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Compressed Sensing and Its Applications pp 1–65Cite as

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An Introduction to Compressed Sensing

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

Abstract

Compressed sensing and many research activities associated with it can be seen as a framework for signal processing of low-complexity structures. A cornerstone of the underlying theory is the study of inverse problems with linear or nonlinear measurements. Whether it is sparsity, low-rankness, or other familiar notions of low complexity, the theory addresses necessary and sufficient conditions behind the measurement process to guarantee signal reconstruction with efficient algorithms. This includes consideration of robustness to measurement noise and stability with respect to signal model inaccuracies. This introduction aims to provide an overall view of some of the most important results in this direction. After discussing various examples of low-complexity signal models, two approaches to linear inverse problems are introduced which, respectively, focus on the recovery of individual signals and recovery of all low-complexity signals simultaneously. In particular, we focus on the former setting, giving rise to so-called nonuniform signal recovery problems. We discuss different necessary and sufficient conditions for stable and robust signal reconstruction using convex optimization methods. Appealing to concepts from non-asymptotic random matrix theory, we outline how certain classes of random sensing matrices, which fully govern the measurement process, satisfy certain sufficient conditions for signal recovery. Finally, we review some of the most prominent algorithms for signal recovery proposed in the literature.

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Notes

  1. 1.

    Technically, a Hilbert space is an inner product space in which every Cauchy sequence converges to a point in the same space.

  2. 2.

    A Steinhaus random variable is a complex random variable distributed uniformly on the complex unit circle.

  3. 3.

    A convex body is a compact convex set with non-empty interior.

  4. 4.

    This idea also extends to signals living on low-dimensional manifolds.

  5. 5.

    This follows from the classical bound \({\left( \frac{d}{k}\right) }^k\le \left( {\begin{array}{c}d\\ k\end{array}}\right) \le {\left( \frac{ed}{k}\right) }^k\).

  6. 6.

    To define the sparse vectors on \(\mathbb {C}^d\), simply replace \({\left\{ \pm \mathbf {e}_n\right\} }\) by \({\left\{ \pm \mathbf {e}_n,\pm i\mathbf {e}_n\right\} }\).

  7. 7.

    We assume that the fundamental frequencies of each complex exponential are integer multiples of the frequency resolution \(f_\mathrm {s} / d\) where \(f_\mathrm {s}\) denotes the sampling rate.

  8. 8.

    A set K is a closed star domain if K is closed, and \(tK \subseteq K \;\forall t \in [0, 1]\).

  9. 9.

    The Grassmann manifold or Grassmannian \(\mathcal {G}(d,s)\) is an abstract Riemannian manifold containing all s-dimensional subspaces of \(\mathbb {R}^d\).

  10. 10.

    A cone \(\mathcal {C}\subset \mathbb {R}^d\) is called polyhedral if it can be expressed as the intersection of finitely many half-spaces.

  11. 11.

    This follows from the fact that \(\ker (\mathbf {A})\) only has a single face on which \({\varPi _{\ker (\mathbf {A})}}\) projects every point \(\mathbf {x}\in \mathbb {R}^d\), namely, \(\ker (\mathbf {A})\) itself.

  12. 12.

    In fact, as we will briefly discuss in Sect. 7, such random constructions are often characterized by more well-behaved scaling constants.

  13. 13.

    The definition can easily be extended to the case where \(\mathcal {D}\subset \mathbb {R}^n\), but we restrict our discussion to the scalar case here.

  14. 14.

    This avoids another issue regarding the so-called instance optimality of pairs \((\mathbf {A},\Delta )\) where \(\Delta :\mathbb {C}^m \rightarrow \mathbb {C}^d\) denotes an arbitrary reconstruction algorithm (see [54, Chap. 11] for details).

  15. 15.

    We are careful not to call this an algorithm class as optimization programs are technically just descriptions of problems which still require specialized algorithms such as interior-point methods to actually solve them.

  16. 16.

    Hence the name shrinkage thresholding.

  17. 17.

    Note that while we used a second-order approximation of h in Eq. (22), we did so by approximating the Hessian \(\nabla ^2 h(\mathbf {x})\) as a scaled identity matrix, thereby ignoring the true second-order information of h.

  18. 18.

    Note that this choice amounts to \(\mathbf {A}\) satisfying the \(\ell _2\)-robust null space property (cf. Definition 10) of order \(k\) with constants \(\rho < 1/3\) and \(\tau > 0\) [54, Theorem 6.13].

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Acknowledgements

We would like to thank the anonymous reviewers and contributors to this book for their invaluable comments regarding this introduction.

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  1. RWTH Aachen Theoretische Informationstechnik, 52056, Aachen, Germany

    Niklas Koep, Arash Behboodi & Rudolf Mathar

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  1. Niklas Koep
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Editors and Affiliations

  1. Department of Electrical and Computer Engineering, Munich Center for Quantum Science and Technology (MCQST), Technical University of Munich, Munich, Germany

    Holger Boche

  2. Institute of Telecommunications Systems, Technical University of Berlin, Berlin, Germany

    Giuseppe Caire

  3. Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA

    Robert Calderbank

  4. Department of Mathematics, Technical University of Berlin, Berlin, Germany

    Gitta Kutyniok

  5. Faculty of Electrical Engineering and Information Technology, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Rudolf Mathar

  6. Mathematical Institute, University of Oxford, Oxford, UK

    Philipp Petersen

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Koep, N., Behboodi, A., Mathar, R. (2019). An Introduction to Compressed Sensing. In: Boche, H., Caire, G., Calderbank, R., Kutyniok, G., Mathar, R., Petersen, P. (eds) Compressed Sensing and Its Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-73074-5_1

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