Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Riemannian gradient descent for spherical area-preserving mappings

Marco Sutti, Mei-Heng Yueh

TL;DR

The paper addresses the challenge of computing spherical area-preserving parameterizations for genus-zero closed surfaces. It formulates the problem as constrained optimization on the power manifold $(S^{2})^{n}$ and solves it with a Riemannian gradient descent method that uses retractions and line-search to minimize the authalic energy $E_A(f)=E_S(f)-\mathcal{A}(f)$. The authors provide convergence guarantees, compare two line-search strategies, and demonstrate the method on multiple meshes, showing improved efficiency over state-of-the-art approaches and a practical landmark-aligned brain surface registration application. The work offers a robust and scalable alternative for geometry processing tasks requiring area-preserving spherical mappings with potential impact in computer graphics and medical imaging.

Abstract

We propose a new Riemannian gradient descent method for computing spherical area-preserving mappings of topological spheres using a Riemannian retraction-based framework with theoretically guaranteed convergence. The objective function is based on the stretch energy functional, and the minimization is constrained on a power manifold of unit spheres embedded in 3-dimensional Euclidean space. Numerical experiments on several mesh models demonstrate the accuracy and stability of the proposed framework. Comparisons with two existing state-of-the-art methods for computing area-preserving mappings demonstrate that our algorithm is both competitive and more efficient. Finally, we present a concrete application to the problem of landmark-aligned surface registration of two brain models.

Riemannian gradient descent for spherical area-preserving mappings

TL;DR

The paper addresses the challenge of computing spherical area-preserving parameterizations for genus-zero closed surfaces. It formulates the problem as constrained optimization on the power manifold and solves it with a Riemannian gradient descent method that uses retractions and line-search to minimize the authalic energy . The authors provide convergence guarantees, compare two line-search strategies, and demonstrate the method on multiple meshes, showing improved efficiency over state-of-the-art approaches and a practical landmark-aligned brain surface registration application. The work offers a robust and scalable alternative for geometry processing tasks requiring area-preserving spherical mappings with potential impact in computer graphics and medical imaging.

Abstract

We propose a new Riemannian gradient descent method for computing spherical area-preserving mappings of topological spheres using a Riemannian retraction-based framework with theoretically guaranteed convergence. The objective function is based on the stretch energy functional, and the minimization is constrained on a power manifold of unit spheres embedded in 3-dimensional Euclidean space. Numerical experiments on several mesh models demonstrate the accuracy and stability of the proposed framework. Comparisons with two existing state-of-the-art methods for computing area-preserving mappings demonstrate that our algorithm is both competitive and more efficient. Finally, we present a concrete application to the problem of landmark-aligned surface registration of two brain models.
Paper Structure (28 sections, 3 theorems, 76 equations, 11 figures, 8 tables, 2 algorithms)

This paper contains 28 sections, 3 theorems, 76 equations, 11 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline of the paper
  4. Notation
  5. Simplicial surfaces, mappings, and objective function
  6. Simplicial surfaces and mappings
  7. The objective function
  8. Riemannian optimization framework
  9. Geometry
  10. Unit sphere
  11. The power manifold TEXT
  12. Projection onto the tangent space
  13. Projection onto TEXT
  14. Retraction onto TEXT
  15. Riemannian gradient descent method
  16. ...and 13 more sections

Key Result

Proposition 2.1

The gradient of ${\mathcal{A}}$ can be explicitly formulated as

Figures (11)

  • Figure 1: An illustration of the cotangent weight defined on the image of $f$.
  • Figure 2: An illustration of the retraction mapping on the unit sphere $S^{2}$.
  • Figure 3: Illustration of the difference between Euclidean and Riemannian gradient.
  • Figure 4: The benchmark triangular mesh models used in this paper.
  • Figure 5: Convergence of the authalic energy for the benchmark mesh models, with the RGD method for minimizing the (normalized) stretch energy $E_{S}$. Line-search strategy: fminbnd.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Proposition 2.1: Formula for $\nabla{\mathcal{A}}$
  • proof
  • Theorem 5.1: AMS:2008
  • Remark 5.1
  • Theorem 5.2: AMS:2008