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Total Variation Minimization in Compressed Sensing

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Book cover Compressed Sensing and its Applications

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

Abstract

This chapter gives an overview over recovery guarantees for total variation minimization in compressed sensing for different measurement scenarios. In addition to summarizing the results in the area, we illustrate why an approach that is common for synthesis sparse signals fails and different techniques are necessary. Lastly, we discuss a generalization of recent results for Gaussian measurements to the subgaussian case.

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References

  1. R.G. Baraniuk, M. Davenport, R.A. DeVore, M. Wakin, A simple proof of the restricted isometry property for random matrices. Constr. Approx. 28(3), 253–263 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Berinde, A. Gilbert, P. Indyk, H. Karloff, M. Strauss, Combining geometry and combinatorics: a unified approach to sparse signal recovery, in 46th Annual Allerton Conference on Communication, Control, and Computing, 2008 (IEEE, New York, 2008), pp. 798–805

    Google Scholar 

  3. J.-F. Cai, W. Xu, Guarantees of total variation minimization for signal recovery. Inf. Infer. 4(4), 328–353 (2015)

    MathSciNet  Google Scholar 

  4. E.J. Candes, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(3), 489–509 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. E.J. Candès, Y.C. Eldar, D. Needell, P. Randall, Compressed sensing with coherent and redundant dictionaries. Appl. Comput. Harmon. Anal. 31(1), 59–73 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Cohen, W. Dahmen, I. Daubechies, R. DeVore, Harmonic analysis of the space BV. Rev. Mat. Iberoam. 19(1), 235–263 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Donoho, High-dimensional centrally-symmetric polytopes with neighborliness proportional to dimension. Technical report, Department of Statistics, Stanford University (2004)

    MATH  Google Scholar 

  8. D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Foucart, H. Rauhut, A Mathematical Introduction to Compressive Sensing (Springer, Berlin, 2013)

    Book  MATH  Google Scholar 

  10. J.J. Fuchs, On sparse representations in arbitrary redundant bases. IEEE Trans. Inf. Theory 50(6), 1341–1344 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Y. Gordon, On Milman’s Inequality and Random Subspaces which Escape Through a Mesh in \(\mathbb {R}^n\) (Springer, Berlin, 1988)

    Google Scholar 

  12. J. Jørgensen, C. Kruschel, D. Lorenz, Testable uniqueness conditions for empirical assessment of undersampling levels in total variation-regularized x-ray CT. Inverse Prob. Sci. Eng. 23(8), 1283–1305 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Kabanava, H. Rauhut, Analysis ℓ 1-recovery with frames and Gaussian measurements. Acta Appl. Math. 140(1), 173–195 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Kabanava, H. Rauhut, H. Zhang, Robust analysis ℓ 1-recovery from Gaussian measurements and total variation minimization. Eur. J. Appl. Math. 26(06), 917–929 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Koltchinskii, S. Mendelson, Bounding the smallest singular value of a random matrix without concentration. Int. Math. Res. Not. 2015(23), 12991–13008 (2015)

    MathSciNet  MATH  Google Scholar 

  16. F. Krahmer, H. Rauhut, Structured random measurements in signal processing. GAMM-Mitteilungen 37(2), 217–238 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Krahmer, R. Ward, New and improved Johnson-Lindenstrauss embeddings via the restricted isometry property. SIAM J. Math. Anal. 43(3), 1269–1281 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Krahmer, R. Ward, Stable and robust sampling strategies for compressive imaging. IEEE Trans. Image Process. 23(2), 612–622 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. F. Krahmer, S. Mendelson, H. Rauhut, Suprema of chaos processes and the restricted isometry property. Commun. Pure Appl. Math. 67(11), 1877–1904 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Krahmer, D. Needell, R. Ward, Compressive sensing with redundant dictionaries and structured measurements. SIAM J. Math. Anal. 47(6), 4606–4629 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. C. Kruschel, Geometrical interpretations and algorithmic verification of exact solutions in compressed sensing. PhD thesis, TU Braunschweig (2015)

    Google Scholar 

  22. S. Mendelson, Learning without concentration, in COLT (2014), pp. 25–39

    Google Scholar 

  23. S. Mendelson, A. Pajor, N. Tomczak-Jaegermann, Reconstruction and subgaussian operators in asymptotic geometric analysis. Geom. Funct. Anal. 17(4), 1248–1282 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Nam, M. Davies, M. Elad, R. Gribonval, The cosparse analysis model and algorithms. Appl. Comput. Harmon. Anal. 34(1), 30–56 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. D. Needell, R. Ward, Near-optimal compressed sensing guarantees for total variation minimization. IEEE Trans. Image Process. 22(10), 3941–3949 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. D. Needell, R. Ward, Stable image reconstruction using total variation minimization. SIAM J. Imag. Sci. 6(2), 1035–1058 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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)

