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New Hybrid Variational Recovery Model for Blurred Images with Multiplicative Noise

Published online by Cambridge University Press:  28 May 2015

Yiqiu Dong  and
Tieyong Zeng
Yiqiu Dong*
Affiliation:
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Denmark
Tieyong Zeng*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
*
Corresponding author. Email Address: yido@dtu.dk
Corresponding author. Email Address: zeng@hkbu.edu.hk
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Abstract

A new hybrid variational model for recovering blurred images in the presence of multiplicative noise is proposed. Inspired by previous work on multiplicative noise removal, an I-divergence technique is used to build a strictly convex model under a condition that ensures the uniqueness of the solution and the stability of the algorithm. A split-Bregman algorithm is adopted to solve the constrained minimisation problem in the new hybrid model efficiently. Numerical tests for simultaneous deblurring and denoising of the images subject to multiplicative noise are then reported. Comparison with other methods clearly demonstrates the good performance of our new approach.

Keywords

52A4165K0565K1590C2590C30Convex modelimage deblurringmultiplicative noiseSplit-Bregman Algorithmtotal variationvariational model
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 4 , November 2013 , pp. 263 - 282
Copyright
Copyright © Global-Science Press 2013

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References

[1]Ambrosio, L., Fusco, N. and Pallara, D., Functions of Bounded Variation and Free Discontinuity Problem, Oxford University Press (2000).CrossRefGoogle Scholar
[2]Aubert, G. and Aujol, J., A variational approach to removing multiplicative noise. SIAM J. Appl. Math. 68, 925–946 (2008).CrossRefGoogle Scholar
[3]Aubert, G. and Kornprobst, P., Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, volume 147, Springer Verlag (2006).Google Scholar
[4]Bertero, M., Boccacci, P., Desiderà, G. and Vicidomini, G., Image deblurring with Poisson data: from cells to galaxies, Inverse Problems 25, 123006 (2009).Google Scholar
[5]Bioucas-Dias, J. and Figueiredo, M., Multiplicative noise removal using variable splitting and constrained optimisation, IEEE Trans. Image Process. 19, 1720–1730 (2010).Google Scholar
[6]Bonettini, S. and Ruggiero, V., An alternating extra gradient method for total variation based image restoration from Poisson data, Inverse Problems 27, 095001 (2011).Google Scholar
[7]Bonettini, S. and Ruggiero, V., On the convergence of primal-dual hybrid gradient algorithms for total variation image restoration, J. Math. Imaging Vis. 44, 236–253 (2012).CrossRefGoogle Scholar
[8]Bovik, A., Handbook of Image and Video Processing, Academic Press (2000).Google Scholar
[9]Boyd, S., Parikh, N., Chu, E., Peleato, B. and Eckstein, J., Distributed optimisation and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learning 3, 1–122 (2010).CrossRefGoogle Scholar
[10]Bregman, L., The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys. 7, 200–217 (1967).Google Scholar
[11]Cai, J., Chan, R. and Shen, Z., A framelet-based image inpainting algorithm, Appl. Comput. Harmon. Anal. 24, 131–149 (2008).Google Scholar
[12]Chambolle, A., An algorithm for total variation minimisation and application, J. Math. Imaging Vis. 20, 89–97 (2004).Google Scholar
[13]Chambolle, A. and Lions, P., Image recovery via total variation minimisation and related problems, Num. Mathematik 76, 167–188 (1997).Google Scholar
[14]Chambolle, A. and Pock, T., A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis. 40, 120–145 (2011).Google Scholar
[15]Chan, T. and Shen, J., Image Processing And Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM (2005).Google Scholar
[16]Chaux, C., Pesquet, J. and Pustelnik, N., Nested iterative algorithms for convex constrained image recovery problems, SIAM J. Imaging Sci. 2, 730–762 (2009).Google Scholar
[17]Combettes, P. and Pesquet, J., A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery, IEEE J. Selected Topics in Signal Processing 1, 564–574 (2007).Google Scholar
[18]Csiszár, I., Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems, Ann. Statist. 19, 2032–2066 (1991).Google Scholar
[19]Dai, Y. and Fletcher, R., New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds, Math. Program. A 106, 403–421(2006).Google Scholar
