Thin plate splines

The thin plate spline method is a popular data fitting technique as it is insensitive to noise in the data. The method has been successfully used in many different applications including; data mining, 3D reconstruction of geometric models, finger print matching, image warping, medical image analysis and optic flow computations.

A problem with the traditional formulation of the thin plate spline is that the resulting system of equations is dense and its size depends on the number of data points. Consequently, when working with large data sets, such as a digital scan of a three dimensional object, the method may break down. It either takes too long to fit the data, or the problem is too large to fit in memory. This is true even on a modern computer. 

We have developed a discrete thin plate spline method that utilise the technology developed for the finite element method. In particular our discrete thin plate spline method uses polynomials with local support defined on finite element grids. The resulting system of equations is sparse and its size depends only on the number of nodes in the finite element grid, so this method is efficient when dealing with large data sets.

In our project we are interested in using finite element analysis to better understand the properties of the discrete thin spline method, as well as incorporating more general numerical techniques to ensure an efficient implementation that will work on real life applications. We are also interested in generalising the finite element formulation so that the method may be applied to examples beyond the traditional application areas of the the standard thin plate spline formulation.