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Spherical Authalic Energy Minimization for Area-Preserving Parameterization

Shu-Yung Liu, Mei-Heng Yueh

TL;DR

The paper tackles the problem of spherical area-preserving parameterization for genus-zero surfaces by introducing the spherical authalic energy $E_\mathbb{A}$ and a reformulation in spherical coordinates. It presents a preconditioned nonlinear conjugate gradient method with a global convergence guarantee to minimize $E_\mathbb{A}$, along with a Riemannian bijective correction as a robust post-processing step. The authors demonstrate that minimizing $E_\mathbb{A}$ significantly reduces folding compared to the original authalic energy $E_A$, and that the bijective correction can resolve hundreds of folding triangles in challenging meshes. Numerical experiments show superior efficiency and area-preservation accuracy relative to state-of-the-art methods such as SDEM and RGD, highlighting the practical impact for high-fidelity spherical mappings in applications like brain morphometry.

Abstract

We propose a new effective method called spherical authalic energy minimization (SAEM) for computing spherical area-preserving parameterizations of genus-zero surfaces. The proposed SAEM has solid theoretical support and guaranteed convergence. In addition, we develop a Riemannian bijective correction method to ensure the bijectivity of the produced mapping under mild assumptions. Numerical experiments showed that the SAEM effectively minimized area distortion with improved bijectivity compared to other state-of-the-art methods.

Spherical Authalic Energy Minimization for Area-Preserving Parameterization

TL;DR

The paper tackles the problem of spherical area-preserving parameterization for genus-zero surfaces by introducing the spherical authalic energy and a reformulation in spherical coordinates. It presents a preconditioned nonlinear conjugate gradient method with a global convergence guarantee to minimize , along with a Riemannian bijective correction as a robust post-processing step. The authors demonstrate that minimizing significantly reduces folding compared to the original authalic energy , and that the bijective correction can resolve hundreds of folding triangles in challenging meshes. Numerical experiments show superior efficiency and area-preservation accuracy relative to state-of-the-art methods such as SDEM and RGD, highlighting the practical impact for high-fidelity spherical mappings in applications like brain morphometry.

Abstract

We propose a new effective method called spherical authalic energy minimization (SAEM) for computing spherical area-preserving parameterizations of genus-zero surfaces. The proposed SAEM has solid theoretical support and guaranteed convergence. In addition, we develop a Riemannian bijective correction method to ensure the bijectivity of the produced mapping under mild assumptions. Numerical experiments showed that the SAEM effectively minimized area distortion with improved bijectivity compared to other state-of-the-art methods.

Paper Structure

This paper contains 21 sections, 3 theorems, 52 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Mathematical background
  4. Simplicial mapping
  5. Authalic mapping
  6. Authalic energy functional
  7. Objective energy functional
  8. Spherical authalic energy functional
  9. Reformulation with spherical coordinate
  10. Preconditioned nonlinear CG method
  11. Algorithmic procedure
  12. Global convergence
  13. Riemannian bijective correction
  14. Numerical experiments
  15. Spherical authalic energy and bijective correction
  16. ...and 6 more sections

Key Result

Theorem 1

Given a simplicial surface $\mathcal{M}$. Let $f: \mathcal{M} \rightarrow \mathbb S^2$ be an orientation-preserving simplicial map and $\varepsilon_f$ be the maximal edge length of $f(\mathcal{M})$. The spherical authalic energy eq:spherical Ea satisfies the estimate where authalic energy $E_A$ and stretch energy $E_S$ are defined as eq:Ea and eq:Es, respectively. Moreover, if $f$ is an area-pres

Figures (8)

  • Figure 1: An illustration of the barycentric coordinates on a triangular face.
  • Figure 2: The illustration of the tetrahedron by $[o, f_i, f_j, f_k]$ constructed by the triangle $[f_i, f_j, f_k]$ and the origin $o$.
  • Figure 3: An illustration of the cotangent weight defined on the surface $f(\mathcal{M})$.
  • Figure 4: An illustration for the tetrahedron $[o, \bf_i, \bf_j, \bf_k]$ formed by the face $[\bf_i, \bf_j, \bf_k]$ and the origin $o$ of $\mathbb{R}^3$.
  • Figure 5: An illustration of the mean value weight Floa03b defined on the surface $f(\mathcal{M})$.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof