Area-preserving mesh parameterization for poly-annulus surfaces based on optimal mass transportation

This work proposes a novel method for computing area-preserving parameterization for genus zero surfaces with multiple boundaries (poly-annuli), which is based on discrete optimal mass transportation and surface Ricci Flow. We first begin with a conformal mapping (which may greatly distort area) by Ricci Flow and then correct the area distortion using the mass transport procedure via a convex optimization. The method is intrinsic and stable, and the resulting parameterization preserves area element and minimizes angle distortion. Comparing with existing algorithms, our method is more general and flexible. It can handle surfaces with more complicated topology, and gives users full control of the target measure, such as the areas of the holes. We have tested the method for applications in various fields. Our experimental results demonstrate the efficiency and efficacy of the proposed method.