Thin plate spline
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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.
The TPS arises from consideration of the integral of the square of the second derivative -- this forms its smoothness measure. In the case where is two dimensional, for interpolation, the TPS fits a mapping function between corresponding point-sets and that minimises the following energy function:
The smoothing variant, correspondingly, uses a tuning parameter to control how non-rigid is allowed for the deformation, balancing the aforementioned criterion with the measure of goodness of fit, thus minimising:
For this variational problem, it can be shown that there exists a unique minimizer (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
The Thin Plate Spline has a natural representation in terms of radial basis functions. Given a set of control points , a radial basis function basically defines a spatial mapping which maps any location in space to a new location , represented by,
where denotes the usual Euclidean norm and is a set of mapping coefficients. The TPS corresponds to the radial basis kernel .
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:
- It produces smooth surfaces, which are infinitely differentiable.
- There are no free parameters that need manual tuning.
- It has closed-form solutions for both warping and parameter estimation.
- There is a physical explanation for its energy function.
- Inverse distance weighting
- Radial basis function
- Subdivision surface (emerging alternative to spline-based surfaces)
- Elastic map (a discrete version of the thin plate approximation for manifold learning)
- Polyharmonic spline (the thin-plate-spline is a special case of a polyharmonic spline)
- 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.