# Thin plate spline

Thin plate splines (TPS) are an interpolation and smoothing technique, the generalisation of splines so that they may be used with two or more dimensions. They were introduced to geometric design by Duchon (Duchon, 1976).

## 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 ${\displaystyle z}$ 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.

## 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 ${\displaystyle x}$ is two dimensional, for interpolation, the TPS fits a mapping function ${\displaystyle f(x)}$ between corresponding point-sets ${\displaystyle \{y_{i}\}}$ and ${\displaystyle \{x_{i}\}}$ that minimises the following energy function:

${\displaystyle E=\iint \left[\left({\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\right)^{2}+2\left({\frac {\partial ^{2}f}{\partial x_{1}\partial x_{2}}}\right)^{2}+\left({\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\right)^{2}\right]{\textrm {d}}x_{1}\,{\textrm {d}}x_{2}}$

The smoothing variant, correspondingly, uses a tuning parameter ${\displaystyle \lambda }$ to control how non-rigid is allowed for the deformation, balancing the aforementioned criterion with the measure of goodness of fit, thus minimising:

${\displaystyle E_{tps}(f)=\sum _{i=1}^{K}\|y_{i}-f(x_{i})\|^{2}+\lambda \iint \left[\left({\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\right)^{2}+2\left({\frac {\partial ^{2}f}{\partial x_{1}\partial x_{2}}}\right)^{2}+\left({\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\right)^{2}\right]{\textrm {d}}x_{1}\,{\textrm {d}}x_{2}}$

For this variational problem, it can be shown that there exists a unique minimizer ${\displaystyle f}$ (Wahba,1990).The finite element discretization of this variational problem, the method of elastic maps, is used for data mining and nonlinear dimensionality reduction.

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The Thin Plate Spline has a natural representation in terms of radial basis functions. Given a set of control points ${\displaystyle \{w_{i},i=1,2,\ldots ,K\}}$, a radial basis function basically defines a spatial mapping which maps any location ${\displaystyle x}$ in space to a new location ${\displaystyle f(x)}$, represented by,

${\displaystyle f(x)=\sum _{i=1}^{K}c_{i}\varphi (\left\|x-w_{i}\right\|)}$

where ${\displaystyle \left\|\cdot \right\|}$ denotes the usual Euclidean norm and ${\displaystyle \{c_{i}\}}$ is a set of mapping coefficients. The TPS corresponds to the radial basis kernel ${\displaystyle \varphi (r)=r^{2}\log r}$.

## Application

TPS has been widely used as the non-rigid transformation model in image alignment and shape matching.

The popularity of TPS comes from a number of advantages:

1. The interpolation is smooth with derivatives of any order.
2. The model has no free parameters that need manual tuning.
3. It has closed-form solutions for both warping and parameter estimation.
4. There is a physical explanation for its energy function.