Fractional PDEs for image restoration and denoising

Abstract

This thesis addresses the fundamental challenge of image denoising: achieving an optimal balance between noise suppres sion and the preservation of essential image structures. It introduces novel fractional partial differential equation (FPDE) models formulated within nonlinear diffusion and nonlinear telegraph diffusion frameworks, employing fractional deriva tives in the Caputo and Caputo-Fabrizio senses to enhance modeling flexibility and denoising performance. Theoretical analysis establishes the well-posedness of the proposed models by proving the existence, uniqueness, and boundedness of weak solutions. Efficient semi-implicit numerical schemes based on additive operator splitting (AOS) are developed and shown to be unconditionally stable and computationally efficient. Comprehensive experiments evaluated using PSNR and SSIM metrics demonstrate that the proposed fractional-order models significantly outperform conventional approaches.