Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Buchtitelbild

2019 | OriginalPaper | Buchkapitel

An Introduction to Compressed Sensing

verfasst von : Niklas Koep, Arash Behboodi, Rudolf Mathar

Erschienen in: Compressed Sensing and Its Applications

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

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.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Nächstes Kapitel Quantized Compressed Sensing: A Survey
Fußnoten
1
Technically, a Hilbert space is an inner product space in which every Cauchy sequence converges to a point in the same space.
 
2
A Steinhaus random variable is a complex random variable distributed uniformly on the complex unit circle.
 
3
A convex body is a compact convex set with non-empty interior.
 
4
This idea also extends to signals living on low-dimensional manifolds.
 
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
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
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
A set K is a closed star domain if K is closed, and \(tK \subseteq K \;\forall t \in [0, 1]\).
 
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
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
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
In fact, as we will briefly discuss in Sect. 7, such random constructions are often characterized by more well-behaved scaling constants.
 
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
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
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
Hence the name shrinkage thresholding.
 
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
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].
 
Literatur
1.
S.I. Adalbjörnsson, A. Jakobsson, M.G. Christensen. Estimating multiple pitches using block sparsity, in 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (May 2013), pp. 6220–6224
2.
R. Adamczak, R. Latała, A.E. Litvak, A. Pajor, N. Tomczak-Jaegermann, Geometry of log-concave ensembles of random matrices and approximate reconstruction. C. R. Math. 349(13), 783–786 (2011)MathSciNetMATH
3.
R. Adamczak, A.E. Litvak, A. Pajor, N. Tomczak-Jaegermann, Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling. Constr. Approx. 34(1), 61–88 (2011)MathSciNetMATH
4.
D. Amelunxen, M. Lotz, M.B. McCoy, J.A. Tropp, Living on the edge: phase transitions in convex programs with random data. Inf. Inference 3(3), 224–294 (2014)MathSciNetMATH
5.
U. Ayaz, S. Dirksen, H. Rauhut, Uniform recovery of fusion frame structured sparse signals. Appl. Comput. Harmon. Anal. 41(2), 341–361 (2016)MathSciNetMATH
6.
W.U. Bajwa, J.D. Haupt, G.M. Raz, S.J. Wright, R.D. Nowak, Toeplitz-structured compressed sensing matrices, in 2007 IEEE/SP 14th Workshop on Statistical Signal Processing (Aug. 2007), pp. 294–298
7.
A.S. Bandeira, M.E. Lewis, D.G. Mixon, Discrete Uncertainty Principles and Sparse Signal Processing. J. Fourier Anal. Appl. 24(4), 935–956 (2018)MathSciNetMATH
8.
R. Baraniuk, M. Davenport, R. DeVore, M. Wakin, A simple proof of the restricted isometry property for random matrices. Constr. Approx. 28(3), 253–263 (2008)MathSciNetMATH
9.
A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)MathSciNetMATH
10.
S. Becker, J. Bobin, E.J. Candès, Nesta: A fast and accurate first-order method for sparse recovery. SIAM J. Imaging Sci. 4, 1–39 (2011)MathSciNetMATH
11.
