Bounds for optimal compressed sensing matrices and practical reconstruction schemes

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dc.contributor.advisorBaraniuk, Richard G.en_US
dc.creatorSarvotham, Shriramen_US
dc.date.accessioned2009-06-03T19:54:21Zen_US
dc.date.available2009-06-03T19:54:21Zen_US
dc.date.issued2008en_US
dc.description.abstractCompressed Sensing (CS) is an emerging field that enables reconstruction of a sparse signal x ∈ Rn that has only k << n non-zero coefficients from a small number m << n of linear projections. The projections can be thought of as a vector that is obtained by multiplying a k-sparse signal in Rn by a matrix (called CS matrix) of size m x n where k < m << n. The central theme of this thesis is to study the role of the CS matrix on robustness in reconstruction as well as the complexity involved in reconstruction schemes. In the first part of the thesis, we explore the impact of the CS matrix on robustness, as measured by the Restricted Isometry Property (RIP). We derive two converse bounds for RIP of the CS matrix in terms of n, m and k. For the first bound (structural bound), ee employ results from algebra of Singular Value Decomposition (SVD) of sub-matrices. The second bound (packing bound) is based on sphere packing arguments which we motivate by showing the equivalence of the RIP measure and codes on grassmannian spaces. The derivation of the two bounds offer rich geometric interpretation and illuminate the relationship between CS matrices and many diverse concepts such as equi-angular tight frames, codes on Euclidean spheres, and the generalized Pythagorean Theorem. In the second part of the thesis, we propose strategies to design the CS matrix so that it lends itself to low-complexity reconstruction schemes. We argue that sparse matrices are a good choice in CS and present two strategies for reconstruction involving group testing and belief propagation respectively.en_US
dc.format.extent135 p.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.callnoTHESIS E.E. 2008 SARVOTHAMen_US
dc.identifier.citationSarvotham, Shriram. "Bounds for optimal compressed sensing matrices and practical reconstruction schemes." (2008) Diss., Rice University. <a href="https://hdl.handle.net/1911/22221">https://hdl.handle.net/1911/22221</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/22221en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectElectronicsen_US
dc.subjectElectrical engineeringen_US
dc.titleBounds for optimal compressed sensing matrices and practical reconstruction schemesen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentElectrical Engineeringen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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