Thin plate energy functional

The exact thin plate energy functional (TPEF) for a function

are the principal curvatures of the surface mapping

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.

[1][3][4] The approximation is derived by assuming that the gradient of

is the identity matrix and the second fundamental form

is We can use the formula for mean curvature

and the formula for Gaussian curvature

{\displaystyle K=f_{xx}f_{yy}-(f_{xy})^{2}.}

[5] 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 The TPEF is rotationally invariant.

This means that if all the points of the surface

The formula for a rotation by an angle

is expressed mathematically by the equation where

and the chain rule implies In equation (2),

Equation (2) 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 (2) since

Swapping the index names

yields Expanding the sum for each pair

yields Computing the TPEF for the rotated surface yields Inserting the coefficients of the rotation matrix

{\displaystyle z_{xx}^{2}+2z_{xy}^{2}+z_{yy}^{2}.}

The approximate thin plate energy functional can be used to fit B-spline surfaces to scattered 1D data on a 2D grid (for example, digital terrain model data).

[6][3] Call the grid points

In order to fit a uniform B-spline

is the "smoothing parameter") is minimized.

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 (5), but it is much more expensive to compute than a B-spline and not as smooth (it is only

at the "centers" and has unbounded second derivatives there).