Compressive Sensing With Wigner D-Functions on Subsets of the Sphere

• Published in: IEEE transactions on signal processing Vol. 70; pp. 5652 - 5667
• Main Authors: Valdez, Marc Andrew, Yuffa, Alex J., Wakin, Michael B.
• Format: Journal Article
• Language: English
• Published: IEEE 2022
• Subjects: Acoustic measurements; Antenna measurements; antenna metrology; Atmospheric measurements; Compressed sensing; Compressive sensing; Particle measurements; Probes; Rotation measurement; slepian functions
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2022.3223848

In this paper, we prove a compressive sensing guarantee for restricted measurement domains on the rotation group, <inline-formula><tex-math notation="LaTeX">\mathrm{SO}(\text{3})</tex-math></inline-formula>. We do so by first defining Slepian functions on a measurement sub-domain <inline-formula><tex-math notation="LaTeX">R</tex-math></inline-formula> of the rotation group <inline-formula><tex-math notation="LaTeX">\mathrm{SO}(\text{3})</tex-math></inline-formula>. Then, we transform the inverse problem from the measurement basis, the bounded orthonormal system of band-limited Wigner <inline-formula><tex-math notation="LaTeX">D</tex-math></inline-formula>-functions on <inline-formula><tex-math notation="LaTeX">\mathrm{SO}(\text{3})</tex-math></inline-formula>, to the Slepian functions in a way that limits increases to signal sparsity. Contrasting methods using Wigner <inline-formula><tex-math notation="LaTeX">D</tex-math></inline-formula>-functions that require measurements on all of <inline-formula><tex-math notation="LaTeX">\mathrm{SO}(\text{3})</tex-math></inline-formula>, we show that the orthogonality structure of the Slepian functions only requires measurements on the sub-domain <inline-formula><tex-math notation="LaTeX">R</tex-math></inline-formula>, which is select-able. Due to the particulars of this approach and the inherent presence of Slepian functions with low concentrations on <inline-formula><tex-math notation="LaTeX">R</tex-math></inline-formula>, our approach gives the highest accuracy when the signal under study is well concentrated on <inline-formula><tex-math notation="LaTeX">R</tex-math></inline-formula>. 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.