J. Bennett, S. Lanning, The netflix prize (2007)
12.
R. Berinde, A.C. Gilbert, P. Indyk, H. Karloff, M.J. Strauss, Combining geometry and combinatorics: a unified approach to sparse signal recovery, in 2008 46th Annual Allerton Conference on Communication, Control, and Computing (Sept. 2008), pp. 798–805
13.
B.N. Bhaskar, G. Tang, B. Recht, Atomic norm denoising with applications to line spectral estimation. IEEE Trans. Signal Process. 61(23), 5987–5999 (2011)MathSciNetMATH
14.
H. Boche, Compressed Sensing and its Applications (Springer Science+Business Media, New York, 2015)
15.
P. Boufounos, G. Kutyniok, H. Rauhut, Sparse recovery from combined fusion frame measurements. IEEE Trans. Inf. Theory 57(6), 3864–3876 (2011)MathSciNetMATH
16.
P.T. Boufounos, L. Jacques, F. Krahmer, R. Saab, Quantization and compressive sensing, in Compressed Sensing and its Applications: MATHEON Workshop 2013, Applied and Numerical Harmonic Analysis, ed. by H. Boche, R. Calderbank, G. Kutyniok, J. Vybíral (Springer International Publishing, Cham, 2015), pp. 193–237MATH
17.
J. Bourgain, An Improved Estimate in the Restricted Isometry Problem, in Geometric Aspects of Functional Analysis, vol. 2116, ed. by B. Klartag, E. Milman (Springer International Publishing, Cham, 2014), pp. 65–70
18.
S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004)
19.
20.
E. Candes, T. Tao, The Dantzig selector: statistical estimation when p is much larger than n. Ann. Stat. 35(6), 2313–2351 (2007)MathSciNetMATH
21.
E.J. Candès, The restricted isometry property and its implications for compressed sensing. C. R. Math. 346(9), 589–592 (2008)MathSciNetMATH
22.
E.J. Candes, D.L. Donoho, Curvelets-a surprisingly effective nonadaptive representation for objects with edges, in Curves and Surfaces Fitting, ed. by L.L. Schumaker, A. Cohen, C. Rabut (Vanderbilt University Press, Nashville, TN, 1999), p. 16
23.
E.J. Candès, D.L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise c2 singularities. Commun. Pure Appl. Math. J. Issued Courant Inst. Math. Sci. 57(2), 219–266 (2004)MATH
24.
E.J. Candes, Y. Plan, Near-ideal model selection by \(\ell _1\) minimization. Ann. Stat. 37, 2145–2177 (2009)MATH
25.
E.J. Candès, J.K. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52, 489–509 (2006)MathSciNetMATH
26.
E.J. Candès, J.K. Romberg, T. Tao, Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)MathSciNetMATH
27.
E.J. Candes, T. Tao, Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005)MathSciNetMATH
28.
E.J. Candès, T. Tao, Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006)MathSciNetMATH
29.
A.Y. Carmi, L. Mihaylova, S.J. Godsill, Compressed Sensing & Sparse Filtering (Springer, 2016)
30.
P.G. Casazza, G. Kutyniok, F. Philipp, Introduction to finite frame theory, in Finite Frames (Springer, 2013), pp. 1–53
31.
V. Chandrasekaran, B. Recht, P.A. Parrilo, A.S. Willsky, The convex geometry of linear inverse problems. Found. Comput. Math. 12(6), 805–849 (2012)MathSciNetMATH
32.
M. Cheraghchi, V. Guruswami, A. Velingker, Restricted isometry of Fourier matrices and list decodability of random linear codes. SIAM J. Comput. 42(5), 1888–1914 (2013)MathSciNetMATH
33.
A. Cohen, W. Dahmen, R. Devore, Compressed sensing and best k-term approximation. J. Am. Math. Soc. 211–231 (2009)
34.
R. Coifman, F. Geshwind, Y. Meyer, Noiselets. Appl. Comput. Harmon. Anal. 10(1), 27–44 (2001)MathSciNetMATH
35.
