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A first-order image restoration model that promotes image contrast preservation. (English) Zbl 1471.94006

Summary: In this paper, we propose a novel first-order variational model for image restoration. The main feature of this model lies in the fact that it helps preserve image contrasts during the image restoration process. To achieve this, we design a new regularizer that presents a lower growth rate than any power function with a positive exponent for large image gradient. Augmented Lagrangian method is employed to minimize this variational model and convergence analysis is established for the proposed algorithm. Numerical experiments are presented to demonstrate the features of the proposed model and also show the efficiency of the proposed numerical method.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
68U10 Computing methodologies for image processing
65R32 Numerical methods for inverse problems for integral equations
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[1] Aubert, G.; Vese, L., A variational method in image recovery, SIAM J. Numer. Anal., 34, 1948-1979 (1997) · Zbl 0890.35033 · doi:10.1137/S003614299529230X
[2] Brito-Loeza, C.; Chen, K., Multigrid algorithm for high order denoising, SIAM J. Imaging Sci., 3, 363-389 (2010) · Zbl 1205.68474 · doi:10.1137/080737903
[3] Bae, E.; Tai, XC; Zhu, W., Augmented Lagrangian method for an Euler’s elastica based segmentation model that promotes convex contours, Inverse Probl. Imag., 11, 1-23 (2017) · Zbl 1416.94013 · doi:10.3934/ipi.2017001
[4] Beck, A.: First-Order Methods in Optimization, vol. 25. SIAM (2017) · Zbl 1384.65033
[5] Bellettini, G.; Caselles, V.; Novaga, M., The total variation flow in \({\mathbb{R}}^n \), J. Differ. Equ., 184, 475-525 (2002) · Zbl 1036.35099
[6] Bertalmio, M.; Vese, L.; Sapiro, G.; Osher, S., Simultaneous structure and texture image inpainting, IEEE Trans. Image Process., 12, 882-889 (2003) · doi:10.1109/TIP.2003.815261
[7] Bredies, K.; Kunisch, K.; Pock, T., Total generalized variation, SIAM J. Imaging Sci., 3, 492-526 (2010) · Zbl 1195.49025 · doi:10.1137/090769521
[8] Chang, QS; Che, ZY, An adaptive algorithm for TV-based model of three norms \(L_q (q=\frac{1}{2},1,2)\) in image restoration, Appl. Math. Comput., 329, 251-265 (2018) · Zbl 1427.94008
[9] Chambolle, A.; Pock, T., A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40, 120-145 (2011) · Zbl 1255.68217 · doi:10.1007/s10851-010-0251-1
[10] Chan, T.; Esedoglu, S., Aspects of total variation regularized \(L^1\) function approximation, SIAM J. Appl. Math., 65, 1817-1837 (2005) · Zbl 1096.94004
[11] Chan, T.; Esedoglu, S.; Park, F.; Yip, MH; Paragios, N.; Chen, Y.; Faugeras, O., Recent developments in total variation image restoration, Handbook of Mathematical Models in Computer Vision (2005), New York: Springer, New York · Zbl 1083.68500
[12] Chambolle, A.; Lions, PL, Image recovery via total variation minimization and related problems, Numer. Math., 76, 167-188 (1997) · Zbl 0874.68299 · doi:10.1007/s002110050258
[13] Chan, T.; Shen, J., Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math., 62, 1019-1043 (2001) · Zbl 1050.68157
[14] Chan, T.; Marquina, A.; Mulet, P., High-order total variation-based image restoration, SIAM J. Sci. Comput., 22, 503-516 (2000) · Zbl 0968.68175 · doi:10.1137/S1064827598344169
[15] Chan, T.; Wong, CK, Total variation blind deconvolution, IEEE Trans. Image Process., 7, 370-375 (1998) · doi:10.1109/83.661187
[16] Goldstein, T.; Osher, S., The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2, 323-343 (2009) · Zbl 1177.65088 · doi:10.1137/080725891
[17] Glowinski, R.; Le Tallec, P., Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics (1989), Philadelphia: SIAM, Philadelphia · Zbl 0698.73001 · doi:10.1137/1.9781611970838
[18] Lysaker, M.; Lundervold, A.; Tai, XC, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12, 1579-1590 (2003) · Zbl 1286.94020 · doi:10.1109/TIP.2003.819229
[19] Lysaker, M.; Osher, S.; Tai, XC, Noise removal using smoothed normals and surface fitting, IEEE Trans. Image Process., 13, 1345-1457 (2004) · Zbl 1286.94022 · doi:10.1109/TIP.2004.834662
[20] Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series, vol 22, American Mathematical Society · Zbl 0987.35003
[21] Mumford, D.; Shah, J., Optimal approximation by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math., 42, 577-685 (1989) · Zbl 0691.49036 · doi:10.1002/cpa.3160420503
[22] Osher, S.; Burger, M.; Goldfarb, D.; Xu, JJ; Yin, WT, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4, 460-489 (2005) · Zbl 1090.94003 · doi:10.1137/040605412
[23] Osher, S.; Sole, A.; Vese, L., Image decomposition and restoration using total variation minimization and the \(H^{-1} norm\), SIAM Multiscale Model. Simul., 1, 349-370 (2003) · Zbl 1051.49026
[24] Rockafellar, RT, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1, 97-116 (1976) · Zbl 0402.90076 · doi:10.1287/moor.1.2.97
[25] Rudin, L.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithm, Physica D, 60, 259-268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[26] Strong, D.; Chan, T., Edge-preserving and scale-dependent properties of total variation regularization, Inverse Prob., 19, 165-187 (2003) · Zbl 1043.94512 · doi:10.1088/0266-5611/19/6/059
[27] Tai, XC; Hahn, J.; Chung, GJ, A fast algorithm for Euler’s elastica model using augmented Lagrangian method, SIAM J. Imaging Sci., 4, 313-344 (2011) · Zbl 1215.68262 · doi:10.1137/100803730
[28] Vese, L., A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Optim., 44, 131-161 (2001) · Zbl 1003.35009 · doi:10.1007/s00245-001-0017-7
[29] Wu, C.; Tai, XC, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3, 300-339 (2010) · Zbl 1206.90245 · doi:10.1137/090767558
[30] Zhu, W.; Chan, T., Image denoising using mean curvature of image surface, SIAM J. Imaging Sci., 5, 1-32 (2012) · Zbl 1258.94021 · doi:10.1137/110822268
[31] Zhu, W.; Tai, XC; Chan, T., Augmented Lagrangian method for a mean curvature based image denoising model, Inverse Probl. Imaging, 7, 1409-1432 (2013) · Zbl 1311.94015 · doi:10.3934/ipi.2013.7.1409
[32] Zhu, W.; Tai, XC; Chan, T., Image segmentation using Euler’s elastica as the regularization, J. Sci. Comput., 57, 414-438 (2013) · Zbl 1282.65037 · doi:10.1007/s10915-013-9710-3
[33] Zhu, W., A numerical study of a mean curvature denoising model using a novel augmented Lagrangian methdod, Inverse Probl. Imaging, 11, 975-996 (2017) · Zbl 1403.94027 · doi:10.3934/ipi.2017045
[34] Zhu, W., A first-order image denoising model for staircase reduction, Adv. Comput. Math., 45, 3217-3239 (2019) · Zbl 1434.94018 · doi:10.1007/s10444-019-09734-5
[35] Zhu, W., Image denoising using Lp-norm of mean curvature of image surface, J. Sci. Comput., 83, 32 (2020) · Zbl 1457.94034 · doi:10.1007/s10915-020-01216-x
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