A rigorous setting for the reinitialization of first order level set equations

Nao Hamamuki (1) , Eleftherios Ntovoris (2)

1 Department of Mathematics [Sapporo]
2 UPE - Université Paris-Est

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.

Mots clés: Viscosity solutions; Level set equations; Distance function; Reinitialization; Homogenization

Domaines: Equations aux dérivées partielles [math.AP]

Nao Hamamuki, Eleftherios Ntovoris. A rigorous setting for the reinitialization of first order level set equations. 2016. ⟨hal-01374544⟩