[20]Durand, S., Fadili, J. and Nikolova, M., Multiplicative noise removal using l1 fidelity on frame coefficients, J. Math. Imaging Vis. 36, 201–226 (2010).Google Scholar
[21]Esser, E., Applications of Lagrangian-based alternating direction methods and connections to split Bregman, CAM report 09-31, UCLA (March 2009).Google Scholar
[22]Esser, E., Zhang, X. and Chan, T., A general framework for a class of first order primal-dual algorithms for convex optimisation in imaging science, SIAM J. Imaging Sci. 3, 1015–1046 (2010).Google Scholar
[23]Figueiredo, M. and Bioucas-Dias, J., Restoration of Poissonian images using alternating direction optimisation, IEEE Trans. Image Process. 19, 3133–3145 (2010).Google Scholar
[24]Gilboa, G. and Osher, S., Nonlocal operators with applications to image processing, Multiscale Model. Sim. 7, 1005–1028 (2009).Google Scholar
[25]Giusti, E., Minimal Surfaces and Functions of Bounded Variation Birkhäuser, Boston (1984).Google Scholar
[26]Goldstein, T. and Osher, S., The split Bregman method for l1 regularized problems, SIAM J. Imaging Sci. 2, 323–343 (2009).Google Scholar
[27]Grimmett, G. and Welsh, D., Probability: An Introduction, Oxford Science Publications (1986).Google Scholar
[28]Huang, Y., Moisan, L., Ng, M. and Zeng, T., Multiplicative noise removal via a learned dictionary, IEEE Trans. Image Process. 21, 4534–4543 (2012).Google Scholar
[29]Huang, Y., Ng, M. and Wen, Y., A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci. 2, 20–40 (2009).CrossRefGoogle Scholar
[30]Huang, Y., Ng, M. and Zeng, T., The convex relaxation method on deconvolution model with multiplicative noise, Commun. Comput. Phys. 13, 1066–1092 (2013). Source: http://www.math.hkbu.edu.hk/~zeng/CiCp-MultiResto2012.pdf.Google Scholar
[31]Kontogiorgis, S. and Meyer, R., A variable-penalty alternating directions method for convex optimisation, Math. Prog. 83, 29–53 (1998).Google Scholar
[32]Kornprobst, P., Deriche, R. and Aubert, G., Image sequence analysis via partial differential equations, J. Math. Imaging Vis. 11, 5–26 (1999).Google Scholar
[33]Lintner, S. and Malgouyres, F., Solving a variational image restoration model which involves L∞ constraints, Inverse Problems 20, 815–831 (2004).Google Scholar
[34]Le, T., Chartrand, R. and Asaki, T., A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vis. 27, 257–263 (2007).Google Scholar
[35]Ng, M., Wang, F. and Yuan, X., Inexact alternating direction methods for image recovery, SIAM J. Sci. Comp. 33, 1643–1668 (2011).CrossRefGoogle Scholar
[36]Nocedal, J. and Wright, S., Numerical Optimisation, Springer Verlag (2006).Google Scholar
[37]Pock, T., Cremers, D., Bischof, H., Cremers, D. and Chambolle, A., An algorithm for minimising the Mumford-Shah functional, in Proc. 12th IEEE Int'l Conf. Computer Vision, pp. 1133–1140 (2009).Google Scholar
[38]Osher, S. and Paragios, N.. Multiplicative denoising and deblurring: theoryand algorithms, in Geometric Level Set Methods in Imaging, Vision, and Graphics, pp. 103–119, Springer Verlag (2003).CrossRefGoogle Scholar
[39]Rudin, L., Osher, S. and Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D 60, 259–268 (1992).Google Scholar
[40]Setzer, S., Operator splittings, Bregman methods and frame shrinkage in image processing, Int. J. Comput. Vis. 92, 265–280 (2011).Google Scholar
[41]Setzer, S., Steidl, G. and Teuber, T., Deblurring Poissonian images by split Bregman techniques, J. Visual Comm. Image Repres. 21, 193–199 (2010).CrossRefGoogle Scholar
[42]Shi, J. and Osher, S., A nonlinear inverse scale space method for a convex multiplicative noise model, SIAM J. Imaging Sci. 1, 294–321 (2008).CrossRefGoogle Scholar
[43]Steidl, G. and Teuber, T., Removing multiplicative noise by Douglas-Rachford splitting methods, J. Math. Imaging Vis. 36, 168–184 (2010).Google Scholar
[44]Teuber, T. and Lang, A., Nonlocal filters for removing multiplicative noise, in Bruckstein, A. M.et al. (Eds.) Scale Space and Variational Methods in Computer Vision, Third International Conference, SSVM 2011, pp. 50–61, Springer Verlag (2012).Google Scholar
[45]Wu, C. and Tai, X., Augmented Lagrangian method, dual methods and split-Bregman iterations for ROF, vectorial TV and higher order, SIAM J. Imaging Sci. 3, 300–339 (2010).Google Scholar
[46]Zeng, T. and Ng, M., On the total variation dictionary model optimisation, IEEE Trans. Image Process. 19, 821–825 (2010).Google Scholar
[47]Zhang, X., Burger, M., Bresson, X. and Osher, S., Bregmanized nonlocal regularisation for decon-volution and sparse reconstruction, SIAM J. Imaging Sci. 3, 253–276 (2010).Google Scholar