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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 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[5] to determine that and the formula for Gaussian curvature[5] (where and are the determinants of the second and first fundamental forms, respectively) to determine that Since and [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
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Transcription
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
(1)
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
(2)
In equation (2), means means means and means 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 is actually the composition
.
Swapping the index names and yields
(3)
Expanding the sum for each pair yields
Computing the TPEF for the rotated surface yields
(4)
Inserting the coefficients of the rotation matrix from equation (1) into the right-hand side of equation (4) 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 (for example, digital terrain model data).[6][3] Call the grid points for (with and ) and the data values In order to fit a uniform B-spline to the data, the equation
(5)
(where is the "smoothing parameter") 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.
Original terrain data
Fitted B-spline surface with large lambda and more smoothing
Fitted B-spline surface with smaller lambda and less smoothing
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).