arXiv subjects

Zhong-Heng Tan

Publications and source records attributed to Zhong-Heng Tan.

7 recordsLinked to original sources

arXiv2026

A Novel Double Periodic Conformal Flattening Algorithm for Genus-One Surfaces

In this paper, we propose novel parameterization methods for genus-one surfaces, called the Double Periodic Conformal Flattening (DPCF) algorithm. The desired conformal map is obtained by minimizing a conformal energy functional under periodic boundary conditions, which is characterized as an easily solvable quadratic functional minimization problem, yielding a sparse linear system. The proposed DPCF algorithm offers several key advantages: (a) the optimal boundary and periodic translation vectors are obtained simultaneously with the conformal map; (b) the resulting map is independent of the chosen cutting path, thus introducing no extra conformal distortion near the cut seams; (c) bijectivity is guaranteed under the positive edge weights condition, which can be satisfied by, e.g., using an intrinsic Delaunay triangulation. Based on this guaranteeing, a simple strategy is employed to ensure bijectivity of the resulting maps for general triangulations. Numerical experiments illustrate that DPCF algorithm exhibits high accuracy and a 5 fold improvement over the state-of-the-art algorithm in terms of efficiency. Applications on texture mapping and medical imaging illustrate the practicality of our developed algorithm.

math.NA↗

arXiv2026

A Novel Bijective Angle and Volume-preservation Balanced Parameterization for $n$-dimensional Manifolds

We propose a unified framework for balanced and bijective parameterizations of $n$-dimensional manifolds. The proposed energy combines conformal and volume-preserving terms to control both local anisotropy and volumetric distortion. At the continuous level, both energies are nonnegative and their zero-energy mappings are characterized. At the discrete level, the conformal, volume-preserving, and logarithmic barrier energies are formulated on oriented simplicial manifolds. A key result is that all their gradients admit a unified cotangent Laplacian-type representation, enabling sparse and dimension-independent computation. Bijectivity is enforced through signed simplex Jacobians, feasibility restoration, and a strictly orientation-preserving logarithmic barrier. The framework applies uniformly to spherical boundary parameterizations and parameterizations of discrete $n$-manifolds onto ball-like canonical domains.

math.NA↗

arXiv2025

Rapid general Electromagnetic Analysis with computational conformal geometry via Conformal Energy Minimization

We recently found that the electromagnetic scattering problem can be very fast in an approach expressing the fields in terms of orthonormal basis functions. In this paper we apply computational conformal geometry with the conformal energy minimization (CEM) algorithm to make possible fast solution of finite-frequency electromagnetic problems involving arbitrarily shaped, simply-connected metallic surfaces. The CEM algorithm computes conformal maps with minimal angular distortion, enabling the transformation of arbitrary simply-connected surfaces into a disk, where orthogonal basis functions can be defined and electromagnetic analysis can be significantly simplified. We demonstrate the effectiveness and efficiency of our method by investigating the resonance characteristics of two metallic surfaces: a square plate and a four-petal plate. Compared to traditional finite element methods (e.g., COMSOL), our approach achieves a three-order-of-magnitude improvement in computational efficiency, requiring only seconds to extract resonant frequencies and fields. Moreover, it reveals low-energy, doubly degenerate resonance modes that are elusive to conventional methods. These findings not only provide a powerful tool for analyzing electromagnetic fields on complex geometries but also pave the way for the design of high-performance electromagnetic devices.

physics.optics↗

arXiv2024

$n$-Dimensional Volumetric Stretch Energy Minimization for Volume-/Mass-Preserving Parameterizations

In this paper, we develop an $n$ dimensional volumetric stretch energy ($n$-VSE) functional for the volume-/mass-preserving parameterization of the $n$-manifolds topologically equivalent to $n$-ball. The $n$-VSE has a lower bound and equal to it if and only if the map is volume-/mass-preserving. This motivates us to minimize the $n$-VSE to achieve the ideal volume-/mass-preserving parameterization. In the discrete case, we also guarantee the relation between the lower bound and the volume-/mass-preservation, and propose the spherical and ball volume-/mass-preserving parameterization algorithms. The numerical experiments indicate the accuracy and robustness of the proposed algorithms. The modified algorithms are applied to the manifold registration and deformation, showing the versatility of $n$-VSE.

math.NA↗

arXiv2023

A Robust Hessian-based Trust Region Algorithm for Spherical Conformal Parameterizations

Surface parameterizations are widely applied in computer graphics, medical imaging and transformation optics. In this paper, we rigorously derive the gradient vector and Hessian matrix of the discrete conformal energy for spherical conformal parameterizations of simply connected closed surfaces of genus-$0$. In addition, we give the sparsity structure of the Hessian matrix, which leads to a robust Hessian-based trust region algorithm for the computation of spherical conformal maps. Numerical experiments demonstrate the local quadratic convergence of the proposed algorithm with low conformal distortions. We subsequently propose an application of our method to surface registrations that still maintains local quadratic convergence.

math.NA↗

arXiv2022

Convergence Analysis of Discrete Conformal Transformation

Continuous conformal transformation minimizes the conformal energy. The convergence of minimizing discrete conformal energy when the discrete mesh size tends to zero is an open problem. This paper addresses this problem via a careful error analysis of the discrete conformal energy. Under a weak condition on triangulation, the discrete function minimizing the discrete conformal energy converges to the continuous conformal mapping as the mesh size tends to zero.

math.NA↗

arXiv2022

Stable Discrete Minimization of Conformal Energy for Disk Conformal Parameterization

Conformal energy minimization is an efficient approach to compute conformal parameterization. In this paper, we develop a stable algorithm to compute conformal parameterization of simply connected open surface, termed Stable Discrete Minimization of Conformal Energy (SDMCE). The stability of SDMCE is reflected in the guarantee of one-to-one and on-to property of computed parameterization and the insensitivity on the initial value. On one hand, SDMCE can avoid degeneration and overlap of solution, also, SDMCE is folding free. On the other hand, even if given poor initial value, it can still correct it in very little computational time. The numerical experiments indicate SDMCE is stable and competitive with state-of-the-art algorithms in efficiency.

math.NA↗