arXiv subjects

Shu-Yung Liu

Publications and source records attributed to Shu-Yung Liu.

6 recordsLinked to original sources

arXiv2026

Area-Preserving Parameterization: Variational Principle, Gradient Flow, and Discrete Approximation

Area-preserving parameterizations are used in applications where relative surface areas must be preserved. We study this problem through the stretch energy. For orientation-preserving diffeomorphisms between compact Riemannian 2-manifolds of equal total area, we show that the stretch energy is characterized by the variance of the area ratio and that its critical points are area-preserving. This variational characterization leads naturally to an $L^2$-gradient flow, which we call the authalic flow. We then develop its simplicial counterpart based on the discrete stretch energy and obtain computational methods for open and closed surfaces of several topological types. To connect the discrete formulation with the smooth theory, we prove the first-order consistency of the stretch energy with respect to mesh refinement and establish a first-order $L^2$ area-distortion bound for discrete global minimizers under the stated geometric approximation assumptions. Numerical experiments on benchmark meshes produce fold-free maps in all reported tests and show competitive area preservation compared with existing methods.

math.NA↗

arXiv2026

Spherical Area-Preserving Parameterization via Energy Minimization

We propose a novel method, called spherical authalic energy minimization (SAEM), for computing spherical area-preserving parameterizations of genus-zero closed surfaces, with strong theoretical foundations. The global convergence of the associated computational algorithm is theoretically guaranteed. In addition, we introduce a Riemannian bijective correction method that ensures the bijectivity of the resulting mapping under mild assumptions. Numerical experiments show that SAEM effectively minimizes area distortion and achieves bijective mappings, outperforming state-of-the-art methods. Finally, we demonstrate the practical utility of SAEM in shape description.

math.NA↗

arXiv2025

Square-Domain Area-Preserving Parameterization for Genus-Zero and Genus-One Closed Surfaces

The parameterization of closed surfaces typically requires either multiple charts or a non-planar domain to achieve a seamless global mapping. In this paper, we propose a numerical framework for the seamless parameterization of genus-zero and genus-one closed simplicial surfaces onto a unit square domain. The process begins by slicing the surface with either the shortest-path or the Reeb graph method. The sliced surface is then mapped onto the unit square using a globally convergent algorithm that minimizes the weighted variance of per-triangle area ratios to achieve area preservation. Numerical experiments on benchmark models demonstrate that our method achieves high accuracy and efficiency. Furthermore, the proposed method enables applications such as geometry images, producing accurate and high-quality surface reconstructions.

math.NA↗

arXiv2024

Energy-Based Distortion-Balancing Parameterization for Open Surfaces

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.

math.NA↗

arXiv2024

Isovolumetric Energy Minimization for Ball-Shaped Volume-Preserving Parameterizations of 3-Manifolds

A volume-preserving parameterization is a bijective mapping that maps a 3-manifold onto a specified canonical domain that preserves the local volume. This paper formulates the computation of ball-shaped volume-preserving parameterizations as an isovolumetric energy minimization (IEM) problem with the boundary points constrained on a unit sphere. In addition, we develop a new preconditioned nonlinear conjugate gradient algorithm for solving the IEM problem with guaranteed theoretical convergence and significantly improved accuracy and computational efficiency compared to other state-of-the-art algorithms. Applications to solid shape registration and deformation are presented to highlight the usefulness of the proposed algorithm.

math.NA↗

arXiv2023

Convergent Authalic Energy Minimization for Disk Area-Preserving Parameterizations

An area-preserving parameterization is a bijective mapping that maps a surface onto a specified domain and preserves the local area. This paper formulates the computation of disk area-preserving parameterization as an authalic energy minimization (AEM) problem and proposes a novel preconditioned nonlinear conjugate gradient method for the AEM with guaranteed theoretical convergence. Numerical experiments indicate that our new approach has significantly improved area-preserving accuracy and computational efficiency compared to another state-of-the-art algorithm. Furthermore, we present an application of surface registration to illustrate the practical utility of area-preserving mappings as parameterizations of surfaces.

math.NA↗