biharmonic spline interpolation

A biharmonic spline interpolation is an interpolation of irregularly spaced two-dimensional data points. The interpolating surface is a linear combination of Green functions centered at each data point. The amplitudes of the Green functions are found by solving a linear system of equations.

The surface s(x) is expressed as

s(x)=Σj=1n wj g(x, xj),

where n is the number of data points xj = (xj, yj), g(x, xj) is Green's function and wj is the weight of data point j. The weights wj are determined by requiring that the surface s(x) passes exactly through the n data points, i.e.

s(xi)=Σj=1n wj g(xi, xj), i = 1, 2, ..., n.

This yields an n×n square linear system of equations which can be solved for the wj.

For twodimensional data Green's function is:

g(xi, xj) = |xi - xj|2 (ln |xi - xj| - 1.0).

See Sandwell (1987) and Deng & Tang (2011) for more information.