Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-16T14:58:46.069Z Has data issue: false hasContentIssue false
  • English
  • Français

The Convex Relaxation Method on Deconvolution Model withMultiplicative Noise

Published online by Cambridge University Press:  03 June 2015

Yumei Huang ,
Michael Ng  and
Tieyong Zeng
Yumei Huang
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Michael Ng
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Tieyong Zeng*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
*
*Corresponding author.Email:zeng@hkbu.edu.hk
Get access

Abstract

In this paper, we consider variational approaches to handle the multiplicative noise removal and deblurring problem. Based on rather reasonable physical blurring-noisy assumptions, we derive a new variational model for this issue. After the study of the basic properties, we propose to approximate it by a convex relaxation model which is a balance between the previous non-convex model and a convex model. The relaxed model is solved by an alternating minimization approach. Numerical examples are presented to illustrate the effectiveness and efficiency of the proposed method.

Keywords

52A4065K1065K1590C26Alternating minimizationconvergencedeblurringmultiplicative noisenon-convex model
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 4 , April 2013 , pp. 1066 - 1092
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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 remove multiplicative noise, SIAM J. Appl. Math., Vol. 68 (2008), pp. 925–946.Google Scholar
[3]Aubert, G. and Kornprobst, P., Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, second edition ed., Ser. Applied Mathematical Sciences. Springer, Vol. 147 (2006).CrossRefGoogle Scholar
[4]Bar, L., Schen, N. and Kiryati, N., Image deblurring in the presence of impulsive noise, Int. J. Computer Vision, Vol. 70 (2006), pp. 279–298.CrossRefGoogle Scholar
[5]Bioucas, J.-Dias and Figueiredo, M., Total variation restoration of speckled images using a split-bregman algorithm, Proc. IEEE ICIP 2009.Google Scholar
[6]Cai, J., Chan, R. and Fiore, C., Minimization of a detail-preserving regularization functional for impulse noise removal, J. Math. Imaging Vis., Vol. 29 (2007), pp. 79–91.Google Scholar
[7]Cai, J., Chan, R. and Nikolova, M., Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise, Inverse Problems and Imaging, Vol. 2 (2008), pp. 187–204.Google Scholar
[8]Chan, R. H., Chan, T. F. and Wong, C. K., Cosine transform based preconditioners for total variation deblurring, IEEE Trans. Image Process., Vol. 8 (1999), pp. 1472–1478.Google Scholar
[9]Chan, R. and Chen, K., Multilevel algorithm for a Poisson noise removal model with totalvariation regularization, Int. J. Comput. Math., Vol. 84 (2007), pp. 1183–1198.CrossRefGoogle Scholar
[10]Cole, E., The removal of unknown image blurs by Homomorphic filtering, Technical Report 0038, Department of Electrical Engineering, University of Toronto, 1973.Google Scholar
[11]Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E., On the Lambert W function, Advances in Computational Mathematics(Berlin, New York: Springer-Verlag), Vol. 5 (1996), pp. 329–359.Google Scholar
[12]Durand, S., Fadili, J. and Nikolova, M., Multiplicative noise removal using L1 fidelity on frame coefficients, J. Math. Imaging Vis., Vol. 36 (2010), pp. 201–226.CrossRefGoogle Scholar
[13]Hadamard, J., Sur les problemes aux derivees partielles et leur signification physique, Princeton University Bulletin, 1902, pp. 49–52.Google Scholar
[14]Huang, Y., Ng, M. and Wen, Y., A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci., Vol. 2 (2009), pp. 20–40.CrossRefGoogle Scholar
[15]Huang, Y., Ng, M. and Wen, Y., Fast image restoration methods for impulse and Gaussian noises removal, IEEE Signal Processing Lett., Vol. 16 (2009), pp. 457–460.Google Scholar
[16]Huang, Y., Ng, M. and Wen, Y., A fast total variation minimization method for image restoration, SIAM Multiscale Model. Simul., Vol. 7 (2008), pp. 774–795.Google Scholar
[17]Kornprobst, P., Deriche, R. and Aubert, G., Image sequence analysis via partial differential equations, J. Math. Imaging Vis., Vol. 11 no. 1 (1999), pp. 5–26.Google Scholar
[18]Liu, C., Szeliski, R., Kang, S., Zitnick, C. and Freeman, W., Automatic estimation and removal of noise from a single image, IEEE Trans. PAMI, Vol. 30 (2008), pp. 299–314.CrossRefGoogle ScholarPubMed
[19]Puetter, R., Gosnell, T. and Amos, Y., Digital image reconstruction: deblurring and denoising, Annual Review of Astronomy & Astrophysics, Vol. 43 (2005), pp. 139–194.Google Scholar
[20]Chartrand, T. R. and Asaki, T., A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vis., Vol. 27 (2007), pp. 257–263.Google Scholar
[21]Rudin, L., Lions, P. and Osher, S., Multiplicative denoising and deblurring: theory and algorithms, In Osher, S. and Paragios, N., editors, Geometric Level Sets in Imaging, Vision, and Graphics, Springer, 2003, pp. 103–119.Google Scholar
[22]Rudin, L., Osher, S. and Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, Vol. 60 (1992), pp. 259–268.Google Scholar
[23]Shi, J. and Osher, S., A nonlinear inverse scale method for a convex multiplicative noise model, SIAM J. Imaging Sci., Vol. 1 (2008), pp. 294–321.CrossRefGoogle Scholar
[24]Steidl, G. and Teuber, T., Removing multiplicative noise by Douglas-Rachford splitting methods, J. Math. Imaging Vis., Vol. 36 (2010), pp. 168–184.Google Scholar
[25]Tsin, Y., Ramesh, V. and Kanade, T., Statistical calibration of CCD imaging process, Proc. IEEE Eighth Int'l Conf. Computer Vision, Vol. 1 (2001), pp. 480–487.Google Scholar
[26]Xiao, Y., Zeng, T., Yu, J. and Ng, M., Restoration of images corrupted by mixed Gaussian-impulse noise via Z1-Z0 minimization, Pattern Recognition, Vol. 44 no. 8 (2011), pp. 1708–1728.Google Scholar
[27]Zeng, T., Li, X. and Ng, M., Alternating minimization method for total variation based wavelet shrinkage model, Communications in Computational Physics, Vol. 8 no. 5 (2010), pp. 976994.Google Scholar
[28]Zeng, T. and Ng, M., On the total variation dictionary model, IEEE Trans. Image Process., Vol. 19 no. 3 (2010), pp. 821–825.CrossRefGoogle ScholarPubMed