inversion-free mapping, Jacobian matrix, distortion, first-order methods, second-order methods
A geometric mapping establishes a correspondence between two domains. Since no real object has zero or negative volume, such a mapping is required to be inversion-free. Computing inversion-free mappings is a fundamental task in numerous computer graphics and geometric processing applications, such as deformation, texture mapping, mesh generation, and others. This task is usually formulated as a non-convex, nonlinear, constrained optimization problem. Various methods have been developed to solve this optimization problem. As well as being inversion-free, different applications have various further requirements. We expand the discussion in two directions to (i) problems imposing specific constraints and (ii) combinatorial problems. This report provides a systematic overview of inversion-free mapping construction, a detailed discussion of the construction methods, including their strengths and weaknesses, and a description of open problems in this research field.
Fu, Xiao-Ming; Su, Jian-Ping; Zhao, Zheng-Yu; Fang, Qing; Ye, Chunyang; and Liu, Ligang
"Inversion-free geometric mapping construction: A survey,"
Computational Visual Media: Vol. 7:
3, Article 2.
Available at: https://dc.tsinghuajournals.com/computational-visual-media/vol7/iss3/2