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Level set methods: an overview and some recent results

The level set method was devised by S. Osher and J. A. Sethian (1988, J. Comput. Phys.79, 12–49) as a simple and versatile method for computing and analyzing the motion of an interface Γ in two or three dimensions. Γ bounds a (possibly multiply connected) region Ω. The goal is to compute and analyze the subsequent motion of Γ under a velocity field v. 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 least Lipschitz continuous) function ϕ (x, t); i.e., Γ(t)={x|ϕ(x, t)=0}. ϕ is positive inside Ω, negative outside Ω, and is zero on Γ(t). Topological merging and breaking are well defined and easily performed. In this review article 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 a user's guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics, such as compressible and incompressible (possibly reacting) 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.