Single image restoration through [formula omitted]-relaxed truncated [formula omitted] analysis-based sparse optimization in tight frames

Abstract

Image restoration problems, i.e., recovery of an original high-quality image from the degraded observation, arise in various science and engineer areas. Over the past decades, the framelet-based methods are particularly investigated and adopted, owing to the excellent ability of sparse approximating the piecewise-smooth functions such as natural images. In this paper, we propose a novel tight frame-based ℓ2-relaxed truncated ℓ0 analysis-sparsity model that simultaneously exploiting the sparsity and support priors. The resulting nonconvex nonsmooth optimization problem is addressed by using the proposed proximal alternating adaptive hard thresholding (PAAHT) method. We also proved that the sequence generated by the proposed algorithm sublinearly converges. Numerical experiments on several typical image restoration problems demonstrate that the proposed method is more effective than the standard sparsity-inducing algorithms and outperforms several state-of-the-art methods in both objective and perceptual quality.