A Second-Order Image Denoising Model for Contrast Preservation

Abstract

In this work, we propose a second-order model for image denoising by employing a novel potential function recently developed in Zhu (J Sci Comput 88: 46, 2021) for the design of a regularization term. Due to this new second-order derivative based regularizer, the model is able to alleviate the staircase effect and preserve image contrast. The augmented Lagrangian method (ALM) is utilized to minimize the associated functional and convergence analysis is established for the proposed algorithm. Numerical experiments are presented to demonstrate the features of the proposed model.

Keywords

Image denoising / Variational model / Image contrast / Augmented Lagrangian method (ALM)

References

[1.] Aubert G, Vese L. A variational method in image recovery. SIAM J. Numer. Anal., 1997, 34: 1948-1979,

[2.] Bae E, Tai XC, Zhu W. Augmented Lagrangian method for an Euler’s elastica based segmentation model that promotes convex contours. Inverse Probl. Imag., 2017, 11: 1-23,

[3.] Bellettini G, Caselles V, Novaga M. The total variation flow in R n \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R} }^{n}$$\end{document}. J. Differential Equations, 2002, 184: 475-525,

[4.] Benning M, Burger M. Modern regularization methods for inverse problems. Acta Numerica, 2018, 27: 1-111,

[5.] Bertalmio M, Vese L, Sapiro G, Osher S. Simultaneous structure and texture image inpainting. IEEE Trans. Image Process., 2003, 12: 882-889,

[6.] Bredies K, Kunisch K, Pock T. Total generalized variation. SIAM J. Imaging Sci., 2010, 3: 492-526,

[7.] Brito-Loeza C, Chen K. Multigrid algorithm for high order denoising. SIAM J. Imaging Sci., 2010, 3: 363-389,

[8.] Chambolle A, Lions PL. Image recovery via total variation minimization and related problems. Numer. Math., 1997, 76: 167-188,

[9.] Chambolle A, Pock T. A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis., 2011, 40: 120-145,

[10.] Chan T, Esedoglu S. Aspects of total variation regularized L 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{1}$$\end{document} function approximation. SIAM J. Appl. Math., 2005, 65: 1817-1837,

[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 Springer Verlag

[12.] Chan T, Marquina A, Mulet P. High-order total variation-based image restoration. SIAM J. Sci. Comput., 2000, 22: 503-516,

[13.] Chan T, Shen J. Mathematical models for local nontexture inpaintings. SIAM J. Appl. Math., 2001, 62: 1019-1043

[14.] Chan T, Wong CK. Total variation blind deconvolution. IEEE Trans. Image Process., 1998, 7: 370-375,

[15.] Chang QS, Che ZY. An adaptive algorithm for TV-based model of three norms L q ( q = 1 2 , 1 , 2 ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{q} (q=\frac{1}{2},1,2)$$\end{document} in image restoration. Appl. Math. Comput., 2018, 329: 251-265

[16.] Glowinski R, Le Tallec P. . Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, 1989 Philadelphia SIAM,

[17.] Goldstein T, Osher S. The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci., 2009, 2: 323-343,

[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., 2003, 12: 1579-1590,

[19.] Lysaker M, Osher S, Tai XC. Noise removal using smoothed normals and surface fitting. IEEE. Trans. Image Process., 2004, 13: 1345-1457,

[20.] Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series, Vol. 22, Amer. Math. Soc. Providence, RI (2001)

[21.] Mumford D, Shah J. Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 1989, 42: 577-685,

[22.] Osher S, Burger M, Goldfarb D, Xu JJ, Yin WT. An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul., 2005, 4: 460-489,

[23.] Osher S, Sole A, Vese L. Image decomposition and restoration using total variation minimization and the H - 1 n o r m \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{-1}norm$$\end{document}. Multiscale Model. Simul., 2003, 1: 349-370,

[24.] Papafitsoros K, Schönlieb CB. A combined first and second order variational approach for image reconstruction. J. Math. Imaging Vis., 2014, 48: 308-338,

[25.] Rockafellar RT. Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res., 1976, 1: 97-116,

[26.] Rudin L, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithm. Phys. D, 1992, 60: 259-268,

[27.] Setzer, S., Steidl, G.: Variational methods with higher order derivatives in image processing. In: Approximation XII, pp. 360–386 (2008)

[28.] Strong D, Chan T. Edge-preserving and scale-dependent properties of total variation regularization. Inverse Problems, 2003, 19: 165-187,

[29.] Tai XC, Hahn J, Chung GJ. A fast algorithm for Euler’s elastica model using augmented Lagrangian method. SIAM J. Imaging Sci., 2011, 4: 313-344,

[30.] Vese L. A study in the BV space of a denoising-deblurring variational problem. Appl. Math. Optim., 2001, 44: 131-161,

[31.] 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., 2010, 3: 300-339,

[32.] Yashtini M, Kang SH, Zhu W. Efficient alternating minimization methods for variational edge-weighted colorization models. Adv. Comput. Math., 2019, 45: 1735-1767,

[33.] Zhu W. A numerical study of a mean curvature denoising model using a novel augmented Lagrangian method. Inverse Probl. Imag., 2017, 11: 975-996,

[34.] Zhu W. A first-order image denoising model for staircase reduction. Adv. Comput. Math., 2019, 45: 3217-3239,

[35.] Zhu W. Image denoising using L ^p-norm of mean curvature of image surface. J. Sci. Comput., 2020, 83: 32,

[36.] Zhu W. A first-order image restoration model that promotes image contrast preservation. J. Sci. Comput., 2021, 88: 46,

[37.] Zhu W, Chan T. Image denoising using mean curvature of image surface. SIAM J. Imaging Sci., 2012, 5: 1-32,

[38.] Zhu W, Tai XC, Chan T. Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Probl. Imag., 2013, 7: 1409-1432,

[39.] Zhu W, Tai XC, Chan T. Image segmentation using Euler’s elastica as the regularization. J. Sci. Comput., 2013, 57: 414-438,