    Article  MathSciNet  MATH  Google Scholar 

  28. C. Poon, On the role of total variation in compressed sensing. SIAM J. Imag. Sci. 8(1), 682–720 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. G. Puy, P. Vandergheynst, Y. Wiaux, On variable density compressive sampling. IEEE Signal Process. Lett. 18, 595–598 (2011)

    Article  Google Scholar 

  30. G. Puy, J. Marques, R. Gruetter, J.-P. Thiran, D. Van De Ville, P. Vandergheynst, Y. Wiaux, Spread spectrum magnetic resonance imaging. IEEE Trans. Med. Imag. 31(3), 586–598 (2012)

    Article  Google Scholar 

  31. H. Rauhut, K. Schnass, P. Vandergheynst, Compressed sensing and redundant dictionaries. IEEE Trans. Inf. Theory 54(5), 2210–2219 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. H. Rauhut, J. Romberg, J.A. Tropp, Restricted isometries for partial random circulant matrices. Appl. Comp. Harmon. Anal. 32(2), 242–254 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1972)

    MATH  Google Scholar 

  34. L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Sandbichler, F. Krahmer, T. Berer, P. Burgholzer, M. Haltmeier, A novel compressed sensing scheme for photoacoustic tomography. SIAM J. Appl. Math. 75(6), 2475–2494 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Tillmann, M. Pfetsch, The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing. IEEE Trans. Inf. Theory 60(2), 1248–1259 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. J. Tropp, Convex recovery of a structured signal from independent random linear measurements, in Sampling Theory, a Renaissance (Springer, Berlin, 2015), pp. 67–101

    MATH  Google Scholar 

  38. R. Vershynin, Estimation in high dimensions: a geometric perspective, in Sampling Theory, a Renaissance (Springer, Berlin, 2015), pp. 3–66

    Book  MATH  Google Scholar 

  39. H. Zhang, Y. Ming, W. Yin, One condition for solution uniqueness and robustness of both ℓ 1-synthesis and ℓ 1-analysis minimizations. Adv. Comput. Math. 42(6), 1381–1399 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

FK and MS acknowledge support by the Hausdorff Institute for Mathematics (HIM), where part of this work was completed in the context of the HIM trimester program “Mathematics of Signal Processing”; FK and CK acknowledge support by the German Science Foundation in the context of the Emmy Noether Junior Research Group “Randomized Sensing and Quantization of Signals and Images” (KR 4512/1-1) and by the German Ministry of Research and Education in the context of the joint research initiative ZeMat. MS has been supported by the Austrian Science Fund (FWF) under Grant no. Y760 and the DFG SFB/TRR 109 “Discretization in Geometry and Dynamics.”

Author information

Authors and Affiliations

  1. Department of Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748, Garching/Munich, Germany

    Felix Krahmer

  2. Georg-August-University Göttingen, Institute for Numerical and Applied Mathematics, Lotzestr. 16-18, 37083, Göttingen, Germany

    Christian Kruschel

  3. IAV GmbH, Development Center, Rockwellstr. 16, 38518, Gifhorn, Germany

    Christian Kruschel

  4. Department of Mathematics, University of Innsbruck, Technikerstraße 13, 6020, Innsbruck, Austria

    Michael Sandbichler

Authors
  1. Felix Krahmer
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  2. Christian Kruschel
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  3. Michael Sandbichler
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Corresponding author

Correspondence to Felix Krahmer .

Editor information

Editors and Affiliations

  1. Fakultät für Elektrotechnik und Informationstechnik, Technische Universität München, Munich, Bavaria, Germany

    Holger Boche

  2. Institut für Telekommunikationssysteme, Technische Universität Berlin, Berlin, Germany

    Giuseppe Caire

  3. Department of Electrical & Computer Engineering, Duke University, Durham, North Carolina, USA

    Robert Calderbank

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

    Maximilian März

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

    Gitta Kutyniok

  6. Lehrstuhl und Institute für Statistik, RWTH Aachen, Aachen, Germany

    Rudolf Mathar

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Krahmer, F., Kruschel, C., Sandbichler, M. (2017). Total Variation Minimization in Compressed Sensing. In: Boche, H., Caire, G., Calderbank, R., März, M., Kutyniok, G., Mathar, R. (eds) Compressed Sensing and its Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-69802-1_11

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