A spectral interpolation scheme on the unit sphere based on the nodes of spherical Lissajous curves

TL;DR

This paper introduces a spectral interpolation and quadrature scheme on the sphere using spherical Lissajous curve nodes, providing mathematical analysis, efficient implementation, and an application to object rotation estimation.

Contribution

It develops a novel spectral interpolation method on the sphere based on spherical Lissajous nodes, with proven unisolvence and efficient computation.

Findings

Interpolation scheme has logarithmic growth in condition number

Scheme is based on a discrete orthogonality structure

Application demonstrated in object rotation estimation

Abstract

For sampling values along spherical Lissajous curves we establish a spectral interpolation and quadrature scheme on the sphere. We provide a mathematical analysis of spherical Lissajous curves and study the characteristic properties of their intersection points. Based on a discrete orthogonality structure we are able to prove the unisolvence of the interpolation problem. As basis functions for the interpolation space we use a parity-modified double Fourier basis on the sphere which allows us to implement the interpolation scheme in an efficient way. We further show that the numerical condition number of the interpolation scheme displays a logarithmic growth. As an application, we use the developed interpolation algorithm to estimate the rotation of an object based on measurements at the spherical Lissajous nodes.