Fast Interpolation and Approximation of Scattered Multidimensional and Dynamic Data Using Radial Basis Functions

Interpolation or approximation of scattered data is very often task in engineering problems. The Radial Basis Functions (RBF) interpolation is convenient for scattered (un-ordered) data sets in k-dimensional space, in general. This approach is convenient especially for a higher dimension k > 2 as the conversion to an ordered data set, e.g. using tessellation, is computationally very expensive. The RBF interpolation is not separable and it is based on distance of two points. It leads to a solution of a Linear System of Equations (LSE) Ax = b. There are two main groups of interpolating functions: ‘global” and “local”. Application of “local” functions, called Compactly Supporting RBF (CSFBF), can significantly decrease computational cost as they lead to a system of linear equations with a sparse matrix. In this paper the RBF interpolation theory is briefly introduced at the “application level” including some basic principles and computational issues and an incremental RBF computation is presented and approximation RBF as well. The RBF interpolation or approximation can be used also for image reconstruction, inpainting removal, for solution of Partial Differential Equations (PDE), in GIS systems, digital elevation model DEM etc. Key-Words: Radial basis function, RBF interpolation, image reconstruction, incremental computation, RBF approximation, fast matrix multiplication