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  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  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  or  coordinates within the plane. Thus, in general, two thin plate splines are needed to specify a two-dimensional coordinate transformation.

REFERENCES:

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.a