Generalized taylor splines

A generalization of polynomial spline theory is presented which allows the construction of splines of any order. The kernel is based on a Taylor series expansion of the governing equations and Gaussian elimination is used for the numerical calculation of the splines coefficients. The general steps are described and tests are performed for the classical cubic and quintic splines and also for an unorthodox 25th order spline.     

An iterative algorithm for spline interpolation

One of the fundamental results in spline interpolation theory is the famous Schoenberg-Whitney Theorem, which completely characterizes those distributions of interpolation points which admit unique interpolation by splines. However, until now there exists no iterative algorithm for the explicit computation of the interpolating spline function, and the only practicable method to obtain this function is to solve explicitly the corresponding system of linear equations. In this paper we suggest a method which computes iteratively the coefficients of the interpolating function in its B-spline basis representation; the starting values of our one-step iteration scheme are quotients of two low order determinants in general, and sometimes even just of two real numbers. Furthermore, we present a generalization of Newton's interpolation formula for polynomials to the case of spline interpolation, which corresponds to a result of G. Mühlbach for Haar spaces.     

散乱数据(2m-1,2n-1)次多项式自然样条插值

考虑对窄间散乱数据(2m-1,2n-1)次多项式自然样条插值,使得插值函数对x的m次偏导数和对y的n次偏导数平方积分极小(带自然边界条件).用希尔伯特空间样条方法,得出其解的结构,解的系数能够用线性方程组确定,方程组系数矩阵对称,可用改进的平方根法解.例子表明方法简单,效果良好.     

Automatic G arc spline interpolation for closed point set

A method for generating an interpolation closed G¹ arc spline on a given closed point set is presented. For the odd case, i.e. when the number of the given points is odd, this paper disproves the traditional opinion that there is only one closed G¹ arc spline interpolating the given points. In fact, the number of the resultant closed G¹ arc splines fulfilling the interpolation condition for the odd case is exactly two. We provide an evaluation method based on the arc length as well such that the choice between those two arc splines is made automatically. For the even case, i.e. when the number of the given points is even, the points are automatically moved based on weight functions such that the interpolation condition for generating closed G¹ arc splines is satisfied, and that the adjustment is small. And then, the G¹ arc spline is constructed such that the radii of the arcs in the spline are close to each other. Examples are given to illustrate the method.     

Multidimensional Spline Interpolation: Theory and Applications

Computing numerical solutions of household’s optimization, one often faces the problem of interpolating functions. As linear interpolation is not very good in fitting functions, various alternatives like polynomial interpolation, Chebyshev polynomials or splines were introduced. Cubic splines are much more flexible than polynomials, since the former are only twice continuously differentiable on the interpolation interval. In this paper, we present a fast algorithm for cubic spline interpolation, which is based on the precondition of equidistant interpolation nodes. Our approach is faster and easier to implement than the often applied B-Spline approach. Furthermore, we will show how to loosen the precondition of equidistant points with strictly monotone, continuous one-to-one mappings. Finally, we present a straightforward generalization to multidimensional cubic spline interpolation.      

Discrete box splines and refinement algorithms

We introduce discrete box splines and use them to give a general knotline refinement algorithm for surfaces which are linear combinations of translates of a box spline. The Lane-Riesenfeld algorithm is obtained as a special case. Since this represents a new algorithm for refining certain nontensor product schemes, it is hoped that this will lead to further applications in computer aided geometric design.     

Convexity preserving splines over triangulations

A general method is given for constructing sets of sufficient linear conditions that ensure convexity of a polynomial in Bernstein-Bézier form on a triangle. Using the linear conditions, computational methods based on macro-element spline spaces are developed to construct convexity preserving splines over triangulations that interpolate or approximate given scattered data.     

The L2-polynomial spline pyramid

The authors are concerned with the derivation of general methods for the L₂ approximation of signals by polynomial splines. The main result is that the expansion coefficients of the approximation are obtained by linear filtering and sampling. The authors apply those results to construct a L₂ polynomial spline pyramid that is a parametric multiresolution representation of a signal. This hierarchical data structure is generated by repeated application of a REDUCE function (prefilter and down-sampler). A complementary EXPAND function (up-sampler and post-filter) allows a finer resolution mapping of any coarser level of the pyramid. Four equivalent representations of this pyramid are considered, and the corresponding REDUCE and EXPAND filters are determined explicitly for polynomial splines of any order n (odd). Some image processing examples are presented. It is demonstrated that the performance of the Laplacian pyramid can be improved significantly by using a modified EXPAND function associated with the dual representation of a cubic spline pyramid     

Interchanging spline curves using IGES

Current CAD/CAM systems employ a variety of spline types, spline representations, and curve design algorithms. The IGES (initial graphic exchange specification) was designed to enable interchange of the more commonly used spline types among CAD/CAM systems, independent of the design algorithms originally used to create them. IGES supports parametric cubic splines through a piecewise polynomial representation. This paper describes the design considerations leading to the interchange common spline curve types. A short introduction to splines and spline representations is included.     

A class of general quartic spline curves with shape parameters

With a support on four consecutive subintervals, a class of general quartic splines are presented for a non-uniform knot vector. The splines have C² continuity at simple knots and include the cubic non-uniform B-spline as a special case. Based on the given splines, piecewise quartic spline curves with three local shape parameters are given. The given spline curves can be C²∩G³ continuous by fixing some values of the curve?s parameters. Without solving a linear system, the spline curves can also be used to interpolate sets of points with C² continuity. The effects of varying the three shape parameters on the shape of the quartic spline curves are determined and illustrated.     

