A discrete uniformization theorem for polyhedral surfaces

Luo, Feng

doi:10.4310/jdg/1527040872

Cited by 84 publications

(168 citation statements)

References 17 publications

(42 reference statements)

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“…Some triangles degenerate along the flow such that the cotangent value in Equation (3) tends to infinity. Inspired by [Luo04], [GLS*18] and [GGL*18], we also incorporate the edge‐flipping surgery to keep M in Delaunay triangulation to avoid singularities along the discrete Calabi flow. See Figure 1, for edge e ij with opposite vertices v k and v l , edge‐flipping replaces e ij with a new e kl if θ k + θ l > π.…”

Section: Algorithmmentioning

confidence: 99%

Discrete Calabi Flow: A Unified Conformal Parameterization Method

Zhou

et al. 2019

Computer Graphics Forum

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Conformal parameterization for surfaces into various parameter domains is a fundamental task in computer graphics. Prior research on discrete Ricci flow provided us with promising inspirations from methods derived via Riemannian geometry, which is rigorous in theory and effective inpractice. In this paper, we propose a unified conformal parameterization approachfor turning triangle meshes into planar and spherical domains using discrete Calabi flow onpiecewise linear metric. We incorporate edge‐flipping surgery to guarantee convergence as well as other significant improvements including approximate Newton's method, optimal step‐lengths, priority embedding and boundary customizing, which achieve better performance and functionality with robustness and accuracy.

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Section: Algorithmmentioning

confidence: 99%

Discrete Calabi Flow: A Unified Conformal Parameterization Method

Zhou

et al. 2019

Computer Graphics Forum

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“…More specifically, we use dynamic Yamabe flow [15] to conformally map the segments onto the respective planar disks. Yamabe flow is a scheme of Ricci flow, which deforms the Riemannian metric proportional to the curvature, such that the curvature evolves according to a non-linear heat diffusion process and becomes constant everywhere.…”

Section: Our Approachmentioning

confidence: 99%

Spherical Parameterization Balancing Angle and Area Distortions

Nadeem

Zeng

et al. 2017

IEEE Trans. Visual. Comput. Graphics

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This work presents a novel framework for spherical mesh parameterization. An efficient angle-preserving spherical parameterization algorithm is introduced, which is based on dynamic Yamabe flow and the conformal welding method with solid theoretic foundation. An area-preserving spherical parameterization is also discussed, which is based on discrete optimal mass transport theory. Furthermore, a spherical parameterization algorithm, which is based on the polar decomposition method, balancing angle distortion and area distortion is presented. The algorithms are tested on 3D geometric data and the experiments demonstrate the efficiency and efficacy of the proposed methods.

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“…Theorem 2.4 (Discrete Uniformization [5]) Given a target curvature K̄ satisfying the Gauss-Bonnet condition in Eqn.4, and for each vertex K̄ i ∈ (−∞, 2 π ), then there exists a solution to the Ricci flow Eqn.8. The solution is unique up to a constant.…”

Section: Theoretic Backgroundmentioning

confidence: 99%

Intrinsic 3D Dynamic Surface Tracking based on Dynamic Ricci Flow and Teichmüller Map

Lei

Wang

et al. 2017

2017 IEEE International Conference on Computer Vision (ICCV)

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3D dynamic surface tracking is an important research problem and plays a vital role in many computer vision and medical imaging applications. However, it is still challenging to efficiently register surface sequences which has large deformations and strong noise. In this paper, we propose a novel automatic method for non-rigid 3D dynamic surface tracking with surface Ricci flow and Teichmüller map methods. According to quasi-conformal Teichmüller theory, the Techmüller map minimizes the maximal dilation so that our method is able to automatically register surfaces with large deformations. Besides, the adoption of Delaunay triangulation and quadrilateral meshes makes our method applicable to low quality meshes. In our work, the 3D dynamic surfaces are acquired by a high speed 3D scanner. We first identified sparse surface features using machine learning methods in the texture space. Then we assign landmark features with different curvature settings and the Riemannian metric of the surface is computed by the dynamic Ricci flow method, such that all the curvatures are concentrated on the feature points and the surface is flat everywhere else. The registration among frames is computed by the Teichmüller mappings, which aligns the feature points with least angle distortions. We apply our new method to multiple sequences of 3D facial surfaces with large expression deformations and compare them with two other state-of-the-art tracking methods. The effectiveness of our method is demonstrated by the clearly improved accuracy and efficiency.

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A discrete uniformization theorem for polyhedral surfaces

Cited by 84 publications

References 17 publications

Discrete Calabi Flow: A Unified Conformal Parameterization Method

Discrete Calabi Flow: A Unified Conformal Parameterization Method

Spherical Parameterization Balancing Angle and Area Distortions

Intrinsic 3D Dynamic Surface Tracking based on Dynamic Ricci Flow and Teichmüller Map

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