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Spherical scattered data Quasi-interpolation by Gaussian Radial Basis Function

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Chinese Annals of Mathematics,Series B 2015年第3期36卷 401-412页

作者： Zhixiang CHEN Feilong CAO Department of Mathematics Shaoxing University Department of Mathematics China Jiliang University

Since the spherical Gaussian radial function is strictly positive definite, the authors use the linear combinations of translations of the Gaussian kernel to interpolate the scattered data on spheres in this article. Seeing that target functions axe usually outside the native spaces, and that one has to solve a large scaled system of linear equations to obtain combinatorial coefficients of interpolant functions, the authors first probe into some problems about interpolation with Gaussian radial functions. Then they construct quasi- interpolation operators by Gaussian radial function, and get the degrees of approximation. Moreover, they show the error relations between quasi-interpolation and interpolation when they have the same basis functions. Finally, the authors discuss the construction and approximation of the quasi-interpolant with a local support function.

关键词： scattered data Approximation Spherical Gaussian radial basis function Modulus of continuity

Geometric Feature Extraction and Model Reconstruction Based on scattered data

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Journal of Donghua University(English Edition) 2004年第4期21卷 86-89页

作者：胡鑫习俊通金烨 Institute of Computer Integrated Manufacturing Shanghai Jiaotong University Shanghai 200030

A method of 3D model reconstruction based on scattered point data in reverse engineering is presented here. The topological relationship of scattered points was established firstly, then the data set was triangulated to reconstruct the mesh surface model. The curvatures of cloud data were calculated based on the mesh surface, and the point data were segmented by edge-based method; Every patch of data was fitted by quadric surface of freeform surface, and the type of quadric surface was decided by parameters automatically, at last the whole CAD model was created. An example of mouse model was employed to confirm the effect of the algorithm.

关键词： Reverse engineering scattered data Feature extraction Model reconstruction.

Trivariate Polynomial Natural Spline for 3D scattered data Hermit Interpolation

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Communications in Mathematical Research 2012年第2期28卷 159-172页

作者： Xu YING-XIANG GUAN LU-TAI XU WEI-ZHI Department of Economics and Trade Xinhua College of Sun Yat-sen University Guangzhou 510275 Department of Scientific Computation and Computer Application Sun Yat-sen University Guangzhou 510275

Consider a kind of Hermit interpolation for scattered data of 3D by trivariate polynomial natural spline, such that the objective energy functional （with natural boundary conditions） is minimal. By the spline function methods in Hilbert space and variational theory of splines, the characters of the interpolation solution and how to construct it are studied. One can easily find that the interpolation solution is a trivariate polynomial natural spline. Its expression is simple and the coefficients can be decided by a linear system. Some numerical examples are presented to demonstrate our methods.

关键词： scattered data Hermit interpolation natural spline

Surface Reconstruction of 3D scattered data with Radial Basis Functions

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Communications in Mathematical Research 2010年第2期26卷 183-192页

作者： Du XIN-WEI YANG XIAO-YING LIANG XUE-ZHANG School of Mathematics Jilin University Changchun 130012 College of Mathematics Changchun University of Technology Changchun 130021

We use Radial Basis Functions （RBFs） to reconstruct smooth surfaces from 3D scattered data. An object＇s surface is defined implicitly as the zero set of an RBF fitted to the given surface data. We propose improvements on the methods of surface reconstruction with radial basis functions. A sparse approximation set of scattered data is constructed by reducing the number of interpolating points on the surface. We present an adaptive method for finding the off-surface normal points. The order of the equation decreases greatly as the number of the off-surface constraints reduces gradually. Experimental results are provided to illustrate that the proposed method is robust and may draw beautiful graphics.

关键词： radial basis function scattered data implicit surface surface reconstruction

Adaptive Hierarchical B-spline Surface Representation of Large-Scale scattered data

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Tsinghua Science and Technology 2000年第4期5卷 457-462页

作者：张伟强唐泽圣唐龙 Department of Computer Science and Technology-Tsinghua University Beijing 100084 China

The representation of large scale scattered data is a difficult problem, especially when various features of the representation, such as C 2-continuity, are required. This paper describes a fast algorithm for large scale scattered data approximation and interpolation. The interpolation algorithm uses a coarse-to-fine hierarchical control lattice to fit the scattered data. The refinement process is only used in the regions where the error between the scattered data and the result in a surface is greater than a specified tolerance. A method to ensure C 2-continuity is introduced to calculate the control lattice under constrained conditions. Experimental results show that this method can quickly represent large scale scattered data set.[

