On Inversion-Free Mapping and Distortion Minimization

This paper addresses a general problem of computing inversion-free maps between continuous and discrete domains that induce minimal geometric distortions. We will refer to this problem as optimal mapping problem. Finding a good solution to the optimal mapping problem is a key part in many applications in geometry processing and computer vision, including: parameterization of surfaces and volumetric domains, shape matching and shape analysis. The first goal of this paper is to provide a self-contained exposition of the optimal mapping problem and to highlight the interrelationship of various aspects of the problem. This includes a formal definition of the problem and of the related unitarily invariant geometric measures, which we call distortions. The second goal is to identify novel properties of distortion measures and to explain how these properties can be used in practice. Our major contributions are: (i) formalization and juxtaposition of key concepts of the optimal mapping problem, which so far have not been formalized in a unified manner; (ii) providing a detailed survey of existing methods for optimal mapping, including exposition of recent optimization algorithms and methods for finding injective mappings between meshes; (iii) providing novel theoretical findings on practical aspects of geometric distortions, including the multi-resolution invariance of geometric energies and the characterization of convex distortion measures. In particular, we introduce a new family of convex distortion measures, and prove that, on meshes, most of the existing distortion energies are non-convex functions of vertex coordinates.