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An Evaluation of the Sparsity Degree for Sparse Recovery with Deterministic Measurement Matrices

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Abstract

The paper deals with the estimation of the maximal sparsity degree for which a given measurement matrix allows sparse reconstruction through ℓ 1-minimization. This problem is a key issue in different applications featuring particular types of measurement matrices, as for instance in the framework of tomography with low number of views. In this framework, while the exact bound is NP hard to compute, most classical criteria guarantee lower bounds that are numerically too pessimistic. In order to achieve an accurate estimation, we propose an efficient greedy algorithm that provides an upper bound for this maximal sparsity. Based on polytope theory, the algorithm consists in finding sparse vectors that cannot be recovered by ℓ 1-minimization. Moreover, in order to deal with noisy measurements, theoretical conditions leading to a more restrictive but reasonable bounds are investigated. Numerical results are presented for discrete versions of tomography measurement matrices, which are stacked Radon transforms corresponding to different tomograph views.

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Acknowledgements

The authors wish to thank the reviewers for their careful reading and relevants remarks that improve significatively the readability of this work.

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Authors and Affiliations

  1. Institut polytechnique de Bordeaux, Université de Bordeaux, IMS, UMR CNRS 5218, 33405, Talence cedex, France

    Y. Berthoumieu & F. Turcu

  2. Université de Bordeaux, IMB, UMR CNRS 5584, 33405, Talence cedex, France

    C. Dossal

  3. Laboratoire de Physique de l’ENS Lyon, UMR CNRS 5672, 69007, Lyon, France

    N. Pustelnik

  4. TOTAL S.A/DG/Direction Scientifique, 92078, Paris la Defense Cedex, France

    P. Ricoux

Authors
  1. Y. Berthoumieu
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  2. C. Dossal
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  3. N. Pustelnik
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  4. P. Ricoux
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  5. F. Turcu
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Corresponding author

Correspondence to N. Pustelnik.

Additional information

Part of this work will appear in the conference proceedings of EUSIPCO 2012 [27].

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Berthoumieu, Y., Dossal, C., Pustelnik, N. et al. An Evaluation of the Sparsity Degree for Sparse Recovery with Deterministic Measurement Matrices. J Math Imaging Vis 48, 266–278 (2014). https://doi.org/10.1007/s10851-013-0453-4

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  • DOI: https://doi.org/10.1007/s10851-013-0453-4

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