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Compressive sensing on the rotation group and sphere with applications to electromagnetic and acoustic field measurements
Valdez, Marc Andrew
Valdez, Marc Andrew
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2024
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Abstract
The focus of this thesis is the development of practical compressive sensing guarantees for device radiation pattern characterizations in electromagnetics and acoustics. Compressive sensing is the acquisition and processing of a signal of interest by under-sampling or measuring a compressed version of the signal. In this thesis, a practical compressive sensing guarantee uses a minimal number of assumptions about the device being characterized, and the guarantee incorporates the limitations of existing measurement systems. The latter of these two criteria is novel in the compressive sensing literature for radiation pattern characterizations.
In broad terms, the work in this thesis develops compressive sensing guarantees where measurements are required to satisfy particular sample domain structures. The direct focus of this work is compressive sensing for one or more signals on the sphere S2 or rotation group SO(3), but the methods and results contained herein may be used more generally. With these facts in mind, this thesis presents four results for radiation pattern characterization: guaranteed compressive signal reconstruction using measurements from a subdomain of S2 or SO(3); two approaches to generate discrete sample patterns that enable compressive sensing guarantees and fast reconstruction algorithms on S2 and SO(3); and compressive measurement and reconstruction approaches for joint signal reconstruction on S2 and SO(3) that surpass the performance of single-signal compressive sensing, i.e., highly sample-efficient broadband radiation pattern characterizations.
The first result addresses the problem of measurement restrictions in spherical field measurements. In many spherical field measurement systems, support structures and measurement devices restrict where measurements can be taken on S2 or SO(3). We develop a compressive sensing approach for this situation. In particular, we express the spherical field measurement linear inverse problem in terms of localized basis functions and prove reconstruction guarantees in this context. Similar to fully sampled restricted measurement approaches, our method can be used when the signal of interest has little energy outside of the measurable region of S2 or SO(3). Moreover, we show that our approach can achieve comparable or better reconstruction accuracy when compared to the fully sampled counterparts.
The second and third results are different approaches to the problem. These results give compressive sensing guarantees using discrete sampling patterns on S2 or SO(3) that enable fast solvers for the compressive version of the linear inverse problem required by spherical field measurements. The sampling patterns used in these guarantees are either identical to patterns already used in spherical field measurements or only slight modifications to existing sampling points. Additionally, the latter of the two approaches achieves, up to constant factors, the same performance as existing guarantees that require the demanding and unrealistic use of continuous random sampling from S2 or SO(3).
The final result establishes compressive sensing approaches for broadband radiation pattern characterizations. This problem amounts to jointly sensing many signals on S2 or SO(3) at once to achieve higher rates of compression than single-frequency sensing. We show that joint compressive sensing can be achieved when all signals of interest on S2 or SO(3) satisfy a joint structure or low-dimensional model. Moreover, we show that common radiation patterns of interest satisfy these structural requirements due to smooth changes in the radiation pattern with frequency, thus achieving compressive sensing guarantees for broadband radiation pattern characterizations that use significantly fewer measurements than many single-frequency compressive characterizations.
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