Blind Deconvolution of Binary and Pattern Images with Pixel Intensity Constraints and Sparse Gradient Prior

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.

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.

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.