The level set method applied to geometrically based motion, materials science, and image processing

The level set method has been used successfully in many areas of applied mathematics. We extend its application for geometrically based motion, materials science, and image processing. For geometrically based motion, we study the motions of codimension two objects such as curves in R3 while allowing merging. We also introduce a level set based representation for constrained problems such as the motion of curves on surfaces. Finally, we compute numerical solutions to the Minkowski Problem using a standard level set approach. Related both to geometrically based motion and materials science, we use a variational based level set method to construct Wulff minimal surfaces through given boundaries. We also run simulations to study a level set method for island dynamics in molecular beam epitaxy. Finally, we modify our algorithm for curves on surfaces to consider image processing of images on surfaces. Along the way, we introduce various applications arising from these methods. Results show that the level set method is very flexible and can easily handle all the problems we look at.