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Energy-Based Distortion-Balancing Parameterization for Open Surfaces

Shu-Yung Liu, Mei-Heng Yueh

TL;DR

The paper addresses balancing angle and area distortions in open surface parameterizations by formulating a constrained optimization that enforces $E_C(f)=E_A(f)$ and proving global convergence of the resulting augmented Lagrangian method. It develops a polar-coordinate reformulation for disk-shaped mappings (with a square-shaped variant), a preconditioned nonlinear conjugate gradient inner solver, and rigorous convergence guarantees to a KKT point. Numerical experiments on diverse triangular meshes show bijective parameterizations with balanced angular and area distortions and illustrate practical geometry-image applications, including a Brain reconstruction and a Bunny case with tunable area emphasis. The approach outperforms prior fixed-point schemes by delivering more uniform distortion balance across meshes, offering a flexible tool for geometry processing tasks where both local shape and area distributions matter.

Abstract

Surface parameterization is a fundamental concept in fields such as differential geometry and computer graphics. It involves mapping a surface in three-dimensional space onto a two-dimensional parameter space. This process allows for the systematic representation and manipulation of surfaces of complicated shapes by simplifying them into a manageable planar domain. In this paper, we propose a new iterative algorithm for computing the parameterization of simply connected open surfaces that achieves an optimal balance between angle and area distortions. We rigorously prove that the iteration in our algorithm converges globally, and numerical results demonstrate that the resulting mappings are bijective and effectively balance angular and area accuracy across various triangular meshes. Additionally, we present the practical usefulness of the proposed algorithm by applying it to represent surfaces as geometry images.

Energy-Based Distortion-Balancing Parameterization for Open Surfaces

TL;DR

The paper addresses balancing angle and area distortions in open surface parameterizations by formulating a constrained optimization that enforces and proving global convergence of the resulting augmented Lagrangian method. It develops a polar-coordinate reformulation for disk-shaped mappings (with a square-shaped variant), a preconditioned nonlinear conjugate gradient inner solver, and rigorous convergence guarantees to a KKT point. Numerical experiments on diverse triangular meshes show bijective parameterizations with balanced angular and area distortions and illustrate practical geometry-image applications, including a Brain reconstruction and a Bunny case with tunable area emphasis. The approach outperforms prior fixed-point schemes by delivering more uniform distortion balance across meshes, offering a flexible tool for geometry processing tasks where both local shape and area distributions matter.

Abstract

Surface parameterization is a fundamental concept in fields such as differential geometry and computer graphics. It involves mapping a surface in three-dimensional space onto a two-dimensional parameter space. This process allows for the systematic representation and manipulation of surfaces of complicated shapes by simplifying them into a manageable planar domain. In this paper, we propose a new iterative algorithm for computing the parameterization of simply connected open surfaces that achieves an optimal balance between angle and area distortions. We rigorously prove that the iteration in our algorithm converges globally, and numerical results demonstrate that the resulting mappings are bijective and effectively balance angular and area accuracy across various triangular meshes. Additionally, we present the practical usefulness of the proposed algorithm by applying it to represent surfaces as geometry images.

Paper Structure

This paper contains 21 sections, 3 theorems, 69 equations, 10 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contribution
  3. Notation
  4. Organization of the Paper
  5. Energy for Simplicial Mapping
  6. Conformal and Authalic Energy
  7. Augmented Lagrangian Method for Distortion-Balancing Parameterization
  8. Reformulation in Polar Coordinate
  9. Preconditioned Nonlinear CG Method
  10. Algorithm of Augmented Lagrangian Method
  11. Modification for Square-shaped
  12. Convergence Analysis
  13. Numerical Experiments
  14. Results of Propose Method
  15. Comparison to the Previous Method
  16. ...and 6 more sections

Key Result

Lemma 1

The Lagrange multiplier $\lambda^* = \max_\lambda \min_\mathbf{f} \mathcal{L}(\mathbf{f}, \lambda)$ is in $[0, 1]$.

Figures (10)

  • Figure 1: An illustration of the areas $|[v,v_{j_s},v_{k_s}]|$, $|[v_{i_s},v,v_{k_s}]|$ and $|[v_{i_s},v_{j_s},v]|$ in barycentric coordinates on a triangular face.
  • Figure 2: An illustration of the angle $\theta_{i,j}^k$ and $\theta_{i,j}^\ell$ defined on the surface $\mathcal{M}$.
  • Figure 3: The benchmark triangular mesh models and associated disk-shaped distortion-balancing parameterization produced by Algorithm \ref{['alg:AugLag']}.
  • Figure 4: The conformal \ref{['eq:Ec']} and authalic \ref{['eq:Ea']} energy during iterations by Algorithm \ref{['alg:AugLag']} among all benchmark triangular meshes.
  • Figure 5: The histogram of angular \ref{['eq:Angle_dist']} and area \ref{['eq:Area_dist']} distortion of the parameterization produced by Algorithm \ref{['alg:AugLag']} among all benchmark triangular meshes.
  • ...and 5 more figures

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof