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Feature Preserving Parameterization for Quadrilateral Mesh Generation Based on ricci flow and Cross Field

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Computer Modeling in Engineering & Sciences 2023年第10期137卷 843-857页

作者： Na Lei Ping Zhang Xiaopeng Zheng Yiming Zhu Zhongxuan Luo International School of Information Science and Engineering Dalian University of TechnologyDalian116000China School of Mathematical Sciences Dalian University of TechnologyDalian116000China School of Software Technology Dalian University of TechnologyDalian116000China

We propose a newmethod to generate surface quadrilateralmesh by calculating a globally defined parameterization with feature constraints.In the field of quadrilateral generation with features,the cross field methods are wellknown because of their superior performance in feature preservation.The methods based on metrics are popular due to their sound theoretical basis,especially the ricci flow algorithm.The cross field methods’major part,the Poisson equation,is challenging to solve in three dimensions directly.When it comes to cases with a large number of elements,the computational costs are expensive while the methods based on metrics are on the contrary.In addition,an appropriate initial value plays a positive role in the solution of the Poisson equation,and this initial value can be obtained from the ricci flow algorithm.So we combine the methods based on metric with the cross field methods.We use the discrete dynamic ricci flow algorithm to generate an initial value for the Poisson equation,which speeds up the solution of the equation and ensures the convergence of the computation.Numerical experiments show that our method is effective in generating a quadrilateral mesh for models with features,and the quality of the quadrilateral mesh is reliable.

关键词： Quadrilateral mesh feature preserving ricci flow cross field

Eguchi–Hanson Singularities in U(2)-Invariant ricci flow

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Peking Mathematical Journal 2023年第1期6卷 1-141页

作者： Alexander Appleton Department of Mathematics UC BerkeleyBerkeleyCA 94720USA

We show that a ricci flow in four dimensions can develop singularities modeled on the Eguchi–Hanson space.In particular,we prove that starting from a class of asymptotically cylindrical U(2)-invariant initial metrics on T S^(2),a Type II singularity modeled on the Eguchi–Hanson space develops in finite time.Furthermore,we show that for these ricci flows the only possible blow-up limits are(i)the Eguchi–Hanson space,(ii)the flat R4/Z2 orbifold,(iii)the 4d Bryant soliton quotiented by Z2,and(iv)the shrinking cylinder R×RP^(3).As a byproduct of our work,we also prove the existence of a new family of Type II singularities caused by the collapse of a two-sphere of self-intersection|k|≥3.

关键词： ricci flow Eguchi–Hanson metric Singularity model Four dimensional Blow-up

Applying ricci flow to high dimensional manifold learning

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Science China(Information Sciences) 2019年第9期62卷 23-36页

作者： Yangyang LI Ruqian LU Key Lab of Management Decision and Information Systems Academy of Mathematics and Systems ScienceChinese Academy of Sciences University of Chinese Academy of Sciences

In machine learning, a high dimensional data set such as the digital image of a human face is often viewed as a point set distributed on a differentiable manifold. In many cases, the intrinsic dimension of this manifold is low but the representation dimension of the data points is high. To ease data processing requirements, manifold learning(ML) techniques can be used to reduce a high dimensional manifold(HDM)to a low dimensional one while keeping the essential geometric properties, such as relative distances between points, unchanged. Traditional ML algorithms often assume that the local neighborhood of any point on an HDM is roughly equal to the tangent space at that point. This assumption leads to the disadvantage that the neighborhoods of points on the manifold, though they have a very different curvature, will be treated equally and will be projected to a lower dimensional space. The curvature is a different way of manifold processing, where traditional dimension reduction is ineffective at preserving the neighborhood.To overcome this obstacle, we perform an "operation" on the HDM using ricci flow before a manifold’s dimension reduction. More precisely, with the ricci flow, we transform each local neighborhood of the HDM to a constant curvature patch. The HDM, as a whole, is then transformed into a subset of a sphere with constant positive curvature. We compare the proposed algorithm with other traditional manifold learning algorithms. Experimental results have shown that the proposed method outperforms other ML algorithms with a better neighborhood preserving rate.

