Cubic Splines with Minimal Norm

Natural cubic interpolatory splines are known to have a minimal L ₂-norm of its second derivative on the C ² (or W ₂ ² ) class of interpolants. We consider cubic splines which minimize some other norms (or functionals) on the class of interpolatory cubic splines only. The cases of classical cubic splines with defect one (interpolation of function values) and of Hermite C ¹ splines (interpolation of function values and first derivatives) with spline knots different from the points of interpolation are discussed.     

L 1 C 1 polynomial spline approximation algorithms for large data sets

In this article, we address the problem of approximating data points by C ¹-smooth polynomial spline curves or surfaces using L ₁-norm. The use of this norm helps to preserve the data shape and it reduces extraneous oscillations. In our approach, we introduce a new functional which enables to control directly the distance between the data points and the resulting spline solution. The computational complexity of the minimization algorithm is nonlinear. A local minimization method using sliding windows allows to compute approximation splines within a linear complexity. This strategy seems to be more robust than a global method when applied on large data sets. When the data are noisy, we iteratively apply this method to globally smooth the solution while preserving the data shape. This method is applied to image denoising.     

Matrix-valued symmetric templates for interpolatory surface subdivisions: I. Regular vertices

The objective of this paper is to introduce a general procedure for deriving interpolatory surface subdivision schemes with “symmetric subdivision templates” (SSTs) for regular vertices. While the precise definition of “symmetry” will be clarified in the paper, the property of SSTs is instrumental to facilitate application of the standard procedure for finding symmetric weights for taking weighted averages to accommodate extraordinary (or irregular) vertices in surface subdivisions, a topic to be studied in a continuation paper. By allowing the use of matrices as weights, the SSTs introduced in this paper may be constructed to overcome the size barrier limited to scalar-valued interpolatory subdivision templates, and thus avoiding the unnecessary surface oscillation artifacts. On the other hand, while the old vertices in a (scalar) interpolatory subdivision scheme do not require a subdivision template, we will see that this is not the case for the matrix-valued setting. Here, we employ the same definition of interpolation subdivisions as in the usual scalar consideration, simply by requiring the old vertices to be stationary in the definition of matrix-valued interpolatory subdivisions. Hence, there would be another complication when the templates are extended to accommodate extraordinary vertices if the template sizes are not small. In this paper, we show that even for C² interpolatory subdivisions, only one “ring” is sufficient in general, for both old and new vertices. For example, for 1-to-4 split C² interpolatory surface subdivisions, we obtain matrix-valued symmetric interpolatory subdivision templates (SISTs) for both triangular and quadrilateral meshes with sizes that agree with those of the Loop and Catmull–Clark schemes, respectively. Matrix-valued SISTs of similar sizes are also constructed for C² interpolatory and subdivision schemes in this paper. In addition to small template sizes, an obvious feature of matrix-valued weights is the flexibility for introducing shape-control parameters. Another significance is that, in contrast to the usual scalar setting, matrix-valued SISTs can be formulated in terms of the coefficient sequence of some vector refinement equation of interpolating bivariate C² splines with small support. For example, by modifying the spline function vectors introduced in our previous work [C.K. Chui, Q.T. Jiang, Surface subdivision schemes generated by refinable bivariate spline function vectors, Appl. Comput. Harmon. Anal. 15 (2003) 147–162; C.K. Chui, Q.T. Jiang, Refinable bivariate quartic and quintic C²-splines for quadrilateral subdivisions, Preprint, 2004], C² symmetric interpolatory subdivision schemes associated with refinement equations of C² cubic and quartic splines on the 6-directional and 4-directional meshes, respectively, are also constructed in this paper.     

The convergence of three spline methods for scattered data interpolation and fitting

The convergences of three L₁ spline methods for scattered data interpolation and fitting using bivariate spline spaces are studied in this paper. That is, L₁ interpolatory splines, splines of least absolute deviation, and L₁ smoothing splines are shown to converge to the given data function under some conditions and hence, the surfaces from these three methods will resemble the given data values.     

On the dimension of trivariate spline spaces with the highest order smoothness on 3D T-meshes

T-meshes are a type of rectangular partitions of planar domains which allow hanging vertices. Because of the special structure of T-meshes, adaptive local refinement is possible for splines defined on this type of meshes, which provides a solution for the defect of NURBS. In this paper, we generalize the definitions to the three-dimensional (3D) case and discuss a fundamental problem – the dimension of trivariate spline spaces on 3D T-meshes. We focus on a special case where splines are C ^d?1 continuous for degree d. The smoothing cofactor method for trivariate splines is explored for this situation. We obtain a general dimension formula and present lower and upper bounds for the dimension. At last, we introduce a type of 3D T-meshes, where we can give an explicit dimension formula.     

