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Random Sampling in Bounded Orthonormal Systems

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A Mathematical Introduction to Compressive Sensing

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

This chapter considers the recovery of signals with a sparse expansion in a bounded orthonormal system. After an inventory of such bounded orthonormal systems including the Fourier systems, theoretical limitations specific to this situation are obtained for the minimal number of samples. Then, using this number of random samples, nonuniform sparse recovery is proved to be possible via ℓ1-minimization. Next, using a slightly higher number of random samples, uniform sparse recovery is also proved to be possible via a variety of algorithms. This is derived via establishing the restricted isometry property for the associated random sampling matrix—the random partial Fourier matrix is a particular case. Finally, a connection to the Λ 1-problem is pointed out.

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Authors and Affiliations

  1. Department of Mathematics, Drexel University, Philadelphia, PA, USA

    Simon Foucart

  2. Lehrstuhl C für Mathematik (Analysis), RWTH Aachen University, Aachen, Germany

    Holger Rauhut

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  1. Simon Foucart
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Foucart, S., Rauhut, H. (2013). Random Sampling in Bounded Orthonormal Systems. In: A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-4948-7_12

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