Technical Explanation of Level Set Methods

Given a moving closed hypersurface G(t), we wish to produce an Eulerian formulation for the motion of the hypersurface propagating along its normal direction with speed F, where F can be a function of various arguments, including the curvature, normal direction, etc. The main idea is to embed this propagating interface as the zero level set of a higher dimensional function phi. Let phi(x,t=0), where x is in n-dimensional space, be defined by

phi(x,t=0) = d(signed)

where d(signed) is the distance from x to G(t=0), and the plus (minus) sign is chosen if the point x is outside (inside) the initial hypersurface G(t=0). Thus, we have an initial function phi(x,t=0) with the property that

G(t=0) = ( x | phi( x, t= 0) = 0 )

Our goal is to now produce an equation for the evolving function phi(x,t) which contains the embedded motion of G(t) as the level set phi = 0. Let x(t) be the path of a point on the propagating front. That is, x (t=0) is a point on the initial front G(t=0), and dx/dt = F (x(t)) with the vector dx/dx normal to the front at x(t). Since the evolving function phi is always zero on the propagating hypersurface, we must have

phi( x ( t) , t ) = 0

By the chain rule,
d(phi)/dt + grad ( x(t,t) ) * dx/dt= 0

where grad is the gradient operator, and the * denotes the dot product. Since F already gives the speed in the outward normal direction, then dx/dt * n = F, where n = grad phi /|grad phi|. Thus, we then have the evolution equation for phi(x,t), namely

d(phi)/dt + F | grad phi | = 0

phi (x,t=0 ) given

We refer to this as a Hamilton-Jacobi ``type'' equation because, for certain forms of the speed function F, we obtain the standard Hamilton-Jacobi equation. There are four major advantages to this Eulerian Hamilton-Jacobi formulation.