Cardinal Spline Research Articles

A local interpolatory cardinal spline method for the determination of eigenstates in quantum-well structures with arbitrary potential profiles

A numerical method based on a variational approach and local interpolatory cardinal spline (LICS) functions is developed to compute the eigenstates of a quantum well with an arbitrary potential profile. The formulation of this method is straightforward, while the numerical solution is more accurate than results obtained by using other methods in the published literature.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

SETOR: Hardware-lighted three-dimensional solid model representations of macromolecules

SETOR is designed to exploit the hardware lighting capabilities of the IRIS-4D series graphics workstations to render high-quality raster images of macromolecules that can undergo rotation and translation interactively. SETOR can render standard all-atom and backbone models of proteins or nucleic acids, but focuses on displaying protein molecules by highlighting elements of secondary structure. The program has a very friendly user interface that minimizes the number of input files by allowing the user to interactively edit parameters, such as colors, lighting coefficients, and descriptions of secondary structure via mouse activated dialogue boxes. The choice of poly mer chain representation can be varied from standard vector models and van der Waal models, to a B-spline fit of polymer backbones that yields a smooth ribbon that approximates the polymer chain, to strict Cardinal splines that interpolate the smoothest curve possible that will precisely follow the polymer chain. The program provides a photograph mode, save/restore facilities, and efficient generation of symmetry-related molecules and packing diagrams. Additionally, SETOR is designed to accept commands and model coordinates from the standard input stream, and to control standard output. Ancillary programs provide a method to interactively edit hardcopy plots of all vector and many solid models generated by SETOR, and to produce standard HPGL or PostScript files. Examples of figures rendered by SETOR of a number of macromolecules of various classes are presented.

Wavelets and pre-wavelets in low dimensions

In Riemenschneider and Shen ( in “Approximation Theory and Functional Analysis” (C. K. Chui, Ed.), pp. 133–149, Academic Press, New York, 1991 ) an explicit orthonormal basis of wavelets for L 2( R s ), s=1,2,3, was constructed from a multiresolution approximation given by box splines. In other words, L 2( R s ) has the orthogonal decomposition ⊕ W ν . (∗) ν ϵ Z Orthonormal bases for the spaces W ν , are given by {2 νs 2 K μ(2 ν · −j)} , j∈ Z s , μ ϵ Z 2 s ⧹ 0, where Z s 2 and the “wavelets” K μ are 2 s − 1 cardinal splines with exponential decay. In this paper, we consider multiresolutions generated by suitable compactly supported and symmetric functions ϑ and explicitly construct 2 s − 1 compactly supported functions ϑ μ , μ ϵ Z 2 s ⧹ 0, such that the translates ϑ μ (· − j), j∈ Z s , are an unconditional basis for W 0. Thus, the functions ϑ μ (2 ν · − j), ν ϵ Z, j∈ Z s , μ ϵ Z 2 s ⧹ 0 comprise a basis for the orthogonal decomposition ( ∗) (the functions are orthogonal for different ν because the decomposition is orthogonal, but neither the translates nor the functions will be orthogonal for given ν). The functions are given as ϑ μ(·/2) 2 s = μ ∗′ β μ with the sequences β μ formed from a single sequence by translation and change in sign pattern. We also discuss various ways to regain some of the orthogonality lost by requiring compact support.

Cardinal spline filters: Stability and convergence to the ideal sinc interpolator

In this paper, we provide an interpretation of polynomial spline interpolation as a continuous filtering process. We prove that the frequency responses of the cardinal spline filters converge to the ideal lowpass filter in all Lfnorms with 1 ~<p < + ~ as the order of the spline tends to infinity. We provide estimates for the resolution errors and the interpolation errors of the various filters. We also derive an upper bound for the error associated with the reconstruction of bandlimited signals using polynomial splines. Zusammenfassung. In dieser Arbeit wird eine Interpretation der Polynom-Spline-Interpolation als kontinuierlicher Filterungs- prozeB vorgestellt. Es wird bewiesen, dab die Frequenzg~inge yon Cardinal-Spline-Filtern in allen LfNormen mit 1 ~<p < oc gegen den idealen TiefpaB konvergieren, wenn die Ordnung der Splines gegen unendlich geht. Wir stellen Absch~itzungen fiir die Aufl6sungsfehler und Interpolationsfehler fiir verschiedene Filter vor. Es wird auBerdem eine obere Grenze fiir den Fehler im Zusammenhang mit der Rekonstruktion von bandbegrenzten Signalen mit Hilfe von Polynom-Splines hergeleitet. Resume. Dans cet article, nous proposons une interpretation de l'interpolation B-spline en terme de filtres continus. Nous prouvons que les r6ponses fr6quentielles des filtres splines cardinaux convergent vers le filtre passe-bas id6al, et ceci dans toutes les normes Lp avec 1 ~<p < ~, quand l'ordre des splines tend vers rinfini. Nous donnons I'estimation des erreurs de r6solution et d'interpolation des differents filtres splines. Nous obtenons une limite superieure de l'erreur associ6e & la reconstruction de signaux a bande limit6e par interpolation spline.

