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Total Variation Based Parameter-Free Model for Impulse Noise Removal

  • Federica Sciacchitano (a1), Yiqiu Dong (a1) and Martin S. Andersen (a1)

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.

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*Corresponding author. Email addresses: (F. Sciacchitano), (Y. Dong), (M. S. Andersen)
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[1] S. Akkoul , R. Ledee , R. Leconge , and R. Harba , A new adaptive switching median filter, IEEE Signal Process. Lett, 17 (2010), pp. 587590.

[3] G. Aubert and J. Aujol , A variational approach to removing multiplicative noise, SIAM J. Appl. Math., 68 (2008), pp. 925946.

[4] L. Bar , N. Kiryati and N. Sochen , Image deblurring in the presence of impulsive noise, International Journal of Computer Vision, 70 (2006), pp. 279298.

[6] D. Brownrigg , The weighted median filter, Comm. ACM, 27 (1984), pp. 807818.

[7] J. Cai , R. Chan , and M. Nikolova , Fast two-phase image deblurring under impulse noise, J. Math. Imaging Vision, 36 (2010), pp. 4653.

[10] A. Chambolle and T. Pock , A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imag. Vis., 40 (2011), pp. 120145.

[11] R. Chan , Y. Dong and M. Hintermüller , An efficient two-phase L1- TV method for restoring blurred images with impulse noise, IEEE Trans. Image Process., 19 (2010), pp. 17311739.

[12] R. Chan , C. Ho and M. Nikolova . Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization, IEEE Trans. Image Process., 14 (2005), pp. 14791485.

[13] T. Chan and S. Esedoglu , Aspects of total variation regularized L1 function approximation, SIAM J. Appl. Math., 65 (2005), pp. 18171837.

[14] T. Chan and J. Shen , Image processing and analysis: variational, PDE, wavelet, and stochastic methods, SIAM, 2005.

[15] T. Chen and H. Wu , Space variant median filters for the restoration of impulse noise corrupted images, IEEE Trans. Circuits Syst. II, 48 (2001), pp. 784789.

[16] T. Chen and H. Wu , Adaptive impulse detection using center-weighted median filters, IEEE Signal Process. Lett., 8 (2001), pp. 13.

[17] Y. Dong , M. Hintermüller and M. Neri An efficient primal dual method for L1- TV image restoration, SIAM J. Imag. Sci., 2 (2009), pp. 11681189.

[18] Y. Dong , R. Chan , and S. Xu , A detection statistic for random-valued impulse noise, IEEE Trans. Image Process., 16 (2007), pp. 11121120.

[19] Y. Dong and T. Zeng , A convex variational model for restoring blurred images with multiplicative noise, SIAM J. Imaging Sci., 6 (2013), pp. 15981625.

[20] M. Elad and M. Aharon , Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15 (2006), pp. 37363745.

[21] B. Figueiredo and J. Bioucas-Dias , Restoration of Poissonian images using alternating direction optimization, IEEE Trans. Image Process., 19 (2010), pp. 31333145.

[22] M. Hintermüller , K. Ito and K. Kunisch , The primal-dual active set strategy as a semismooth Newton method, SIAM J. Opt., 13 (2002), pp. 865888.

[23] Y. Huang , D. Lu and T. Zeng , A Two-Step Approach for the Restoration of Images Corrupted by Multiplicative, SIAM J. Sci. Comput., 35 (2013), pp. A2856A2873.

[24] H. Hwang and R. Haddad Adaptive median filters: new algorithms and results, IEEE Trans. Image Process., 4 (1995), pp. 499502.

[25] S. Ko and Y. Lee , Center weighted median filters and their applications to image enhancement, IEEE Trans. Circuits Syst., 38 (1991), pp. 984993.

[26] T. Le , T. Chartrand , and T. Asaki , A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vis., 27 (2007), pp. 257263.

[27] Y. Li , L. Shen , D. Dai and B. Suter , Framelet algorithms for de-blurring images corrupted by impulse plus Gaussian noise, IEEE Trans. Image Process., 20 (2011), pp. 18221837.

[28] L. Ma , J. Yu , and T. Zeng , Sparse Representation Prior and Total Variation–Based Image Deblurring Under Impulse Noise, SIAM J. Imag Sci, 6 (2013), pp. 22582284.

[29] L. Ma , M. Ng , J. Yu , and T. Zeng , Efficient box-constrained TV-type-l1 Algorithms for Restoring Images with Impulse Noise, J. Comp. Math., 31 (2013), pp. 249270.

[30] M. Nikolova , Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers, SIAM J. Numer. Anal., 40 (2002), pp. 965994.

[31] M. Nikolova , A variational approach to remove outliers and impulse noise, J. Math. Imag. Vis., 20 (2004), pp. 99120, .

[32] J. Nocedal and S. Wright , Numerical Optimization, New York: Springer, Second edition, 2006.

[33] L. Qi and J. Sun , A nonsmooth version of Newton's method, Math. Programm., 58 (1993), pp 353367.

[35] L. Rudin , P. Lions , and S. Osher , Multiplicative denoising and deblurring: theory and algorithms, Geometric Level Sets in Imaging, Vision and Graphics, S. Osher and N. Paragios , Eds. New York: Springer, pp. 103119, 2003.

[36] L. Rudin , S. Osher , and E. Fatemi , Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259268.

[37] S. Setzer and G. Steidl and T. Teuber , Deblurring Poissonian images by split Bregman techniques, J. Visual Commun. and Image Represent., 21 (2010), pp. 193199.

[38] Y. Xiao , T. Zeng , J. Yu and M. Ng , Restoration of Images Corrupted by Mixed Gaussian-Impulse Noise via l1-l0 Minimization, Pattern Recogn., 44 (2011), pp. 17081728.

[39] J. Yang , Y. Zhang and W. Yin , An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31 (2009), pp. 28422865.

[40] W. Yin , D. Goldfarb and S. Osher , The total variation regularized L1 model for multiscale decomposition, Multiscale Model. Simul., 6 (2007), pp. 190211.

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Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
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