Fast Marching Methods

Abstract

The fast marching method was introduced by Sethian [190, 191, 192] as a computationally efficient solution to eikonal equations on flat domains. A related method was presented by Tsitsiklis in [205]. The fast marching method was extended to triangulated surfaces by Kimmel and Sethian in [112]. The extended method solves eikonal equations on flat rectangular or curved triangulated domains in O(M) steps, where M is the number of vertices. In other words, the computational complexity of finding the solution is optimal. Here, we will present an O(M log M) method that is simple to implement. Using this technique, one can efficiently compute distances on curved manifolds with local weights. In this chapter we start with a simple example of distance computation in 1D that simplifies the notion of viscosity solutions of eikonal equations. Next, the fast marching method is applied to path planning of a robot navigating in nontrivial configuration space with a small number of degrees of freedom. Finally, we explore the power of efficient distance computation on curved domains, and present applications like minimal geodesic computation on triangulated surfaces, geodesic Voronoi diagrams, and curve offset calculations on weighted surfaces.