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Xianfeng Gu

Publications and source records attributed to Xianfeng Gu.

44 records · Page 3Linked to original sources
arXiv2014

A discrete uniformization theorem for polyhedral surfaces II

A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss-Bonnet formula. Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery. In particular, each hyperbolic polyhedral metric on a closed surface with negative Euler characteristic is discrete conformal to a unique hyperbolic metric.

math.GT↗
arXiv2013

A discrete uniformization theorem for polyhedral surfaces

A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a discrete Yamabe flow with surgery.

math.GT↗
arXiv2013

A Variational Principle for Improving 2D Triangle Meshes based on Hyperbolic Volume

In this paper, we consider the problem of improving 2D triangle meshes tessellating planar regions. We propose a new variational principle for improving 2D triangle meshes where the energy functional is a convex function over the angle structures whose maximizer is unique and consists only of equilateral triangles. This energy functional is related to hyperbolic volume of ideal 3-simplex. Even with extra constraints on the angles for embedding the mesh into the plane and preserving the boundary, the energy functional remains well-behaved. We devise an efficient algorithm for maximizing the energy functional over these extra constraints. We apply our algorithm to various datasets and compare its performance with that of CVT. The experimental results show that our algorithm produces the meshes with both the angles and the aspect ratios of triangles lying in tighter intervals.

cs.CG↗
arXiv2013

Rigidity of Infinite Hexagonal Triangulation of the Plane

In the paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in [5]. Our result shows that if a geometric hexagonal triangulation of the plane is PL conformal to the regular hexagonal triangulation and all inner angles are in $[δ, π/2 -δ]$ for any constant $δ> 0$, then it is the regular hexagonal triangulation. This partially solves a conjecture of Luo [4]. The proof uses the concept of \emph{quasi-harmonic} functions to unfold the properties of the mesh.

math.GT↗
arXiv2012

A Conformal Approach for Surface Inpainting

We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the incomplete shape by filling in the surface holes is necessary. In this paper, we propose a novel conformal approach to inpaint surface holes on a Riemann surface based on its surface geometry. The basic idea is to represent the Riemann surface using its conformal factor and mean curvature. According to Riemann surface theory, a Riemann surface can be uniquely determined by its conformal factor and mean curvature up to a rigid motion. Given a Riemann surface $S$, its mean curvature $H$ and conformal factor $λ$ can be computed easily through its conformal parameterization. Conversely, given $λ$ and $H$, a Riemann surface can be uniquely reconstructed by solving the Gauss-Codazzi equation on the conformal parameter domain. Hence, the conformal factor and the mean curvature are two geometric quantities fully describing the surface. With this $λ$-$H$ representation of the surface, the problem of surface inpainting can be reduced to the problem of image inpainting of $λ$ and $H$ on the conformal parameter domain. Once $λ$ and $H$ are inpainted, a Riemann surface can be reconstructed which effectively restores the 3D surface with missing holes. Since the inpainting model is based on the geometric quantities $λ$ and $H$, the restored surface follows the surface geometric pattern. We test the proposed algorithm on synthetic data as well as real surface data. Experimental results show that our proposed method is an effective surface inpainting algorithm to fill in surface holes on an incomplete 3D models based their surface geometry.

math.DG↗
arXiv2012

Teichmüller extremal mapping and its applications to landmark matching registration

Registration, which aims to find an optimal 1-1 correspondence between shapes, is an important process in different research areas. Conformal mappings have been widely used to obtain a diffeomorphism between shapes that minimizes angular distortion. Conformal registrations are beneficial since it preserves the local geometry well. However, when landmark constraints are enforced, conformal mappings generally do not exist. This motivates us to look for a unique landmark matching quasi-conformal registration, which minimizes the conformality distortion. Under suitable condition on the landmark constraints, a unique diffeomporphism, called the Teichmüller extremal mapping between two surfaces can be obtained, which minimizes the maximal conformality distortion. In this paper, we propose an efficient iterative algorithm, called the Quasi-conformal (QC) iterations, to compute the Teichmüller mapping. The basic idea is to represent the set of diffeomorphisms using Beltrami coefficients (BCs), and look for an optimal BC associated to the desired Teichmüller mapping. The associated diffeomorphism can be efficiently reconstructed from the optimal BC using the Linear Beltrami Solver(LBS). Using BCs to represent diffeomorphisms guarantees the diffeomorphic property of the registration. Using our proposed method, the Teichmüller mapping can be accurately and efficiently computed within 10 seconds. The obtained registration is guaranteed to be bijective. The proposed algorithm can also be extended to compute Teichmüller mapping with soft landmark constraints. We applied the proposed algorithm to real applications, such as brain landmark matching registration, constrained texture mapping and human face registration. Experimental results shows that our method is both effective and efficient in computing a non-overlap landmark matching registration with least amount of conformality distortion.

cs.CG↗
arXiv2002

Computing Conformal Structure of Surfaces

This paper solves the problem of computing conformal structures of general 2-manifolds represented as triangle meshes. We compute conformal structures in the following way: first compute homology bases from simplicial complex structures, then construct dual cohomology bases and diffuse them to harmonic 1-forms. Next, we construct bases of holomorphic differentials. We then obtain period matrices by integrating holomorphic differentials along homology bases. We also study the global conformal mapping between genus zero surfaces and spheres, and between general meshes and planes. Our method of computing conformal structures can be applied to tackle fundamental problems in computer aid design and computer graphics, such as geometry classification and identification, and surface global parametrization.

cs.GR↗