Analysis and an Interior-Point Approach for TV Image Reconstruction Problems on Smooth Surfaces

Lai and Chan [Computer Vis. Image Underst., 115 (2011), pp. 1647--1661] introduced an analogue of the total variation image reconstruction approach of Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259--268] for images on smooth surfaces. The problem is defined in terms of quantities intrinsic to the surface and is therefore independent of the parametrization. In this paper, a rigorous analytical framework is developed for this model and its Fenchel predual. It is shown that the predual of the total variation problem is a quadratic optimization problem for the predual vector field ${q} \in {H}(\operatorname{div};S)$ with pointwise inequality constraints on the surface. As in the flat case, ${q}$ serves as an edge detector. A function space interior-point method is proposed for the predual problem, which is discretized by conforming Raviart--Thomas finite elements on a triangulation of the surface. Well-posedness of the barrier problems is established. Numerical examples including denoising and inpainti...