An Efficient TVL1 Algorithm for Deblurring Multichannel Images Corrupted by Impulsive Noise

Abstract

We extend the alternating minimization algorithm recently proposed in [Y. Wang, J. Yang, W. Yin, and Y. Zhang, SIAM J. Imag. Sci., 1 (2008), pp. 248-272]; [J. Yang, W. Yin, Y. Zhang, and Y. Wang, SIAM J. Imag. Sci., 2 (2009), pp. 569-592] to the case of recovering blurry multichannel (color) images corrupted by impulsive rather than Gaussian noise. The algorithm minimizes the sum of a multichannel extension of total variation and a data fidelity term measured in the $\ell_1$-norm, and is applicable to both salt-and-pepper and random-valued impulsive noise. We derive the algorithm by applying the well-known quadratic penalty function technique and prove attractive convergence properties, including finite convergence for some variables and $q$-linear convergence rate. Under periodic boundary conditions, the main computational requirements of the algorithm are fast Fourier transforms and a low-complexity Gaussian elimination procedure. Numerical results on images with different blurs and impulsive noise are presented to demonstrate the efficiency of the algorithm. In addition, it is numerically compared to the least absolute deviation method [H. Y. Fu, M. K. Ng, M. Nikolova, and J. L. Barlow, SIAM J. Sci. Comput., 27 (2006), pp. 1881-1902] and the two-phase method [J. F. Cai, R. Chan, and M. Nikolova, AIMS J. Inverse Problems and Imaging, 2 (2008), pp. 187-204] for recovering grayscale images. We also present results of recovering multichannel images.