Higher Order Nonlinear PDEs Based On Image Processing

In the past 20 years, the method by employing systematically Partial Differential Equations(PDEs) has been developed a rising field on image processing and computer vision. So far it has got fruitful results and demonstrated strong energy. Currently, a lot of prolific and systematic results about theoretical study for second order nonlinear PDEs in image processing have been obtained. However, the theoretical analysis for higher order nonlinear PDEs on this aspect has been known very little. In addition, it is also a very new field on multiplicative noise removal based on variation. The theoretical investigation for this aspect has not been developed perfectly. This thesis mainly study three aspects as follows:First, the existence and uniqueness of weak solutions for fourth order Regular-ized YK Equation and strong solutions for Generalized PM Equation in image denoising are discussed; Second, the existence, uniqueness and positivity of solutions for Viscous BSCB Equation and Modified BSCB Systems in image inpainting are studied; Finally, the existence, uniqueness and boundedness of BV solutions for the variational problem and the corresponding evolution equation on multiplicative noise removal are discussed.In Chapter 1, we recall the backgrounds and give a survey of the study on higher order nonlinear PDEs in image processing and variational model for multiplicative noise removal. Then we give an outline of this thesis including our main problems and results.In Chapter 2, we propose the Regularized YK Equation based on the ill-posed prop-erties for YK Equation in image denoising. The existence and uniqueness of weak so-lutions for the regularized equation are established and some numerical examples are presented. Additionally, we prove the existence and uniqueness of strong solutions for Generalized YK Equation on 1-D bounded intervals.In Chapter 3, at first, we obtain the existence and uniqueness of smooth solutions for viscous BSCB Equation and Modified BSCB System on 2-D torus. Then the exis-tence and uniqueness of weak solutions for a BSCB System with nonlinear diffusion term are established on 2-D bounded domains. Finally, by combining the principle of image inpainting, we present proper initial-boundary value conditions, and prove the existence and positivity of weak solutions for a Modified BSCB System on 2-D bounded domains. In Chapter 4, firstly, we introduce the known variational models for multiplica-tive noise removal. Then we propose a new variational model, and prove the existence, uniqueness and boundness of BV solutions for the new variational model and the associ-ated evolution equation. Finally, some numerical examples are given to show that the new model can remove effectively the multiplicative noise of images by comparing the new model with a known model.In this thesis, we mainly use the methods including Galerkin approximation, vanish-ing viscosity, a prior estimate and fixed point method.