Partial Differential Equations In Several Applications In Image Processing

The use of Partial Differential Equations in image processing is a new research topic. Its mathematical model can represent image accurately and can solve many complicated problems. Now, PDEs have been widely used in many aspects of image processing.In this thesis the PDE method is introduced and some of the applications are presented. First the PDEs are deduced independently from both image regular space model and the anisotropic diffusion equations. This deduction also shows the general method to analysis image processing problems through PDEs: convert the problem to an optimal estimation problem - express the optimum mathematically - get the energy function - solve the problem using variation calculus method.Numerical solutions to PDEs in image processing are very complicated. In this thesis, a discretization scheme and several methods to solve these descrete equations are discussed. These methods include gradient descent method, steepest descent method and conjugate gradient method.The PDEs are applied to some image processing problems such as image denoising, deblurring, inpainting and context transplanting. The PDEs are competent in solving these problems and their results are satisfactory.