共查询到20条相似文献,搜索用时 31 毫秒
1.
Günther Nürnberger Vera Rayevskaya Larry L. Schumaker Frank Zeilfelder 《Constructive Approximation》2005,23(1):33-59
We describe a method which can be used to interpolate
function values at a set of scattered points
in a planar domain using bivariate polynomial splines
of any prescribed smoothness.
The method starts with an arbitrary given triangulation
of the data points, and involves refining some of the
triangles with Clough-Tocher splits.
The construction of the interpolating splines requires
some additional function values at selected points in
the domain, but no derivatives are needed at any point.
Given n data points and a corresponding
initial triangulation, the interpolating spline can be
computed in just O(n) operations.
The interpolation method is local
and stable, and provides optimal order approximation of smooth
functions. 相似文献
2.
3.
加密网格点二元局部基插值样条函数 总被引:1,自引:0,他引:1
1.简介 由于在理论以及应用两方面的重要性,多元样条引起了许多人的注意([6],[7]),紧支撑光滑分片多项式函数对于曲面的逼近是一个十分有效的工具。由于它们的局部支撑性,它们很容易求值;由于它们的光滑性,它们能被应用到要满足一定光滑条件的情况下;由于它们是紧支撑的,它们的线性包有很大的逼近灵活性,而且用它们构造逼近方法来解决的系统是 相似文献
4.
Hendrik Speleers 《Constructive Approximation》2013,37(1):41-72
We construct a suitable B-spline representation for a family of bivariate spline functions with smoothness r≥1 and polynomial degree 3r?1. They are defined on a triangulation with Powell–Sabin refinement. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction involves the determination of triangles that must contain a specific set of points. We further consider a number of CAGD applications. We show how to define control points and control polynomials (of degree 2r?1), and we provide an efficient and stable computation of the Bernstein–Bézier form of such splines. 相似文献
5.
Algorithms are presented for fitting a Powell-Sabin spline toa set of scattered data. Both the detemination of least-squaresand smoothing splines are considered. For the latter we adoptthe philosophy of an existing tensor product spline algorithm.The triangulation is determined in an automatic and adaptiveway. The algorithm employs a single parameter to control thetradeoff between closeness of fit and smoothness of fit. The Powell-Sabin splines are represented in terms of locallysupported basis functions. The use of the Bernstein-Bzier ordinatesof these B-splines results in efficient calculations. Numericalexamples illustrate the usefulness of the given algorithms. 相似文献
6.
It is shown that bivariate interpolatory splines defined on a rectangleR can be characterized as being unique solutions to certain variational problems. This variational property is used to prove the uniform convergence of bivariate polynomial splines interpolating moderately smooth functions at data which includes interpolation to values on a rectangular grid. These results are then extended to bivariate splines defined on anL-shaped region.This research was supported by a University of Kansas General Research Grant. 相似文献
7.
Zhi-qiangXu Ren-hongWang 《计算数学(英文版)》2004,22(6):807-816
Super splines are bivariate splines defined on triangulations, where the smoothness enforced at the vertices is larger than the smoothness enforced across the edges. In this paper, the smoothness conditions and conformality conditions for super splines are presented. Three locally supported super splines on type-1 triangulation are presented. Moreover, the criteria to select local bases is also givenBy using local supported super spline function, a variation-diminishing operator is built. The approximation properties of the operator are also presented. 相似文献
8.
This article presents and compares two approaches of principal component (PC) analysis for two-dimensional functional data on a possibly irregular domain. The first approach applies the singular value decomposition of the data matrix obtained from a fine discretization of the two-dimensional functions. When the functions are only observed at discrete points that are possibly sparse and may differ from function to function, this approach incorporates an initial smoothing step prior to the singular value decomposition. The second approach employs a mixed effects model that specifies the PC functions as bivariate splines on triangulations and the PC scores as random effects. We apply the thin-plate penalty for regularizing the function estimation and develop an effective expectation–maximization algorithm for calculating the penalized likelihood estimates of the parameters. The mixed effects model-based approach integrates scatterplot smoothing and functional PC analysis in a unified framework and is shown in a simulation study to be more efficient than the two-step approach that separately performs smoothing and PC analysis. The proposed methods are applied to analyze the temperature variation in Texas using 100 years of temperature data recorded by Texas weather stations. Supplementary materials for this article are available online. 相似文献
9.
Bivariate Polynomial Natural Spline Interpolation Algorithms with Local Basis for Scattered Data 总被引:3,自引:0,他引:3
Lutai Guan 《Journal of Computational Analysis and Applications》2003,5(1):77-101
Because of its importance in both theory and applications, multivariate splines have attracted special attention in many fields. Based on the theory of spline functions in Hilbert spaces, bivariate polynomial natural splines for interpolating, smoothing or generalized interpolating of scattered data over an arbitrary domain are constructed with one-sided functions. However, this method is not well suited for large scale numerical applications. In this paper, a new locally supported basis for the bivariate polynomial natural spline space is constructed. Some properties of this basis are also discussed. Methods to order scattered data are shown and algorithms for bivariate polynomial natural spline interpolating are constructed. The interpolating coefficient matrix is sparse, and thus, the algorithms can be easily implemented in a computer. 相似文献
10.
