Examples
The next figure shows the interpolation through four points (marked by "circles") using different types of polyharmonic splines. The "curvature" of the interpolated curves grows with the order of the spline and the extrapolation at the left boundary (x < 0) is reasonable. The figure also includes the radial basis functions phi = exp(-r2) which gives a good interpolation as well. Finally, the figure includes also the non-polyharmonic spline phi = r2 to demonstrate, that this radial basis function is not able to pass through the predefined points (the linear equation has no solution and is solved in a least squares sense).
The next figure shows the same interpolation as in the first figure, with the only exception that the points to be interpolated are scaled by a factor of 100 (and the case phi = r2 is no longer included). Since phi = (scale*r)k = (scalek)*rk, the factor (scalek) can be extracted from matrix A of the linear equation system and therefore the solution is not influenced by the scaling. This is different for the logarithmic form of the spline, although the scaling has not much influence. This analysis is reflected in the figure, where the interpolation shows not much differences. Note, for other radial basis functions, such as phi = exp(-k*r2) with k=1, the interpolation is no longer reasonable and it would be necessary to adapt k.
The next figure shows the same interpolation as in the first figure, with the only exception that the polynomial term of the function is not taken into account (and the case phi = r2 is no longer included). As can be seen from the figure, the extrapolation for x < 0 is no longer as "natural" as in the first figure for some of the basis functions. This indicates, that the polynomial term is useful if extrapolation occurs.
Read more about this topic: Polyharmonic Spline
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