220 research outputs found
Advances in radial and spherical basis function interpolation
The radial basis function method is a widely used technique for interpolation of scattered data. The method is meshfree, easy to implement independently of the number of dimensions, and for certain types of basis functions it provides spectral accuracy. All these properties also apply to the spherical basis function method, but the class of applicable basis functions, positive definite functions on the sphere, is not as well studied and understood as the radial basis functions for the Euclidean space. The aim of this thesis is mainly to introduce new techniques for construction of Euclidean basis functions and to establish new criteria for positive definiteness of functions on spheres.
We study multiply and completely monotone functions, which are important for radial basis function interpolation because their monotonicity properties are in some cases necessary and in some cases sufficient for the positive definiteness of a function. We enhance many results which were originally stated for completely monotone functions to the bigger class of multiply monotone functions and use those to derive new radial basis functions. Further, we study the connection of monotonicity properties and positive definiteness of spherical basis functions. In the processes several new sufficient and some new necessary conditions for positive definiteness of spherical radial functions are proven. We also describe different techniques of constructing new radial and spherical basis functions, for example shifts. For the shifted versions in the Euclidean space we prove conditions for positive definiteness, compute their Fourier transform and give integral representations. Furthermore, we prove that the cosine transforms of multiply monotone functions are positive definite under some mild extra conditions. Additionally, a new class of radial basis functions which is derived as the Fourier transforms of the generalised
Gaussian φ(t) = e−tβ is investigated.
We conclude with a comparison of the spherical basis functions, which we derived in this thesis and those spherical basis functions well known. For this numerical test a set of test functions as well as recordings of electroencephalographic data are used to evaluate the performance of the different basis functions
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 scattered sites, the standard basis for such a space utilizes
\emph{globally} supported kernels; computing with it is prohibitively expensive
for large . 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
kernels, which makes the construction computationally
cheap. We prove that the new basis is 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
approximation orders. Although our focus is on , much of the
theory applies to other manifolds - , 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 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
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