arXiv subjects

Mei-Heng Yueh

Publications and source records attributed to Mei-Heng Yueh.

16 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↗

arXiv2026

DiffRGD: An Inference-Time Diffusion Guidance Through Riemannian Gradient Descent

Recently, diffusion models have been widely adopted in generative modeling and have served as foundational models for many image generation tasks. To control the generation without costly re-training or fine-tuning, many works seek inference-time guidance methods to steer the latent via a differentiable objective at inference time. However, these methods cannot effectively preserve the original Gaussian distribution because they introduce distributional drift, thereby degrading the sample quality. To address this gap, we propose DiffRGD, a distribution-aware guidance framework that explicitly preserves the latent Gaussian structure. DiffRGD formulates each sampling step as a constrained optimization problem on a spherical manifold induced by the latent Gaussian distribution, and solves it efficiently via Riemannian Gradient Descent (RGD). DiffRGD is a plug-and-play method that can be seamlessly integrated into any pre-trained diffusion model. Extensive experiments demonstrate that DiffRGD outperforms previous methods in most image restoration and conditional generation tasks. Our project page is available at https://diffrgd.github.io/.

cs.CV↗

arXiv2026

An Improved Variational Method for Image Denoising

The total variation (TV) method is an image denoising technique that aims to reduce noise by minimizing the total variation of the image, which measures the variation in pixel intensities. The TV method has been widely applied in image processing and computer vision for its ability to preserve edges and enhance image quality. In this paper, we propose a Mixed-norm TV (MixTV) model for image denoising and the associated numerical algorithm to carry out the procedure, which is particularly effective in removing several types of noise and their combinations. Our MixTV admits a unique solution and the associated numerical algorithm guarantees convergence. Numerical experiments are demonstrated to show improved effectiveness and denoising quality compared to other TV models. Such encouraging results further enhance the utility of the TV method in image processing. Our project page is available at https://jing-en-huang.github.io/MixTV.

cs.CV↗

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↗

arXiv2025

Toroidal area-preserving parameterizations of genus-one closed surfaces

We consider the problem of computing toroidal area-preserving parameterizations of genus-one closed surfaces. We propose four algorithms based on Riemannian geometry: the projected gradient descent method, the projected conjugate gradient method, the Riemannian gradient method, and the Riemannian conjugate gradient method. Our objective function is based on the stretch energy functional, and the minimization is constrained on a power manifold of ring tori embedded in three-dimensional Euclidean space. Numerical experiments on several mesh models demonstrate the effectiveness of the proposed framework. Finally, we show how to use the proposed algorithms in the context of surface registration and texture mapping applications.

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↗

arXiv2024

Riemannian gradient descent for spherical area-preserving mappings

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.

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↗

arXiv2022

Convergence Analysis of Volumetric Stretch Energy Minimization and its Associated Optimal Mass Transport

The volumetric stretch energy has been widely applied to the computation of volume-/mass-preserving parameterizations of simply connected tetrahedral mesh models. However, this approach still lacks theoretical support. In this paper, we provide the theoretical foundation for volumetric stretch energy minimization (VSEM) to compute volume-/mass-preserving parameterizations. In addition, we develop an associated efficient VSEM algorithm with guaranteed asymptotic R-linear convergence. Furthermore, based on the VSEM algorithm, we propose a projected gradient method for the computation of the volume/mass-preserving optimal mass transport map with a guaranteed convergence rate of $\mathcal{O}(1/m)$, and combined with Nesterov-based acceleration, the guaranteed convergence rate becomes $\mathcal{O}(1/m^2)$. Numerical experiments are presented to justify the theoretical convergence behavior for various examples drawn from known benchmark models. Moreover, these numerical experiments show the effectiveness and accuracy of the proposed algorithm, particularly in the processing of 3D medical MRI brain images.

math.NA↗

arXiv2022

Convergence Analysis of Dirichlet Energy Minimization for Spherical Conformal Parameterizations

