[IPOL announce] new article: Thin-plate Splines on the Sphere for Interpolation, Computing Spherical Averages, and Solving Inverse Problems

[announcements about the IPOL journal]

A new article is available in IPOL: https://www.ipol.im/pub/art/2026/451/

Max Dunitz,
Thin-plate Splines on the Sphere for Interpolation, Computing Spherical 
Averages, and Solving Inverse Problems,
Image Processing On Line, 16 (2026), pp. 1–112.
https://doi.org/10.5201/ipol.2026.451

Abstract
In many applications, planar spline interpolations of scattered data on 
the sphere are unsatisfactory; 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 as well as its 'wiggliness'. This 
latter term is the square of a seminorm penalty based on the 
Laplace-Beltrami operator. The minimization problem is posed in a 
reproducing kernel Hilbert space (RKHS) of functions of finite 
wiggliness, whose reproducing kernel is isotropic and, due to a result 
by Schoenberg (1942), given by a Legendre series. A closed-form 
expression (in terms of the polylogarithm) for the kernel was 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 resolution 
of inverse problems, more tractable in fields where scattered data and 
spherical models are common, such as remote sensing, geostatistics, 
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 with 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 interpolant wiggliness without concern for 
application-specific factors like atmospheric winds - affects the 
computation of global averages.