Thin plate spline

Thin plate splines (TPS) are a spline-based technique for data interpolation and smoothing. They were introduced to geometric design by Duchon. They are an important special case of a polyharmonic spline.

The name thin plate spline refers to a physical analogy involving the bending of a thin sheet of metal. Just as the metal has rigidity, the TPS fit resists bending also, implying a penalty involving the smoothness of the fitted surface. In the physical setting, the deflection is in the ${\displaystyle z}$ 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 ${\displaystyle x}$ or ${\displaystyle y}$ coordinates within the plane. In 2D cases, given a set of ${\displaystyle K}$ corresponding points, the TPS warp is described by ${\displaystyle 2(K+3)}$ parameters which include 6 global affine motion parameters and ${\displaystyle 2K}$ 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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