A nonlocal model for image restoration corrupted by multiplicative noise

In the field of image processing, addressing multiplicative noise, particularly when it follows the Gamma distribution, has been the subject of numerous studies. In this work, we introduce a novel nonlocal model using a nonlocal variable exponent norm designed to remove noise from images affected by this type of noise. The non-linear nature, lack of differentiability and non-convexity of the proposed model pose mathematical and numerical challenges. To tackle these obstacles, we examine the well-posedness of the nonlinear model using optimization techniques. Additionally, to overcome the non-convexity numerically, we introduce an auxiliary variable within the Alternating Direction Method of Multipliers (ADMM) framework, which incorporates another auxiliary variable. Notably, our approach incorporates the non-local p-norm with a variable exponent, the proposed model can be interpreted as an extension of the traditional non-local total variation (TV) regularization, enabling the utilization of additional information and allowing for fine details preservation. This approach not only ensures expeditious processing but also exhibits robustness, offering resilience against the sensitivity inherent in conventional classical methods. By introducing this method, we aim to provide a solution that not only effectively tackles multiplicative noise but also surpasses the limitations associated with existing techniques, thereby contributing to advancements in image denoising methodologies.