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

f(x,y)

is :

\int_^ \int_^ (\kappa_1^2 + \kappa_2^2) \sqrt \,dx \,dy

where

\kappa_1

and

\kappa_2

are the

principal curvatures In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by d ...

of the surface mapping

f

at the point

(x,y).

This is the

surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...

\kappa_1^2 + \kappa_2^2,

hence the

\sqrt

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

f

is 0. At any point where

f_x = f_y =0,

the

first fundamental form In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of . It permits the calculation of curvature and me ...

g_

of the surface mapping

f

is the identity matrix and the second fundamental form

b_

is :

\begin f_ & f_ \\ f_ & f_ \end

. We can use the formula for mean curvature

H=b_g^/2

to determine that

H = (f_+f_)/2

and the formula for

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...

K=b/g

(where

b

and

g

are the determinants of the second and first fundamental forms, respectively) to determine that

K=f_f_ - (f_)^2.

Since

H=(k_1+k_2)/2

and

K=k_1k_2,

the integrand of the exact TPEF equals

4H^2 - 2K.

The expressions we just computed for the mean curvature and Gaussian curvature as functions of partial derivatives of

f

show that the integrand of the exact TPEF is :

4H^2 - 2K = (f_ + f_)^2 - 2(f_f_ - f_^2) = f_^2 + 2f_^2 + f_^2.

So the approximate thin plate energy functional is :

= \int_^ \int_^ f_^2 + 2f_^2 + f_^2 \,dx \,dy.

Rotational invariance

The TPEF is rotationally invariant. This means that if all the points of the surface

z(x,y)

are rotated by an angle

\theta

about the

z

-axis, the TPEF at each point

(x,y)

of the surface equals the TPEF of the rotated surface at the rotated

(x,y).

The formula for a rotation by an angle

\theta

about the

z

-axis is The fact that the

z

value of the surface at

(x,y)

equals the

z

value of the rotated surface at the rotated

(x,y)

is expressed mathematically by the equation :

Z(X,Y) = z(x,y) = (z\circ xy)(X,Y)

where

xy

is the inverse rotation, that is,

xy(X,Y) = R^(X, Y)^ = R^(X,Y)^.

Z = z\circ xy

and the chain rule implies In equation (),

Z_0

means

Z_X,

Z_1

means

Z_Y,

z_0

means

z_x,

and

z_1

means

z_y.

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

z_j

is actually the composition

z_j \circ xy:

Z_ = z_R_ R_

. Swapping the index names

j

and

k

yields Expanding the sum for each pair

i,j

yields :

\begin Z_ & = &  R_^2 z_ + 2R_R_z_ + R_^2 z_, \\ Z_ & = & R_R_z_ + (R_R_ + R_R_)z_ + R_R_z_, \\ Z_ & = & R_^2 z_ + 2R_R_z_ + R_^2 z_. \end

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

R

from equation () into the right-hand side of equation () simplifies it to

z_^2 + 2 z_^2 + z_^2.

Data fitting

The approximate thin plate energy functional can be used to fit

B-spline In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expresse ...

surfaces to scattered 1D data on a 2D grid (for example, digital terrain model data). Call the grid points

(x_i,y_i)

for

i=1\dots N

(with

x_i \in,b /math> and y_i \in,d /math>) and the data values z_i. In order to fit a uniform B-spline f(x,y) to the data, the equation

 

(where \lambda is the "smoothing parameter") is minimized.  Larger values of \lambda 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. File:Original terrain data1.png, Original terrain data
File:Fitted bspline large lambda.png, Fitted B-spline surface with large lambda and more smoothing
File:Fitted bspline smaller lambda.png, Fitted B-spline surface with smaller lambda and less smoothing The thin plate smoothing spline also minimizes equation (), but it is much more expensive to compute than a B-spline and not as smooth (it is only C^1 at the "centers" and has unbounded second derivatives there).

References

{{reflist Splines (mathematics)