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 .
Suppose the points are in 2 dimensions (). One can use homogeneous coordinates for the point-set where a point is represented as a vector . The unique minimizer is parameterized by which comprises two matrices and ().
where d is a matrix representing the affine transformation (hence is a vector) and c is a warping coefficient matrix representing the non-affine deformation. The kernel function is a vector for each point , where each entry for each () dimensions. Note that for TPS, the control points are chosen to be the same as the set of points to be warped , so we already use in the place of the control points.
If one substitutes the solution for , becomes:
where and are just concatenated versions of the point coordinates and , and is a matrix formed from the . Each row of each newly formed matrix comes from one of the original vectors. The matrix represents the TPS kernel. Loosely speaking, the TPS kernel contains the information about the point-set's internal structural relationships. When it is combined with the warping coefficients , a non-rigid warping is generated.
A nice property of the TPS is that it can always be decomposed into a global affine and a local non-affine component. Consequently, the TPS smoothness term is solely dependent on the non-affine components. This is a desirable property, especially when compared to other splines, since the global pose parameters included in the affine transformation are not penalized.
The separation of the affine and non-affine warping space is done through a QR decomposition (Wahba,1990).
where Q1 and Q2 are and orthonormal matrices, respectively. The matrix is upper triangular. With the QR decomposition in place, we have
where is a matrix. Setting (which in turn implies that ) enables us to cleanly separate the first term in last third equation into a non-affine term and an affine term (first and second terms last equation respectively).
The least-squares energy function in the last equation can be first minimized w.r.t and then w.r.t. . By applying Tikhonov regularization we have
The minimum value of the TPS energy function obtained at the optimum is
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.