Thin plate spline

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Thin plate splines (TPS) are a spline-based technique for data interpolation and smoothing. They were introduced to geometric design by Duchon. [1]

Physical analogy[edit]

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 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.

Smoothness measure[edit]

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 is two dimensional, for interpolation, the TPS fits a mapping function f(x) between corresponding point-sets \{y_i\} and \{x_i\} that minimises the following energy function:


	E_{tps}(f) = \sum_{i=1}^K \|y_i - f(x_i) \|^2

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


	E_{tps,smooth}(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 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.

Radial basis function[edit]

Main article: Radial basis function

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


	f(x) = \sum_{i = 1}^K c_{i}\varphi(\left\| x - w_{i}\right\|)

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


Application[edit]

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

The Thin-plate-spline has a number of properties which have contributed to its popularity:

  1. It produces smooth surfaces, which are infinitely differentiable.
  2. There are 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.

See also[edit]

References[edit]

  1. ^ 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
  • Haili Chui: Non-Rigid Point Matching: Algorithms, Extensions and Applications. PhD Thesis, Yale University, May 2001.
  • G. Wahba, 1990, Spline models for observational data. Philadelphia: Society for Industrial and Applied Mathematics.

External links[edit]