Skip to main content

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.

Corresponding author
*Corresponding author. Email addresses: (F. Sciacchitano), (Y. Dong), (M. S. Andersen)
Hide All
[1] Akkoul S., Ledee R., Leconge R., and Harba R., A new adaptive switching median filter, IEEE Signal Process. Lett, 17 (2010), pp. 587590.
[2] Astola J. and Kuosmanen P., Fundamentals of Nonlinear Digital Filtering, vol. 8, CRC, Boca Raton, FL, 1997.
[3] Aubert G. and Aujol J., A variational approach to removing multiplicative noise, SIAM J. Appl. Math., 68 (2008), pp. 925946.
[4] Bar L., Kiryati N. and Sochen N., Image deblurring in the presence of impulsive noise, International Journal of Computer Vision, 70 (2006), pp. 279298.
[5] Bovik A., Handbook of Image and Video Processing, New York: Academic, 2010.
[6] Brownrigg D., The weighted median filter, Comm. ACM, 27 (1984), pp. 807818.
[7] Cai J., Chan R., and Nikolova M., Fast two-phase image deblurring under impulse noise, J. Math. Imaging Vision, 36 (2010), pp. 4653.
[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. 187204.
[9] Chambolle A., An algorithm for total variation minimization and applications, J. Math. Imag. Vis., 20 (2004), pp. 8997.
[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. 120145.
[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. 17311739.
[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. 14791485.
[13] Chan T. and Esedoglu S., Aspects of total variation regularized L1 function approximation, SIAM J. Appl. Math., 65 (2005), pp. 18171837.
[14] Chan T. and Shen J., Image processing and analysis: variational, PDE, wavelet, and stochastic methods, SIAM, 2005.
[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. 784789.
[16] Chen T. and Wu H., Adaptive impulse detection using center-weighted median filters, IEEE Signal Process. Lett., 8 (2001), pp. 13.
[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. 11681189.
[18] Dong Y., Chan R., and Xu S., A detection statistic for random-valued impulse noise, IEEE Trans. Image Process., 16 (2007), pp. 11121120.
[19] Dong Y. and Zeng T., A convex variational model for restoring blurred images with multiplicative noise, SIAM J. Imaging Sci., 6 (2013), pp. 15981625.
[20] Elad M. and Aharon M., Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15 (2006), pp. 37363745.
[21] Figueiredo B. and Bioucas-Dias J., Restoration of Poissonian images using alternating direction optimization, IEEE Trans. Image Process., 19 (2010), pp. 31333145.
[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. 865888.
[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. A2856A2873.
[24] Hwang H. and Haddad R. Adaptive median filters: new algorithms and results, IEEE Trans. Image Process., 4 (1995), pp. 499502.
[25] Ko S. and Lee Y., Center weighted median filters and their applications to image enhancement, IEEE Trans. Circuits Syst., 38 (1991), pp. 984993.
[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. 257263.
[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. 18221837.
[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. 22582284.
[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. 249270.
[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. 965994.
[31] Nikolova M., A variational approach to remove outliers and impulse noise, J. Math. Imag. Vis., 20 (2004), pp. 99120, .
[32] Nocedal J. and Wright S., Numerical Optimization, New York: Springer, Second edition, 2006.
[33] Qi L. and Sun J., A nonsmooth version of Newton's method, Math. Programm., 58 (1993), pp 353367.
[34] Pratt W., Median Filtering, Technical report, Image Processing Institute, University of Southern California, Los Angeles, CA, 1975.
[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. 103119, 2003.
[36] Rudin L., Osher S., and Fatemi E., Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259268.
[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. 193199.
[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. 17081728.
[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. 28422865.
[40] Yin W., Goldfarb D. and Osher S., The total variation regularized L1 model for multiscale decomposition, Multiscale Model. Simul., 6 (2007), pp. 190211.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 57 *
Loading metrics...

Abstract views

Total abstract views: 313 *
Loading metrics...

* Views captured on Cambridge Core between 20th February 2017 - 24th January 2018. This data will be updated every 24 hours.