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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. 〔J. Duchon, 1976, Splines minimizing rotation invariant semi-norms in Sobolev spaces. pp 85–100, In: Constructive Theory of Functions of Several Variables, Oberwolfach 1976, W. Schempp and K. Zeller, eds., Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977〕
==Physical analogy==
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 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 or coordinates within the plane. In 2D cases, given a set of corresponding points, the TPS warp is described by parameters which include 6 global affine motion parameters and coefficients for correspondences of the control points. These parameters are computed by solving a linear system, in other words, TPS has closed-form solution.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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