Shape Analysis System Based On Laplace-Beltrami Operator And Research On Some Properties Of Manifold Harmonic Basis

In recent years,the spectral method has been widely used in scientific computing visualization,machine learning,computer graphics and other fields because of its powerful ability in denoising,dimension reduction,and intrinsic feature extraction.The core idea of the spectral method is to solve the feature by solving the eigenmatrix of the three-dimensional graphics,and projecting the obtained eigenvalues and eigenvectors into the eigenspace for analysis or processing.The eigenvalues and eigenvectors obtained are the manifold harmonic basis of the object of this paper.In this paper,the key techniques of transforming the mesh model from geometric space to frequency space are studied.The shape analysis system based on Laplace-Beltrami operator is realized,and the properties of manifold harmonic basis are deeply studied.It has both theoretical significance and practical engineering value in network transmission and geometric compression.The main work and results of this paper are as follows:1.The shape analysis system based on Laplace-Beltrami operator is designed and implemented,which realizes the system function of converting the model file in tri and vert format into PLY format and calculating matrix eigenvalue and eigenvector.In this paper,the finite element modeling method is used to obtain the discretized form of Laplace-Beltrami operator.The eigenfunctions of Laplace operator are solved by finite element approximation and lumped mass matrix approximation,which simplifies the eigenproblem of matrix.The three-dimensional model correlation matrix of PLY format is calculated,and the stiffness matrix Q and mass matrix D are obtained.The eigenvalues and eigenvectors of the matrix are calculated by ARPACK++,and the manifold harmonic basis of the research object is obtained.2.Several properties of manifold harmonic basis are studied.The transformation of the mesh model from geometric space to frequency space is realized by manifold harmonic transform using manifold harmonic basis.By calculating the manifold harmonic transform and the inverse manifold harmonic transform,the relationship between the number m of the manifold harmonic basis functions and the reconstructed model and the original model vertex error is obtained.According to the size relationship of the reconstruction quantity in the model reconstruction process,the high and low frequency components in the three-dimensional model are distinguished,and three specific properties of the manifold harmonic basis are obtained,including:(1)The amount of reconstruction increases with the increase of m,which is an increasing process that is constantly approaching;(2)The reconstructed model must have errors with the original model,even if m takes the maximum value,the model can not be reconstructed completely;(3)The above properties of the manifold harmonic basis do not change with the model.