Approximation Methods for Thin Plate Spline Mappings and Principal Warps

The thin plate spline (TPS) is an effective tool for modeling coordinate transformations that has been applied successfully in several computer vision applications. Unfortunately the solution requires the inversion of a p × p matrix, where p is the number of points in the data set, thus making it impractical for large scale applications. In practical applications, however, a surprisingly good approximate solution is often possible using only a small subset of corresponding points. We begin by discussing the obvious approach of using this subset to estimate a transformation that is then applied to all the points, and we show the drawbacks of this method. We then proceed to borrow a technique from the machine learning community for function approximation using radial basis functions (RBFs) and adapt it to the task at hand. Using this method, we demonstrate a significant improvement over the naive method. One drawback of this method, however, is that is does not allow for principal warp analysis, a technique for studying shape deformations introduced by Bookstein based on the eigenvectors of the p× p bending energy matrix. To address this, we describe a third approximation method based on a classic matrix completion technique that allows for principal warp analysis as a by-product. By means of experiments on real and synthetic data, we demonstrate the pros and cons of these different approximations so as to allow the reader to make an informed decision suited to his or her application.