A Fourth-Order Nonlinear Directional Diffusion Model Arising in Image Processing

This article presents a theoretical analysis of a new fourth-order nonlinear directional diffusion model, making a significant contribution to the field of image processing and analysis. It provides a robust mathematical framework for further exploration and application of advanced diffusion techniques in tasks such as image restoration and enhancement. The study employs implicit-time discretization to approximate the nonlinear parabolic problem arising from the fourth-order diffusion model by converting it into a series of nonlinear elliptic problems related to energy minimization. To address these elliptic problems, we use a topological variational approach, establishing weak solutions within an appropriate functional framework, including the Orlicz space. Using these variational solutions, we systematically construct an approximate solution for the original nonlinear parabolic problem. The weak convergence of this approximate solution is rigorously demonstrated using a priori estimates, which provide essential bounds on the solution's behavior, and the monotone method, ensuring its stability and uniqueness.