BPM: Blended Piecewise Moebius Maps

TL;DR

This paper introduces a new Moebius-based interpolator for discrete maps between triangle meshes, producing smooth, low-distortion mappings suitable for texture transfer and surface parameterization.

Contribution

The paper presents a closed-form, local Moebius interpolator that improves smoothness and reduces distortion in discrete mesh mappings, applicable to various mesh types.

Findings

Produces smoother texture transfer than existing methods

Achieves lower quasi-conformal distortion

Works effectively even on coarse triangulations

Abstract

We propose a novel Moebius interpolator that takes as an input a discrete map between the vertices of two planar triangle meshes, and outputs a smooth map on the input domain. The output map interpolates the discrete map, is continuous between triangles, and has low quasi-conformal distortion when the input map is discrete conformal. Our map leads to considerably smoother texture transfer compared to the alternatives, even on very coarse triangulations. Furthermore, our approach has a closed-form expression, is local, applicable to any discrete map, and leads to smooth results even for extreme deformations. Finally, by working with local intrinsic coordinates, our approach is easily generalizable to discrete maps between a surface triangle mesh and a planar mesh, i.e., a planar parameterization. We compare our method with existing approaches, and demonstrate better texture transfer results, and lower quasi-conformal errors.