Composite analysis‐based model for image restoration solved by alternating direction method of multipliers fast iterative shrinkage‐thresholding algorithm

Abstract

A composite analysis‐based model for image restoration by utilising the image edge's well preserving property of the total variation (TV) and the sparsely representation capability of the wavelet frame‐based image restoration model is proposed. This model is an analysis‐based approach that the wavelet frame coefficient is quite often able to be linked with the smoothness of the image and is more in accord with the model's essence of TV model. This model is solved using the alternating direction method of multipliers fast iterative shrinkage‐thresholding algorithm and a faster alternating minimisation algorithm, and numerical experiments show that the proposed method is more suitable for image restoration.

Coupled image restoration model with non‐convex non‐smooth ℓ p wavelet frame and total variation regularisation

“…Comparing with the composite analysis‐based ℓ 1 ‐norm model in our pervious work [11], the coupled non‐smooth and non‐convex ℓ p (0 < p < 1) quasi‐norm model in this paper is comparably easy to be implemented and to be solved, even with arbitrary p (0 < p ≤ 1) values. Moreover, when p = 1, the GISA will degenerate to the soft shrinkage/thresholding method, then converges to the global minimum.…”

“…In [10], the authors used the augmented Langranian to solve the combined TV and framelet synthesis‐based model, in which the model was split into several easily solvable subproblems; then these subproblems were solved by the simple soft shrinkage/thresholding method and FISTA [21]. In our previous work [11], we have solved an analysis‐based combined model that different from the hybrid models in [9, 10] using the ADMM [22], FISTA and a faster alternating minimisation algorithms.…”

“…A theoretical explanation for this phenomenon was provided in [1]. Motivated by the attractive properties of the ℓ p (0 < p < 1)‐norm regularisation to the image processing, in this paper, we improve the model in our previous work [11] by replacing the ℓ 1 ‐norm with the non‐convex non‐smooth ℓ p (0 < p < 1) quasi‐norm as follows: where is the discrete gradient and W is the discrete tight wavelet frame transform satisfying W T W = I .…”

“…Moreover, the combination of TV regulariser and the wavelet frame analysis‐based regulariser in objection function also is a challenging task for the optimal solution. Following the ideas of our previous work [11], we first use the ADMM and the AL methods to divide the non‐smooth, non‐convex model into three subproblems. After that, the GISA in company with the FISTA, which are more easily to be implemented and have the excellent convergence rate, are applied to solve the non‐convex non‐smooth subproblems.…”

“…It should be pointed out that the proposed image restoration models [9, 10] are synthesis‐based hybrid models. In our previous paper [11], an analysis‐based hybrid model also has been proposed as follows: The discussion of the merits between the synthesis‐based model and the analysis‐based model involving only the wavelet frame regularisation can track back to the literature [3] and the references therein.…”