Thinplate Splines on the Sphere

Symmetry Integrability and Geometry-Methods and Applications

article

Rick K. Beatson¹  Wolfgang zu Castell² 

[1] School of Mathematics and Statistics, University of Canterbury, Private Bag 4800;Scientific Computing Research Unit

关键词: positive definite functions;    zonal functions;    thinplate splines;    ultraspherical expansions;    Gegenbauer polynomials;   

DOI  :  10.3842/SIGMA.2018.083

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown