Thin plate energy functional


The exact thin plate energy functional for a function is
where and are the principal curvatures of the surface mapping at the point This is the surface integral of hence the in the integrand.
Minimizing the exact thin plate energy functional would result in a system of non-linear equations. So in practice, an approximation that results in linear systems of equations is often used. The approximation is derived by assuming that the gradient of is 0. At any point where the first fundamental form of the surface mapping is the identity matrix and the second fundamental form is
We can use the formula for mean curvature to determine that and the formula for Gaussian curvature to determine that Since and the integrand of the exact TPEF equals The expressions we just computed for the mean curvature and Gaussian curvature as functions of partial derivatives of show that the integrand of the exact TPEF is
So the approximate thin plate energy functional is

Rotational invariance

The TPEF is rotationally invariant. This means that if all the points of the surface are rotated by an angle about the -axis, the TPEF at each point of the surface equals the TPEF of the rotated surface at the rotated The formula for a rotation by an angle about the -axis is
The fact that the value of the surface at equals the value of the rotated surface at the rotated is expressed mathematically by the equation
where is the inverse rotation, that is, So and the chain rule implies
In equation, means means means and means Equation and all subsequent equations in this section use non-tensor summation convention, that is, sums are taken over repeated indices in a term even if both indices are subscripts. The chain rule is also needed to differentiate equation since is actually the composition
Swapping the index names and yields

Expanding the sum for each pair yields
Computing the TPEF for the rotated surface yields

Inserting the coefficients of the rotation matrix from equation into the right-hand side of equation simplifies it to

Data fitting

The approximate thin plate energy functional can be used to fit B-spline surfaces to scattered 1D data on a 2D grid. Call the grid points for and the data values In order to fit a uniform B-spline to the data, the equation
is minimized. Larger values of result in a smoother surface and smaller values result in a more accurate fit to the data. The following images illustrate the results of fitting a B-spline surface to some terrain data using this method.
The thin plate smoothing spline also minimizes equation, but it is much more expensive to compute than a B-spline and not as smooth.