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The Quest for Optimal Sampling: Computationally Efficient, Structure-Exploiting Measurements for Compressed Sensing

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Compressed Sensing and its Applications

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

Abstract

An intriguing phenomenon in many instances of compressed sensing is that the reconstruction quality is governed not just by the overall sparsity of the object to recover, but also on its structure. This chapter is about understanding this phenomenon, and demonstrating how it can be fruitfully exploited by the design of suitable sampling strategies in order to outperform more standard compressed sensing techniques based on random matrices.

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Notes

  1. 1.

    In actual fact, the sensing device takes measurements of the continuous Fourier transform of a function \(x \in L^{2}(\mathbb{R}^{d})\). As discussed in [1, 4], modelling continuous Fourier measurements as discrete Fourier measurements can lead to inferior reconstructions, as well as inverse crimes. To avoid this, one must consider an infinite-dimensional compressed sensing approach, as in (5.2). See [2, 4] for details, as well as [29] for implementation in MRI. For simplicity, we shall continue to work with the finite-dimensional model in the remainder of this chapter.

  2. 2.

    For brevity, we do not give the proof of this lemma or the later recovery results for Haar wavelets, Theorem 2. Details of the proof can be found in the short note [3].

  3. 3.

    For a proof, we refer to [3].

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Acknowledgements

The authors thank Andy Ellison from Boston University Medical School for kindly providing the MRI fruit image, and General Electric Healthcare for kindly providing the brain MRI image. BA acknowledges support from the NSF DMS grant 1318894. ACH acknowledges support from a Royal Society University Research Fellowship. ACH and BR acknowledge the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L003457/1.

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada, V5A 1S6

    Ben Adcock

  2. DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WA, UK

    Anders C. Hansen & Bogdan Roman

Authors
  1. Ben Adcock
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  2. Anders C. Hansen
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  3. Bogdan Roman
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Correspondence to Ben Adcock .

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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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Adcock, B., Hansen, A.C., Roman, B. (2015). The Quest for Optimal Sampling: Computationally Efficient, Structure-Exploiting Measurements for Compressed Sensing. 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_5

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