Blind Deconvolution with Non-smooth Regularization via Bregman Proximal DCAs

TL;DR

This paper introduces a Bregman proximal DC algorithm with extrapolation for blind deconvolution, providing theoretical convergence guarantees and demonstrating superior performance in image deblurring tasks.

Contribution

It reformulates blind deconvolution as a difference of convex functions satisfying the $L$-smad property and develops an efficient algorithm with convergence guarantees.

Findings

Successfully recovered original images in deblurring experiments

Outperformed existing algorithms in accuracy and speed

Provided theoretical convergence analysis for the proposed method

Abstract

Blind deconvolution is a technique to recover an original signal without knowing a convolving filter. It is naturally formulated as a minimization of a quartic objective function under some assumption. Because its differentiable part does not have a Lipschitz continuous gradient, existing first-order methods are not theoretically supported. In this paper, we employ the Bregman-based proximal methods, whose convergence is theoretically guaranteed under the $L$-smooth adaptable ($L$-smad) property. We first reformulate the objective function as a difference of convex (DC) functions and apply the Bregman proximal DC algorithm (BPDCA). This DC decomposition satisfies the $L$-smad property. The method is extended to the BPDCA with extrapolation (BPDCAe) for faster convergence. When our regularizer has a sufficiently simple structure, each iteration is solved in a closed-form expression, and thus our algorithms solve large-scale problems efficiently. We also provide the stability analysis of the equilibrium and demonstrate the proposed methods through numerical experiments on image deblurring. The results show that BPDCAe successfully recovered the original image and outperformed other existing algorithms.