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Compressed Sensing, Sparse Inversion, and Model Mismatch

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Compressed Sensing and its Applications

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

Abstract

The advent of compressed sensing theory has revolutionized our view of imaging, as it demonstrates that subsampling has manageable consequences for image inversion, provided that the image is sparse in an apriori known dictionary. For imaging problems in spectrum analysis (estimating complex exponential modes), and passive and active radar/sonar (estimating Doppler and angle of arrival), this dictionary is usually taken to be a DFT basis (or frame) constructed for resolution of 2π∕n, with n a window length, array length, or pulse-to-pulse processing length. However, in reality no physical field is sparse in a DFT frame or in any apriori known frame. No matter how finely we grid the parameter space (e.g., frequency, delay, Doppler, and/or wavenumber) the sources may not lie in the center of the grid cells and consequently there is always mismatch between the assumed and the actual frames for sparsity. But what is the sensitivity of compressed sensing to mismatch between the physical model that generated the data and the mathematical model that is assumed in the sparse inversion algorithm? In this chapter, we study this question. The focus is on the canonical problem of DFT inversion for modal analysis.

The authors were supported in part by the NSF by Grants CCF-1018472, CCF-1017431, CCF-0916314, CCF-0915299, and CCF-1422658.

©2011 IEEE. Reprinted, with permission, from Y. Chi, L. L. Scharf, A. Pezeshki, A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Transactions on Signal Processing, vol. 59, no. 5, pp. 2182–2195, May 2011.

©2011 IEEE. Reprinted, with permission, from L. L. Scharf, E. K. P. Chong, A. Pezeshki, and J. R. Luo, “Sensitivity considerations in compressed sensing,” Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR), Pacific Grove, CA, 6–9 Nov. 2011, pp. 744–748.

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Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, and Department of Mathematics, Colorado State University, Fort Collins, CO, 80523, USA

    Ali Pezeshki & Edwin K. P. Chong

  2. Department of Electrical and Computer Engineering, and Department of Biomedical Informatics, The Ohio State University, Columbus, OH, 43210, USA

    Yuejie Chi

  3. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523, USA

    Louis L. Scharf

Authors
  1. Ali Pezeshki
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  2. Yuejie Chi
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  3. Louis L. Scharf
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  4. Edwin K. P. Chong
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Corresponding author

Correspondence to Ali Pezeshki .

Editor information

Editors and Affiliations

  1. Lehrstuhl für Theoretische Informationstechnik, Technische Universität München, München, Bayern, Germany

    Holger Boche

  2. Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina, USA

    Robert Calderbank

  3. Institut für Mathematik, Technische Universität Berlin, Berlin, Berlin, Germany

    Gitta Kutyniok

  4. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Jan Vybíral

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Pezeshki, A., Chi, Y., Scharf, L.L., Chong, E.K.P. (2015). Compressed Sensing, Sparse Inversion, and Model Mismatch. In: Boche, H., Calderbank, R., Kutyniok, G., Vybíral, J. (eds) Compressed Sensing and its Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-16042-9_3

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