In this work we describe the algorithms to construct the skeletons, simplified 1D representations for a 3D surface depicted by a mesh of points, given the respective eigenfunctions of the Discrete Laplace-Beltrami Ope...In this work we describe the algorithms to construct the skeletons, simplified 1D representations for a 3D surface depicted by a mesh of points, given the respective eigenfunctions of the Discrete Laplace-Beltrami Operator (LBO). These functions are isometry invariant, so they are independent of the object’s representation including parameterization, spatial position and orientation. Several works have shown that these eigenfunctions provide topological and geometrical information of the surfaces of interest [1] [2]. We propose to make use of that information for the construction of a set of skeletons, associated to each eigenfunction, which can be used as a fingerprint for the surface of interest. The main goal is to develop a classification system based on these skeletons, instead of the surfaces, for the analysis of medical images, for instance.展开更多
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
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
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
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
文摘In this work we describe the algorithms to construct the skeletons, simplified 1D representations for a 3D surface depicted by a mesh of points, given the respective eigenfunctions of the Discrete Laplace-Beltrami Operator (LBO). These functions are isometry invariant, so they are independent of the object’s representation including parameterization, spatial position and orientation. Several works have shown that these eigenfunctions provide topological and geometrical information of the surfaces of interest [1] [2]. We propose to make use of that information for the construction of a set of skeletons, associated to each eigenfunction, which can be used as a fingerprint for the surface of interest. The main goal is to develop a classification system based on these skeletons, instead of the surfaces, for the analysis of medical images, for instance.
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.
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)}
