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,uₓ ,uy ) = 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.