Thin Plate Spline

The thin plate spline is the two-dimensional analog of the cubic spline in one dimension. It is the fundamental solution to the biharmonic equation, and has the form


Given a set of data points, a weighted combination of thin plate splines centered about each data point gives the interpolation function that passes through the points exactly while minimizing the so-called "bending energy." Bending energy is defined here as the integral over R^2 of the squares of the second derivatives,


Regularization may be used to relax the requirement that the interpolant pass through the data points exactly.

The name "thin plate spline" refers to a physical analogy involving the bending of a thin sheet of metal. In the physical setting, the deflection is in the z direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the x or y coordinates within the plane. Thus, in general, two thin plate splines are needed to specify a two-dimensional coordinate transformation.

See also

Cubic Spline, Spline

This entry contributed by Serge Belongie

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Bookstein, F. L. "Principal Warps: Thin Plate Splines and the Decomposition of Deformations." IEEE Trans. Pattern Anal. Mach. Intell. 11, 567-585, 1989.Duchon, J. "Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces." RAIRO Analyse Numérique 10, 5-12, 1976.Meinguet, J. "Multivariate Interpolation at Arbitrary Points Made Simple." J. Appl. Math. Phys. 30, 292-304, 1979.Wahba, G. Spline Models for Observational Data. Philadelphia, PA: SIAM, 1990.

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Thin Plate Spline

Cite this as:

Belongie, Serge. "Thin Plate Spline." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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