三维空间中的有理样条插值

对三维空间某个多面体区域的四面体剖分，通过在每个四面体胞腔的棱和顶点设置适当的插值结点．本文给出了（１，１）型Ｃ⁰及Ｃ¹光滑的非奇异有理样条存在的充分必要条件．     

Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected C ^∞ Riemannian manifolds, including the important cases of spheres and SO(3), and using techniques involving differential geometry and Lie groups, we establish that the kernels obtained as fundamental solutions of certain partial differential operators generate Lagrange functions that are uniformly bounded and decay away from their center at an algebraic rate, and in certain cases, an exponential rate. An immediate corollary is that the corresponding Lebesgue constants for interpolation as well as for L ₂ minimization are uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The kernels considered here include the restricted surface splines on spheres, as well as surface splines for SO(3), both of which have elementary closed-form representations that are computationally implementable. In addition to obtaining bounded Lebesgue constants in this setting, we also establish a “zeros lemma” for domains on compact Riemannian manifolds—one that holds in as much generality as the corresponding Euclidean zeros lemma (on Lipschitz domains satisfying interior cone conditions) with constants that clearly demonstrate the influence of the geometry of the boundary (via cone parameters) as well as that of the Riemannian metric.     

Positive interpolation with rational splines

A method is presented for the construction of positive rational splines of continuity classC ².     

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.     

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.     

S-convex,monotone, and positive interpolation with rational bicubic splines of C2-continuity

In this paper we deal with shape preserving interpolation of data sets given on rectangular grids. The aim is to show that there exist spline interpolants of the continuity classC ² which areS-convex, monotone, or positive if the data sets have these properties. This is done by using particular rational bicubic splines defined on the grids introduced by the data. Interpolants of the desired type can be constructed by a simple search procedure.     

Convergence of local variational spline interpolation

In this paper we first revisit a classical problem of computing variational splines. We propose to compute local variational splines in the sense that they are interpolatory splines which minimize the energy norm over a subinterval. We shall show that the error between local and global variational spline interpolants decays exponentially over a fixed subinterval as the support of the local variational spline increases. By piecing together these locally defined splines, one can obtain a very good C⁰ approximation of the global variational spline. Finally we generalize this idea to approximate global tensor product B-spline interpolatory surfaces.     

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.     

Construction of spherical spline quasi-interpolants based on blossoming

A general theory of quasi-interpolants based on quadratic spherical Powell-Sabin splines on spherical triangulations of a sphere-like surface S is developed by using polar forms. As application, various families of discrete and differential quasi-interpolants reproducing quadratic spherical Bézier-Bernstein polynomials or the whole space of the spherical Powell-Sabin quadratic splines of class C¹ are presented.     

On uniform Lebesgue constants of local exponential splines with equidistant knots

For a linear differential operator L _r of arbitrary order r with constant coefficients and real pairwise different roots of the characteristic polynomial, we study Lebesgue constants (the norms of linear operators from C to C) of local exponential splines corresponding to this operator with a uniform arrangement of knots; such splines were constructed by the authors in earlier papers. In particular, for the third-order operator L ₃ = D(D ² ? β ²) (β > 0), we find the exact values of Lebesgue constants for two types of local splines and compare these values with Lebesgue constants of exponential interpolation splines.     

Convex cubic HERMITE-spline interpolation

We present a method to construct convex cubic C¹-splines which interpolate a given convex data set. The problem is reduced to the solution of a system of linear inequalities. The existence of such convex interpolation splines is assured if the data fulfill slight additional conditions. For stronger conditions some easier methods are developed. Finally, error estimations are given.     

The Nöther and Riemann-Roch type theorems for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, the Nöther type theorems for C ^µ piecewise algebraic curves are obtained. The theory of the linear series of sets of places on the piecewise algebraic curve is also established. In this theory, singular cycles are put into the linear series, and a complete series of the piecewise algebraic curves consists of all effective ordinary cycles in an equivalence class and all effective singular cycles which are equivalent specifically to any effective ordinary cycle in the equivalence class. This theory is a generalization of that of linear series of the algebraic curve. With this theory and the fundamental theory of multivariate splines on smoothing cofactors and global conformality conditions, and the results on the general expression of multivariate splines, we get a formula on the index, the order and the dimension of a complete series of the irreducible C ^µ piecewise algebraic curves and the degree, the genus and the smoothness of the curves, hence the Riemann-Roch type theorem of the C ^µ piecewise algebraic curve is established.     

Lagrange Interpolation by Bivariate C 1-Splines with Optimal Approximation Order

We develop the first local Lagrange interpolation scheme for C ¹-splines of degree q≥3 on arbitrary triangulations. For doing this, we use a fast coloring algorithm to subdivide about half of the triangles by a Clough–Tocher split in an appropriate way. Based on this coloring, we choose interpolation points such that the corresponding fundamental splines have local support. The interpolating splines yield optimal approximation order and can be computed with linear complexity. Numerical examples with a large number of interpolation points show that our method works efficiently.     

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.     

Convergence en moyenne quadratique de l'estimateur de la régression par splines hybrides

In this work, we consider the estimation of a smooth regression function, belonging to C^m [0, 1], by hybrid splines. We give the asymptotic behavior of the integrated mean square error by considering two different assumptions on the noise.     

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.     

Filtrations of Simplicial Functors and the Novikov Conjecture

We show that the Strong Novikov Conjecture for the maximal C^*-algebra C^*(π) of a discrete group π is equivalent to a statement in topological K-theory for which the corresponding statement in algebraic K-theory is always true. We also show that for any group π, rational injectivity of the full assembly map for K_*^t(C^*(π)) follows from rational injectivity of the restricted assembly map. (Received: February 2006)     

Scattered Data Fitting by Direct Extension of Local Polynomials to Bivariate Splines

We present a new scattered data fitting method, where local approximating polynomials are directly extended to smooth (C ¹ or C ²) splines on a uniform triangulation Δ (the four-directional mesh). The method is based on designing appropriate minimal determining sets consisting of whole triangles of domain points for a uniformly distributed subset of Δ. This construction allows to use discrete polynomial least squares approximations to the local portions of the data directly as parts of the approximating spline. The remaining Bernstein–Bézier coefficients are efficiently computed by extension, i.e., using the smoothness conditions. To obtain high quality local polynomial approximations even for difficult point constellations (e.g., with voids, clusters, tracks), we adaptively choose the polynomial degrees by controlling the smallest singular value of the local collocation matrices. The computational complexity of the method grows linearly with the number of data points, which facilitates its application to large data sets. Numerical tests involving standard benchmarks as well as real world scattered data sets illustrate the approximation power of the method, its efficiency and ability to produce surfaces of high visual quality, to deal with noisy data, and to be used for surface compression.     

On the Approximation Order of Splines on Spherical Triangulations

Bounds are provided on how well functions in Sobolev spaces on the sphere can be approximated by spherical splines, where a spherical spline of degree d is a C ^r function whose pieces are the restrictions of homogeneous polynomials of degree d to the sphere. The bounds are expressed in terms of appropriate seminorms defined with the help of radial projection, and are obtained using appropriate quasi-interpolation operators.