Infinite-dimensional widths in the spaces of functions, II

The concepts of three ∞-widths are proposed and some of their properties are studied in this paper. The main result is that we obtain the exact values of the three ∞-widths of Sobolev function classes B r p( R ) in L p( R ) (1 ⩽ p ⩽ ∞) and find the optimal subspaces and the optimal linear operator. An application of the ∞-widths to optimal recovery is given. New extremal properties of cardinal splines and cardinal spline interpolation are discovered.

High levelm-harmonic cardinal B-splines

We generalize the notion ofm-harmonic cardinal B-spline defined in [Rabut, [6c]] to obtain “B-splines” on an infinite regular grid, which are halfway between “elementary B-splines” and the Lagrangean cardinal spline function. We give the main properties of these functions: Fourier transform, decay when ∥x∥ → ∞, integration,P k -reproduction (fork<-2m−1) of the associated B-spline approximation, etc. We show that, in some sense, “high levelm-harmonic B-splines” may be considered as a finer regular approximation of the Dirac distribution than the elementarym-harmonic B-splines are.

Polynomial spline signal approximations: filter design and asymptotic equivalence with Shannon's sampling theorem

The least-squares polynomial spline approximation of a signal g(t) in L/sub 2/(R) is obtained by projecting g(t) on S/sup n/(R) (the space of polynomial splines of order n). It is shown that this process can be linked to the classical problem of cardinal spline interpolation by first convolving g(t) with a B-spline of order n. More specifically, the coefficients of the B-spline interpolation of order 2n+1 of the sampled filtered sequence are identical to the coefficients of the least-squares approximation of g(t) of order n. It is shown that this approximation can be obtained from a succession of three basic operations: prefiltering, sampling, and postfiltering, which confirms the parallel with the classical sampling/reconstruction procedure for bandlimited signals. The frequency responses of these filters are determined for three equivalent spline representations using alternative sets of shift-invariant basis functions of S/sup n/(R): the standard expansion in terms of B-spline coefficients, a representation in terms of sampled signal values, and a representation using orthogonal basis functions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A Cardinal Spline Approach to Wavelets

A Cardinal Spline Approach to Wavelets

Theory of exponential splines

Pruess [12, 14] has shown that exponential splines can produce co-convex and co-monotone interpolants. These results justify the further study of the mathematical properties of exponential splines as they pertain to their utility as numerical approximations. They also warrant the generalization of the exponential spline in fruitful directions. Herein, we present convergence rates and extremal properties for exponential spline approximation, cardinal spline and B-spline bases for the space of exponential splines, and generalizations to higher order tension splines and Hermite tension interpolants.

The homotopy model: a generalized model for smooth surface generation from cross sectional data

A generalized model, called the homotopy model, is presented to reconstruct surfaces from cross-sectional data of objects using a homotopy to generate surfaces connecting consecutive contours. The homotopy model consists of continuous toroidal graph representation and homotopic generation of surfaces from the representation. It is shown that the homotopy model includes triangulation as a special case and generates smooth parametric surfaces from contour-line definitions using homotopy. The model can be applied to contours represented by parametric curves as well as linear line segments. First, a heuristic method that finds the optimal path on the toroidal graph is presented. Then the toroidal graph is expanded to a continuous version. Finally, homotopy is used for reconstructing parametric surfaces from the toroidal graph representation. A loft surface is also a special case of homotopy, a straight-line homotopy. Homotopy that corresponds to the cardinal spline surface is also introduced. Three-dimensional surface reconstruction of human auditory surface reconstruction of human auditory ossicles illustrates the advantages of the homotopy model over the others.

