Skip to main content
SpringerLink
Log in
Book cover

Sparse and Redundant Representations pp 169–184Cite as

Sparsity-Seeking Methods in Signal Processing

  • Chapter
  • First Online:

Abstract

All the previous chapters have shown us that the problem of finding a sparse solution to an underdetermined linear system of equation, or approximation of it, can be given a meaningful definition and, contrary to expectation, can also be computationally tractable. We now turn to discuss the applicability of these ideas to signal and image processing. As we argue in this chapter, modeling of informative signals is possible using their sparse representation over a well-chosen dictionary. This will give rise to linear systems and their sparse solution, as dealt with earlier.

This is a preview of subscription content, 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   49.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   99.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Further Reading

  1. J. Bobin, Y. Moudden, J.-L. Starck, and M. Elad, Morphological diversity and source separation, IEEE Signal Processing Letters, 13(7):409–412, July 2006.

    Article  Google Scholar 

  2. J. Bobin, J.-L. Starck, M.J. Fadili, and Y. Moudden, SZ and CMB reconstruction using generalized morphological component analysis, Astronomical Data Analysis ADA’06, Marseille, France, September, 2006.

    Google Scholar 

  3. R.W. Buccigrossi and E.P. Simoncelli, Image compression via joint statistical characterization in the wavelet domain, IEEE Trans. on Image Processing, 8(12):1688–1701, 1999.

    Article  Google Scholar 

  4. E.J. Candès and D.L. Donoho, Recovering edges in ill-posed inverse problems: optimality of curvelet frames, Annals of Statistics, 30(3):784–842, 2000.

    Google Scholar 

  5. E.J. Candès and D.L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise-C2 singularities, Comm. Pure Appl. Math., 57:219–266, 2002.

    Article  Google Scholar 

  6. E.J. Candès and J. Romberg, Practical signal recovery from random projections, in Wavelet XI, Proc. SPIE Conf. 5914, 2005.

    Google Scholar 

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

    Article  Google Scholar 

  8. E. Candès, J. Romberg, and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, to appear in Communications on Pure and Applied Mathematics.

    Google Scholar 

  9. E.J. Candès and T. Tao, Decoding by linear programming, IEEE Trans. on Information Theory, 51(12):4203–4215, December 2005.

    Article  Google Scholar 

  10. V. Chandrasekaran, M. Wakin, D. Baron, R. Baraniuk, Surflets: a sparse representation for multidimensional functions containing smooth discontinuities, IEEE Symposium on Information Theory, Chicago, IL, 2004.

    Google Scholar 

  11. S.S. Chen, D.L. Donoho, and M.A. Saunders, Atomic decomposition by basis pursuit, SIAM Journal on Scientific Computing, 20(1):33–61 (1998).

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. R. Coifman and D.L. Donoho, Translation-invariant denoising. Wavelets and Statistics, Lecture Notes in Statistics, 103:120–150, 1995.

    Google Scholar 

  14. R.R. Coifman, Y. Meyer, S. Quake, and M.V. Wickerhauser, Signal processing and compression with wavelet packets. In Progress in wavelet analysis and applications (Toulouse, 1992), pp. 77–93.

    Google Scholar 

  15. R.R. Coifman and M.V.Wickerhauser, Adapted waveform analysis as a tool for modeling, feature extraction, and denoising, Optical Engineering, 33(7):2170– 2174, July 1994.

    Article  Google Scholar 

  16. R.A. DeVore, B. Jawerth, and B.J. Lucier, Image compression through wavelet transform coding, IEEE Trans. Information Theory, 38(2):719–746, 1992.

    Article  MathSciNet  Google Scholar 

  17. M.N. Do and and M. Vetterli, The finite ridgelet transform for image representation, IEEE Trans. on Image Processing, 12(1):16–28, 2003.

    Article  MathSciNet  Google Scholar 

  18. M.N. Do and and M. Vetterli, Framing pyramids, IEEE Trans. on Signal Processing, 51(9):2329–2342, 2003.

    Article  MathSciNet  Google Scholar 

  19. M.N. Do and M. Vetterli, The contourlet transform: an e_cient directional multiresolution image representation, IEEE Trans. Image on Image Processing, 14(12):2091–2106, 2005.

