Thinplate Splines on the Sphere

TL;DR

This paper derives explicit closed-form formulas for semi-reproducing kernels of thinplate spline interpolation on the sphere, improving computational efficiency and practical applicability in spherical data analysis.

Contribution

It provides recurrence relations and closed-form expressions for the kernels, advancing the computational methods for spherical thinplate splines.

Findings

Closed-form kernel expressions are faster to evaluate than series.

Recurrence relations enable efficient computation of kernels.

Enhanced practicality of thinplate splines on the sphere for applications.

Abstract

In this paper we give explicit closed forms for the semi-reproducing kernels associated with thinplate spline interpolation on the sphere. Polyharmonic or thinplate splines for ${\mathbb R}^d$ were introduced by Duchon and have become a widely used tool in myriad applications. The analogues for ${\mathbb S}^{d-1}$ are the thin plate splines for the sphere. The topic was first discussed by Wahba in the early 1980's, for the ${\mathbb S}^2$ case. Wahba presented the associated semi-reproducing kernels as infinite series. These semi-reproducing kernels play a central role in expressions for the solution of the associated spline interpolation and smoothing problems. The main aims of the current paper are to give a recurrence for the semi-reproducing kernels, and also to use the recurrence to obtain explicit closed form expressions for many of these kernels. The closed form expressions will in many cases be significantly faster to evaluate than the series expansions. This will enhance the practicality of using these thinplate splines for the sphere in computations.