UBC Theses and Dissertations
Compressed sensing with deterministic measurement matrices Arian, Arman
Compressed sensing (CS) is a signal acquisition paradigm to simultaneously acquire and reduce dimension of signals that admit sparse representations. This is achieved by collecting linear, non-adaptive measurements of a signal, which can be formalized as multiplying the signal with a “measurement matrix". If the measurement matrix satisfies the so-called restricted isometry property (RIP), then it will be appropriate for compressed sensing. While wide classes of random matrices provably satisfy the RIP with high probability, explicit and deterministic constructions have been shown (so far) to satisfy the RIP only in a significantly suboptimal regime. In this thesis, our focus is on deterministic measurement matrices in compressed sensing. In a nutshell, we investigate quantization methods for a class of deterministic matrices (Chapter 2); introduce a novel deterministic construction of a family of binary, circulant measurement matrices using the Legendre symbol (Chapter 3); and propose two novel approaches for improving the RIP constant estimates based on Gershgorin circle theorem, obtaining an improvement over the Gershgorin bounds by a multiplicative constant. One of our approaches here, together with a conjecture we make regarding the distribution of quadratic residues in a finite field provides a potential path to break the so-called “square-root barrier"–we give a proof based on the assumption that the conjecture holds.
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