Discrete thin plate spline smoothing in 3D

Abstract

The thin plate spline method is often used to fit data in high dimensions. Standard thin plate splines require the solution of a dense linear system of equations whose size increases with the number of data points and can be expensive when used on large data sets. In this paper we present a discrete thin plate spline method that 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. Theory is developed for general $d$-dimensional data sets and several example results are given for 3D models.