Abstract
We consider minimization of functions that are compositions of functions having closed-form proximity operators with linear transforms. A wide range of image processing problems including image deblurring can be formulated in this way. We develop proximity algorithms based on the fixed point characterization of the solution to the minimization problems . We further refine the proposed algorithms when the outer functions of the composed objective functions are separable. The convergence analysis of the developed algorithms is established. Numerical experiments in comparison with the well-known Chambolle-Pock algorithm and Zhang-Burger-Osher scheme for image deblurring are given to demonstrate that the proposed algorithms are efficient and robust.
Highlights
In this paper, we study minimization problems of the form min{f1(A1x) + f2(A2x) : x ∈ Rn}, (1)where Ai are mi × n matrices and the functions fi : Rmi →
By Theorem 3.3, the convergence analysis for Algorithm 2 with the parameters given by Equation (55) is missing currently, our numerical experiments presented in the rest of the paper indicate that Algorithm 2 converges and usually produces better recovered images than that with the parameters given by Equation (53) in terms of the CPU times consumed
In problems of image deblurring with the L2-total variation (TV) model, a noisy image is obtained by blurring an ideal image with a-GBM followed by adding white Gaussian noise
Summary
We study minimization problems of the form min{f1(A1x) + f2(A2x) : x ∈ Rn},. (7) 5, We identify the L2-TV and L1-TV models as special cases of the general problem (3) and demonstrate that the where f ∗ represents the Fenchel conjugate of f whose definition proximity operators of the corresponding functions can be will be given . In section and a solution of Equation (7) can be derived from a stationary 6, we apply the proposed algorithms to solve L2-TV and point of the Lagrangian function of Equation (7). The dual formulation of model (1) has a we shall see that a saddle point of the Lagrangian form of function of Equation (7) will yield a solution of Equation (3) and a solution of Equation (7). We identify the saddle point of the Lagrangian function of Equation (7) as a solution of a fixed-point equation in terms of proximity operator and propose an iterative scheme to solve this fixed-point equation.
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