Surface Spline Interpolation Research Articles

The L p -Approximation Order of Surface Spline Interpolation for 1 \leq p \leq 2

We show that the Lp-approximation order of surface spline interpolation equals m+1/p for p in the range 1 \leq p \leq 2, where m is an integer parameter which specifies the surface spline. Previously it was known that this order was bounded below by m + ½ and above by m+1/p. With h denoting the fill-distance between the interpolation points and the domain Ω, we show specifically that the Lp(Ω)-norm of the error between f and its surface spline interpolant is O(hm + 1/p) provided that f belongs to an appropriate Sobolev or Besov space and that Ω \subset Rd is open, bounded, and has the C2m-regularity property. We also show that the boundary effects (which cause the rate of convergence to be significantly worse than O(h2m)) are confined to a boundary layer whose width is no larger than a constant multiple of h |log h|. Finally, we state numerical evidence which supports the conjecture that the Lp-approximation order of surface spline interpolation is m + 1/p for 2 < p \leq \infty.

Scattered Date Interpolation from Principal Shift-Invariant Spaces

Under certain assumptions on the compactly supported function φ∈C(Rd), we propose two methods of selecting a function s from the scaled principal shift-invariant space Sh(φ) such that s interpolates a given function f at a scattered set of data locations. For both methods, the selection scheme amounts to solving a quadratic programming problem and we are able to prove error estimates similar to those obtained by Duchon for surface spline interpolation.

ON THE ACCURACY OF SURFACE SPLINE APPROXIMATION AND INTERPOLATION TO BUMP FUNCTIONS

AbstractLet $\sOm$ be the closure of a bounded open set in $\mathbb{R}^d$, and, for a sufficiently large integer $\kappa$, let $f\in C^\kappa(\sOm)$ be a real-valued ‘bump’ function, i.e. $\supp(f)\subset\textint(\sOm)$. First, for each $h&gt;0$, we construct a surface spline function $\sigma_h$ whose centres are the vertices of the grid $\mathcal{V}_h=\sOm\cap h\zd$, such that $\sigma_h$ approximates $f$ uniformly over $\sOm$ with the maximal asymptotic accuracy rate for $h\rightarrow0$. Second, if $\ell_1,\ell_2,\dots,\ell_n$ are the Lagrange functions for surface spline interpolation on the grid $\mathcal{V}_h$, we prove that $\max_{x\in\sOm}\sum_{j=1}^n\ell_j^2(x)$ is bounded above independently of the mesh-size $h$. An interesting consequence of these two results for the case of interpolation on $\mathcal{V}_h$ to the values of a bump data function $f$ is obtained by means of the Lebesgue inequality.AMS 2000 Mathematics subject classification: Primary 41A05; 41A15; 41A25; 41A63

Approximation in L p (R d ) from a space spanned by the scattered shifts of a radial basis function

A new multivariate approximation scheme on R d using scattered translates of the “shifted” surface spline function is developed. The scheme is shown to provide spectral L p -approximation orders with 1 ≤ p ≤ ∞, i.e., approximation orders that depend on the smoothness of the approximands. In addition, it applies to noisy data as well as noiseless data. A numerical example is presented with a comparison between the new scheme and the surface spline interpolation method.

The $L_{2}$-approximation order of surface spline interpolation

We show that if the open, bounded domain Ω C R d has a sufficiently smooth boundary and if the data function, f is sufficiently smooth, then the L p (Ω)-norm of the error between f and its surface spline interpolant is O (δ γp+1/2 ) (1 ≤ p ≤ ∞), where γp:= min{m,m - d/2 + d/p} and m is an integer parameter specifying the surface spline. In case p = 2, this lower bound on the approximation order agrees with a previously obtained upper bound, and so we conclude that the L 2 -approximation order of surface spline interpolation is m + 1/2.

