Optimal-Dimensionality Sampling on the Sphere: Improvements and Variations

TL;DR

This paper introduces methods to improve the stability and accuracy of the spherical harmonic transform for band-limited signals on the sphere by optimizing sample placement and iterative computation techniques.

Contribution

It proposes a new sample placement strategy and a multi-pass algorithm to enhance the conditioning and accuracy of the optimal-dimensionality spherical harmonic transform.

Findings

Improved stability of linear system inversion in SHT computation.

Enhanced accuracy of SHT through iterative multi-pass algorithm.

Maintained computational complexity with significant accuracy gains.

Abstract

For the accurate representation and reconstruction of band-limited signals on the sphere, an optimal-dimensionality sampling scheme has been recently proposed which requires the optimal number of samples equal to the number of degrees of freedom of the signal in the spectral (harmonic) domain. The computation of the spherical harmonic transform (SHT) associated with the optimal-dimensionality sampling requires the inversion of a series of linear systems in an iterative manner. The stability of the inversion depends on the placement of iso-latitude rings of samples along co-latitude. In this work, we have developed a method to place these iso-latitude rings of samples with the objective of improving the well-conditioning of the linear systems involved in the computation of the SHT. We also propose a multi-pass SHT algorithm to iteratively improve the accuracy of the SHT of band-limited signals. Furthermore, we review the changes in the computational complexity and improvement in accuracy of the SHT with the embedding of the proposed methods. Through numerical experiments, we illustrate that the proposed variations and improvements in the SHT algorithm corresponding to the optimal-dimensionality sampling scheme significantly enhance the accuracy of the SHT.