Bicubic B-Spline

The Bicubic B-Spline procedure in AutoSignal is a subset of the algorithm used in the Estimate Gridded Data option in the Non-Parametric menu of the TableCurve 3D surface fitting product.

Author

Carl de Boor

References

Carl de Boor, "A Practical Guide to Splines", Springer-Velag, 1978, p332-346

Description

This option offers the standard two-dimensional tensor product B-Spline. For AutoSignal, the orders are fixed at a bicubic (order 3 in both X and Y). The algorithm does not offer extrapolation.

Interpolation

A bicubic order 3/3 spline offers a smooth surface, smooth first partial derivatives, and continuous second derivatives. The splines exactly interpolate the data. The x and y knots will equal the number of x and y mesh values.

Extrapolation

The algorithm offers no extrapolation. Points outside the rectangular bounds of the data are mapped to the bounds and evaluated there.

Estimated Volumes

The algorithm computes exact integrals within the bounds of the data. If any of the integration limits are outside the bounds, the spline is used to interpolate a uniform grid of 10,000 points. A B-spline is fitted to this generated data and this second B-spline is then integrated. The precision error reported in the 3D Evaluation procedure will be the fractional difference with a similar 2,500 node integration. Note that such integrations will involve values that have been forced to the bounds. A repeat evaluation with the same limits will use a numeric double integration procedure. A very fast double Gaussian quadrature procedure is first attempted to 1E-5 precision. If this is unsuccessful, a double adaptive quadrature procedure is then used. This second evaluation is offered as a verification of accuracy.