A LEVEL SET FORMULATION FOR THE SOLUTION OF THE DIRICHLET PROBLEM FOR HAMILTON-JACOBI EQUATIONS

A level set formulation for the solution of the Hamilton-Jacobi equation F(x, y, u, u(x), u(y)) = 0 is Presented, where u is prescribed on a set of closed bounded noncharacteristic curves. A time dependent Hamilton-Jacobi equation is derived such that the zero level set at various time t of this solution is precisely the set of points (x, y) for which u(x, y) = t. This gives a fast and simple numerical method for generating the viscosity solution to F = 0. The level set capturing idea was first introduced by Osher and Sethian [J. Comput. Phys., 79 (1988), pp. 12-49], and the observation that this is useful for an important computer vision problem of this type was then made by Kimmel and Bruckstein in [Technion (Israel) Computer Science Report, CIS #9209, 1992] following Bruckstein [Comput. Vision Graphics Image Process, 44 (1988), pp. 139-154]. Finally, it is noted that an extension to many space dimensions is immediate.