High Quality Content by WIKIPEDIA articles! This is a brief derivation for the closed form solutions for smoothing Thin Plate Spline. Details about these splines can be found in (Wahba, 1990). Thin plate splines (TPS) were introduced to geometric design by Duchon (Duchon, 1976). The name thin plate spline refers to a physical analogy involving the bending of a thin sheet of metal. In the physical setting, the deflection is in the z direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the x or y coordinates within the plane. In 2D cases, given a set of K corresponding points, the TPS warp is described by 2(K + 3) parameters which include 6 global affine motion parameters and 2K coefficients for correspondences of the control points. These parameters are computed by solving a linear system, in other words, TPS has closed-form solution.

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