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Wikipedia / 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. They are an important special case of a polyharmonic spline. Robust Point Matching (RPM) is a common extension and shortly known as the TPS-RPM algorithm.

Physical analogy

The name thin plate spline refers to a physical analogy involving the bending of a plate coefficients for correspondences of the control points. These parameters are computed by solving a linear system, in other words, TPS has a closed-form solution.

Smoothness measure

that minimizes the following energy function:

to control the rigidity of the deformation, balancing the aforementioned criterion with the measure of goodness of fit, thus minimizing:

. The finite element discretization of this variational problem, the method of elastic maps, is used for data mining and nonlinear dimensionality reduction." It is in a general case needed to make the mapping unique.

Radial basis function

, represented by

denotes the usual Euclidean norm.

Spline

). One can use homogeneous coordinates).

matrix representing the affine transformation warping coefficient matrix in the place of the control points.

becomes:

, 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.

Applications

TPS has been widely used as the non-rigid transformation model in image alignment and shape matching. An additional application is the analysis and comparisons of archaeological findings in 3D and was implemented for triangular meshes in the GigaMesh Software Framework.

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.

However, note that splines already in one dimension can cause severe "overshoots". In 2D such effects can be much more critical, because TPS are not objective.

See also

References