Framelet-Based Blind Image Restoration with Minimax Concave Regularization

Images are important tools in various sciences. Despite the development of photo-taking tools, creating clear and image without noise remains challenging in practice. In particular, Poisson noise has an effect on medical and astronomical images, and reduces their quality. Additionally, blur is another factor that has an effect on image quality. The problem of image restoration becomes very complicated when we have no information about the Point Spread Function (PSF). These types of problems are known as blind case. However, in some images, such as some astronomical images, the type of PSF can be specified, and these types of problems are known as nonblind problems. Total Variation (TV) is a widely used method for solving such inverse problems, where the selection of the penalty function is the most critical factor that affects the method's performance. In this paper, to improve edge preservation, we employ a reweighted $\ell_1$-regularization of the fractional order derivative. Furthermore, we propose a nonblind and blind image deblurring approach under Poisson noise using the Minimax Concave Penalty (MCP), which is a continuous, sparsity promoting, and nearly unbiased regularizer. This formulation leads to a nonconvex optimization model. To solve the proposed model, we introduce an efficient numerical algorithm based on the Alternating Direction Method of Multipliers (ADMM) and provide an analysis of its convergence. Finally, the effectiveness of the proposed algorithm are demonstrated through extensive experiments on various images.

The total scaled-gradient variation (TSGV) regularizer, derived from sparse modeling of piecewise-linear structures, has been shown to preserve edges and corners in image restoration. However, its highly nonconvex and nonlinear nature poses severe computational challenges, as existing methods often suffer from parameter sensitivity or lack convergence guarantees. To overcome this, we propose a tailored bilinear decomposition that decouples the nonlinear weighted gradient in the TSGV regularizer. This approach yields an equivalent optimization problem governed by cone or sphere constraints, depending on the chosen scaling function. In particular, the cone constraint plays a central role in characterizing edge- and corner-preserving behavior. We solve this reformulation using the alternating minimization method (AMM) equipped with a majorization--minimization strategy, ensuring a monotonic decrease in energy without step-size tuning. Furthermore, we provide a geometric interpretation of the edge-preserving properties of these constraints by analyzing their asymptotic behavior near image singularities. We establish the global convergence of the proposed method to a critical point within the Kurdyka--Łojasiewicz framework. Extensive numerical experiments on Gaussian denoising and non-line-of-sight (NLOS) imaging show that the proposed method achieves PSNR and SSIM competitive with or superior to representative variational methods, especially at high noise levels, and improves the structural reconstruction under dense and sparse scanning.

MG-SpaIR is a training-data-free framework for restoring a clean image from a single observation corrupted by a mixture of blur, downsampling, noise, and missing pixels. Building on implicit neural representations (INRs), we introduce a multi-grade coarse-to-fine residual hierarchy that progressively refines the reconstruction across resolution grades, improving representational fidelity and mitigating spectral limitations. To stabilize reconstruction optimization and suppress INR-induced artifacts, we further propose an explicit sparse proximal regularization (e.g., $\ell_0$-type) applied directly in the high-resolution image domain, which discourages spurious high-frequency patterns while preserving sharp structures. The resulting optimization is solved efficiently via a multi-grade proximal alternating scheme, and we establish convergence guarantees for the associated updates under standard regularity conditions. Experiments on mixed-degradation benchmarks demonstrate that MG-SpaIR consistently outperforms strong training-data-free baselines such as Deep Image Prior, providing a stable, interpretable, and data-efficient alternative to conventional learning-based restoration methods.