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.