Geometry-based distortion measures for space deformation.

We present a framework for optimizing a rich family of geometry-based energies defined on planes, surfaces and volumetric domains. Our approach is based on the concept of first order distortion measures and on the steepest descent optimization. Specifically, we present an algorithm for inducing optimal deformations for triangular and tetrahedral meshes. The resulting techniques apply to a variety of geometry processing problems, including ones that are highly non-convex. Among these problems are deformation and parametrization of planes, surfaces and volumetric meshes, surface mapping using volumetric textures, generation of triangular and tetrahedral meshes. In particular, the proposed techniques can be employed to devise “as close to being conformal as possible” mappings and other deformations that are nearly optimal with respect to related distortion measures, such as the isometric distortion and the distortion of a local volume. Tests, carried out on 2D and 3D data, show that the optimization process is numerically stable and fast-converging. Our approach is general and it can be run in parallel processes.