A Novel Algorithm for Periodic Conformal Flattening of Genus-one and Multiply Connected Genus-zero Surfaces

TL;DR

This paper introduces a new algorithm for conformal flattening of genus-one and multiply connected genus-zero surfaces, emphasizing efficiency, accuracy, and bijectivity without dependence on cut path selection.

Contribution

The paper presents DPCF and SPCF algorithms that improve conformal flattening by ensuring bijectivity and independence from cut path choice, with significant efficiency gains.

Findings

Achieves 4-5 times faster performance than existing methods.

Ensures bijectivity for intrinsic Delaunay triangulations.

Demonstrates high accuracy in numerical experiments.

Abstract

In this paper, we propose a novel parameterization method for genus-one and multiply connected genus-zero surfaces, called periodic conformal flattening. The conformal energy minimization technique is utilized to compute the desired conformal map, which is characterised as an easily solvable quadratic functional minimization problem, yielding a sparse linear system. The advantages of the proposed algorithms DPCF and SPCF are a) independence from the cut path selection, which introduces no additional conformal distortion near the cut seams; b) bijectivity guaranteeing for intrinsic Delaunay triangulations. The numerical experiments illustrate that DPCF and SPCF express high accuracy and a 4-5 times improvement in terms of efficiency compared with state-of-the-art algorithms.Based on the theoretical proof of the bijectivity guaranteeing, a simple strategy is applied for to guarantee the bijectivity of the resulting maps for non-Delaunay triangulations. The application on texture mapping illustrates the practicality of our developed algorithms.