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. Robust Point Matching (RPM) is a common extension and shortly known as the TPS-RPM algorithm.
Physical analogy The name thin plate spline refers to a physical analogy involving the bending of a plate or 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 z {\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 x {\displaystyle x} or y {\displaystyle y} coordinates within the plane. In 2D cases, given a set of K {\displaystyle K} corresponding control points (knots), the TPS warp is described by 2 ( K + 3 ) {\displaystyle 2(K+3)} parameters which include 6 global affine motion parameters and 2 K {\displaystyle 2K} coefficients for correspondences of the control points. These parameters are computed by solving a linear system, in other words, TPS has a closed-form solution.
Smoothness measure The TPS arises from consideration of the integral of the square of the second derivative—this forms its smoothness measure. In the case where x {\displaystyle x} is two dimensional, for interpolation, the TPS fits a mapping function f ( x ) {\displaystyle f(x)} between corresponding point-sets { y i } {\displaystyle \{y_{i}\}} and { x i } {\displaystyle \{x_{i}\}} that minimizes the following energy function:
E t p s ( f ) = ∑ i = 1 K ‖ y i − f ( x i ) ‖ 2 {\displaystyle E_{\mathrm {tps} }(f)=\sum _{i=1}^{K}\|y_{i}-f(x_{i})\|^{2}}
The smoothing variant, correspondingly, uses a tuning parameter λ {\displaystyle \lambda } to control the rigidity of the deformation, balancing the aforementioned criterion with the measure of goodness of fit, thus minimizing:
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