Thin plate splines (TPS) are a spline-based technique for data interpolation and smoothing.
"A spline is a function defined by polynomials in a piecewise manner.
[3] 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.
[4] The name thin plate spline refers to a physical analogy involving the bending of a plate or 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 order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the
corresponding control points (knots), the TPS warp is described by
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.
The TPS arises from consideration of the integral of the square of the second derivative—this forms its smoothness measure.
is two dimensional, for interpolation, the TPS fits a mapping function
that minimizes the following energy function: The smoothing variant, correspondingly, uses a tuning parameter
to control the rigidity of the deformation, balancing the aforementioned criterion with the measure of goodness of fit, thus minimizing:[1][2] For this variational problem, it can be shown that there exists a unique minimizer
[5] The finite element discretization of this variational problem, the method of elastic maps, is used for data mining and nonlinear dimensionality reduction.
"[1][2] It is in a general case needed to make the mapping unique.
The thin plate spline has a natural representation in terms of radial basis functions.
The TPS corresponds to the radial basis kernel
One can use homogeneous coordinates for the point-set where a point
warping coefficient matrix representing the non-affine deformation.
are chosen to be the same as the set of points to be warped
are just concatenated versions of the point coordinates
Each row of each newly formed matrix comes from one of the original vectors.
Loosely speaking, the TPS kernel contains the information about the point-set's internal structural relationships.
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.
TPS has been widely used as the non-rigid transformation model in image alignment and shape matching.
[6] An additional application is the analysis and comparisons of archaeological findings in 3D[7] and was implemented for triangular meshes in the GigaMesh Software Framework.
[8] The thin plate spline has a number of properties which have contributed to its popularity: However, note that splines already in one dimension can cause severe "overshoots".