Thomas A. Foley
Abstract
The weighted bicubic spline that is a C1 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.
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Index Terms
- Weighted bicubic spline interpolation to rapidly varying data
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Reviews
Richard Franke
This paper discusses the important problem of interpolation of bivariate gridded data where the data imply large slopes. In this case the usual bicubic spline methods yield surfaces with undershoot/overshoot. Generalizing the work of Salkauskas [1] for the univariate case, the author introduces a weighted version of the usual bicubic pseudonorm and minimizes this over interpolating bicubics, with appropriate boundary conditions. The weight function is constant over each subrectangle implied by the data. An automatic weight selection scheme is given and works well in the examples. This appears to be an effective scheme for dealing with the problem and could be useful in many cases.
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