Partial Differential Equation And Its Applications In Image Denoising And Segmentation

Image denoising and segmentation are fundamental problems in both image processing and computer vision with numerous applations. The use of ordinary and partial differential equations for image processing became a major research topic in the past years. We divide the dissertation into three parts. In the first part, we mainly discuss the curvature flow driven diffusion model for image denoising in the first part. Next, in the second part, we are concerned with the active contour models which are based on the region for image segmentation. Finally, the exitinction and positivity of the solutions for a p-laplacian equation with absorption on graphs is considred. The details of my work are as follows:In the first part (Chapter 2), we discuss a weighted average of the maximal and minimal principle curvature as the diffusion coefficient for denoising digital images with additive noise. The main advantage of this approach is that it preserves important structures similar to Gauss curvature-driven diffusion, and it has a stable and fast numerical algorithm. Moreover, the proposed model helps gaining better denoising results than Gauss curvature-driven diffusion in the areas of SNR and the smoothness of the denoised image.The second part is Chapter 3 to Chapter 5 of this dissertation. Firstly, in Chapter 3, we propose a CV active contour model with a constraint condition for the bimodal image segmentation problem. Through adding a suitable constraint condition, such that the CV model has a global optimal solution and the level set function can be initialized as a non-zero constant function. The improved model is treated rigorously. Based on techniques in calculus of variations, the existence and uniqueness of the global optimal solution of the improved model is established in a constrained BV space. Also, we give an effective algorithm to compute the global optimal solution. We illustrate this algorithm on several image segmentation problems, including a synthetic example and some real images such as medical, remote sensing and natural images.In Chapter 4, we study the nonlocal p -Laplacian evolution equation with a nonlinear reaction term and Neumann boundary conditions for the bimodal image segmentation. The existence and uniqueness of the local solution is obtained by using Banach's fixed point theorem. Moreover, for p = 1, the change of the energies (?)dx are established with time t. We give an effective algorithm to compute the local solution. Finally, we include some numerical experiments for a synthetic example and some natural images, which illustrate our model's capability.In Chapter 5, we improve Lee-Seo's bimodal image segmentation model using a regularization term. This regularization term will maintain the smoothness of the level set function and decrease the level set function's oscillations around the desired steady state when the noise level is larger. Furthermore, we also provide a rigorous study of the modified model. Based on techniques in calculus of variations, the existence of solutions of the modified model in a BV space is established. Based on the theory we present (see Lemmas 5.2, 5.3), we construct a fast convergent algorithm to process images. It turns out our method is twice as fast in processing an image than Lee-Seo's algorithm with the same constant valued initial level set function.In the last part, we deal with the extinction of the solutions of the initial-boundary value problem of the discrete p -Laplacian equation with absorption ut =Δp,ωu ? uq, where p > 1, q > 0,which is said to be the discrete p -Laplacian equation on weighted graphs. For 0 < q< 1, we show that the nontrivial solution becomes extinct in finite time. While it remains strictly positive for p≥2, q≥1and q≥p? 1. Finally, a numerical experiment on a simple graph with standard weight is given.