1996, 1999, 2006
J.A. Sethian

Stationary Formulation: The Boundary Value PDE View
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The central idea of level set methods is to track a propagating interface by embedding it as the zero level set of a higher-dimensional function. This is an initial value partial differential equation; the initial front gamma(t=0) provides initial data as the zero level set of the level set function phi, and tracking the evolution of phi is identified with the evolution of the interface under its normal speed F. In the case where the front is moving with a speed F which is always positive, front propagation problems can be recast as a stationary boundary value problem. In the special case where this speed depends only on position, the resulting Eikonal equation is a static Hamilton-Jacobi equation with a long history; it has applications in such areas as geometric optics, seismology, and electromagnetics. More precisely, given a interface initially at gamma(t=0), let T(x,y) be the time T at which the interface crosses the point (x,y). Then a static Hamilton-Jacobi equation of the form

F | nabla T | = 1 with T=0 on gamma(t=0)

( A more detailed explanation of this stationary boundary value formulation and Fast Marching Methods)

can be given. There are several numerical techniques to solve this equation. However, using the previous work on the theory of curve and surface evolution, viscosity theory, upwind schemes, and sorting/causality relationships borrowed from computer science, a particular fast optimal technique, known as the Fast Marching Method can be developed.

New Book and Resource on Level Set and Fast Marching Methods

References for the Eikonal Equation:

  1. Partial Differential Equations : Garabedian, P., Partial Differential Equations, Wiley, New York, 1964.
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