W. Dai, O. Milenkovic, Subspace pursuit for compressive sensing signal reconstruction. IEEE Trans. Inf. Theory 55, 2230–2249 (2009)MathSciNetMATH
36.
S. Dasgupta, A. Gupta, An elementary proof of a theorem of Johnson and Lindenstrauss. Random Struct. Algorithms 22(1), 60–65 (2003)MathSciNetMATH
37.
R.A. DeVore, Nonlinear approximation. Acta Numer. 7, 51–150 (1998)MATH
38.
S. Diamond, S. Boyd, Cvxpy: a python-embedded modeling language for convex optimization. J. Mach. Learn. Res. 17(1), 2909–2913 (2016)MathSciNetMATH
39.
S. Dirksen, G. Lecué, H. Rauhut, On the gap between restricted isometry properties and sparse recovery conditions. IEEE Trans. Inf. Theory 64(8), 5478–5487 (2018)MathSciNetMATH
40.
41.
D.L. Donoho, M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via \(\ell _1\) minimization. Proc. Natl. Acad. Sci. 100(5), 2197–2202 (2003)MathSciNetMATH
42.
D.L. Donoho, M. Elad, V.N. Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inf. Theory 52, 6–18 (2006)MathSciNetMATH
43.
D.L. Donoho, I. Johnstone, A. Montanari, Accurate prediction of phase transitions in compressed sensing via a connection to minimax denoising. IEEE Trans. Inf. Theory 59, 3396–3433 (2013)MathSciNetMATH
44.
D.L. Donoho, A. Maleki, A. Montanari, Message passing algorithms for compressed sensing. Proc. Natl. Acad. Sci. U. S. A. 106(45), 18914–9 (2009)
45.
D.L. Donoho, J. Tanner, Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 367 (1906), 4273–4293 (2009)
46.
M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. (Springer, New York, 2010). OCLC: ocn646114450
47.
Y.C. Eldar, G. Kutyniok (eds.), Compressed Sensing: Theory and Applications (Cambridge University Press, Cambridge, 2012)
48.
E. Elhamifar, R. Vidal, Sparse subspace clustering, in 2009 IEEE Conference on Computer Vision and Pattern Recognition (June 2009), pp. 2790–2797
49.
H.G. Feichtinger, T. Strohmer, Gabor Analysis and Algorithms: Theory and Applications (Springer Science & Business Media, 2012)
50.
M. Fornasier, S. Peter, An overview on algorithms for sparse recovery, in Sparse Reconstruction and Compressive Sensing in Remote Sensing, ed. by X. Zhu, R. Bamler (Springer, June 2015), p. 76
51.
52.
S. Foucart, Flavors of compressive sensing, in Approximation Theory XV: San Antonio 2016, ed. by G.E. Fasshauer, L.L. Schumaker (Springer International Publishing, Cham, 2017), pp. 61–104
53.
S. Foucart, A. Pajor, H. Rauhut, T. Ullrich, The Gelfand widths of \(\ell _p\)-balls for \(0<p\le 1\). J. Complex. 26(6), 629–640 (2010)MATH
54.
S. Foucart, H. Rauhut, A Mathematical Introduction to Compressive Sensing (Birkhäuser, Basel, 2013)MATH
55.
R. Foygel, L.W. Mackey, Corrupted sensing: novel guarantees for separating structured signals. IEEE Trans. Inf. Theory 60, 1223–1247 (2014)MathSciNetMATH
56.
D. Goldberg, D. Nichols, B.M. Oki, D. Terry, Using collaborative filtering to weave an information tapestry. Commun. ACM 35(12), 61–70 (1992)
57.
Y. Gordon, On milman’s inequality and random subspaces which escape through a mesh in \(\mathbb{R}^n\), in Geometric Aspects of Functional Analysis, ed. by J. Lindenstrauss, V.D. Milman (Springer, Berlin, 1988), pp. 84–106
58.
J. Gouveia, P.A. Parrilo, R.R. Thomas, Theta bodies for polynomial ideals. SIAM J. Optim. 20, 2097–2118 (2010)MathSciNetMATH
59.
M. Grant, S. Boyd, Y. Ye, CVX: Matlab software for disciplined convex programming (2008)
60.
Z. Han, H. Li, W. Yin, Compressive Sensing for Wireless Networks (Cambridge University Press, 2013)
