaréchal, C., Convex Analysis and Minimization Algorithms: Part 1: Fundamentals (1996), Springer-Verlag: Springer-Verlag Berlin |
| [37] | Guo, M.; Han, C.; Wang, W.; Zhong, S.; Lv, R.; Liu, Z., A novel truncated nonconvex nonsmooth variational method for SAR image despeckling, Remote Sens. Lett., 12, 2, 122-131 (2021) |
| [38] | Wang, Z.; Bovik, A. C.; Sheikh, H. R.; Simoncelli, E. P., Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process., 13, 4, 600-612 (2004) |
| [39] | Zhang, L.; Zhang, L.; Mou, X.; Zhang, D., FSIM: a feature similarity index for image quality assessment, IEEE Trans. Image Process., 20, 8, 2378-2386 (2011) · Zbl 1373.62333 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
