Analogue of the Total Variation Denoising Model in the Context of Geometry Processing

We propose a new variational model for surface processing. It is the natural analogue, in the context of geometry processing, of the total variation based image denoising model of Rudin, Osher, and Fatemi. In particular, it admits as solutions surfaces that may have discontinuities in their normals and can reconstruct shapes with sharp ridges and corners. As the analogue of a perfectly well-posed image model, the proposed surface model differs from those based on ill-posed equations of image processing. Since the proposed geometry regularization is variational, it can be incorporated into a variety of applications, such as surface fairing, three-dimensional object reconstruction from multiple viewpoints, or shape optimization. Drawing on important previous work of geometers who were interested in extending certain classical theorems from smooth to polyhedral manifolds, we describe in detail the appropriate numerical implementation of the proposed model in the case of triangulated surfaces. We then demonstrate the efficacy of the model with several surface fairing examples.