61.
I. Haviv, O. Regev, The restricted isometry property of subsampled fourier matrices, in Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics (Springer, Cham, 2017), pp. 163–179
62.
W.B. Johnson, J. Lindenstrauss, Extensions of lipschitz mappings into a hilbert space. Contemp. Math. 26(189–206), 1 (1984)MathSciNetMATH
63.
V. Koltchinskii, Oracle inequalities in empirical risk minimization and sparse recovery problems: École d’été de probabilités de Saint-Flour XXXVIII-2008. Number 2033 in Lecture notes in mathematics. (Springer, Berlin, 2011). OCLC: ocn733246860
64.
F. Krahmer, S. Mendelson, H. Rauhut, Suprema of chaos processes and the restricted isometry property. Commun. Pure Appl. Math. 67(11), 1877–1904 (2014)MathSciNetMATH
65.
G. Kutyniok, D. Labate (eds.), Shearlets: multiscale analysis for multivariate data. Applied and Numerical Harmonic Analysis (Birkhäuser, New York, 2012). OCLC: ocn794844320
66.
C. Liaw, A. Mehrabian, Y. Plan, R. Vershynin, A simple tool for bounding the deviation of random matrices on geometric sets (2016). CoRR, arXiv:​1603.​00897
67.
G.G. Lorentz, M.V. Golitschek, Y. Makovoz, Constructive Approximation: Advanced Problems (Springer, Berlin, 2005). OCLC: 903339623
68.
S.G. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, 3rd edn. (Elsevier/Academic Press, Amsterdam, 2009)
69.
C.A. Metzler, A. Maleki, R.G. Baraniuk, From denoising to compressed sensing. IEEE Trans. Inf. Theory 62, 5117–5144 (2016)MathSciNetMATH
70.
M. Mishali, Y.C. Eldar, Blind multiband signal reconstruction: compressed sensing for analog signals. IEEE Trans. Signal Process. 57(3), 993–1009 (2009)MathSciNetMATH
71.
72.
B.K. Natarajan, Sparse approximate solutions to linear systems. SIAM J. Comput. 24(2), 227–234 (1995)MathSciNetMATH
73.
S. Nathan, A. Shraibman, Rank, trace-norm and max-norm, in COLT (2005)
74.
J. Nelson, E. Price, M. Wootters, New constructions of rip matrices with fast multiplication and fewer rows, in Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics (2014), pp. 1515–1528
75.
Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, 1st edn. (Springer Publishing Company, Incorporated, 2014)
76.
S. Oymak, B. Hassibi, New null space results and recovery thresholds for matrix rank minimization (Nov. 2010). arXiv:​1011.​6326 [cs, math, stat]
77.
N. Parikh, S.P. Boyd, Proximal algorithms. Found. Trends Optim. 1, 127–239 (2014)
78.
F. Parvaresh, H. Vikalo, S. Misra, B. Hassibi, Recovering sparse signals using sparse measurement matrices in compressed dna microarrays. IEEE J. Sel. Top. Signal Process. 2(3), 275–285 (2008)
79.
Y. Plan, R. Vershynin, Robust 1-bit compressed sensing and sparse logistic regression: a convex programming approach. IEEE Trans. Inf. Theory 59(1), 482–494 (2013)MathSciNetMATH
80.
Y. Plan, R. Vershynin, The generalized Lasso with non-linear observations. IEEE Trans. Inf. Theory 62(3), 1528–1537 (2016)MathSciNetMATH
81.
Y.L. Polo, Y. Wang, A. Pandharipande, G. Leus, Compressive wide-band spectrum sensing, in 2009 IEEE International Conference on Acoustics, Speech and Signal Processing (Apr. 2009), pp. 2337–2340
82.
S. Rangan, Generalized approximate message passing for estimation with random linear mixing, in2011 IEEE International Symposium on Information Theory Proceedings (2011), pp. 2168–2172
83.
