Abstract

Variational models for image segmentation and fracture are similar mathematically, and difficult to solve and to approximate numerically. These difficulties are illustrated by the Mumford-Shah problem, where the main issue is finding the unknown free surface. There are numerical methods for "solving" this problem, but these tend (at best) to find only local minimizers (the exception being the method of Bourdin-Chambolle). Recently, L. Vese and T. Chan introduced numerical schemes based on the level set methods of Osher and Sethian. While being simpler than $\Gamma$ convergence-based methods, these methods are known to not find minimizers for at least two reasons. First, although it is known that only triple junctions can occur in local minimizers, the Vese-Chan algorithms generally result in quadruple junctions. The second, more serious, limitation of these methods is that they are incapable of finding crack tips, which is critical in models for fracture. In this talk, I will describe a new level set-based method that naturally finds triple junctions and crack tips. This is joint work with C. Richardson and M. Sarkis (WPI).

TUESDAY, March 8, 2005
Time: 1:30 P.M.
Location: DH 4303