A Novel Double Periodic Conformal Flattening Algorithm for Genus-One Surfaces

Abstract

In this paper, we propose novel parameterization methods for genus-one surfaces, called the Double Periodic Conformal Flattening (DPCF) algorithm. The desired conformal map is obtained by minimizing a conformal energy functional under periodic boundary conditions, which is characterized as an easily solvable quadratic functional minimization problem, yielding a sparse linear system. The proposed DPCF algorithm offers several key advantages: (a) the optimal boundary and periodic translation vectors are obtained simultaneously with the conformal map; (b) the resulting map is independent of the chosen cutting path, thus introducing no extra conformal distortion near the cut seams; (c) bijectivity is guaranteed under the positive edge weights condition, which can be satisfied by, e.g., using an intrinsic Delaunay triangulation. Based on this guaranteeing, a simple strategy is employed to ensure bijectivity of the resulting maps for general triangulations. Numerical experiments illustrate that DPCF algorithm exhibits high accuracy and a 5 fold improvement over the state-of-the-art algorithm in terms of efficiency. Applications on texture mapping and medical imaging illustrate the practicality of our developed algorithm.