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Total Variation Based Parameter-Free Model for Impulse Noise Removal

Part of: Communication, information General convexity Existence theories Numerical methods in calculus of variations and optimal control

Published online by Cambridge University Press:  20 February 2017

Federica Sciacchitano ,
Yiqiu Dong  and
Martin S. Andersen
Federica Sciacchitano*
Affiliation:
Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
Yiqiu Dong*
Affiliation:
Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
Martin S. Andersen*
Affiliation:
Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
*
*Corresponding author. Email addresses:feds@dtu.dk (F. Sciacchitano), yido@dtu.dk (Y. Dong), mskan@dtu.dk (M. S. Andersen)
*Corresponding author. Email addresses:feds@dtu.dk (F. Sciacchitano), yido@dtu.dk (Y. Dong), mskan@dtu.dk (M. S. Andersen)
*Corresponding author. Email addresses:feds@dtu.dk (F. Sciacchitano), yido@dtu.dk (Y. Dong), mskan@dtu.dk (M. S. Andersen)
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Abstract

We propose a new two-phase method for reconstruction of blurred images corrupted by impulse noise. In the first phase, we use a noise detector to identify the pixels that are contaminated by noise, and then, in the second phase, we reconstruct the noisy pixels by solving an equality constrained total variation minimization problem that preserves the exact values of the noise-free pixels. For images that are only corrupted by impulse noise (i.e., not blurred) we apply the semismooth Newton's method to a reduced problem, and if the images are also blurred, we solve the equality constrained reconstruction problem using a first-order primal-dual algorithm. The proposed model improves the computational efficiency (in the denoising case) and has the advantage of being regularization parameter-free. Our numerical results suggest that the method is competitive in terms of its restoration capabilities with respect to the other two-phase methods.

Keywords

Image deblurringimage denoisingimpulse noisenoise detectorprimal-dual first-order algorithmsemismooth Newton methodtotal variation regularization