S. Rangan, P. Schniter, A.K. Fletcher, Vector approximate message passing, in 2017 IEEE International Symposium on Information Theory (ISIT) (2017), pp. 1588–1592
84.
N.S. Rao, B. Recht, R.D. Nowak, Universal measurement bounds for structured sparse signal recovery, in AISTATS (2012)
85.
H. Rauhut, Circulant and Toeplitz matrices in compressed sensing, in SPARS 09-Signal Processing with Adaptive Sparse Structured Representations (Saint Malo, France, Apr. 2009), p. 7
86.
H. Rauhut, K. Schnass, P. Vandergheynst, Compressed sensing and redundant dictionaries. IEEE Trans. Inf. Theory 54(5), 2210–2219 (2008)MathSciNetMATH
87.
H. Rauhut, R. Ward, Sparse recovery for spherical harmonic expansions, in Proceedings of the SampTA 2011 (2011)
88.
R.T. Rockafellar, Convex Analysis (Princeton University Press, 2015)
89.
M. Rudelson, R. Vershynin, On sparse reconstruction from Fourier and Gaussian measurements. Commun. Pure Appl. Math. 61(8), 1025–1045 (2008)MathSciNetMATH
90.
S. Sarvotham, D. Baron, R.G. Baraniuk, Measurements vs. bits: compressed sensing meets information theory, in Allerton Conference on Communication, Control and Computing (2006)
91.
M. Stojnic, \(\ell _1\) optimization and its various thresholds in compressed sensing, in 2010 IEEE International Conference on Acoustics, Speech and Signal Processing (2010), pp. 3910–3913
92.
G. Tang, B.N. Bhaskar, P. Shah, B. Recht, Compressed sensing off the grid. IEEE Trans. Inf. Theory 59(11), 7465–7490 (2013)MathSciNetMATH
93.
R. Tibshirani, M. Saunders, S. Rosset, J. Zhu, K. Knight, Sparsity and smoothness via the fused lasso. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 67(1), 91–108 (2005)MathSciNetMATH
94.
R.J. Tibshirani, The lasso problem and uniqueness (2012)
95.
A.M. Tillmann, M.E. Pfetsch, The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing. IEEE Trans. Inf. Theory 60, 1248–1259 (2014)MathSciNetMATH
96.
J.A. Tropp, Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004)MathSciNetMATH
97.
E. van den Berg, M.P. Friedlander, Spgl1: a solver for large-scale sparse reconstruction (2007)
98.
E. van den Berg, M.P. Friedlander, Probing the pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2), 890–912 (2008)MathSciNetMATH
99.
R. Vershynin, Introduction to the non-asymptotic analysis of random matrices, in Compressed Sensing, Theory and Applications (Cambridge University Press, Cambridge, 2012), pp. 210–268
100.
R. Vershynin, Estimation in High Dimensions: A Geometric Perspective (Springer International Publishing, Cham, 2015), pp. 3–66
101.
L. Welch, Lower bounds on the maximum cross correlation of signals (corresp.). IEEE Trans. Inf. Theory 20(3), 397–399 (1974)
102.
J. Wright, A.Y. Yang, A. Ganesh, S.S. Sastry, Y. Ma, Robust face recognition via sparse representation. IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 210–227 (2009)
103.
S.J. Wright, R.D. Nowak, M.A.T. Figueiredo, Sparse reconstruction by separable approximation. IEEE Trans. Signal Process. 57, 2479–2493 (2008)MathSciNetMATH
104.
H. Zhang, W. Yin, L. Cheng, Necessary and sufficient conditions of solution uniqueness in 1-norm minimization. J. Optim. Theory Appl. 164, 109–122 (2015)MathSciNetMATH
105.
Metadaten
Titel
An Introduction to Compressed Sensing
verfasst von
Niklas Koep
Arash Behboodi
Rudolf Mathar
Copyright-Jahr
2019
Buch
Compressed Sensing and Its Applications

Print ISBN: 978-3-319-73073-8

Electronic ISBN: 978-3-319-73074-5

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-319-73074-5_1