A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects

The theory of splines is a well studied topic, but the kinship of splines with fractals is novel. We introduce a simple explicit construction for a -cubic Hermite Fractal Interpolation Function (FIF). Under some suitable hypotheses on the original function, we establish a priori estimates (with respect to the L _p -norm, 1≤p≤∞) for the interpolation error of the -cubic Hermite FIF and its first derivative. Treating the first derivatives at the knots as free parameters, we derive suitable values for these parameters so that the resulting cubic FIF enjoys global smoothness. Consequently, our method offers an alternative to the standard moment construction of -cubic spline FIFs. Furthermore, we identify appropriate values for the scaling factors in each subinterval and the derivatives at the knots so that the graph of the resulting -cubic FIF lies within a prescribed rectangle. These parameters include, in particular, conditions for the positivity of the cubic FIF. Thus, in the current article, we initiate the study of the shape preserving aspects of fractal interpolation polynomials. We also provide numerical examples to corroborate our results.     

A constructive approach to nodal splines

In the context of local spline interpolation methods, nodal splines have been introduced as possible fundamental functions by de Villiers and Rohwer in 1988. The corresponding local spline interpolation operator possesses the desirable property of reproducing a large class of polynomials. However, it was remarked that their definition is rather intricate so that it seems desirable to reveal the actual origin of these splines. The real source can be found in the Martensenoperator which can be obtained by two-point Hermite spline interpolation problem posed and proved by Martensen [Darstellung und Entwicklung des Restgliedes der Gregoryschen Quadraturformel mit Hilfe von Spline-Funktionen, Numer. Math. 21(1973)70–80]. On the one hand, we will show how to represent the Hermite Martensen spline recursively and, on the other hand, explicitly in terms of the B-spline by using the famous Marsden identity. Having introduced the Martensenoperator, we will show that the nodal spline interpolation operator can be obtained by a special discretization of the occurring derivatives. We will consider symmetric nodal splines of odd degree that can be obtained by our methods in a natural way.     

An O(h6) cubic spline interpolating procedure for harmonic functions

An O(h⁶) method for the interpolation of harmonic functions in rectangular domains is described and analyzed, The method is based on an earlier cubic spline technique [N. Papamichael and J.R. Whiteman, BIT 14 , 452–459 (1974)], and makes use of recent results concerning the a posteriori correction of interpolatory cubic splines.     

A General Multiresolution Method for Fitting Functions on the Sphere

In [7], Lyche and Schumaker have described a method for fitting functions of class C ¹ on the sphere which is based on tensor products of quadratic polynomial splines and trigonometric splines of order three associated with uniform knots. In this paper, we present a multiresolution method leading to C ²-functions on the sphere, using tensor products of polynomial and trigonometric splines of odd order with arbitrary simple knot sequences. We determine the decomposition and reconstruction matrices corresponding to the polynomial and trigonometric spline spaces. We describe the general tensor product decomposition and reconstruction algorithms in matrix form which are convenient for the compression of surfaces. We give the different steps of the computer implementation of these algorithms and, finally, we present a test example.     

Integral representation of slowly growing equidistant splines

In this paper we consider polynomial splines S(x) with equidistant nodes which may grow as O (|x|^s). We present an integral representation of such splines with a distribution kernel. This representation is related to the Fourier integral of slowly growing functions. The part of the Fourier exponentials herewith play the so called exponential splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines. First, it allows us to construct a rich library of splines possessing the property that translations of any such spline form a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline spaces and of spline interpolation of a growing function. We derive formulas for approximate evaluation of splines projecting a function onto the spline space and establish therewith exact estimations of the approximation errors.     

A sinusoidal polynomial spline and its Bezier blended interpolant

Functional polynomials composed of sinusoidal functions are introduced as basis functions to construct an interpolatory spline. An interpolant constructed in this way does not require solving a system of linear equations as many approaches do. However there are vanishing tangent vectors at the interpolating points. By blending with a Bezier curve using the data points as the control points, the blended curve is a proper smooth interpolant. The blending factor has the effect similar to the “tension” control of tension splines. Piecewise interpolants can be constructed in an analogous way as a connection of Bezier curve segments to achieve C¹ continuity at the connecting points. Smooth interpolating surface patches can also be defined by blending sinusoidal polynomial tensor surfaces and Bezier tensor surfaces. The interpolant can very efficiently be evaluated by tabulating the sinusoidal function.     

