Some Nonlinear Diffusion Models And Numerical Approaches In Image Processing

Author: Lu BiboMajor: Applied MathematicsSupervisor: Professor Yin Jingxue, Professor Zou YongkuiIn this monograph, we study the PDE method for image processing, especially noise removal and image segmentation with nonlinear diffusion equations.Some major approaches to image processing include Fourier analysis, morphological approach, wavelet analysis, stochastic modeling, variational methods and partial differentialequations (PDEs). The successful applications of partial differential equations in image processing can be credited to two main factors. First, many variational problems or their regularized approximations can often be effectively computed from their Euler-Langrange equations. Second, PDEs as in classical mathematical physical are powerful tolls to describe, model, and simulate many dynamic as well as equilibrium phenomena, including diffusion, advection or transport, reaction, and so on. Third, the theory results and numerical results related to PDEs are available for image processing.The theory in PDEs can provide rigorous mathematical foundations related to PDEs in image processing. Diffusion equation come from a variety of diffusion phenomena appearedwidely in nature. They are suggested to be a powerful tool for image denoising. To describe the denoising procedure of image with edge, jump solution of diffusion equation is introduced in theory. In fact, to study discontinuous solutions, BV space is chosen as the reasonable class of solutions. Yin, Li and Pang consider the following equation:whereThey study discontinuous solutions of a partial differential equation of strongly degenerate parabolic type. A notion of weak solutions of BV class is proposed, and existence and uniqueness results are obtained. Pang , Wang, and Yin considered the free boundary problem of the equation. The equation cover many PDEs which has been used in image processing successfully.Two order PDEs has been used in image denoising with great success, but staircase effect is undesired for image recovery. As the solution of two order equation convergence to piecewise constant, staircase effect appears in some region of the image. Some researchers try to solve this problem with high order nonlinear diffusion equation. You proposed the following four order equation:Some other four order models have been devised to overcome this disadvantage with a promising result.The goal of image segmentation is to partition a given image into a collection region, and the points in each region share with some similar property.Kass, Witkin and Terzopoulos proposed the active contour model for image segmentation.The essential idea of active contour model is to detect the object in the image with curve evolution under some constrains. A closed boundary of the object is obtained with this model. Osher and Sethian represented a planar curve as zero level set and captured the evolution of the curve. The curve is represented as the interface of a plane and a surface. The level set function evolute smoothly in R³, but the curve represented as the zero level set may change topology, break, merge.Chan and Vese proposed a model: active contour without edge(CV model). They consider such energy functional:whereμ> 0 is a weight coefficient, Length(C^ls) denotes the length of curve C^ls. The edge obtained in CV model depends on the global structure of the image instead of the local gradient. So CV model can detect some weak edges and inner edge inside the object.In the first chapter, we consider a diffusion-convection equation, which is involved in image processing.where (?)(s) satisfiesThe equation (1) is strongly degenerate. In fact, to study discontinuous solutions, we should choose BV space as the reasonable class of solutions.As we know, the coefficient of the convection term of the equation (1) can control the orientation and the magnitude of the convection. Therefore, it is possible to get a BV solution with a vertical jump line by controlling the coefficient of the convection term. If it is the case, the discontinuous point of the solution remains unmoved. Thus we can prevent the edge from shifting in the procedure of diffusion. That is to say, it is possible to find a suitable convection such that the discontinuous point of the solution remains unmoved. By controlling the coefficient of the convection term, we get a solution with a vertical jump line to the problem. This reveals a new phenomenon:, the anti-shifting property of convection. That is to say, it is possible to find a suitable convection such that the discontinuous point of the solution remains unmoved. The result motive us to devise a diffusion model with convection to remain the edge unmoved.In the second chapter, we propose four order geometric nonlinear diffusion model for noise removal. To remove staircase effect, with geometric method of image processing, we develop two four order noise removal models: four order geometric model and four order Gauss curvature model.We consider the second fundamental formΠof surface (x, y, u(x, y)):whereThe second fundamental coefficients are:The second fundamental form is approximate to the twice distance between the surface and the tangent plane. It describes the degree of the surface from the tangent plane, that is to say, it measure the degree of bending of the surface. So, we consider the following energy functionalwhereλ> 0 is a weight coefficient. When this energy functional reaches the minimum, the the surface is near to the tangent plane, and the surface convergence to a piecewise linear function. As it is known that, the piecewise linear function can avoid the staircase effect. The Euler-Lagrange equation of above-mentioned functional is: where A_g = |u_xx| + 2|u_xy|+ |u_yy|, g = 1 + u_x²+ u_y².Parameterizing the descent direction by an artificial time t, we get the four order geometric model as follows:This equation shares the double features of diffusion and advection. With a boundary detection operation g, the diffusion speed varies with the positions of the boundary and the region. Compared with previous four order model, the adaptive diffusion of four order geometric model can preserve edges while denoising. As a four order model, four order geometric model can remove staircase effect. We extend four order geometric model to signal denoising and color image denoising with a satisfied result. In the last part of this chapter, we give an energy functional involved in the Gauss curvature K of surface (x,y,u(x,y)):where K = u_xxu_yy-u_xy²/1+u_x²+u_y².To solve the associated Euler-Lagrange equation by an artificial time t, we obtain the four order Gauss curvature model:where B_g = u_xxu_yy - u_xy². Four order Gauss curvature model can also preserve edges while avoiding staircase effect.In the third chapter, we extract the boundary of the grain from petrographic image and the boundary of the road from remote sensing image with active contour without edge model. We can not obtain the boundary of the interesting object from the segmentation result as the structure of the image is very complexed. As the contour of the boundary of the object is a closed curve, we analysis the features of every object. The interesting objects are selected after the comparision the extracted features with the feature from the background of the task. We extract the boundary of the grain from petrographic image and the boundary of the road from remote sensing image with satisfied results.