A Duchon framework for the sphere

In his fundamental paper (RAIRO Anal. Numer. 12 (1978) 325) Duchon presented a strategy for analysing the accuracy of surface spline interpolants to sufficiently smooth target functions. In the mid-1990s Duchon's strategy was revisited by Light and Wayne (J. Approx. Theory 92 (1992) 245) and Wendland (in: A. Le Méhauté, C. Rabut, L.L. Schumaker (Eds.), Surface Fitting and Multiresolution Methods, Vanderbilt Univ. Press, Nashville, 1997, pp. 337–344), who successfully used it to provide useful error estimates for radial basis function interpolation in Euclidean space. A relatively new and closely related area of interest is to investigate how well radial basis functions interpolate data which are restricted to the surface of a unit sphere. In this paper we present a modified version Duchon's strategy for the sphere; this is used in our follow up paper (Lp-error estimates for radial basis function interpolation on the sphere, preprint, 2002) to provide new Lp error estimates (p[1,∞]) for radial basis function interpolation on the sphere

An axiomatic approach to scalar data interpolation on surfaces

We discuss possible algorithms for interpolating data given on a set of curves in a surface of ℝ^3. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The Absolutely Minimizing Lipschitz Extension model (AMLE) is singled out and studied in more detail. We study the correctness of our numerical approach and we show experiments illustrating the interpolation of data on some simple test surfaces like the sphere and the torus

A climatically-derived global soil moisture data set for use in the GLAS atmospheric circulation model seasonal cycle experiment

Algorithms for point interpolation and contouring on the surface of the sphere and in Cartesian two-space are developed from Shepard's (1968) well-known, local search method. These mapping procedures then are used to investigate the errors which appear on small-scale climate maps as a result of the all-too-common practice of of interpolating, from irregularly spaced data points to the nodes of a regular lattice, and contouring Cartesian two-space. Using mean annual air temperatures field over the western half of the northern hemisphere is estimated both on the sphere, assumed to be correct, and in Cartesian two-space. When the spherically- and Cartesian-approximted air temperature fields are mapped and compared, the magnitudes (as large as 5 C to 10 C) and distribution of the errors associated with the latter approach become apparent

Spherical basis function approximation with particular trend functions

summary:The paper is concerned with the measurement of scalar physical quantities at nodes on the $(d-1)$-dimensional unit sphere surface in the \hbox{$d$-dimensional} Euclidean space and the spherical RBF interpolation of the data obtained. In particular, we consider $d=3$. We employ an inverse multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 defined in Cartesian coordinates. We prove the existence of the interpolation formula of the type considered. The formula can be useful in the interpretation of many physical measurements. We show an example concerned with the measurement of anisotropy of magnetic susceptibility having extensive applications in geosciences and present numerical difficulties connected with the high condition number of the matrix of the system defining the interpolation

Localized bases for kernel spaces on the unit sphere

Approximation/interpolation from spaces of positive definite or conditionally positive definite kernels is an increasingly popular tool for the analysis and synthesis of scattered data, and is central to many meshless methods. For a set of $N$ scattered sites, the standard basis for such a space utilizes $N$ \emph{globally} supported kernels; computing with it is prohibitively expensive for large $N$. Easily computable, well-localized bases, with "small-footprint" basis elements - i.e., elements using only a small number of kernels -- have been unavailable. Working on \sphere, with focus on the restricted surface spline kernels (e.g. the thin-plate splines restricted to the sphere), we construct easily computable, spatially well-localized, small-footprint, robust bases for the associated kernel spaces. Our theory predicts that each element of the local basis is constructed by using a combination of only $\mathcal{O}((\log N)^2)$ kernels, which makes the construction computationally cheap. We prove that the new basis is $L_p$ stable and satisfies polynomial decay estimates that are stationary with respect to the density of the data sites, and we present a quasi-interpolation scheme that provides optimal $L_p$ approximation orders. Although our focus is on $\mathbb{S}^2$, much of the theory applies to other manifolds - $\mathbb{S}^d$, the rotation group, and so on. Finally, we construct algorithms to implement these schemes and use them to conduct numerical experiments, which validate our theory for interpolation problems on $\mathbb{S}^2$ involving over one hundred fifty thousand data sites.Comment: This article supersedes arXiv:1111.1013 "Better bases for kernel spaces," which proved existence of better bases for various kernel spaces. This article treats a smaller class of kernels, but presents an algorithm for constructing better bases and demonstrates its effectiveness with more elaborate examples. A quasi-interpolation scheme is introduced that provides optimal linear convergence rate

Surface Reconstruction from Scattered Point via RBF Interpolation on GPU

In this paper we describe a parallel implicit method based on radial basis functions (RBF) for surface reconstruction. The applicability of RBF methods is hindered by its computational demand, that requires the solution of linear systems of size equal to the number of data points. Our reconstruction implementation relies on parallel scientific libraries and is supported for massively multi-core architectures, namely Graphic Processor Units (GPUs). The performance of the proposed method in terms of accuracy of the reconstruction and computing time shows that the RBF interpolant can be very effective for such problem.Comment: arXiv admin note: text overlap with arXiv:0909.5413 by other author