What is now known as the Gibbs phenomenon was first observed in the context of truncated Fourier expansions, but other versions of it arise also in situations such as truncated integral transforms and for different interpolation methods. Radial basis functions (RBF) is a modern interpolation technique which includes both splines and trigonometric interpolations as special cases in 1-D, and it g...

We review characterizations of (conditional) positive definiteness and show how they apply to the theory of radial basis functions. We then give complete proofs for the (conditional) positive definiteness of all practically relevant basis functions. Furthermore, we show how some of these characterizations may lead to construction tools for positive definite functions. Finally, we give new const...

High-dimensional visualization is usually connected with large data processing. Because of dimensionality, it is nearly impossible to make a tessellation, like the Delaunay tessellation in E, followed by data interpolation. One possibility of data interpolation is the use of the Radial Basis Functions (RBF) interpolation. The RBF interpolation supports the interpolation of scattered data in d-d...

In the context of regression analysis with penalised linear models (such as RBF networks) certain model selection criteria can be diierentiated to yield a re-estimation formula for the regularisation parameter such that an initial guess can be iteratively improved until a local minimum of the criterion is reached. In this paper we discuss some enhancements of this general approach including imp...

Data acquired from ultrasound examinations is of interest not only for the physician, but also for the patient. While the physician uses the ultrasound data for diagnostic purposes the patient might be more interested in beautiful images in the case of prenatal imaging. Ultrasound data is noisy by nature and visually compelling 3D renderings are not always trivial to produce. This paper present...

The average diffusion propagator (ADP) obtained from diffusion MRI (dMRI) data encapsulates important structural properties of the underlying tissue. Measures derived from the ADP can be potentially used as markers of tissue integrity in characterizing several mental disorders. Thus, accurate estimation of the ADP is imperative for its use in neuroimaging studies. In this work, we propose a sim...

We examine the ability of radial basis functions (RBFs) to generalize. We compare the performance of several types of RBFs. We use the inverse dynamics of an idealized two-joint arm as a test case. We find that without a proper choice of a norm for the inputs, RBFs have poor generalization properties. A simple global scaling of the input variables greatly improves performance. We suggest some e...

Radial basis functions are tools for reconstruction of mul-tivariate functions from scattered data. This includes, for instance, reconstruction of surfaces from large sets of measurements, and solving partial diierential equations by collocation. The resulting very large linear N N systems require eecient techniques for their solution, preferably of O(N) or O(N log N) computational complexity. ...

A radial basis function (RBF) has the general form

In this paper, we have implemented a radial basis function (RBF) based method for solving the Black–Scholes partial differential equation. The application we have chosen is the valuation of European call options based on several underlying assets. We have shown that by appropriate choices of the RBF shape parameter and the node point placement, the accuracy of the results can be improved by at ...