Level Set Methods

Level Set Methods

The level set method was devised by Osher and Sethian as a simple and versatile method for computing and analyzing the motion of the interface in two or three dimensions. The goal is to compute and analyze the subsequent motion of the interface under a velocity field. This velocity can depend on position, time, the geometry of the interface and the external physics. The interface is captured for later time as the zero level set of a smooth (at lest Lipschitz continuous) function. Topological merging and breaking are well defined and easily performed.

We discuss recent variants and extensions, including the motion of curves in three dimensions, the Dynamic Surface Extension method, fast methods for steady state problems, diffusion generated motion and the variational level set approach. We also give user's guide to the level set dictionary and technology, couple the method to a wide variety of problems involving external physics, such as compressible and incompressible (possibly reaction) flow, Stefan problems, kinetic crystal growth, epitaxial growth of thin films, vortex dominated flows and extensions to multiphase motion. We conclude with a discussion of applications to computer vision and image processing.(from CAM 00-08 Abstract, Feb 2000) Report by Osher and Fedkiw

Variational Problems and PDE on Implicit Surfaces

A novel framework for solving variational problems and partial differential equations defined on surfaces in introduced. The key idea is to implicitly represent the surface as the level-set of a higher dimensional function, and solve the problem in a fixed Cartesian coordinate system using this new embedding function. When this is combined with the general theory of harmonic maps, we can address such problems as intrinsic regularization of data defined on the surface and pattern generation using the intrinsic surface geometry. More precisely, we address the problems of isotropic and anisotropic regularization of images and direction maps defined on the surface and the problem of intrinsic texture synthesis. (from CAM 00-23 Abstract, June 2000). Report by Bertalmio,Cheng, Osher and Sapiro