NONLINEAR EVOLUTIONARY PDEs IN IMAGE PROCESSING AND COMPUTER VISION

Abstract

Evolutionary PDE-based methods are widely used in image processing and computer vision. For many of these evolutionary PDEs, there is little or no theory on the existence and regularity of solutions, thus there is little or no understanding on how to implement them effectively to produce the desired effects. In this thesis work, we study one class of evolutionary PDEs which appear in the literature and are highly degenerate. The study of such second order parabolic PDEs has been carried out by using semi-group theory and maximum monotone operator in case that the initial value is in the space of functions of bounded variation. But the noisy initial image is usually not in this space, it is desirable to know the solution property under weaker assumption on initial image. Following the study of time dependent minimal surface problem, we study the existence and uniqueness of generalized solutions of a class of second order parabolic PDEs. Second order evolutionary PDE-based methods preserve edges very well but sometimes they have undesirable staircase effect. In order to overcome this drawback, fourth order evolutionary PDEs were proposed in the literature. Following the same approach, we study the existence and regularity of generalized solutions of one class of fourth order evolutionary PDEs in space of functions of bounded Hessian and bounded Laplacian. Finally, we study some evolutionary PDEs which do not satisfy the parabolicity condition by adding a regularization term. Through the rigorous study of evolutionary PDEs which appear in the literature of image processing and computer vision, we provide a solid theoretical foundation for them which helps us better understand the behaviors and properties of them. The existence and regularity theory is the first step toward effective numerical scheme. The regularity results also answer the questions to which function spaces the solutions of evolutionary PDEs belong and the questions if the processing results have the desired properties.