An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise. (English) Zbl 1195.68110

Summary: We extend the alternating minimization algorithm recently proposed by Y. Wang, J. Yang, W. Yin and Y. Zhang [SIAM J. Imaging Sci. 1, No. 3, 248–272 (2008; Zbl 1187.68665); SIAM J. Imaging Sci. 2, No. 2, 569–592 (2009; Zbl 1181.68304)] 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. Fu, M. K. Ng, M. Nikolova and J. L. Barlow, SIAM J. Sci. Comput. 27, No. 6, 1881–1902 (2006; Zbl 1103.65044)] and the two-phase method [J.-F. Cai, R. Chan and M. Nikolova, Inverse Probl. Imaging 2, No. 2, 187–204 (2008; Zbl 1154.94306)] for recovering grayscale images. We also present results of recovering multichannel images.


68U10 Computing methodologies for image processing
65K10 Numerical optimization and variational techniques
65T50 Numerical methods for discrete and fast Fourier transforms
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory


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