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, planar spline interpolations based on projections of the sphere onto a plane are unsatisfactory, and spherical splines are desired. Wahba (1981) defined the thin-plate splines on the sphere by analogy with the polynomial splines on the circle and 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 its 'wiggliness'. 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) of functions determined by this wiggliness penalty and to which corresponds an isotropic kernel, for which closed-form expressions (in terms of the polylogarithm) were found by Wendelberger (1982) and re-discovered by Beatson and zu Castell (2018). These closed-form expressions make not just spline interpolation but also downstream signal-processing tasks, such as cubature or solving inverse problems, more tractable in fields where scattered data and spherical models are common, such as remote sensing, geostatistics, oceanography, meteorology, motion planning, graphics, and medical imaging. In this paper, we present a tutorial on spline methods in RKHSs and show how they can be used to interpolate, smooth, and numerically integrate scattered data on the sphere and solve related inverse problems. The accompanying demo compares thin-plate spline interpolation over the sphere using these closed-form expressions for the kernel, thin-plate splines on an equirectangular projection, and natural cubic splines on a one-dimensional latitudinal projection used in greenhouse gas monitoring. Global mean values of the interpolation surfaces are presented as well, to illustrate how this isotropic spherical kernel - which penalizes wiggliness without concern for application-specific factors like atmospheric winds - affects the computation of global averages.