Mathematics, Computation and Geometry of Data

Volume 3 (2023)

Number 1

A two-stage algorithm for combined quasiconformal and optimal mass transportation spherical parameterization

Pages: 29 – 57



Zhiyuan Lyu (Chinese University of Hong Kong)

Qiguang Chen (Chinese University of Hong Kong)

Lok Ming Lui (Chinese University of Hong Kong)


This paper addresses the problem of surface parameterization. Surface parameterization aims to transform a general surface onto a simple parameter domain, so that surface processing can be easily carried out on the parameter domain. It has a vast variety of applications in various fields, such as computer graphics, computer visions and medical imaging. Depending on the application, surface parameterization is required to satisfy certain properties. Existing parameterization methods usually enforce one given property, such as conformal or angle-preserving, on the entire surface. In some practical applications, it may be more desirable to have a surface parameterization that satisfies different properties at different regions. We are therefore motivated to develop an effective and efficient algorithm to compute a multi-property surface parameterization, which satisfies different geometric properties at different regions of the surface. More specifically, we propose a method to compute the parameterization of a genus‑0 closed surface that is area-preserving in some regions and quasiconformal in the other regions. The obtained parameterization is more applicable for practical uses. Our proposed method is a two-stage algorithm that involves the computation of the optimal control tranportation map and the quasiconformal map. Using the sterographic projection, an optimal control transportation map can be computed in the two-dimensional plane through the Brenier approach. In the second stage, another stereographic projection is then used and a landmark-matching quasiconformal parameterization is computed in the 2D plane. The quasiconformal parameterization welds the parameterizations with different properties smoothly. Also, the computation is carried out in he 2D plane, which makes the computation efficient. Experiments on synthetic and real data have been carried out to validate the proposed method. Results show the efficacy of our proposed algorithm to compute the multi-property parameterization of general genus‑0 closed surfaces.

L.M. Lui is supported by HKRGC GRF (Reference: 14306721).

Received 12 July 2022

Published 16 May 2023