Riemannian gradient descent for spherical area-preserving mappings

Marco Sutti; Mei-Heng Yueh; Marco Sutti; Mei-Heng Yueh

doi:10.3934/math.2024946

AIMS Mathematics

2024, Volume 9, Issue 7: 19414-19445. doi: 10.3934/math.2024946

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Riemannian gradient descent for spherical area-preserving mappings

Marco Sutti ^{1
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Mei-Heng Yueh ^{2
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1.
Mathematics Division, National Center for Theoretical Sciences, Taipei, Taiwan
2.
Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan

Received: 18 March 2024 Revised: 25 May 2024 Accepted: 05 June 2024 Published: 12 June 2024
MSC : 68U05, 65K10, 65D18, 65D19

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 three-dimensional Euclidean space. Numerical experiments on several mesh models demonstrate the accuracy and stability of the proposed framework. Comparisons with three 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.
- stretch-energy functional,
- area-preserving mapping,
- Riemannian optimization,
- Riemannian gradient descent,
- matrix manifolds
Citation: Marco Sutti, Mei-Heng Yueh. Riemannian gradient descent for spherical area-preserving mappings[J]. AIMS Mathematics, 2024, 9(7): 19414-19445. doi: 10.3934/math.2024946

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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 three-dimensional Euclidean space. Numerical experiments on several mesh models demonstrate the accuracy and stability of the proposed framework. Comparisons with three 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.

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