We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embedding approach for solving the surface eikonal eq...We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embedding approach for solving the surface eikonal equation.We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood.Our proposed algorithm is easy to implement and efficient.We will give some two-and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.展开更多
This contribution gives results on the action of the Laplace-Beltrami derivative on suffi- ciently smooth kernels on the sphere, those defined by absolutely and uniformly expansions generated by a family of at least c...This contribution gives results on the action of the Laplace-Beltrami derivative on suffi- ciently smooth kernels on the sphere, those defined by absolutely and uniformly expansions generated by a family of at least continuous functions. Among other things, the results show that convenient Laplace-Beltrami derivatives of positive definite kernels on the sphere are positive definite too. We also include similar results on the action of the Laplace-Beltrami derivative on condensed spherical harmonic expansions.展开更多
Non-rigid shape deformation without tearing or stretching is called isometry. There are many difficulties to research non-rigid shape in Euclidean space. Therefore, non-rigid shapes are firstly embedded into a none-Eu...Non-rigid shape deformation without tearing or stretching is called isometry. There are many difficulties to research non-rigid shape in Euclidean space. Therefore, non-rigid shapes are firstly embedded into a none-Euclidean space. Spectral space is chosen in this paper. Then three descriptors are proposed based on three spectral distances. The existence of zero-eigenvalue has negative effects on computation of spectral distance, Therefore the spectral distance should be computed from the first non-zcro-eigenvalue. Experiments show that spectral distance distributions are very effective to describe the non-rigid shapes.展开更多
Partial similarity of shapes is a challenging problem arising in many important applications in computer vision,shape analysis,and graphics,e.g.when one has to deal with partial information and acquisition artifacts.T...Partial similarity of shapes is a challenging problem arising in many important applications in computer vision,shape analysis,and graphics,e.g.when one has to deal with partial information and acquisition artifacts.The problem is especially hard when the underlying shapes are non-rigid and are given up to a deformation.Partial matching is usually approached by computing local descriptors on a pair of shapes and then establishing a point-wise non-bijective correspondence between the two,taking into account possibly different parts.In this paper,we introduce an alternative correspondence-less approach to matching fragments to an entire shape undergoing a non-rigid deformation.We use region-wise local descriptors and optimize over the integration domains on which the integral descriptors of the two parts match.The problem is regularized using the Mumford-Shah functional.We show an efficient discretization based on the Ambrosio-Tortorelli approximation generalized to triangular point clouds and meshes,and present experiments demonstrating the success of the proposed method.展开更多
Let M be an n-dimensional compact Riemannian manifold with or without boundary,and its Ricci curvature Ric M≥n-1.The paper obtains an inequality for the first eigenvalue η 1 of M with mixed boundary condition,whic...Let M be an n-dimensional compact Riemannian manifold with or without boundary,and its Ricci curvature Ric M≥n-1.The paper obtains an inequality for the first eigenvalue η 1 of M with mixed boundary condition,which is a generalization of the results of Lichnerowicz,Reilly,Escobar and Xia.It is also proved that η 1≥n for certain n-dimensional compact Riemannian manifolds with boundary,which is an extension of the work of Cheng,Li and Yau.展开更多
We consider optimal control problems of elliptic PDEs on hypersurfaces F in R^n for n = 2, 3. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhed...We consider optimal control problems of elliptic PDEs on hypersurfaces F in R^n for n = 2, 3. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhedral approximation of F. The discrete optimal control problem is formulated on the approximating surface and is solved numerically with a semi-smooth Newton algorithm. We derive optimal a priori error estimates for problems including control constraints and provide numerical examples confirming our analytical findings.展开更多
With the aid of Plancherel-Godement Theorem, we prove that every positive distributionT onSO (3, 1) which is bi-invariant underSO(3) corresponds to a measure μ on ω=∝σC|s(2-s)>=0∝, and μ can be decomposed int...With the aid of Plancherel-Godement Theorem, we prove that every positive distributionT onSO (3, 1) which is bi-invariant underSO(3) corresponds to a measure μ on ω=∝σC|s(2-s)>=0∝, and μ can be decomposed intoμ=μ 1+μ 2, whereμ 1 is a bounded measure on 0<=s<=2 andμ 2 is slowly increasing measure on (sχC|Re(s)=1)}展开更多
