Numerical Analysis of Geometry-Driven Diffusion Equations

Abstract

The various equations described in the previous chapters are of the parabolic and hyperbolic types. They arise when evolutionary processes are modeled: the solution of the equations can be interpreted as evolving from an initial state as time is increasing. In our analysis the initial state will be n image acquired at a certain resolution, and the evolution will generate a multi-scale representation. The nonlinear partial differential equations in the previous chapters cannot be solved analytically, so we have to resort to methods of numerical analysis. Some approximation schemes have already been mentioned by the various authors. This chapter provides a general framework for the numerical approximation of evolution equations. The organization of the chapter is as follows: In section 2.1 we treat the numerical approximation of the linear diffusion equation to introduce the concepts of explicit and implicit methods and discuss their stability criteria. We then (section 2.2) deal with the implementation of the hyperbolic advective equation, and introduce the concept of upwind schemes. In section 3 we will show how these methods can be used to find finite difference schemes for the nonlinear equations that were proposed. Since some of the equations are very anisotropic, their implementation on a fixed grid is problematic. We therefore introduce a scheme in which the solution is approximated using Gaussian derivative operators which are rotationally invariant. This scheme is applicable to all diffusion (parabolic type) equations. For hyperbolic terms we need to use an upwind scheme as proposed by Osher & Sethian [285]. In section 5 we show several results on test images and medical images.