Nonlinear diffusion filtering on surfaces

Abstract

Nonlinear diffusion filtering is a PDE-based method to remove noise from images that has found much success. This dissertation looks at whether nonlinear diffusion filtering can be combined with the closest point method, a relatively new and novel method for solving partial differential equations on surfaces. The closest point method is an embedding method that uses a simple representation of surfaces. The theory and implementation of for the closest point method is presented. We perform convergence studies that show good agreement with theory.\ud \ud We discuss the use of linear and nonlinear diffusion for image processing, in particular the Perona{Malik and Gaussian schemes. We show that they can be combined with the closest point method to produce impressive results, visualised beautifully using an OpenGL raytracer designed for use with the closest point method. Some surprising and unexpected eects were discovered when moving from a plane to a three-dimensional surface. These effects are described and investigated