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Inertial s-iteration forward-backward algorithm for a family of nonexpansive operators with applications to image restoration problems

Bussaban Limpapat (Faculty of Science, Chiang Mai University, Chiang Mai, Thailand), lim.bussaban@gmail.com
Kaewkhao Attapol (Research Center in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand), akaewkhao@yahoo.com
Suantai Suthep (Research Center in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand + Data Science Research Center, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand), suthep.s@cmu.ac.th

Image restoration is an important branch of image processing which has been studied extensively while there are several methods to solve this problem by many authors with the challenges of computational speed and accuracy of algorithms. In this paper, we present two methods, called ”Inertial S-iteration forward-backward algorithm (ISFBA)” and ”A fast iterative shrinkage-thresholding algorithm-Siteration (FISTA-S)”, for finding an approximate solution of least absolute shrinkage and selection operator problem by using a special technique in fixed point theory and prove weak convergence of the proposed methods under some suitable conditions. Moreover, we apply our main results to solve image restoration problems. It is shown by some numerical examples that our algorithms have a good behavior compared with forward-backward algorithm (FBA), a new accelerated proximal gradient algorithm (nAGA) and a fast iterative shrinkage-thresholding algorithm (FISTA).

Keywords: S-iteration, forward-backward algorithm, nonexpansive operator, image restoration problem