Radial basis functions (RBFs) are real valued functions whose domain is in a real or complex vector space. The term “radial” refers to the fact that their value at a point depends only on the distance of that point from a fixed center. Linear combinations of RBFs with different centers are useful for interpolation and approximation in applications ranging from the representation of signals and images to modeling the motion of animated figures. In most cases, the functions are smooth, and the interpolation problem yields a positive definite matrix.
This paper deals only with Gaussian RBF of the form exp[-(||x-c||/s)2] to interpolate test images as models for image recognition. The authors describe a method for solving the interpolation problem using a finite Fourier transform with the added efficiency of the fast Fourier transform (FFT). Unfortunately, even the FFT method results in a linear system with a full N × N matrix, where N is the number of points in the test image. The authors show, however, that the larger off-diagonal elements of the inverse of this matrix are in the corners, and the inverse is well approximated by the identity plus a matrix with small diagonal blocks in the corner. They present results showing that the threshold used in approximating the inverse relates directly to the errors in the solution.
Except for some inconsistent notation, these interesting results are clearly presented. Indeed, they might apply to other RBF problems using functions other than the Gaussian basis functions. Another possible area of investigation is whether the approximation method they use for the inverse of their FFT matrix could be applied to the N × N matrix of the original interpolation problem with a similar reduction in operation count.
