ℓ0-minimization methods for image restoration problems based on wavelet frames. (29th May 2019)

• Record Type: Journal Article
• Authors: Lu, Jian; Qiao, Ke; Li, Xiaorui; Lu, Zhaosong; Zou, Yuru
• Is Part Of: Inverse problems. Volume 35:Number 6(2019)
• Publication Date: 2019-05-29
• Subjects: ℓ0 minimization -- wavelet frame -- PD algorithm -- image restoration -- image deblurring -- image denoising; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ab08de
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: In this paper we consider a class of -minimization and wavelet frame-based models for image deblurring and denoising. Mathematically, they can be formulated as minimizing the sum of a data fidelity term and the l 0 -'norm' of the framelet coefficients of the underlying image, and we are particularly interested in three different types of data fidelity forms for image restoration problems. We first study the first-order optimality conditions for these models. We then propose a penalty decomposition (PD) method for solving these problems in which a sequence of penalty subproblems are solved by a block coordinate descent (BCD) method. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the PD method satisfies the first-order optimality conditions of these problems. Moreover, for the problems in which the data fidelity term is convex, we show that such an accumulation point is a local minimizer of the problems. In addition, we show that any accumulation point of the sequence generated by the BCD method is a block coordinate minimizer of the penalty subproblem. Furthermore, under some convexity assumptions on the data fidelity term, we prove that such an accumulation point is a local minimizer of the penalty subproblem. Numerical simulations show that the proposed -minimization methods enjoy great potential for image deblurring and denoising in terms of solution quality and/or speed.