Radial Basis Function

Radial Basis Function interpolation is a diverse group of data interpolation methods. All of the Radial Basis Function methods are exact interpolators, so they attempt to honor your data. You can introduce a smoothing factor to all the methods in an attempt to produce a smoother surface.

Specifying Radial Basis Function Options

In the Property Manager Gridding page, select Radial Basis Function as the Gridding method and then click the Advanced Options

button to display the Radial Basis Options dialog.

Function Types

The Basis function list specifies the basis kernel function to use during gridding. This defines the optimal weights applied to the data points during the interpolation. In terms of the ability to fit your data and to produce a smooth surface, the Multiquadric method is considered by many to be the best. Successful use of the Thin Plate Spline basis function is also reported regularly in the technical literature.

The basis kernel functions define the optimal set of weights to apply to the data points when interpolating a grid node. The available basis kernel functions are listed in the Basis function drop-down list in the Radial Basis Function Options dialog.

R2 Value

The R2 parameter is a shaping or smoothing factor. The larger the R2 parameter shaping factor, the rounder and smoother the results. There is no universally accepted method for computing an optimal value for this factor. A reasonable trial value for R2 parameter is between the average sample spacing and one-half the average sample spacing.

The default value for R2 in the Radial Basis Function gridding algorithm is calculated as follows:

Anisotropy

A concise and readable introduction to Radial Basis Function interpolation can be found in Carlson and Foley (1991a). Given the clarity of presentation and the numerous examples, Hardy (1990) provides an excellent overview of the method, although this paper focuses exclusively on the special case of multiquadrics.

Carlson, R.E., and Foley, T.A. (1991a), Radial Basis Interpolation Methods on Track Data, Lawrence Livermore National Laboratory, UCRL-JC-1074238.

Carlson, R. E., and Foley, T. A. (1991b), The Parameter R2 in Multiquadric Interpolation, Computers Math. Applic, v. 21, n. 9, p. 29-42.

Franke, R. (1982), Scattered Data Interpolation: Test of Some Methods, Mathematics of Computations, v. 33, n. 157, p. 181-200.

Hardy, R. L. (1990), Theory and Applications of the Multiquadric-BiHarmonic Method, Computers Math. Applic, v. 19, n. 8/9, p. 163-208.

Powell, M.J.D. (1990), The Theory of Radial Basis Function Approximation in 1990, University of Cambridge Numerical Analysis Reports, DAMTP 1990/NA11.

Type	Equation
Inverse Multiquadric	$image\img00028.gif$
Multilog	$image\img00029.gif$
Multiquadric	$image\img00030.gif$
Natural Cubic Spline	$image\img00031.gif$
Thin Plate Spline	$image\img00032.gif$