Thin-plate Splines on the Sphere for Interpolation, Computing Global Averages, and Solving Inverse Problems
Max Dunitz
⚠ This is a preprint. It may change before it is accepted for publication.


In many applications, most notably in the geosciences, planar spline interpolations based on projections of the sphere onto a plane are unsatisfactory, and spherical splines are desired. In the early 1980s, Wahba defined the thin-plate splines on the sphere as an extension of the one-dimensional periodic polynomial splines and by analogy with the thin-plate splines in Rd. The thin-plate spline fit to a scattered data set on the sphere is the solution to an empirical risk minimization problem that penalizes the infidelity of the fit to the data and the 'wiggliness' of the fit; this latter term is the square of a seminorm based on the Laplace-Beltrami operator. The minimization problem is posed in a reproducing kernel Hilbert space (RKHS) that is determined by this wiggliness penalty. Since the kernel associated with this RKHS takes the form of an infinite series that converges slowly, Wahba proposed a 'pseudo-spline', found by tweaking the wiggliness penalty so that a reproducing kernel in closed form could be obtained. In addition to facilitating spline interpolation, closed-form expressions for the reproducing kernel can make many signal processing tasks, such as cubature or solving inverse problems, tractable. In the intervening years, Wendelberger (1982), Martinez-Morales (2005), and Beatson and zu Castell (2018) have found closed forms (in terms of special functions) of certain thin-plate splines on the sphere. In this paper, we present a tutorial on spline methods in reproducing kernel Hilbert spaces (RKHSs) and show how they can be used to interpolate, smooth, and numerically integrate scattered data on the sphere and solve related inverse problems. Spherical models are common in many fields such as remote sensing, oceanography, meteorology, motion planning, graphics, and medicine, and the applications of splines to scattered data problems on the sphere is facilitated by a little-known result by Wendelberger, later rediscovered and extended by Beatson and zu Castell.