On geometric variational models for inpainting surface holes

Geometric approaches for filling-in surface holes are introduced and studied in this paper. The basic principle is to choose the completing surface as one which minimizes a power of the mean curvature. We interpret this principle in a level set formulation, that is, we represent the surface of interest in implicit form and we construct an energy functional for the embedding function u. We first explore two different formulations (which can be considered as alternative) inspired by the above principle: in the first one we write the mean curvature as the divergence of the normal vector field θ to the isosurfaces of u; in the second one we used the signed distance function D to the surface as embedding function and we write the mean curvature in terms of it. Then we solve the Euler-Lagrange equations of these functionals which consist of a system of second order partial differential equations (PDEs) for u and θ, in the first case, or a fourth order PDE for D in the second case. Then, simpler methods based on second order elliptic PDEs, like Laplace equation or the absolutely minimizing Lipschitz extension, are also proposed and compared with the above higher order methods. The theoretical and computational framework, as well as examples with synthetic and real data, are presented in this paper.