In this paper,a basic estimate for the conditional Riemannian Brownian motion on a complete manifold with non-negative Ricci curvature is established.Applying it to the heat kernel estimate of the operator 1/2△+b,we ...In this paper,a basic estimate for the conditional Riemannian Brownian motion on a complete manifold with non-negative Ricci curvature is established.Applying it to the heat kernel estimate of the operator 1/2△+b,we obtain the Aronson′s estimate for the operator 1/2△+b,which can be regarded as an extension of Peter Li and S.T.Yau's heat kernel estimate for the Laplace-Beltrami operator.展开更多
Partial differential equations(PDE)on manifolds arise in many areas,including mathematics and many applied fields.Due to the complicated geometrical structure of the manifold,it is difficult to get efficient numerical...Partial differential equations(PDE)on manifolds arise in many areas,including mathematics and many applied fields.Due to the complicated geometrical structure of the manifold,it is difficult to get efficient numerical method to solve PDE on manifold.In the paper,we propose a method called point integral method(PIM)to solve the Poisson-type equations from point clouds.Among different kinds of PDEs,the Poisson-type equations including the standard Poisson equation and the related eigenproblem of the Laplace-Beltrami operator are one of the most important.In PIM,the key idea is to derive the integral equations which approximates the Poisson-type equations and contains no derivatives but only the values of the unknown function.This featuremakes the integral equation easy to be discretized frompoint cloud.In the paper,we explain the derivation of the integral equations,describe the point integral method and its implementation,and present the numerical experiments to demonstrate the convergence of PIM.展开更多
Feature analysis plays a significant role in computer vision and computer graphics.In the task of shape retrieval,shape descriptor is indispensable.In recent years,feature extraction based on deep learning becomes ver...Feature analysis plays a significant role in computer vision and computer graphics.In the task of shape retrieval,shape descriptor is indispensable.In recent years,feature extraction based on deep learning becomes very popular,but the design of geometric shape descriptor is still meaningful due to the contained intrinsic information and interpretability.This paper proposes an effective and robust descriptor of 3D models.The descriptor is constructed based on the probability distribution of the normalized eigenfunctions of the Laplace–Beltrami operator on the surface,and a spectrum method for dimensionality reduction.The distance metric of the descriptor space is learned by utilizing the joint Bayesian model,and we introduce a matrix regularization in the training stage to re-estimate the covariance matrix.Finally,we apply the descriptor to 3D shape retrieval on a public benchmark.Experiments show that our method is robust and has good retrieval performance.展开更多
Riemannian流形和Khler流形上Laplace-Beltrami算子谱的下界的估计是微分几何研究领域的热点问题.针对LiS和Tran M A得到的关于Laplace-Beltrami算子谱的下界的估计,利用华罗庚先生和陆启铿先生关于有界对称典型域的研究结论,得出了...Riemannian流形和Khler流形上Laplace-Beltrami算子谱的下界的估计是微分几何研究领域的热点问题.针对LiS和Tran M A得到的关于Laplace-Beltrami算子谱的下界的估计,利用华罗庚先生和陆启铿先生关于有界对称典型域的研究结论,得出了第一类有界对称典型域上Laplace-Beltrami算子谱的下界估计.展开更多
基金supported in part by the Hong Kong RGC 16302223.
文摘We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embedding approach for solving the surface eikonal equation.We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood.Our proposed algorithm is easy to implement and efficient.We will give some two-and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.
基金supported by CAPES-Brasilsupported by FAPESP-Brasil(Grant No.2010/19734-6)
文摘This contribution gives results on the action of the Laplace-Beltrami derivative on suffi- ciently smooth kernels on the sphere, those defined by absolutely and uniformly expansions generated by a family of at least continuous functions. Among other things, the results show that convenient Laplace-Beltrami derivatives of positive definite kernels on the sphere are positive definite too. We also include similar results on the action of the Laplace-Beltrami derivative on condensed spherical harmonic expansions.
基金Partly Supported by NKBRPC(2004CB318006)NNSFC(60873164 and 60533090)
文摘Non-rigid shape deformation without tearing or stretching is called isometry. There are many difficulties to research non-rigid shape in Euclidean space. Therefore, non-rigid shapes are firstly embedded into a none-Euclidean space. Spectral space is chosen in this paper. Then three descriptors are proposed based on three spectral distances. The existence of zero-eigenvalue has negative effects on computation of spectral distance, Therefore the spectral distance should be computed from the first non-zcro-eigenvalue. Experiments show that spectral distance distributions are very effective to describe the non-rigid shapes.
基金The author would like to thank the referees for the helpful suggestionsThis work has been supported in part by the Israeli Science Foundation grant 615/11+1 种基金the German-Israeli Foundation grant 2269/2010and the Swiss High Performance and High Productivity Computing(HP2C)grant.