Overcoming the boundary effects in surface spline interpolation

Surface spline interpolation when the domain is all of R d is known to converge much faster to the data function f than in the case when the domain is the unit ball. This difference is understood to be due to boundary effects which, as will be shown, also affect the size of the surface spline's coefficients. We propose a modified form of surface spline interpolation which, to a great extent, overcomes these boundary effects. This modified surface spline interpolant uses only the values of f at the given interpolation points.

Mapping of scalp potentials by surface spline interpolation

Evoked potentials and EEGs record punctuate electrical activity at electrode sites. To represent the overall potential distribution on the entire scalp it is necessary to interpolate between these sampled values. Surface splines are mathematical tools for interpolating functions of two variables. In comparison to the classical methods of interpolation, based on linear combination of the potentials of the 4 nearest electrodes, spline methods are smoother, give more precisely located extrema and converge faster toward the ‘true’ potential surface when the number of recording electrodes is increased. These advantages are at the expense of lengthier computation time.

Spherical spline interpolation—basic theory and computational aspects

The purpose of the paper is to adapt to the spherical case the basic theory and the computational method known from surface spline interpolation in Euclidean spaces. Spline functions are defined on the sphere. The solution process is made simple and efficient for numerical computation. In addition, the convergence of the solution obtained by spherical spline interpolation is developed using estimates for Legendre polynomials.

Antarctic Ice Elevation Maps From Balloon Altimetry

Antarctic ice-elevation maps are presented, based on 4 000 measurement points, analysed from traverses made by constant-density balloons. These balloons were launched during the Tropical Wind, Energy Conversion and Reference Level Experiment (TWERLE). The TWERLE data yielded daily maps of the elevation of the 150 mbar pressure surface in the southern hemisphere. These maps serve as a reference level from which the altimeter readings are subtracted to yield the ice elevation. The accuracy of the ice-elevation data is estimated at better than 60 m. The accuracy of the contouring depends on the density of measurements in the particular area. Two contouring techniques were used. One is based on generating elevation data at fixed grid-points from arbitrarily distributed measurements, using Cressman's method. The second is a surface spline interpolation technique by Duchon. The elevation data are of special value on the inland plateau areas. There, the typical distance between balloon measurements is sufficiently small, relative to the typical distance between contour lines at 100 m intervals. Furthermore, the inland areas are the least accessible to other mapping techniques.

Antarctic Ice Elevation Maps From Balloon Altimetry

Antarctic ice-elevation maps are presented, based on 4 000 measurement points, analysed from traverses made by constant-density balloons. These balloons were launched during the Tropical Wind, Energy Conversion and Reference Level Experiment (TWERLE). The TWERLE data yielded daily maps of the elevation of the 150 mbar pressure surface in the southern hemisphere. These maps serve as a reference level from which the altimeter readings are subtracted to yield the ice elevation.The accuracy of the ice-elevation data is estimated at better than 60 m. The accuracy of the contouring depends on the density of measurements in the particular area. Two contouring techniques were used. One is based on generating elevation data at fixed grid-points from arbitrarily distributed measurements, using Cressman's method. The second is a surface spline interpolation technique by Duchon.The elevation data are of special value on the inland plateau areas. There, the typical distance between balloon measurements is sufficiently small, relative to the typical distance between contour lines at 100 m intervals. Furthermore, the inland areas are the least accessible to other mapping techniques.

Multivariate interpolation at arbitrary points made simple

The concrete method of ‘surface spline interpolation’ is closely connected with the classical problem of minimizing a Sobolev seminorm under interpolatory constraints; the intrinsic structure of surface splines is accordingly that of a multivariate extension of natural splines. The proper abstract setting is a Hilbert function space whose reproducing kernel involves no functions more complicated than logarithms and is easily coded. Convenient representation formulas are given, as also a practical multivariate extension of the Peano kernel theorem. Owing to the numerical stability of Cholesky factorization of positive definite symmetric matrices, the whole construction process of a surface spline can be described as a recursive algorithm, the data relative to the various interpolation points being exploited in sequence.