An energy-minimization framework for monotonic cubic spline interpolation

This paper describes the use of cubic splines for interpolating monotonic data sets. Interpolating cubic splines are popular for fitting data because they use low-order polynomials and have C² continuity, a property that permits them to satisfy a desirable smoothness constraint. Unfortunately, that same constraint often violates another desirable property: monotonicity. It is possible for a set of monotonically increasing (or decreasing) data points to yield a curve that is not monotonic, i.e., the spline may oscillate. In such cases, it is necessary to sacrifice some smoothness in order to preserve monotonicity.The goal of this work is to determine the smoothest possible curve that passes through its control points while simultaneously satisfying the monotonicity constraint. We first describe a set of conditions that form the basis of the monotonic cubic spline interpolation algorithm presented in this paper. The conditions are simplified and consolidated to yield a fast method for determining monotonicity. This result is applied within an energy minimization framework to yield linear and nonlinear optimization-based methods. We consider various energy measures for the optimization objective functions. Comparisons among the different techniques are given, and superior monotonic C² cubic spline interpolation results are presented. Extensions to shape preserving splines and data smoothing are described.     

Local modification of Pythagorean-hodograph quintic spline curves using the B-spline form

The problems of determining the B–spline form of a C ² Pythagorean–hodograph (PH) quintic spline curve interpolating given points, and of using this form to make local modifications, are addressed. To achieve the correct order of continuity, a quintic B–spline basis constructed on a knot sequence in which each (interior) knot is of multiplicity 3 is required. C ² quintic bases on uniform triple knots are constructed for both open and closed C ² curves, and are used to derive simple explicit formulae for the B–spline control points of C ² PH quintic spline curves. These B-spline control points are verified, and generalized to the case of non–uniform knots, by applying a knot removal scheme to the Bézier control points of the individual PH quintic spline segments, associated with a set of six–fold knots. Based on the B–spline form, a scheme for the local modification of planar PH quintic splines, in response to a control point displacement, is proposed. Only two contiguous spline segments are modified, but to preserve the PH nature of the modified segments, the continuity between modified and unmodified segments must be relaxed from C ² to C ¹. A number of computed examples are presented, to compare the shape quality of PH quintic and “ordinary” cubic splines subject to control point modifications.     

Exact Lebesgue constants for interpolatory ℒ-splines of third order

In this paper, we obtain the Lebesgue constants for interpolatory ?-splines of third order with uniform nodes, i.e., the norms of interpolation operators from C to C describing the process of interpolation of continuous bounded and continuous periodic functions by ?-splines of third order with uniform nodes on the real line. As a corollary, we obtain exact Lebesgue constants for interpolatory polynomial parabolic splines with uniform nodes.     

关于具局部插值性质的样条

引言 插值样条作为逼近工具有许多优点,但也受到一些限制。例如大部分样条都只限于多项式样条。又如样条插值带有整体性,即一插值点上的任何变化将波及整个样条的所有各点。此外高阶样条的计算较复杂。 本文给出一种新的构造样条的方法,它将不限于多项式样条,并且主要是它具有局部插值性,即这种样条在一个子区间上的值只与其邻近的几个插值点有关。我们称这种样条为局部插值样条。 与通常的多项式样条相比,局部样条的计算比较简单,并且一个插值点上的数值变动只影响其邻近的局部范围。     

On the variational characterization and convergence of bivariate splines

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.     

Range restricted interpolation using Gregory's rational cubic splines

The construction of range restricted univariate and bivariate interpolants to gridded data is considered. We apply Gregory's rational cubic C¹ splines as well as related rational quintic C² splines. Assume that the lower and upper obstacles are compatible with the data set. Then the tension parameters occurring in the mentioned spline classes can be always determined in such a way that range restricted interpolation is successful.     

Some extremal properties of multivariate polynomial splines in the metric Lp(ℝd)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (?^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (?^d in the metric L_p((?^d).     

Smoothing splines using compactly supported, positive definite, radial basis functions

In this paper, we develop a fast algorithm for a smoothing spline estimator in multivariate regression. To accomplish this, we employ general concepts associated with roughness penalty methods in conjunction with the theory of radial basis functions and reproducing kernel Hilbert spaces. It is shown that through the use of compactly supported radial basis functions it becomes possible to recover the band structured matrix feature of univariate spline smoothing and thereby obtain a fast computational algorithm. Given n data points in R ², the new algorithm has complexity O(n ²) compared to O(n ³), the order for the thin plate multivariate smoothing splines.     

Exponential box splines

In a recent paper by Nira Dyn and the author, univariate cardinal exponential B-splines are shown to have a representation similar to the wellknown box spline representation of the univariate cardinal polynomialB-splines. Motivated by this, we construct, for a set ofn directions inZ ^s and a vector of constants λ ?R ⁿ, an “exponential box spline” which has the same smoothness and support as the polynomial box spline, and is a positive piecewise exponential in its support. We derive recurrence relations for the exponential box splines which are simpler than those for the polynomial case. A relatively simple structure of the space spanned by the translates of an exponential box spline is obtained for λ in a certain open dense set ofR ⁿ—the “simple” λ. In this case, the characterization of the local independence of the translates and related topics, as well as the proofs involved, are quite simple when compared with the polynomial case (corresponding toλ = 0).