Skip to main content
SpringerLink
Log in

Sparsity and Compressed Sensing in Inverse Problems

  • Chapter
  • First Online:
Extraction of Quantifiable Information from Complex Systems

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 102))

  • 1277 Accesses

Abstract

This chapter is concerned with two important topics in the context of sparse recovery in inverse and ill-posed problems. In first part we elaborate conditions for exact recovery. In particular, we describe how both ℓ 1-minimization and matching pursuit methods can be used to regularize ill-posed problems and moreover, state conditions which guarantee exact recovery of the support in the sparse case. The focus of the second part is on the incomplete data scenario. We discuss extensions of compressed sensing for specific infinite dimensional ill-posed measurement regimes. We are able to establish recovery error estimates when adequately relating the isometry constant of the sensing operator, the ill-posedness of the underlying model operator and the regularization parameter. Finally, we very briefly sketch how projected steepest descent iterations can be applied to retrieve the sparse solution.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Adcock, B., Hansen, A.C.: Generalized sampling and infinite dimensional compressed sensing. Technical report NA2011/02, DAMTP, University of Cambridge (2012)

    Google Scholar 

  2. Adcock, B., Hansen, A.C., Herrholz, E., Teschke, G.: Generalized sampling, infinite-dimensional compressed sensing, and semi-random sampling for asymptotically incoherent dictionaries (2012, submitted)

    Google Scholar 

  3. Adcock, B., Hansen, A.C., Herrholz, E., Teschke, G.: Generalized sampling: extension to frames and inverse and ill-posed problems. Inverse Probl. 29(1), 015,008 (2013)

    Article  MathSciNet  Google Scholar 

  4. Adcock, B., Hansen, A.C., Roman, B., Teschke, G.: Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum. Adv. Imaging Electron Phys. 182, 1–51 (2008)

    Google Scholar 

  5. Bonesky, T., Bredies, K., Lorenz, D.A., Maass, P.: A generalized conditional gradient method for nonlinear operator equations with sparsity constraints. Inverse Probl. 23, 2041–2058 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bredies, K., Lorenz, D., Maass, P.: A generalized conditional gradient method and its connection to an iterative shrinkage method. Comput. Optim. Appl. 42(2), 173–193 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bredies, K., Lorenz, D.A.: Linear convergence of iterative soft-thresholding. J. Fourier Anal. Appl. 14(5–6), 813–837 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Candès, E., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)

    Article  MATH  Google Scholar 

  9. Candès, E., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)

    Article  MATH  Google Scholar 

  10. Candès, E., Tao, T.: Near optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006)

    Article  Google Scholar 

  11. Candès, E.J.: The restricted isometry property and its implications for compressed sensing. C. R. Acad. Sci. 346(9–10), 589–592 (2008)

    Article  MATH  Google Scholar 

  12. Candès, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005)

    Article  MATH  Google Scholar 

  13. Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM Rev. 43(1), 129–159 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)

    Article  MATH  Google Scholar 

  15. Daubechies, I., Teschke, G., Vese, L.: Iteratively solving linear inverse problems with general convex constraints. Inverse Probl. Imaging 1(1), 29–46 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Daubechies, I., Teschke, G., Vese, L.: On some iterative concepts for image restoration. Adv. Imaging Electron Phys. 150, 1–51 (2008)

    Article  Google Scholar 

  17. Denis, L., Lorenz, D.A., Dennis, T.: Greedy solution of ill-posed problems: error bounds and exact inversion. Inverse Probl. 25(11), 115,017 (24pp) (2009)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Donoho, D.L., Elad, M.: Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization. Proc. Natl. Acad. Sci. 100, 2197–2202 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Donoho, D.L., Hou, X.: Uncertainty priciples and ideal atomic decomposition. IEEE Trans. Inf. Theory 47, 2845–2862 (2001)

    Article  MATH  Google Scholar 

  21. Dossal, C., Mallat, S.: Sparse spike deconvolution with minimum scale. In: Proceedings of SPARS05, Rennes (2005)

    Google Scholar 

  22. Eldar, Y.C.: Compressed sensing of analog signals in shift-invariant spaces. IEEE Trans. Signal Process. 57(8), 2986–2997 (2009)

    Article  MathSciNet  Google Scholar 

  23. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)

    Book  MATH  Google Scholar 

  24. Feuer, A., Nemirovski, A.: On sparse representations in unions of bases. IEEE Trans. Inf. Theory 49, 1579–1581 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Fornasier, M.: Domain decomposition methods for linear inverse problems with sparsity constraints. Inverse Probl. 23, 2505–2526 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Fornasier, M., Rauhut, H.: Recovery algorithms for vector valued data with joint sparsity constraint. SIAM J. Numer. Anal. 46(2), 577–613 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Grasmair, M., Haltmeier, M., Scherzer, O.: Sparse regularization with ℓ q penalty term. Inverse Probl. 24(5), 055,020 (13pp) (2008)

