Accurate Reconstruction of Finite Rate of Innovation Signals on the Sphere

TL;DR

This paper presents a new method for accurately reconstructing finite rate of innovation signals, specifically Dirac functions on the sphere, using fewer samples and achieving significantly higher accuracy than existing techniques.

Contribution

The authors introduce an annihilating filter-based approach for sphere signals that improves conditioning and reduces sampling requirements compared to prior methods.

Findings

Achieves more accurate reconstruction by a factor of 10^7 or more.

Requires the same or fewer samples than existing methods.

Demonstrates superior performance through numerical experiments.

Abstract

We develop a method for the accurate reconstruction of non-bandlimited finite rate of innovation signals on the sphere. For signals consisting of a finite number of Dirac functions on the sphere, we develop an annihilating filter based method for the accurate recovery of parameters of the Dirac functions using a finite number of observations of the bandlimited signal. In comparison to existing techniques, the proposed method enables more accurate reconstruction primarily due to better conditioning of systems involved in the recovery of parameters. For the recovery of $K$ Diracs on the sphere, the proposed method requires samples of the signal bandlimited in the spherical harmonic~(SH) domain at SH degree equal or greater than $ K + \sqrt{K + \frac{1}{4}} - \frac{1}{2}$. In comparison to the existing state-of-the art technique, the required bandlimit, and consequently the number of samples, of the proposed method is the same or less. We also conduct numerical experiments to demonstrate that the proposed technique is more accurate than the existing methods by a factor of $10^{7}$ or more for $2 \le K\le 20$.