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 U(r) = r^2 ln r. 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 R^2 of the squares of the second derivatives, I[f(x, y)] = integral integral_R^2(f_(x x)^2 + 2f_(x y)^2 + f_(y y)^2) d x d y.
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