The penalized spline method has been widely used for estimating univariate smooth functions based on noisy data. This paper studies its extension to the two-dimensional case. To accommodate the need of handling data distributed on irregular regions, we consider bivariate splines defined on triangulations. Penalty functions based on the second-order derivatives are employed to regularize the spline fit and generalized cross-validation is used to select the penalty parameters. A simulation study shows that the penalized bivariate spline method is competitive to some well-established two-dimensional smoothers. The method is also illustrated using a real dataset on Texas temperature. 相似文献
11.
The piecewise algebraic curve is a kind generalization of the classical algebraic curve. Nöther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the Nöther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented. 相似文献
12.
Jean-Daniel Boissonnat Ramsay Dyer Arijit Ghosh 《Foundations of Computational Mathematics》2018,18(2):399-431
We present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact manifold, a manifold Delaunay complex is produced for a perturbed point set provided the transition functions are bi-Lipschitz with a constant close to 1, and the original sample points meet a local density requirement; no smoothness assumptions are required. If the transition functions are smooth, the output is a triangulation of the manifold. The output complex is naturally endowed with a piecewise-flat metric which, when the original manifold is Riemannian, is a close approximation of the original Riemannian metric. In this case the output complex is also a Delaunay triangulation of its vertices with respect to this piecewise-flat metric. 相似文献
13.
The piecewise algebraic curve is a kind generalization of the classical algebraic curve.N(o)ther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the N(o)ther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented. 相似文献
14.
F. Jeeawock-Zedek 《分析论及其应用》1994,10(2):1-16
The aim of this paper is to give upper bounds of the norm of the operator and associated error for a Lagrange interpolation
problem by C1 quadratic splines. The domain is rectangular and the type-2 triangulation is non-uniform. Moreover the location of data points
allows a very simple computation of the interpolant. 相似文献
15.
Summary.
We describe algorithms for constructing point sets at which interpolation by
spaces of bivariate splines of arbitrary degree and smoothness is
possible. The splines are defined on rectangular partitions adding
one or two diagonals to each rectangle. The interpolation sets
are selected in such a way that the grid points of the partition
are contained in these sets, and no large linear systems have to be solved.
Our method is to generate a net of line segments and to choose point sets in
these segments which satisfy the Schoenberg-Whitney condition for
certain univariate spline spaces such that a principle of degree
reduction can be applied. In order to include the grid points in the
interpolation sets, we give a sufficient Schoenberg-Whitney type
condition for interpolation by bivariate splines supported in certain cones.
This approach is completely different
from the known interpolation methods for bivariate splines of degree at most
three. Our method is illustrated by some numerical examples.
Received
October 5, 1992 / Revised version received May 13, 1994 相似文献
16.
Charles Kooperberg 《Journal of computational and graphical statistics》2013,22(3):322-341
Abstract A procedure for estimating a bivariate density based on data that may be censored is described. After the data are transformed to the unit square, the bivariate density is estimated using linear splines and their tensor products. The combined procedure yields an estimate of the bivariate density on the original scale, which may provide insight about the dependence structure. The procedure can also be used to estimate densities that are known to be symmetric and to test for independence. 相似文献
17.
一类分层三角剖分下三次样条空间的维数 总被引:1,自引:0,他引:1
本文定义了平面单连通多边形域的一类较任意的三角剖分-分层三角剖分,并通过分析二元样条的积分协调条件,确定了分层三角剖分卜三次C1作条函数空间的维数. 相似文献
18.
Jia-Chang Sun 《计算数学(英文版)》1994,12(3):195-202
1.Introducti0llIthasbeenshownl1]thatbivariateB-splineisaveryusefult0olf0rdesitwngsurfacemodeiling.Onemalndifficultyinpractice,however,istodevelopanefficientalgorithmforevalatinganddisplayingtheresultingsurface.Infact,foragivenpartitionflabivaxiatesplineinthespaces:(fl)isapiecewisebivariatepolyn0mialoft0taldegreekwithglobalcontinuitydegreep.ItmeansthatineachsubdomainthesurfacecanberepresentedasaBernsteinBezierform.Byusingwell-knownsubdivisi0nteclmiqueI2]onemaygiveanalgorithmf0rB-Bsurfaceinea… 相似文献
19.
Based on polyhedral splines, some multivariate splines of different orders
with given supports over arbitrary topological meshes are developed. Schemes for
choosing suitable families of multivariate splines based on pre-given meshes are
discussed. Those multivariate splines with inner knots and boundary knots from
the related meshes are used to generate rational spline shapes with related control
points. Steps for up to $C^2$-surfaces over the meshes are designed. The relationship
among the meshes and their knots, the splines and control points is analyzed. To
avoid any unexpected discontinuities and get higher smoothness, a heart-repairing
technique to adjust inner knots in the multivariate splines is designed.With the theory above, bivariate $C^1$-quadratic splines over rectangular meshes are
developed. Those bivariate splines are used to generate rational $C^1$-quadratic surfaces over the meshes with related control points and weights. The properties of
the surfaces are analyzed. The boundary curves and the corner points and tangent
planes, and smooth connecting conditions of different patches are presented. The $C^1$−continuous connection schemes between two patches of the surfaces are presented. 相似文献
20.
We have studied the numerical integration 2D based on bivariate C
1 local polynomial splines with a criss–cross triangulation of nonuniform rectangular partition. We have constructed the cubature formula and proved the convergence properties and error bounds. The paper includes some numerical tests that illustrate the performance of the corresponding algorithm. In the appendix there are explicit expressions of the quadratic polynomial restrictions of the B-splines related to every triangular cell. 相似文献