Compressive Sensing with Wigner $D$-functions on Subsets of the Sphere

TL;DR

This paper establishes a compressive sensing framework on the rotation group $ ext{SO}(3)$ using Slepian functions on measurement subsets, enabling accurate signal reconstruction with fewer measurements than traditional methods.

Contribution

It introduces a novel approach transforming the inverse problem to Slepian functions, reducing measurement requirements and improving accuracy for signals concentrated on specific sub-domains of $ ext{SO}(3)$.

Findings

Outperforms classical compressive sensing methods in reconstruction quality.

Requires fewer measurements while maintaining accuracy.

Effective for signals well concentrated on the measurement sub-domain.

Abstract

In this paper, we prove a compressive sensing guarantee for restricted measurement domains on the rotation group, $\mathrm{SO}(3)$. We do so by first defining Slepian functions on a measurement sub-domain $R$ of the rotation group $\mathrm{SO}(3)$. Then, we transform the inverse problem from the measurement basis, the bounded orthonormal system of band-limited Wigner $D$-functions on $\mathrm{SO}(3)$, to the Slepian functions in a way that limits increases to signal sparsity. Contrasting methods using Wigner $D$-functions that require measurements on all of $\mathrm{SO}(3)$, we show that the orthogonality structure of the Slepian functions only requires measurements on the sub-domain $R$, which is select-able. Due to the particulars of this approach and the inherent presence of Slepian functions with low concentrations on $R$, our approach gives the highest accuracy when the signal under study is well concentrated on $R$. We provide numerical examples of our method in comparison with other classical and compressive sensing approaches. In terms of reconstruction quality, we find that our method outperforms the other compressive sensing approaches we test and is at least as good as classical approaches but with a significant reduction in the number of measurements.