Surface parameterization based on polar factorization

A mapping between measurable subsets of Euclidean space can be uniquely factorized to the composition of a measure-preserving mapping and an optimal transportation map, the later is also the gradient map of a convex function. This work introduces an algorithm based on variational approach to compute this type of polar factorization for mappings between planar domains and some direct applications. Our method greatly increases the flexibility for surface parameterizations by balancing between area distortion and angle distortion, and improves the accuracy and numerical stability for down steam geometric processing tasks.