Variational principles for Minkowski type problems, discrete optimal transport, and discrete Monge–Ampère equations

Gu, Xianfeng; Luo, Feng; Sun, Jian; Yau, Shing–Tung

doi:10.4310/ajm.2016.v20.n2.a7

Cited by 95 publications

(106 citation statements)

References 5 publications

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“…The proof of Theorem 3.3 is reported in [4]. With this theory, the global minimum can be obtained efficiently using Newton's method due to the convexity of the energy.…”

Section: Theorem 33 (Discrete Optimal Mass Transport [4])mentioning

confidence: 94%

“…We refer readers to [23] for Kantorovich's approach, [27] and [30] for Breniner's approach, [4] for more detailed proofs of the proposed method.…”

Section: Theoretic Backgroundmentioning

confidence: 99%

“…Brenier proved polar factorization in [27] using probability measures, his theory encompasses both the continuous, semi-discrete and discrete settings. Recently, Gu et al [4] gave a novel proof for the existence and uniqueness based on the variational principle, which formulates the optimal mass transportation problem as a convex optimization, the Hessian of the energy has explicit geometric meanings.…”

Section: Discrete Optimal Mass Transportmentioning

confidence: 99%

“…In our framework, we take MongeBrenier's approach. However, our work is based on the newly discovered variational principle [4] which is the underspinning of Monge-Brenier's approach. Our framework is general and works with any valid measures, µ and ν, defined on two surfaces.…”

Section: Optimal Mass Transportationmentioning

confidence: 99%

“…This work proposes a novel algorithm to compute volumetric parameterization with a prescribed measure for a simply connected tetrahedral mesh with a single boundary surface, furthermore the method is capable of handling solids with voids inside. The algorithm is based on the recently developed discrete optimal mass transportation theory [4], which is rigorous and easy to implement. This work mainly focuses on the algorithmic design and implementation.…”

Section: Introductionmentioning

confidence: 99%

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Measure controllable volumetric mesh parameterization

Chen

Lei

et al. 2016

Computer-Aided Design

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Volumetric parameterization is a fundamental problem in solid and physical modeling. In practice, it is highly desirable to control the volumes of the regions of interests in the parameter domain. This work introduces a novel volumetric parameterization method, which allows users to prescribe the target volumetric measure of the input solid.Given a simply connected tetrahedral mesh with a single boundary surface, we first compute a volumetric harmonic map to parameterize the solid onto the unit solid ball; then we compute an optimal mass transportation map from the unit solid ball with the push-forward volume element induced by the harmonic map onto the parameter domain with the user prescribed volumetric measure. The composition of the volumetric harmonic map and the optimal mass transportation map gives a measure controllable volumetric parameterization. Furthermore, this method can handle solids with empty voids inside.The method has solid theoretic foundation, and is based on conventional algorithms in computational geometry, and easy to implement. The experimental results demonstrate the efficiency and efficacy of the proposed method.

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“…The proof of Theorem 3.3 is reported in [4]. With this theory, the global minimum can be obtained efficiently using Newton's method due to the convexity of the energy.…”

Section: Theorem 33 (Discrete Optimal Mass Transport [4])mentioning

confidence: 94%

“…We refer readers to [23] for Kantorovich's approach, [27] and [30] for Breniner's approach, [4] for more detailed proofs of the proposed method.…”

Section: Theoretic Backgroundmentioning

confidence: 99%

Section: Discrete Optimal Mass Transportmentioning

confidence: 99%

Section: Optimal Mass Transportationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Measure controllable volumetric mesh parameterization

Chen

Lei

et al. 2016

Computer-Aided Design

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A 3D face registration algorithm based on conformal mapping

Qian

Zhang

et al. 2018

Concurrency and Computation

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Summary Recently, 3D facial datasets are more easily available, and at the same time, the research of 3D face becomes more and more important. One of the most important research fields is 3D face registration, which plays an important role in face recognition, face shape analysis, and face animation. However, one of the most challenging issues in 3D face registration is to obtain a unique mapping for faces with different expression and landmark constraints. In this paper, we propose a novel conformal mapping algorithm to deal with the 3D face registration. Besides, the calculation is about harmonic energy, which makes our method applicable to low‐quality meshes. We begin with a harmonic mapping, then minimize the harmonic energy by a specific boundary condition on surfaces, and to obtain the conformal mapping, finally, we use a landmark‐constrained surface registration algorithm to register faces. Numerical experiments on various surfaces demonstrate the efficiency and robustness of our method.

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Variational Wasserstein Clustering

Liang

Zhang

et al. 2018

Computer Vision – ECCV 2018

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We propose a new clustering method based on optimal transportation. We discuss the connection between optimal transportation and k-means clustering, solve optimal transportation with the variational principle, and investigate the use of power diagrams as transportation plans for aggregating arbitrary domains into a fixed number of clusters. We drive cluster centroids through the target domain while maintaining the minimum clustering energy by adjusting the power diagram. Thus, we simultaneously pursue clustering and the Wasserstein distance between the centroids and the target domain, resulting in a measure-preserving mapping. We demonstrate the use of our method in domain adaptation, remeshing, and learning representations on synthetic and real data.

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Variational principles for Minkowski type problems, discrete optimal transport, and discrete Monge–Ampère equations

Cited by 95 publications

References 5 publications

Measure controllable volumetric mesh parameterization

Measure controllable volumetric mesh parameterization

A 3D face registration algorithm based on conformal mapping

Variational Wasserstein Clustering

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