Abstract
In this paper, a new accelerated fixed point algorithm for solving a common fixed point of a family of nonexpansive operators is introduced and studied, and then a weak convergence result and the convergence behavior of the proposed method is proven and discussed. Using our main result, we obtain a new accelerated image restoration algorithm, called the forward-backward modified W-algorithm (FBMWA), for solving a minimization problem in the form of the sum of two proper lower semi-continuous and convex functions. As applications, we apply the FBMWA algorithm to solving image restoration problems. We analyze and compare convergence behavior of our method with the others for deblurring the image. We found that our algorithm has a higher efficiency than the others in the literature.
Highlights
This problem can be transformed to an optimization problem using the least absolute shrinkage and selection operator (LASSO) model
We aim to prove a weak convergence theorem of Algorithm 1 (MWA) to a common fixed point of
We proposed a modified W-algorithm for solving a common fixed point problem of a family of nonexpansive operators and proved the weak convergence result of the proposed method under some control conditions
Summary
It is well-known that fixed point theory has relevant applications in many branches of analysis [1,2,3,4,5,6,7,8,9] and it can be applied to solving many areas of science and applied science, engineering, economics and medicine, such as image/signal processing [10,11,12,13,14,15,16,17] and modeling intensity modulated radiation theory treatment planning [18,19,20]. Many fixed point algorithms have been introduced and studied to solve various kinds of real world problems, such as Mann iteration [7], Ishikawa iteration [4], SP-iteration [21] and W-iteration [22]. The image restoration problem is an important topic in image processing. This problem can be transformed to an optimization problem using the least absolute shrinkage and selection operator (LASSO) model. We recall the background of a mathematical model for the image restoration problem and some related algorithms used to solving this problem. In order to solve the problem (1), Tibshirani in [29], introduced the least absolute shrinkage and selection operator (LASSO) for solving the following minimization problem:. The classical forward-backward splitting (FBS) algorithm [30] for problem (3) is given by the following iterative formula:
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