Weighted bicubic spline interpolation to rapidly varying data

Abstract

The weighted bicubic spline that is a C 1 piecewise bicubic interpolant to three-dimensional gridded data is introduced. This is a generalization of the univariate weighted spline, developed by Salkauskas, in that a weighted minimization problem is solved. The minimization problem solved is a weighted version of the problem that the natural bicubic spline and Gordon's spline-blended interpolants minimize. The surface is represented as a piecewise bicubic Hermite interpolant whose derivatives are the solution of a linear system of equations. For computer-aided-design applications, the shape of the surface is controlled by weighting the variation over the individual patches, whereas many other shape-control methods weight the discrete data points. A method for selecting the weights is presented so that the weighted bicubic spline effectively solves the important and often difficult problem of interpolating rapidly varying data.