Fast B-spline transforms for continuous image representation and interpolation

Efficient algorithms for the continuous representation of a discrete signal in terms of B-splines (direct B-spline transform) and for interpolative signal reconstruction (indirect B-spline transform) with an expansion factor m are described. Expressions for the z-transforms of the sampled B-spline functions are determined and a convolution property of these kernels is established. It is shown that both the direct and indirect spline transforms involve linear operators that are space invariant and are implemented efficiently by linear filtering. Fast computational algorithms based on the recursive implementations of these filters are proposed. A B-spline interpolator can also be characterized in terms of its transfer function and its global impulse response (cardinal spline of order n). The case of the cubic spline is treated in greater detail. The present approach is compared with previous methods that are reexamined from a critical point of view. It is concluded that B-spline interpolation correctly applied does not result in a loss of image resolution and that this type of interpolation can be performed in a very efficient manner.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A cardinal spline approach to wavelets

While it is well known that the mth order B-spline N m (x) with integer knots generates a multiresolution analysis, ...⊂V −1 ⊂V 0 m ⊂..., with the mth order of approximation, we prove that ψ(x):=L 2m (m) (2x−1), where L 2m (x) denotes the (2m)th order fundamental cardinal interpolatory splines, generates the orthogonal complementary wavelet spaces W k . Note that for m=1, when the B-spline N 1 (x) is the characteristic function of the unit interval [0,1], our basic wavelet L 2 '(2x−1) is simply the well-known Haar wavelet. In proving that V k+1 =V k ○+W k , we give the exact formulation of N m (2x−j), j∈Z, in terms of integer translates of N m (x) and ψ(x)

Using the refinement equation for the construction of pre-wavelets

A variety of methods have been proposed for the construction of wavelets. Among others, notable contributions have been made by Battle, Daubechies, Lemarie, Mallat, Meyer, and Stromberg. This effort has led to the attractive mathematical setting of multiresolution analysis as the most appropriate framework for wavelet construction. The full power of multiresolution analysis led Daubechies to the construction ofcompactly supported orthonormal wavelets with arbitrarily high smoothness. On the other hand, at first sight, it seems some of the other proposed methods are tied to special constructions using cardinal spline functions of Schoenberg. Specifically, we mention that Battle raises some doubt that his block spin method “can produce only the Lemarie Ondelettes”. A major point of this paper is to extend the idea of Battle to the generality of multiresolution analysis setup and address the easier job of constructingpre-wavelets from multiresolution.

Polyharmonic cardinal splines: A minimization property

Polyharmonic cardinal splines are distributions which are annihilated by iterates of the Laplacian in the complement of a lattice in Euclidean n-space and satisfy certain continuity conditions. Some of the basic properties were recorded in our earlier paper on the subject. Here we show that such splines solve a variational problem analogous to the univariate case considered by I. J. Schoenberg.

Non-linear arrival time inversion: constraining velocity anomalies by seeking smooth models in 3-D

SUMMARY The problem of constraining 3-D seismic anomalies using arrival times from a regional network is examined. The non-linear dependence of arrival times on the hypocentral parameters of the earthquakes and the 3-D velocity field leads to a multiparameter-type non-linear inverse problem, and the distribution of sources and receivers from a typical regional network results in an enormous 3-D variation in data constraint. To ensure computational feasibility, authors have tended to neglect the non-linearity of the problem by linearizing about some best-guess discretized earth model. One must be careful in interpreting 3-D structure from linearized inversions because the inadequacy of the data window may combine with non-linear effects to produce artificial or phantom ‘structure’. To avoid the generation of artificial velocity gradients we must determine only those velocity variations which are necessary to fit the data rather than merely estimating local velocities in different parts of the model, which is the more common practice. We present a series of inversion algorithms which seek to inhibit the generation of unnecessary structure while performing efficiently within the framework of a large-scale inversion. This is achieved by extending the subspace method of Kennett, Sambridge & Williamson (1988) and incorporating the smoothing strategy proposed by Constable, Parker & Constable (1987). A flexible model parametrization involving Cardinal spline functions is used, and full 3-D ray tracing performed. A comparison between linear and non-linear inversions shows that if a breakdown in the linearizing approximation occurs spurious velocity models may be obtained which would appear acceptable in a linear inversion. Application of the techniques to a SE Australian data set show that unnecessary structure can be suppressed. As the smoothing power of the algorithm is improved a robust low-velocity anomaly dipping to the north becomes the most dominant feature of the P-wave model and much of the complex structure of pure data-fitting models is removed.

