Discrete weighted cubic splines

This paper presents methods for shape preserving spline interpolation. These methods are based on discrete weighted cubic splines. The analysis results in two algorithms with automatic selection of the shape control parameters: one to preserve the data monotonicity and other to retain the data convexity. Discrete weighted cubic B-splines and control point approximation are also considered.     

Optimal smoothing and interpolating splines with constraints

This paper considers the problem for designing optimal smoothing and interpolating splines with equality and/or inequality constraints. The splines are constituted by employing normalized uniform B-splines as the basis functions, namely as weighted sum of shifted B-splines of degree k. Then a central issue is to determine an optimal vector of the so-called control points. By employing such an approach, it is shown that various types of constraints are formulated as linear function of the control points, and the problems reduce to quadratic programming problems. We demonstrate the effectiveness and usefulness by numerical examples including approximation of probability density functions, approximation of discontinuous functions, and trajectory planning.     

Orthogonal splines based on B-splines — with applications to least squares,smoothing and regularisation problems

Orthogonal linear and cubic splines are introduced, based on a simple recurrence procedure using B-splines. Stable formulae are obtained for explicit least squares approximation. An application to the smoothing of noisy data is given, in which the approximation is essentially a second integral of an orthogonal linear spline and this leads to an efficient solution procedure. An application to the regularisation of integral transforms, and specifically to the finite moment problem, is also described.     

Local cubic splines on non-uniform grids and real-time computation of wavelet transform

In this paper, local cubic quasi-interpolating splines on non-uniform grids are described. The splines are designed by fast computational algorithms that utilize the relation between splines and cubic interpolation polynomials. These splines provide an efficient tool for real-time signal processing. As an input, the splines use either clean or noised arbitrarily-spaced samples. Formulas for the spline’s extrapolation beyond the sampling interval are established. Sharp estimations of the approximation errors are presented. The capability to adapt the grid to the structure of an object and to have minimal requirements to the operating memory are of great advantages for offline processing of signals and multidimensional data arrays. The designed splines serve as a source for generating real-time wavelet transforms to apply to signals in scenarios where the signal’s samples subsequently arrive one after the other at random times. The wavelet transforms are executed by six-tap weighted moving averages of the signal’s samples without delay. On arrival of new samples, only a couple of adjacent transform coefficients are updated in a way that no boundary effects arise.     

Non-uniform exponential tension splines

We describe explicitly each stage of a numerically stable algorithm for calculating with exponential tension B-splines with non-uniform choice of tension parameters. These splines are piecewisely in the kernel of D ²(D ²–p ²), where D stands for ordinary derivative, defined on arbitrary meshes, with a different choice of the tension parameter p on each interval. The algorithm provides values of the associated B-splines and their generalized and ordinary derivatives by performing positive linear combinations of positive quantities, described as lower-order exponential tension splines. We show that nothing else but the knot insertion algorithm and good approximation of a few elementary functions is needed to achieve machine accuracy. The underlying theory is that of splines based on Chebyshev canonical systems which are not smooth enough to be ECC-systems. First, by de Boor algorithm we construct exponential tension spline of class C ¹, and then we use quasi-Oslo type algorithms to evaluate classical non-uniform C ² tension exponential splines.      

Monotone and convex interpolation by weighted quadratic splines

In this paper we discuss the design of algorithms for interpolating discrete data by using weighted C ¹ quadratic splines in such a way that the monotonicity and convexity of the data are preserved. The analysis culminates in two algorithms with automatic selection of the shape control parameters: one to preserve the data monotonicity and other to retain the data convexity. Weighted C ¹ quadratic B-splines and control point approximation are also considered.     

On the Interpolation by Discrete Splines with Equidistant Nodes

In this paper we consider equidistant discrete splines S(j), j , which may grow as O(|j|^s) as |j|→∞. Such splines are relevant for the purposes of digital signal processing. We give the definition of the discrete B-splines and describe their properties. Discrete splines are defined as linear combinations of shifts of the B-splines. We present a solution to the problem of discrete spline cardinal interpolation of the sequences of power growth and prove that the solution is unique within the class of discrete splines of a given order.     

Bivariate interpolating polynomials and splines (I)

The multivariate splines which were first presented by de Boor as a complete theoretical system have intrigued many mathematicians who have devoted many works in this field which is still in the process of development. The author of this paper is interested in the area of interpolation with special emphasis on the interpolation methods and their approximation orders. But such B-splines (both univariate and multivariate) do not interpolated directly, so I approached this problem in another way which is to extend my interpolating spline of degree 2n-1 in univariate case (See[7]) to multivariate case. I selected triangulated region which is inspired by other mathematician’s works (e.g. [2] and [3]) and extend the interpolating polynomials from univariate to m-variate case (See [10])In this paper some results in the case m=2 are discussed and proved in more concrete details. Based on these polynomials, the interpolating splines (it is defined by me as piecewise polynomials in which the unknown partial derivatives are determined under certain continuous conditions) are also discussed. The approximation orders of interpolating polynomials and of cubic interpolating splines are inverstigated. We limited our discussion on the rectangular domain which is partitioned into equal right triangles. As to the case in which the rectangular domain is partitioned into unequal right triangles as well as the case of more complicated domains, we will discuss in the next paper.     

一个新的时间序列的插值模型

在这篇文章里,通过使用二次B样条,给出了一个用于数据挖掘的新的插值法方法,给出了时间序列的局部插值模型,插值曲线是C1连续的,该方法具有不需要解线性方程组的优点,应用上海股票指数进行了数值实验,实验性结果表明方法是有效的.     

