Discrete Calabi Flow: A Unified Conformal Parameterization Method

Abstract

Conformal parameterization for surfaces into various parameter domains is a fundamental task in computer graphics. Prior research on discrete Ricci flow provided us with promising inspirations from methods derived via Riemannian geometry, which is rigorous in theory and effective inpractice. In this paper, we propose a unified conformal parameterization approachfor turning triangle meshes into planar and spherical domains using discrete Calabi flow onpiecewise linear metric. We incorporate edge‐flipping surgery to guarantee convergence as well as other significant improvements including approximate Newton's method, optimal step‐lengths, priority embedding and boundary customizing, which achieve better performance and functionality with robustness and accuracy.

Inversion-free geometric mapping construction: A survey

“…After the convergence of the intrinsic flows, we achieve a new length l i for each edge e i = v a v b to reconstruct a parameterized mesh. The first reconstruction method assembles the triangle one by one [12]; however, it accumulates the numerical error so that the resulting parameterized mesh breaks. To distribute the numerical errors evenly, a novel extrinsic shape optimization procedure is proposed in Ref.…”

“…where K( v i ) is the Gaussian curvature at v i and K t i is the target Gaussian curvature at v i . The Calabi energy can be minimized by the Calabi flow [12].…”

Texture Integrationvia Discrete Optimal Mass Transportation

“…任选网格 G 的某个三角形面元 f 作为起点, 广度优先遍历网格 G 中的所有面元, 并将遍历过的面 元和面元包含的边进行标记. 映射 [15] 、调和映射 [16] 、Ricci 流 [17] 和 Calabi 流 [18] 等. 本文实验中选择使用调和映射计算最优传输 的初始映射.…”