关键词： manifold learning ricci flow ricci curvature dimension reduction curvature estimation

Survey on Discrete Surface ricci flow

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Journal of Computer Science & Technology 2015年第3期30卷 598-613页

作者：章敏曾薇郭任罗锋顾险峰 Department of Computer Science State University of New York at Stony Brook Stony Brook NY 11794-4400 U.S.A School of Computing and Infor Tnation Sciences Florida International University Miami Florida 33199 U.S.A. Department of Mathematics Oregon State University Corvallis OR 97331-4605 U.S.A. Department of Mathematics Rutgers University Piscataway NJ 08854 U.S.A.

ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a nonlinear heat diffusion process, and becomes constant eventually. ricci flow is a powerful computational tool to design Riemannian metrics by prescribed curvatures This work surveys the theory of discrete surface ricci flow registration and shape analysis. Surface ricci flow has been generalized to the discrete setting. its computational algorithms, and the applications for surface

关键词： ricci flow discrete Riemannian metric ricci energy uniformization theory

Injectivity radius bound of ricci flow with positive ricci curvature and applications

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Frontiers of Mathematics in China 2013年第5期8卷 1129-1137页

作者： Li MA Anqiang ZHU Zhongyuan Institute of Mathematics and Department of Mathematics Henan Normal University Xinxiang 453007 China Department of Mathematics Wuhan University Wuhan 430072 China

We study the injectivity radius bound for 3-d ricci flow with bounded curvature. As applications, we show the long time existence of the ricci flow with positive ricci curvature and with curvature decay condition at i... 详细信息

关键词： Injectivity radius bound ricci flow positive ricci curvature global solution

Relative volume comparison of ricci flow

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Science China Mathematics 2021年第9期64卷 1937-1950页

作者： Gang Tian Zhenlei Zhang School of Mathematical Sciences and Beijing International Center for Mathematical Research Peking UniversityBeijing 100871China School of Mathematical Sciences Capital Normal UniversityBeijing 100048China

In this paper we derive a relative volume comparison of ricci flow under a certain local curvature condition.It is a refinement of Perelman’s no local collapsing theorem in Perelman(2002).

关键词： ricci flow relative volume comparison local entropy

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The Harnack Estimate for a Nonlinear Parabolic Equation under the ricci flow

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Acta Mathematica Sinica,English Series 2011年第10期27卷 1935-1940页

作者： Song Bo HOU Department of Applied Mathematics College of Science East Campus China Agricultural University Beijing 100083 P. R. China

Let （M,g（t））, 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative ricci c for some constant C 〉 0 and g（t） evolving by the ricci flow curvature, │Rc│ ≤C/t for some constant C 〉 0 and g（t） evolving by the ricci flow gij/ t=-2Rij.In this paper, we derive a differential Harnack estimate for positive solutions to parabolic equations of the type u~ = /△u - aulogu - bu on M x （0,T], where a 〉 0 and b ∈ R.

关键词： Closed manifold ricci flow nonlinear parabolic equation Harnack estimate

On Gradient Solitons of the ricci–Harmonic flow

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Acta Mathematica Sinica,English Series 2015年第11期31卷 1798-1804页

作者： Hong Xin GUO Robert PHILIPOWSKI Anton THALMAIER School of Mathematics and Information Science Wenzhou University Mathematics Research Unit University of Luxembourg

In this paper, we study gradient solitons to the ricci flow coupled with harmonic map heat flow. We derive new identities on solitons similar to those on gradient solitons of the ricci flow. When the soliton is compac... 详细信息

关键词： ricci flow harmonic map heat flow gradient solitons quasi-Einstein manifold

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First Eigenvalue Monotonicity for the p-Laplace Operator under the ricci flow

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Acta Mathematica Sinica,English Series 2011年第8期27卷 1591-1598页

作者： Jia Yong WU Department of Mathematics Shanghai Maritime University Shanghai 201306 P. R. China

In this note, we discuss the monotonicity of the first eigenvalue of the p-Laplace operator （p ≥2） along the ricci flow on closed Riemannian manifolds. We prove that the first eigenvalue of the p-Laplace operator is nondecreasing along the ricci flow under some different curvature assumptions, and therefore extend some parts of Ma＇s results [Ann. Glob. Anal.Geom,29,287-292（2006）]

关键词： ricci flow first eigenvalue p-Laplace operator monotonicity