文摘Partial similarity of shapes is a challenging problem arising in many important applications in computer vision,shape analysis,and graphics,e.g.when one has to deal with partial information and acquisition artifacts.The problem is especially hard when the underlying shapes are non-rigid and are given up to a deformation.Partial matching is usually approached by computing local descriptors on a pair of shapes and then establishing a point-wise non-bijective correspondence between the two,taking into account possibly different parts.In this paper,we introduce an alternative correspondence-less approach to matching fragments to an entire shape undergoing a non-rigid deformation.We use region-wise local descriptors and optimize over the integration domains on which the integral descriptors of the two parts match.The problem is regularized using the Mumford-Shah functional.We show an efficient discretization based on the Ambrosio-Tortorelli approximation generalized to triangular point clouds and meshes,and present experiments demonstrating the success of the proposed method.
基金Research supported by the National Natural Science Foundation of China( 1 0 2 31 0 1 0 ) Trans- CenturyTraining Programme Foundation for Talents by the Ministry of Education of ChinaNatural ScienceFoundation of Zhejiang provinc
文摘Let M be an n-dimensional compact Riemannian manifold with or without boundary,and its Ricci curvature Ric M≥n-1.The paper obtains an inequality for the first eigenvalue η 1 of M with mixed boundary condition,which is a generalization of the results of Lichnerowicz,Reilly,Escobar and Xia.It is also proved that η 1≥n for certain n-dimensional compact Riemannian manifolds with boundary,which is an extension of the work of Cheng,Li and Yau.
文摘We consider optimal control problems of elliptic PDEs on hypersurfaces F in R^n for n = 2, 3. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhedral approximation of F. The discrete optimal control problem is formulated on the approximating surface and is solved numerically with a semi-smooth Newton algorithm. We derive optimal a priori error estimates for problems including control constraints and provide numerical examples confirming our analytical findings.
基金the National Natural Science F oundation of China (198710 65 ) and Hua Cheng Mathematics Science Foundation
文摘With the aid of Plancherel-Godement Theorem, we prove that every positive distributionT onSO (3, 1) which is bi-invariant underSO(3) corresponds to a measure μ on ω=∝σC|s(2-s)>=0∝, and μ can be decomposed intoμ=μ 1+μ 2, whereμ 1 is a bounded measure on 0<=s<=2 andμ 2 is slowly increasing measure on (sχC|Re(s)=1)}
文摘In this paper,a basic estimate for the conditional Riemannian Brownian motion on a complete manifold with non-negative Ricci curvature is established.Applying it to the heat kernel estimate of the operator 1/2△+b,we obtain the Aronson′s estimate for the operator 1/2△+b,which can be regarded as an extension of Peter Li and S.T.Yau's heat kernel estimate for the Laplace-Beltrami operator.
基金This research was partial supported by NSFC Grant(11201257 to Z.S.,11371220 to Z.S.and J.S.and 11271011 to J.S.)National Basic Research Program of China(973 Program 2012CB825500 to J.S.).
文摘Partial differential equations(PDE)on manifolds arise in many areas,including mathematics and many applied fields.Due to the complicated geometrical structure of the manifold,it is difficult to get efficient numerical method to solve PDE on manifold.In the paper,we propose a method called point integral method(PIM)to solve the Poisson-type equations from point clouds.Among different kinds of PDEs,the Poisson-type equations including the standard Poisson equation and the related eigenproblem of the Laplace-Beltrami operator are one of the most important.In PIM,the key idea is to derive the integral equations which approximates the Poisson-type equations and contains no derivatives but only the values of the unknown function.This featuremakes the integral equation easy to be discretized frompoint cloud.In the paper,we explain the derivation of the integral equations,describe the point integral method and its implementation,and present the numerical experiments to demonstrate the convergence of PIM.
基金the National Natural Science Foundation of China under Grant Nos.61872316,61932018.
文摘Feature analysis plays a significant role in computer vision and computer graphics.In the task of shape retrieval,shape descriptor is indispensable.In recent years,feature extraction based on deep learning becomes very popular,but the design of geometric shape descriptor is still meaningful due to the contained intrinsic information and interpretability.This paper proposes an effective and robust descriptor of 3D models.The descriptor is constructed based on the probability distribution of the normalized eigenfunctions of the Laplace–Beltrami operator on the surface,and a spectrum method for dimensionality reduction.The distance metric of the descriptor space is learned by utilizing the joint Bayesian model,and we introduce a matrix regularization in the training stage to re-estimate the covariance matrix.Finally,we apply the descriptor to 3D shape retrieval on a public benchmark.Experiments show that our method is robust and has good retrieval performance.
文摘Riemannian流形和Khler流形上Laplace-Beltrami算子谱的下界的估计是微分几何研究领域的热点问题.针对LiS和Tran M A得到的关于Laplace-Beltrami算子谱的下界的估计,利用华罗庚先生和陆启铿先生关于有界对称典型域的研究结论,得出了第一类有界对称典型域上Laplace-Beltrami算子谱的下界估计.