Almost cardinal spline interpolation

Interpolation of a doubly infinite sequence of data by spline functions is studied. When the interpolation points and the knots of the interpolating splines are characterized by a periodic behavior, the interpolating problem is called Cardinal Interpolation. This work extends known results on Cardinal Interpolation to the “almost cardinal” case, where the interpolation is cardinal except for a finite number of interpolation points and knots. In passing from the cardinal to the “almost cardinal” case, the “invariance under translation” property of the interpolating spaces is lost. Thus classical arguments used in solving the cardinal case do not apply. Instead we use the intimate connection between the interpolating “almost cardinal splines” and Oscillatory Matrices. The main conclusion of this work is that a wide range of Almost Cardinal Interpolation Problems have the same type of solution as the corresponding Cardinal Interpolation Problem.

Polyharmonic cardinal splines

Polyharmonic splines, sometimes called thin plate splines, are distributions which are annihilated by iterates of the Laplacian in the complement of a discrete set in Euclidean n-space and satisfy certain continuity conditions. The term cardinal is often used when the set of knots is a lattice. Here, in addition to developing certain basic properties of polyharmonic cardinal splines, it is shown that such splines interpolate numerical data on the lattice uniquely.

Cardinal interpolation by polynomial splines: Interpolation of data with exponential growth

Let m ∈ N and define S m to be the class of functions ƒ ∈ C m− 1 ( R which, in each [ j − 1, j] (j ∈ Z , coincide with some real polynomial of degree ⩽ m. We study the cardinal spline interpolation problem of constructing an element sϵS m with a( j − λ) = y j ,j ∈ Z where λ∈(0,1] is a translation parameter. Under some natural conditions on λ and m, ter Morsche (in “Spline Functions” ( K. Böhmer, G. Meinardus, and W. Schempp, Eds.), pp. 210–219, Springer-Verlag, Berlin/Heidelberg/New York, 1976) and Schoenberg ( J. Approx. Theory 6 (1972) , 404–420; in “Studies in Spline-Functions and Approximation Theory” ( S. Karlin, Ch. A. Micchelli, A. Pinkus, I. J. Schoenberg, Eds.), pp. 251–276, Academic Press, New York, 1976) have proved that this problem has a unique solution of power growth, provided that the interpolation data are of power growth and that this solution can be given by a series of Lagrangian splines converging locally uniformly. In what follows we prove an analogous result for exponential growth conditions instead of power growth conditions. Moreover, we extend the concept of extremal bases, given by Reimer (in “Approximation Theory III” ( E. Cheney, Ed.), pp. 723–728, Academic Press, New York, 1980), to topological bases of normed spaces with infinite dimension and apply this concept to the subspace of all bounded functions of S m .

Convergence of cardinal series

The result of this paper is a generalization of our characterization of the limits of multivariate cardinal splines. Let M n {M_n} denote the n n -fold convolution of a compactly supported function M ∈ L 2 ( R d ) M \in {L_2}({{\mathbf {R}}^d}) and denote by \[ S n := { ∑ j ∈ Z d c ( j ) M n ( ⋅ − j ) : c ∈ l 2 ( Z d ) } {S_n}: = \left \{ {\sum \limits _{j \in {{\mathbf {Z}}^d}} {c(j){M_n}( \cdot - j):c \in {l_2}({{\mathbf {Z}}^d})} } \right \} \] the span of the translates of M n {M_n} . We prove that there exists a set Ω \Omega with vol d ( Ω ) = ( 2 π ) d {\operatorname {vol} _d}(\Omega ) = {(2\pi )^d} such that for any f ∈ L 2 ( R d ) f \in {L_2}({{\mathbf {R}}^d}) , \[ dist ⁡ ( f , S n ) → 0 as n → ∞ , \operatorname {dist} (f,{S_n}) \to 0\quad {\text {as }}n \to \infty , \] if and only if the support of the Fourier transform of f f is contained in Ω \Omega .

An elementary algebraic representation of polynomial spline interpolants for equidistant lattices and its condition

A customary representation formula for periodic spline interpolants contains redundance which, however, can be eliminated by a transcendent method. We use an elementary indentity for the generalized Euler-Frobenius-polynomials, which seems to be unknown until now, in order to derive the theory by purely algebraic arguments. The general cardinal spline interpolation theory can be obtained from the periodic case by a simple approach to the limit. Our representation has minimum condition for odd/even degree if the interpolation points are the lattice (mid-)points. We evaluate the corresponding condition numbers and give an asymptotic representation for them.