Framelet-Based Blind Image Restoration with Minimax Concave Regularization

Recovering corrupted images is one of the most challenging problems in image processing. Among various restoration tasks, blind image deblurring has been extensively studied due to its practical importance and inherent difficulty.

In this problem, both the point spread function (PSF) and the underlying latent sharp image must be estimated simultaneously. This problem cannot be solved directly due to its ill-posed nature.

One powerful tool for solving such problems is total variation (TV) regularization. The $\ell_0$-norm regularization within the TV framework has been widely adopted to promote sparsity in image gradients or transform domains, leading to improved preservation of edges and fine structures.

However, the use of the $\ell_0$-norm results in a highly nonconvex and computationally intractable optimization problem, which limits its practical applicability. To overcome these difficulties, we employ the minimax concave penalty (MCP), which promotes enhanced sparsity and provides a closer approximation to the $\ell_0$-norm.

In addition, a reweighted $\ell_1$-norm regularization is incorporated to further reduce estimation bias and improve the preservation of fine image details and textures. After introducing the proposed model, a numerical algorithm is developed to solve the resulting optimization problem.

The effectiveness of the proposed approach is then demonstrated through experimental evaluations on several test images.