Compressed Sensing and ΣΔ-Quantization

von Joe-Mei Feng
Dissertation
Datum der mündl. Prüfung:2018-02-12
Erschienen:2019-02-11
Betreuer:Prof. Dr. Felix Krahmer
Gutachter:Prof. Dr. Gerlind Plonka-Hoch
Gutachter:Dr. Sjoerd Dirksen
Gutachter:Prof. Dr. Anita Schöbel
Gutachter:Prof. Dr. Russell Luke
Gutachter:Dr. Timo Aspelmeier
Zum Verlinken/Zitieren: http://dx.doi.org/10.53846/goediss-7272

 

 

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Zusammenfassung

Englisch

The main issue of my thesis is to bound the error while recovering signals from their compressed and quantized form. Especially my central contribution is that, together with my co-authors, we provide the first analysis of a non-trivial quantization scheme for compressed sensing measurements arising from structured measurements. Specifically, our analysis studies compressed sensing matrices consisting of rows selected at random, without replacement, from a circulant matrix generated by a random subgaussian vector. We quantize the measurements using stable, possibly one-bit, Sigma-Delta schemes, and use a reconstruction method based on convex optimization. We show that the part of the reconstruction error due to quantization decays polynomially in the number of measurements. This is in-line with analogous results on Sigma-Delta quantization associated with random Gaussian or subgaussian matrices, and significantly better than results associated with the widely assumed memoryless scalar quantization. Moreover, we prove that our approach is stable and robust; i.e., the reconstruction error degrades gracefully in the presence of non-quantization noise and when the underlying signal is not strictly sparse. The analysis relies on results concerning subgaussian chaos processes as well as a variation of McDiarmid's inequality. Also my co- author and I provide a new approach to estimating the error of re- construction from ΣΔ-quantized compressed sensing measurements. Our method is based on the restricted isometry property (RIP) of a certain projection of the measurement matrix. Our result yields simple proofs and a slight generalization of the best-known reconstruction error bounds for Gaussian and subgaussian measurement matrices.
Keywords: Compressed; Sensing; Quantization; ΣΔ-Quantization
 
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