31 research outputs found
On spherical averages of radial basis functions
A radial basis function (RBF) has the general form
where the coefficients a 1,âŠ,a n are real numbers, the points, or centres, b 1,âŠ,b n lie in â d , and Ï:â d ââ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when Ï is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm Ï(x)=âxâ when d is an odd positive integer, the thin plate spline Ï(x)=âxâ2log ââxâ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243â264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the PaleyâWiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserlesâs study of geometric integration
Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators
In this paper we introduce a generalized Sobolev space by defining a
semi-inner product formulated in terms of a vector distributional operator
consisting of finitely or countably many distributional operators
, which are defined on the dual space of the Schwartz space. The types of
operators we consider include not only differential operators, but also more
general distributional operators such as pseudo-differential operators. We
deduce that a certain appropriate full-space Green function with respect to
now becomes a conditionally positive
definite function. In order to support this claim we ensure that the
distributional adjoint operator of is
well-defined in the distributional sense. Under sufficient conditions, the
native space (reproducing-kernel Hilbert space) associated with the Green
function can be isometrically embedded into or even be isometrically
equivalent to a generalized Sobolev space. As an application, we take linear
combinations of translates of the Green function with possibly added polynomial
terms and construct a multivariate minimum-norm interpolant to data
values sampled from an unknown generalized Sobolev function at data sites
located in some set . We provide several examples, such
as Mat\'ern kernels or Gaussian kernels, that illustrate how many
reproducing-kernel Hilbert spaces of well-known reproducing kernels are
isometrically equivalent to a generalized Sobolev space. These examples further
illustrate how we can rescale the Sobolev spaces by the vector distributional
operator . Introducing the notion of scale as part of the
definition of a generalized Sobolev space may help us to choose the "best"
kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D.
thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}
Reconstruction from circular and spherical mean data
AbstractCircular and spherical mean data arise in various models of thermoacoustic and photoacoustic tomography which are rapidly developing modalities for in vivo imaging. We describe variants of a summability type reconstruction method adapted to this type of data. Among the highlights of the resulting algorithms, suggested by the results of numerical experiments, is the feature that the detectors need not lie on a regular curve or surface, such as a circle or a sphere. Several such numerical examples are included here
Summability kernels for circular and spherical mean data.
The reconstruction of images from data modeled by circular or spherical mean Radon transforms plays an important role in thermoacoustic and photoacoustic tomography. We consider a modification of a summability-type approximate reconstruction method described in earlier work and show that in the limit it leads to exact reconstruction. Among the consequences of this development are certain two- and three-dimensional inversion-type formulas in which the detectors lie on ellipses or ellipsoids respectively
Polyharmonic multiresolution analysis : an overview and some new results
International audienc