The Osher—Sethian Level Set Method

Abstract

In this chapter we introduce the level set approach for curve and surface evolution. One way to represent a curve is as a level set or an equal-height contour of a given function. The intersection between this function and a plane parallel to the coordinate plane yields the curve. This function is an implicit representation of its level set, and actually of all its level set curves. This obvious relation between explicit curves and their implicit representation was used by Osher and Sethian [161] to introduce a powerful way for numerically tracking evolving interfaces; see also [155]. Consider the planar curve evolution Ct = V N N. Let ¢(x, y; t) be an implicit representation of the curve so that C(s, t) = {(x, y)I¢(x, y; t) = O}, that is, the zero level set of a time-varying surface function ¢(x, y; t). Then, the propagation rule for ¢ that yields the correct curve propagation equation is given by [161] we will prove this relation in the next section. In some problems it is natural to use a given image I as initialization for the implicit function ¢(x, y; 0) = I. The level set formulation for curve evolution thus provides a beautiful link between dynamic curves and scalar (gray-level) evolving images. Tracking the zero level set of the bivariate function ¢(x, y) propagating in time overcomes numerical and topological problems of the propagating level set curve in an elegant way. The implicit formulation for the implementation of propagating curves was explored by Osher and Sethian [161]. Sethian called it the Eulerian formulation for curve evolution [189].