MSC classification

Secondary: 68U10: Image processing 94A08: Image processing (compression, reconstruction, etc.) 49J40: Variational methods including variational inequalities 52A41: Convex functions and convex programs 65K10: Optimization and variational techniques 90C47: Minimax problems 49M15: Newton-type methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 1 , February 2017 , pp. 186 - 204
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Akkoul, S., Ledee, R., Leconge, R., and Harba, R., A new adaptive switching median filter, IEEE Signal Process. Lett, 17 (2010), pp. 587–590.Google Scholar
[2] Astola, J. and Kuosmanen, P., Fundamentals of Nonlinear Digital Filtering, vol. 8, CRC, Boca Raton, FL, 1997.Google Scholar
[3] Aubert, G. and Aujol, J., A variational approach to removing multiplicative noise, SIAM J. Appl. Math., 68 (2008), pp. 925–946.Google Scholar
[4] Bar, L., Kiryati, N. and Sochen, N., Image deblurring in the presence of impulsive noise, International Journal of Computer Vision, 70 (2006), pp. 279–298.Google Scholar
[5] Bovik, A., Handbook of Image and Video Processing, New York: Academic, 2010.Google Scholar
[6] Brownrigg, D., The weighted median filter, Comm. ACM, 27 (1984), pp. 807–818.CrossRefGoogle Scholar
[7] Cai, J., Chan, R., and Nikolova, M., Fast two-phase image deblurring under impulse noise, J. Math. Imaging Vision, 36 (2010), pp. 46–53.Google Scholar
[8] Cai, J., Chan, R., and Nikolova, M., Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise, Inverse Probl. Imaging, 2 (2008), pp. 187–204.Google Scholar
[9] Chambolle, A., An algorithm for total variation minimization and applications, J. Math. Imag. Vis., 20 (2004), pp. 89–97.Google Scholar
[10] Chambolle, A. and Pock, T., A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imag. Vis., 40 (2011), pp. 120–145.Google Scholar
[11] Chan, R., Dong, Y. and Hintermüller, M., An efficient two-phase L1- TV method for restoring blurred images with impulse noise, IEEE Trans. Image Process., 19 (2010), pp. 1731–1739.Google Scholar
[12] Chan, R., Ho, C. and Nikolova, M.. Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization, IEEE Trans. Image Process., 14 (2005), pp. 1479–1485.Google Scholar
[13] Chan, T. and Esedoglu, S., Aspects of total variation regularized L1 function approximation, SIAM J. Appl. Math., 65 (2005), pp. 1817–1837.CrossRefGoogle Scholar
[14] Chan, T. and Shen, J., Image processing and analysis: variational, PDE, wavelet, and stochastic methods, SIAM, 2005.Google Scholar
[15] Chen, T. and Wu, H., Space variant median filters for the restoration of impulse noise corrupted images, IEEE Trans. Circuits Syst. II, 48 (2001), pp. 784–789.Google Scholar
[16] Chen, T. and Wu, H., Adaptive impulse detection using center-weighted median filters, IEEE Signal Process. Lett., 8 (2001), pp. 1–3.Google Scholar
[17] Dong, Y., Hintermüller, M. and Neri, M. An efficient primal dual method for L1- TV image restoration, SIAM J. Imag. Sci., 2 (2009), pp. 1168–1189.CrossRefGoogle Scholar
[18] Dong, Y., Chan, R., and Xu, S., A detection statistic for random-valued impulse noise, IEEE Trans. Image Process., 16 (2007), pp. 1112–1120.Google Scholar
[19] Dong, Y. and Zeng, T., A convex variational model for restoring blurred images with multiplicative noise, SIAM J. Imaging Sci., 6 (2013), pp. 1598–1625.CrossRefGoogle Scholar
[20] Elad, M. and Aharon, M., Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15 (2006), pp. 3736–3745.Google Scholar
[21] Figueiredo, B. and Bioucas-Dias, J., Restoration of Poissonian images using alternating direction optimization, IEEE Trans. Image Process., 19 (2010), pp. 3133–3145.Google Scholar
[22] Hintermüller, M., Ito, K. and Kunisch, K., The primal-dual active set strategy as a semismooth Newton method, SIAM J. Opt., 13 (2002), pp. 865–888.Google Scholar
[23] Huang, Y., Lu, D. and Zeng, T., A Two-Step Approach for the Restoration of Images Corrupted by Multiplicative, SIAM J. Sci. Comput., 35 (2013), pp. A2856–A2873.Google Scholar
[24] Hwang, H. and Haddad, R. Adaptive median filters: new algorithms and results, IEEE Trans. Image Process., 4 (1995), pp. 499–502.CrossRefGoogle ScholarPubMed
[25] Ko, S. and Lee, Y., Center weighted median filters and their applications to image enhancement, IEEE Trans. Circuits Syst., 38 (1991), pp. 984–993.CrossRefGoogle Scholar
[26] Le, T., Chartrand, T., and Asaki, T., A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vis., 27 (2007), pp. 257–263.Google Scholar
[27] Li, Y., Shen, L., Dai, D. and Suter, B., Framelet algorithms for de-blurring images corrupted by impulse plus Gaussian noise, IEEE Trans. Image Process., 20 (2011), pp. 1822–1837.Google Scholar
[28] Ma, L., Yu, J., and Zeng, T., Sparse Representation Prior and Total Variation–Based Image Deblurring Under Impulse Noise, SIAM J. Imag Sci, 6 (2013), pp. 2258–2284.CrossRefGoogle Scholar
[29] Ma, L., Ng, M., Yu, J., and Zeng, T., Efficient box-constrained TV-type-l1 Algorithms for Restoring Images with Impulse Noise, J. Comp. Math., 31 (2013), pp. 249–270.Google Scholar
[30] Nikolova, M., Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers, SIAM J. Numer. Anal., 40 (2002), pp. 965–994.Google Scholar
[31] Nikolova, M., A variational approach to remove outliers and impulse noise, J. Math. Imag. Vis., 20 (2004), pp. 99–120, .Google Scholar
[32] Nocedal, J. and Wright, S., Numerical Optimization, New York: Springer, Second edition, 2006.Google Scholar
[33] Qi, L. and Sun, J., A nonsmooth version of Newton's method, Math. Programm., 58 (1993), pp 353–367.Google Scholar
[34] Pratt, W., Median Filtering, Technical report, Image Processing Institute, University of Southern California, Los Angeles, CA, 1975.Google Scholar
[35] Rudin, L., Lions, P., and Osher, S., Multiplicative denoising and deblurring: theory and algorithms, Geometric Level Sets in Imaging, Vision and Graphics, Osher, S. and Paragios, N., Eds. New York: Springer, pp. 103–119, 2003.Google Scholar
[36] Rudin, L., Osher, S., and Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259–268.Google Scholar
[37] Setzer, S. and Steidl, G. and Teuber, T., Deblurring Poissonian images by split Bregman techniques, J. Visual Commun. and Image Represent., 21 (2010), pp. 193–199.Google Scholar
[38] Xiao, Y., Zeng, T., Yu, J. and Ng, M., Restoration of Images Corrupted by Mixed Gaussian-Impulse Noise via l1-l0 Minimization, Pattern Recogn., 44 (2011), pp. 1708–1728.Google Scholar
[39] Yang, J., Zhang, Y. and Yin, W., An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31 (2009), pp. 2842–2865.Google Scholar
[40] Yin, W., Goldfarb, D. and Osher, S., The total variation regularized L1 model for multiscale decomposition, Multiscale Model. Simul., 6 (2007), pp. 190–211.Google Scholar