New spline basis functions for sampling approximations

We describe a method of constructing a new kind of splines with compact support on . These basis functions consisting of a linear combination of the cardinal B-splines of mixed orders enable us to achieve simultaneously a good sampling approximation and an interpolation of any smooth function.     

Quantum B-splines

Quantum splines are piecewise polynomials whose quantum derivatives (i.e. certain discrete derivatives or equivalently certain divided differences) agree up to some order at the joins. Just like classical splines, quantum splines admit a canonical basis with compact support: the quantum B-splines. These quantum B-splines are the q-analogues of classical B-splines. Here quantum B-spline bases and quantum B-spline curves are investigated, using a new variant of the blossom: the q (quantum)-blossom. The q-blossom of a degree d polynomial is the unique symmetric, multiaffine function in d variables that reduces to the polynomial along the q-diagonal. By applying the q-blossom, algorithms and identities for quantum B-spline bases and quantum B-spline curves are developed, including quantum variants of the de Boor algorithms for recursive evaluation and quantum differentiation, knot insertion procedures for converting from quantum B-spline to piecewise quantum Bézier form, and a quantum variant of Marsden’s identity.     

Algorithms for surface fitting using Powell-Sabin splines

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.     

Wavelets and framelets from dual pseudo splines

Dual pseudo splines constitute a new class of refinable functions with B-splines as special examples.In this paper,we shall construct Riesz wavelet associated with dual pseudo splines.Furthermore,we use dual pseudo splines to construct tight frame systems with desired approximation order by applying the unitary extension principle.     

Generalized conic curves and their applications in curve approximation

The appriximation properties of generalized conic curves are studied in this paper. A generalized conic curve is defined as one of the following curves or their affine and translation equivalent curves:

1. conic curves, including parabolas, hyperbolas and ellipses;
2. generalized monomial curves, including curves of the form x=y^γ, γ∈R, γ≠0,1, in the x?y Cartesian coordinate system;
3. exponential spiral curves of the form ρ(?)=Ae^γ?, A>0, γ≠0, in the ρ-? polar coordinate system.

This type of curves has many important properties such as convexity, approximation property, effective numerical computation property and the subdivision property etc. Applications of these curves in both interpolation and approximations using piecewise generalized conic segment are also developed. It is shown that these generalized conic splines are very similar to the cubic polynomial splines and the best error of approximation isO(h ⁵) or at leastO(h ⁴) in general provided appropriate procedures are used. Finally some numerical examples of interpolation and approximations with generalized conic splines are given.     

Computing ECT-B-splines recursively

ECT-spline curves for sequences of multiple knots are generated from different local ECT-systems via connection matrices. Under appropriate assumptions there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized to form a nonnegative partition of unity. The basic functions can be defined by generalized divided differences [24]. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines. For sequences of simple knots and connection matrices that are nonsingular, lower triangular and totally positive the spline weights are identified as Neville–Aitken weights of certain generalized interpolation problems. For multiple knots they are limits of Neville–Aitken weights. In many cases the spline weights can be computed easily by recurrence. Our approach covers the case of Bézier-ECT-splines as well. They are defined by different local ECT-systems on knot intervals of a finite partition of a compact interval [a,b] connected at inner knots all of multiplicities zero by full connection matrices A ^[i] that are nonsingular, lower triangular and totally positive. In case of ordinary polynomials of order n they reduce to the classical Bézier polynomials. We also present a recursive algorithm of de Boor type computing ECT-spline curves pointwise. Examples of polynomial and rational B-splines constructed from given knot sequences and given connection matrices are added. For some of them we give explicit formulas of the spline weights, for others we display the B-splines or the B-spline curves. *Supported in part by INTAS 03-51-6637.     

凸四边形上一类具有C~2-连续的插值问题的样条解及其应用

本文讨论了一类凸四边形上的插值问题.指出这类插值问题是可解的,其解是分片二元三次多项式,且在凸四边形上是C~2-连续的.我们证明了这类插值问题的解的存在性和唯一性,给出了解样条的分片表达式及其逼近度的估计.最后还给出了一个应用实例和图形显示来说明本方法是可行的.     

样条空间S_3~1（△_（mn）~（1））上的插值与逼近

本文讨论样条空间Ｓ１３（△_（１）~ｍｎ）上的插值问题，导出了一类插值条件下样条插值的存在性与唯一性结论以及计算插值样条的递推格式．其主要结论是对四阶光滑的函数，插值排条可达２阶（相对网格长度）逼近度．     

A local Lagrange interpolation method based on cubic splines on Freudenthal partitions

A trivariate Lagrange interpolation method based on cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation method is local and stable, provides optimal order approximation, and has linear complexity.

     

On computational aspects of simplicial splines

Some new results on multivariate simplex B-splines and their practical application are presented. New recurrence relations are derived based on [2] and [15]. Remarks on boundary conditions are given and an example of an application of bivariate quadratic simplex splines is presented. The application concerns the approximation of a surface which is constrained by a differential equation.Communicated by Charles Micchelli.     

Methods for solving singular boundary value problems using splines: a review

This paper surveys and reviews papers of spline solution of singular boundary value problems. Among a number of numerical methods used to solve two-point singular boundary value problems, spline methods provide an efficient tool. Techniques collected in this paper include cubic splines, non-polynomial splines, parametric splines, B-splines and TAGE method.