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.

本文解决了计算连续域和离散域之间的无反演映射的一般问题，这些映射导致最小的几何失真。我们将把这个问题称为最优映射问题。找到最优映射问题的良好解决方案是几何处理和计算机视觉中许多应用的关键部分，包括：表面和体积域的参数化、形状匹配和形状分析。本文的第一个目标是提供对最优映射问题的独立阐述，并突出问题各个方面的相互关系。这包括问题的正式定义和相关的幺正几何测度，我们称之为畸变. 第二个目标是确定失真度量的新特性，并解释如何在实践中使用这些特性。我们的主要贡献是： (i) 最优映射问题的关键概念的形式化和并列，迄今为止尚未以统一的方式形式化；(ii) 提供对现有最佳映射方法的详细调查，包括介绍最近的优化算法和寻找网格之间的单射映射的方法；(iii) 提供关于几何失真实际方面的新理论发现，包括几何能量的多分辨率不变性和凸失真度量的表征。特别是，我们引入了一系列新的凸失真度量，并证明，在网格上，