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.

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 Numérique10, 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.