    Google Scholar 

  28. Gribonval, R., Nielsen, M.: Beyond sparsity: recovering structured representations by ℓ 1 minimization and greedy algorithms. Adv. Comput. Math. 28(1), 23–41 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston (1984)

    MATH  Google Scholar 

  30. Groetsch, C.W.: Inverse Problems in the Mathematical Sciences. Vieweg, Braunschweig (1993)

    Book  MATH  Google Scholar 

  31. Herrholz, E., Teschke, G.: Compressive sensing principles and iterative sparse recovery for inverse and ill-posed problems. Inverse Probl. 26, 125,012 (2010)

    Article  MathSciNet  Google Scholar 

  32. Klann, E., Kuhn, M., Lorenz, D.A., Maass, P., Thiele, H.: Shrinkage versus deconvolution. Inverse Probl. 23, 2231–2248 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  33. Laska, J., Kirolos, S., Duarte, M., Ragheb, T., Baraniuk, R., Massoud, Y.: Theory and implementation of an analog-to-information converter using random demodulation. In: Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS), New Orleans (2007)

    Google Scholar 

  34. Lorenz, D.A.: Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. J. Inverse Ill-Posed Probl. 16(5), 463–478 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  35. Lorenz, D.A., Schiffler, S., Trede, D.: Beyond convergence rates: exact inversion with Tikhonov regularization with sparsity constraints. Inverse Probl. 27, 085,009 (2011)

    Article  MathSciNet  Google Scholar 

  36. Mallat, S., Davis, G., Zhang, Z.: Adaptive time-frequency decompositions. SPIE J. Opt. Eng. 33(7), 2183–2191 (1994)

    Article  Google Scholar 

  37. Mallat, S.G., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)

    Article  MATH  Google Scholar 

  38. Mishali, M., Eldar, Y.C.: Blind multi-band signal reconstruction: compressed sensing for analog signals. IEEE Trans. Signal Process. 57(3), 993–1009 (2009)

    Article  MathSciNet  Google Scholar 

  39. Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Springer, New York (1984)

    Book  Google Scholar 

  40. Pati, Y., Rezaiifar, R., Krishnaprasad, P.: Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition. In: Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, vol. 1, pp. 40–44 (1993)

    Google Scholar 

  41. Ramlau, R., Teschke, G.: A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints. Numer. Math. 104(2), 177–203 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  43. Soulez, F., Denis, L., Fournier, C., Thiébaut, É., Goepfert, C.: Inverse problem approach for particle digital holography: accurate location based on local optimisation. J. Opt. Soc. Am. A 24(4), 1164–1171 (2007)

    Article  Google Scholar 

  44. Teschke, G.: Multi-frame representations in linear inverse problems with mixed multi-constraints. Appl. Computat. Harmon. Anal. 22, 43–60 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  45. Teschke, G., Borries, C.: Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints. Inverse Probl. 26, 025,007 (2010)

    Article  MathSciNet  Google Scholar 

  46. Tropp, J.A.: Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Neubrandenburg University of Applied Sciences, Brodaer Straße 2, 17033, Neubrandenburg, Germany

    Evelyn Herrholz & Gerd Teschke

  2. Technical University of Braunschweig, Pockelsstr. 14, 38106, Braunschweig, Germany

    Dirk Lorenz

  3. University of Bremen, Bibliothekstrae 1, Gebäude MZH, 28359, Bremen, Germany

    Dennis Trede

Authors
  1. Evelyn Herrholz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dirk Lorenz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gerd Teschke
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dennis Trede
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gerd Teschke .

Editor information

Editors and Affiliations

  1. FB 12 Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Stephan Dahlke

  2. Inst. Geometrie & Praktische Mathemathik, RWTH Aachen University, Aachen, Germany

    Wolfgang Dahmen

  3. Universität Bonn Institut für Numerische Simulation, Bonn, Germany

    Michael Griebel

  4. Max-Planck-Inst. f. Mathem. in d. Naturwissenschaften, Leipzig, Sachsen, Germany

    Wolfgang Hackbusch

  5. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Klaus Ritter

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

    Reinhold Schneider

  7. Seminar für Angewandte Mathematik, ETH Zürich, Zürich, Switzerland

    Christoph Schwab

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

    Harry Yserentant

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Herrholz, E., Lorenz, D., Teschke, G., Trede, D. (2014). Sparsity and Compressed Sensing in Inverse Problems. In: Dahlke, S., et al. Extraction of Quantifiable Information from Complex Systems. Lecture Notes in Computational Science and Engineering, vol 102. Springer, Cham. https://doi.org/10.1007/978-3-319-08159-5_18

Download citation

Publish with us

Policies and ethics

Search

Navigation