关键词： scattered data hierarchical B-spline interpolation data approximatiox

Large scattered data Fitting Based on Radial Basis Functions

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Computer Aided Drafting,Design and Manufacturing 2007年第1期17卷 66-72页

作者： FENG Ren-zhong XU Liang Department of Mathematics Beijing University of Aeronautics and Astronautics Beijing 100083 China Key Laboratory of Mathematics Informatics and Behavioral Semantics Ministry of Education

Solving large radial basis function （RBF） interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficulties, we present a fitting based on radial basis functions satisfying side conditions by least squares, although compared with interpolation the method loses some accuracy, it reduces the computational cost largely. Since the fitting accuracy and the non-singularity of coefficient matrix in normal equation are relevant to the uniformity of chosen centers of the fitted RBE we present a choice method of uniform centers. Numerical results confirm the fitting efficiency.

关键词： scattered data radial basis functions interpolation least squares fitting uniform centers

Rational Quasi-Interpolation Approximation of scattered data in R^(3)

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Numerical Mathematics(Theory,Methods and Applications) 2018年第1期11卷 169-186页

作者： Renzhong Feng Lifang Song School of Mathematics and Systematic Science&Key Laboratory of Mathematics Informatics and Behavioral SemanticsMinistry of EducationBeijing University of Aeronautics and AstronauticsBeijing 100191China

This paper is concerned with a piecewise smooth rational quasi-interpolation with algebraic accuracy of degree(n+1)to approximate the scattered data in R *** firstly use the modified Taylor expansion to expand the mean value coordinates interpolation with algebraic accuracy of degree one to one with algebraic accuracy of degree(n+1).Then,based on the triangulation of the scattered nodes in R^(2),on each triangle a rational quasi-interpolation function is *** constructed rational quasi-interpolation is a linear combination of three different expanded mean value coordinates interpolations and it has algebraic accuracy of degree(n+1).By comparing accuracy,stability,and efficiency with the C^(1)-Tri-interpolation method of Goodman[16]and the MQ Shepard method,it is observed that our method has some computational advantages.

关键词： scattered data mean value coordinates interpolation modified Taylor expansion rational quasi-interpolation algebraic accuracy

Reduction of distortion and improvement of efficiency for gridding of scattered gravity and magnetic data

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Applied Geophysics 2012年第4期9卷 378-390,494页

作者：张晨姚长利谢永茂郑元满关胡良洪东明地质过程与矿产资源国家重点实验室北京100083 地下信息探测技术与仪器教育部重点实验室北京100083 中国地质大学(北京)地球物理与信息技术学院北京100083

This paper presents a reasonable gridding-parameters extraction method for setting the optimal interpolation nodes in the gridding of scattered observed data. The method can extract optimized gridding parameters based on the distribution of features in raw data. Modeling analysis proves that distortion caused by gridding can be greatly reduced when using such parameters. We also present some improved technical measures that use human- machine interaction and multi-thread parallel technology to solve inadequacies in traditional gridding software. On the basis of these methods, we have developed software that can be used to grid scattered data using a graphic interface. Finally, a comparison of different gridding parameters on field magnetic data from Ji Lin Province, North China demonstrates the superiority of the proposed method in eliminating the distortions and enhancing gridding efficiency.

关键词： scattered data gridding parameters analysis of distribution features human-machine interaction multi-thread parallel technology

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散乱数据的数值微分及其误差估计

高等学校计算数学学报 2003年第1期25卷 81-90页

作者：贾现正王彦博程晋复旦大学数学系复旦大学数学系上海 200433

1 背景及问题的提出导数是数学分析中的一个基本的慨念。对于数学工作者来讲,计算导数不是一项特别困难的工作。但是,对于研究实际问题的科学工作者来讲,这项工作就不是一件简单的工作了,首先,求导数的问题是一个典型的Hadamard意义下... 详细信息

1 背景及问题的提出导数是数学分析中的一个基本的慨念。对于数学工作者来讲,计算导数不是一项特别困难的工作。但是,对于研究实际问题的科学工作者来讲,这项工作就不是一件简单的工作了,首先,求导数的问题是一个典型的Hadamard意义下的不适定问题([5],[12]等)

关键词： numerical differentiation scattered data nonuniform grid natural cubic spline.