In this paper, we first derive a theoretical basis for spherical conformal parameterizations between a simply connected closed surface $\mathcal{S}$ and a unit sphere $\mathbb{S}^2$ by minimizing the Dirichlet energy on $\overline{\mathbb{C}}$ by stereographic projection. The Dirichlet energy can be rewritten as the sum of the energies associated with the southern and northern hemispheres and can be decreased under an equivalence relation by alternatingly solving the corresponding Laplacian equations. Based on this theoretical foundation, we develop a modified Dirichlet energy minimization with nonequivalence deflation for the computation of the spherical conformal parameterization between $\mathcal{S}$ and $\mathbb{S}^2$. In addition, under some mild conditions, we verify the asymptotically R-linear convergence of the proposed algorithm. Numerical experiments on various benchmarks confirm that the assumptions for convergence always hold and indicate the efficiency, reliability and robustness of the developed modified Dirichlet energy minimization.

math.NA↗

arXiv2022

Theoretical Foundation of the Stretch Energy Minimization for Area-Preserving Mappings

The stretch energy is a fully nonlinear energy functional that has been applied to the numerical computation of area-preserving mappings. However, this approach lacks theoretical support and the analysis is complicated due to the full nonlinearity of the functional. In this paper, we provide a theoretical foundation of the stretch energy minimization for the computation of area-preserving mappings, including a neat formulation of the gradient of the functional, and the proof of the minimizers of the functional being area-preserving mappings. In addition, the geometric interpretation of the stretch energy is also provided to better understand this energy functional. Furthermore, numerical experiments are demonstrated to validate the effectiveness and accuracy of the stretch energy minimization for the computation of square-shaped area-preserving mappings of simplicial surfaces.

math.NA↗

arXiv2017

A Novel Stretch Energy Minimization Algorithm for Equiareal Parameterizations

Surface parameterizations have been widely applied to computer graphics and digital geometry processing. In this paper, we propose a novel stretch energy minimization (SEM) algorithm for the computation of equiareal parameterizations of simply connected open surfaces with a very small area distortion and a highly improved computational efficiency. In addition, the existence of nontrivial limit points of the SEM algorithm is guaranteed under some mild assumptions of the mesh quality. Numerical experiments indicate that the efficiency, accuracy, and robustness of the proposed SEM algorithm outperform other state-of-the-art algorithms. Applications of the SEM on surface remeshing and surface registration for simply connected open surfaces are demonstrated thereafter. Thanks to the SEM algorithm, the computations for these applications can be carried out efficiently and robustly.

cs.GR↗

arXiv2015

Graph invariants from ideas in physics and number theory

We study free scalar field theory on a graph, which gives rise to a modified version of discrete Green's function on a graph studied in \cite{CY}. We show that this gives rise to a graph invariant, which is closely related to the 2-dim Weisfeiler-Lehman algorithm for graph isomorphism testing. We complement this invariant by another type of graph invariants, coming from viewing graphs as quadratic forms over the integers. We explain that the combination of these two ideas give rise to an interesting approach to the graph isomorphism problem.

math.CO↗

arXiv2015

Conformal Surface Morphing with Applications on Facial Expressions

Morphing is the process of changing one figure into another. Some numerical methods of 3D surface morphing by deformable modeling and conformal mapping are shown in this study. It is well known that there exists a unique Riemann conformal mapping from a simply connected surface into a unit disk by the Riemann mapping theorem. The dilation and relative orientations of the 3D surfaces can be linked through the Möbius transformation due to the conformal characteristic of the Riemann mapping. On the other hand, a 3D surface deformable model can be built via various approaches such as mutual parameterization from direct interpolation or surface matching using landmarks. In this paper, we take the advantage of the unique representation of 3D surfaces by the mean curvatures and the conformal factors associated with the Riemann mapping. By registering the landmarks on the conformal parametric domains, the correspondence of the mean curvatures and the conformal factors for each surfaces can be obtained. As a result, we can construct the 3D deformation field from the surface reconstruction algorithm proposed by Gu and Yau. Furthermore, by composition of the Möbius transformation and the 3D deformation field, the morphing sequence can be generated from the mean curvatures and the conformal factors on a unified mesh structure by using the cubic spline homotopy. Several numerical experiments of the face morphing are presented to demonstrate the robustness of our approach.

cs.GR↗