Scientific Computation and Applied & Industrial Mathematics: Roland Herzog

We present an analog of the total variation image reconstruction approach by Rudin, Osher, Fatemi (1992) for images defined on smooth surfaces, together with a proper analytical framework. The problem is defined in terms of quantities intrinsic to the surface and it is therefore independent of the parametrization. It is shown that the Fenchel predual of the total variation problem is a quadratic optimization problem for the vector-valued predual variable with pointwise constraints on the surface. The predual problem is solved using a function space interior point method, and discretized by conforming Raviart-Thomas finite elements on a triangulation of the surface. As in the flat case, the predual variable serves as an edge detector. Numerical examples including denoising and un-erasing problems with both gray-scale and color images on complex 3D geometries are presented.