  In this paper we set up a rigorous justification for the reinitialization
algorithm. Using the theory of viscosity solutions, we propose a well-posed
Hamilton-Jacobi equation with a parameter, which is derived from homogenization
for a Hamiltonian discontinuous in time which appears in the reinitialization.
We prove that, as the parameter tends to infinity, the solution of the initial
value problem converges to a signed distance function to the evolving
interfaces. A locally uniform convergence is shown when the distance function
is continuous, whereas a weaker notion of convergence is introduced to
establish a convergence result to a possibly discontinuous distance function.
In terms of the geometry of the interfaces, we give a necessary and sufficient
condition for the continuity of the distance function. We also propose another
simpler equation whose solution has a gradient bound away from zero.
