A variational approach to Eulerian geometry processing

Abstract

We present a purely Eulerian framework for geometry processing of surfaces and foliations. Contrary to current Eulerian methods used in graphics, we use conservative methods and a variational interpretation, offering a unified framework for routine surface operations such as smoothing, offsetting, and animation. Computations are performed on a fixed volumetric grid without recourse to Lagrangian techniques such as triangle meshes, particles, or path tracing. At the core of our approach is the use of the Coarea Formula to express area integrals over isosurfaces as volume integrals. This enables the simultaneous processing of multiple isosurfaces, while a single interface can be treated as the special case of a dense foliation. We show that our method is a powerful alternative to conventional geometric representations in delicate cases such as the handling of high-genus surfaces, weighted offsetting, foliation smoothing of medical datasets, and incompressible fluid animation.