一类加权有理四次插值样条曲线的形状控制

将插值曲线约束于给定的区域之内是曲线形状控制中的重要问题。利用带导数的和不带导数的分母为线性的有理四次插值样条构造了一类新的加权有理四次插值样条函数,插值函数具有简单的显示表示,这类新的插值样条中含有权系数,因而增加了处理问题的灵活性,给约束控制带来了方便。给出了将该种插值曲线约束于给定的折线、二次曲线之上、之下或之间的充分条件。证明了满足约束条件的加权有理样条的存在性。     

一类四次有理样条的形状控制及其逼近性质

本文基于一类带控制参数包含极点的(4,2)~k(k=1,2)阶有理插值样条,研究了它的约束插值问题,给出了将该种插值曲线约束于给定折线、二次曲线之上、之下或之间的充分条件.并讨论了该插值的逼近性质,最后给出了数值例子.     

Smoothing of cubic parametric splines

The most common curve representation in CADCAM systems of today is the cubic parametric spline. Unfortunately this curve will sometimes oscillate and cause unwanted inflexions which are difficult to deal with. This paper has developed from the need to eliminate oscillations and remove inflexions from such splines, a need which may occur for example when interpolating data measured from a model. A method for interactive smoothing is outlined and a smoothing algorithm is described which is mathematically comparable to manual smoothing with a physical spline.     

Spline-approximation of thermodynamic properties of solutions

Fitting of splines to the thermodynamic functions of solutions in the presence of “singularities” have been considered. The spline are shown to enable a successful combination of the advantages of a tabular and an analytical methods of data representation. Examples of cubic spline application for smoothing the primary experimental data, as well as for compiling the standard thermodynamic tables and their use (interpolating, integrating the Gibbs-Duhem and Duhem-Margules equations and other calculations) are given.     

Identification of IIR Wiener systems with spline nonlinearities that have variable knots

An algorithm is developed for the identification of Wiener systems, linear dynamic elements followed by static nonlinearities. In this case, the linear element is modeled using a recursive digital filter, while the static nonlinearity is represented by a spline of arbitrary but fixed degree. The primary contribution in this note is the use of variable knot splines, which allow for the use of splines with relatively few knot points, in the context of Wiener system identification. The model output is shown to be nonlinear in the filter parameters and in the knot points, but linear in the remaining spline parameters. Thus, a separable least squares algorithm is used to estimate the model parameters. Monte-Carlo simulations are used to compare the performance of the algorithm identifying models with linear and cubic spline nonlinearities, with a similar technique using polynomial nonlinearities.     

Subdivision algorithms for the generation of box spline surfaces

This paper proposes a general approach to subdivision algorithms used in interactive computer aided design for splines which are linear combinations of translates of any box splines. We show how these algorithms can be used for efficient generation of the corresponding spline surfaces. Our results extend several known special cases.     

C2 interpolation of spatial data subject to arc-length constraints using Pythagorean–hodograph quintic splines

In order to reconstruct spatial curves from discrete electronic sensor data, two alternative C² Pythagorean–hodograph (PH) quintic spline formulations are proposed, interpolating given spatial data subject to prescribed constraints on the arc length of each spline segment. The first approach is concerned with the interpolation of a sequence of points, while the second addresses the interpolation of derivatives only (without spatial localization). The special structure of PH curves allows the arc-length conditions to be expressed as algebraic constraints on the curve coefficients. The C² PH quintic splines are thus defined through minimization of a quadratic function subject to quadratic constraints, and a close starting approximation to the desired solution is identified in order to facilitate efficient construction by iterative methods. The C² PH spline constructions are illustrated by several computed examples.     

Estimation of Discontinuous Functions of Two Variables with Unknown Discontinuity Lines (Rectangular Elements)

We construct and analyze discontinuous interpolating splines for the approximation of discontinuous functions. We develop an algorithm to estimate the discontinuous function whose unknown discontinuities lie on the lines parallel to the coordinate axes, by approximating it by the discontinuous interpolating spline. We also develop an algorithm to find the discontinuities of the discontinuous function on the basis of the concept of ε-continuity of functions of two variables and present the examples.     

Splines over regular triangulations in numerical simulation

We investigate the use of smooth spline spaces over regular triangulations as a tool in (isogeometric) Galerkin methods. In particular, we focus on box splines over three-directional meshes. Box splines are multivariate generalizations of univariate cardinal B-splines sharing the same properties. Tensor-product B-splines with uniform knots are a special case of box splines. The use of box splines over three-directional meshes has several advantages compared with tensor-product B-splines, including enhanced flexibility in the treatment of the geometry and stiffness matrices with stronger sparsity. Boundary conditions are imposed in a weak form to avoid the construction of special boundary functions. We illustrate the effectiveness of the approach by means of a selection of numerical examples.     

带形状参数的三次三角Hermite插值样条曲线

给出了一种带形状参数的三次三角Hermite插值样条曲线,具有标准三次Hermite插值样条曲线完全相同的性质。给定插值条件时,样条曲线的形状可通过改变形状参数的取值进行调控。在适当条件下,该样条曲线对应的Ferguson曲线可精确表示椭圆、抛物线等工程曲线。通过选择合适的形状参数,该插值样条曲线能达到[C2]连续,而且其整体逼近效果要好于标准三次Hermite插值样条曲线。