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I am not sure I understand what Thin-Plate-Splines are.

I thought it was the name of a regularization technique for b-splines surface fitting (i.e. approximation smoothing, not exact interpolation). But reading papers, I understand they are a new type of splines (i.e. not b-splines), built out of the linear combination of RBF (Radial Basis Functions) - and the addition of a plane, to offset things.

I am a bit lost. Can anyone shed some light on this?

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Splines that are derived from minimizing the integral of squared 2nd derivatives can be called thin-plate splines. Whether the surface is of B-spline basis or composed of radial basis functions does not matter.

However, it does seem conventionally that only surfaces interpolating a given set of points (while minimizing the integral of squared 2nd derivatives) is referred as thin-plate splines. For surface fitting from minimizing a weighted sum of least squared error and the integral of squared 2nd derivatives, they are typically not referred as thin-plate splines probably because the least squared error term is typically the dominant error to be minimized.

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