Study On Generalized Barycentric Coordinates Based On Sparse Combination And Its Optimization

Because the generalized barycentric coordinate is not unique to the sequence of points,this paper calculates and optimizes the generalized barycentric coordinate based on sparse combination.A method is given for calculating a few points of barycentic coordinate combination to obtain the generalized barycenter coordinate.Two different types of weighted functions with parameters are constructed.Contour and curvature analysis,which are commonly used in barycenter coordinate,are applied to shape-preserving optimization for two-dimensional and three-dimensional cases.This paper begins with a introduction of the research background at home and abroad and related basic knowledge of mathematics.Secondly,a 2D generalized barycenter coordinate calculation and optimization algorithm on sparse combination is proposed.The generalized barycentric coordinate of 2D point series is obtained through their sparse solution combination of barycentric coordinate.Four sets of parameters of two weight functions are selected to map the plane figure to the plane,and contour line is used for shape-preserving comparison.Moreover,numerical examples and effect analysis are provided for practical application of two-dimensional plane.Furthermore,a 3D generalized barycenter coordinate calculation and optimization related to sparse combination are investigated.By finding the barycentric coordinate of a few points,that is,the sparse solutions of barycenter coordinate,and then combining them to calculate the generalized barycenter coordinate of 3D point arrays.Four groups of parameters of two weight functions are selected to map the plane figure to the space surface,and the contour is used for shape-preserving comparison.Moreover,numerical examples and effect analysis are presented for the practical application of 3D space.Next,the sparse combination method is compared with MV method,discrete harmonic method and Wachspress method.In order to analyze these four generalized barycentic coordinates methods,Lagrange interpolation algorithm is used to fit the curve by selecting an appropriate number of points on the contour.The curvature of points on the curve are calculated and curvature plots are made to analyze the four generalized barycentic coordinates methods.The corresponding diagrams and result analysis are given.Finally,the research results and future research directions are summarized.