Implemented radial basis functions (RBFs) for surface spline interpolation/approximation. More...

Detailed Description

The symbols used in the following table are:

phi(r) The radial basis function evaluated at a point (x,y) r Distance to center, r^2 = (x-xc)^2 + (y-yc)^2 (xc,yc) Center point eps Shape parameter

RadialBasisFunction::Unknown	Unknown or unsupported function.
RadialBasisFunction::VariableOrder	phi(r) = (r^2)^(m-1) * Ln( r^2 )
RadialBasisFunction::ThinPlateSpline	phi(r) = r^2 * Ln( r )
RadialBasisFunction::Gaussian	phi(r) = Exp( -(eps*r)^2 )
RadialBasisFunction::Multiquadric	phi(r) = Sqrt( 1 + (eps*r)^2 )
RadialBasisFunction::InverseMultiquadric	phi(r) = 1/Sqrt( 1 + (eps*r)^2 )
RadialBasisFunction::InverseQuadratic	phi(r) = 1/( 1 + (eps*r)^2 )
RadialBasisFunction::DDMVariableOrder	Variable-order polyharmonic DDM-RBF.
RadialBasisFunction::DDMThinPlateSpline	Thin plate spline DDM-RBF.
RadialBasisFunction::DDMMultiquadric	Multiquadric DDM-RBF.

DDM (Domain Decomposition Method) RBF interpolation/approximation algorithms require O(n^2) time for spline construction, in contrast to standard implementations, which have O(n^3) time complexity. This makes the DDM-RBF algorithms capable of handling large-scale problems with tens of thousands of data points, while standard algorithms are limited to less than about 3000 points with current hardware. DDM-RBF algorithms require a running PixInsight core application. Standard algorithms are directly implemented in PCL, so they are suitable for standalone applications.