Knot selection for least squares approximation using thin plate splines.

Abstract

Given a large set of scattered data (x(i),y(i) ,f(i)), a method for selecting a significantly smaller set of knot points which will represent the larger set is described, leading to a package of FORTRAN subroutines. The selection of the knot point locations is based on the minimization of the sum of the squares of the difference between the average number of points per Dirichlet tile and the actual number of points in each tile, subject to the constraint that each knot is located at the centroid of its tile. The pertinent theoretical and computational aspects of the subroutines are introduced and described in detail. Using the least squares thin plate spline approximation method for constructing surfaces, various test surfaces are examined and compared to surfaces obtained using smoothing splines and the bicubic Hermite approximation method. The FORTRAN subroutines are made available to prospective users through a point of contact.