    Article  MathSciNet  Google Scholar 

  20. D.L. Donoho, Compressed sensing, IEEE Trans. on Information Theory, 52(4):1289–306, April 2006.

    Article  MathSciNet  Google Scholar 

  21. D.L. Donoho, De-noising by soft thresholding, IEEE Trans. on Information Theory, 41(3):613–627, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  22. D.L. Donoho, For most large underdetermined systems of linear equations, the minimal ℓ1-norm solution is also the sparsest solution, Communications on Pure and Applied Mathematics, 59(6):797–829, June 2006.

    Article  MATH  MathSciNet  Google Scholar 

  23. D.L. Donoho, For most large underdetermined systems of linear equations, the minimal ℓ1-norm near-solution approximates the sparsest near-solution, Communications on Pure and Applied Mathematics, 59(7):907–934, July 2006.

    Article  MathSciNet  Google Scholar 

  24. D.L. Donoho and Y. Tsaig, Extensions of compressed sensing, Signal Processing, 86(3):549–571, March 2006.

    Article  MATH  Google Scholar 

  25. M. Elad, B. Matalon, and M. Zibulevsky, Image denoising with shrinkage and redundant representations, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), NY, June 17–22, 2006.

    Google Scholar 

  26. M. Elad, J.-L. Starck, P. Querre, and D.L. Donoho, Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA), Journal on Applied and Comp. Harmonic Analysis, 19:340–358, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  27. R. Eslami and H. Radha, The contourlet transform for image de-noising using cycle spinning, in Proceedings of Asilomar Conference on Signals, Systems, and Computers, pp. 1982–1986, November 2003.

    Google Scholar 

  28. R. Eslami and H. Radha, Translation-invariant contourlet transform and its application to image denoising, IEEE Trans. on Image Processing, 15(11):3362–3374, November 2006.

    Article  Google Scholar 

  29. O.G. Guleryuz, Nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising - Part I: Theory, IEEE Trans. On Image Processing, 15(3):539–554, 2006.

    Article  Google Scholar 

  30. O.G. Guleryuz, Nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising - Part II: Adaptive algorithms, IEEE Trans. on Image Processing, 15(3):555–571, 2006.

    Article  Google Scholar 

  31. W. Hong, J. Wright, K. Huang, and Y. Ma, Multi-scale hybrid linear models for lossy image representation, IEEE Transactions on Image Processing, 15(12):3655–3671, December 2006.

    Article  MathSciNet  Google Scholar 

  32. A.K. Jain, Fundamentals of Digital Image Processing, Englewood Cli_s, NJ, Prentice-Hall, 1989.

    MATH  Google Scholar 

  33. M. Jansen, Noise Reduction by Wavelet Thresholding, Springer-Verlag, New York 2001.

    MATH  Google Scholar 

  34. E. Kidron, Y.Y. Schechner, and M. Elad, Cross-modality localization via sparsity, to appear in the IEEE Trans. on Signal Processing.

    Google Scholar 

  35. M. Lang, H. Guo, and J.E. Odegard, Noise reduction using undecimated discrete wavelet transform, IEEE Signal Processing Letters, 3(1):10–12, 1996.

    Article  Google Scholar 

  36. M. Lustig, D.L. Donoho, and J.M. Pauly, Sparse MRI: The application of compressed sensing for rapid MR imaging, submitted to Magnetic Resonance in Medicine.

    Google Scholar 

  37. M. Lustig, J.M. Santos, D.L. Donoho, and J.M. Pauly, k-t SPARSE: High frame rate dynamic MRI exploiting spatio-temporal sparsity, Proceedings of the 13th Annual Meeting of ISMRM, Seattle, 2006.

    Google Scholar 

  38. Y. Ma, A. Yang, H. Derksen, and R. Fossum, Estimation of subspace arrangements with applications in modeling and segmenting mixed data, SIAM Review, 50(3):413–458, August 2008.

    Article  MATH  MathSciNet  Google Scholar 

  39. D.M. Malioutov, M. Cetin, and A.S. Willsky, Sparse signal reconstruction perspective for source localization with sensor arrays, IEEE Trans. on Signal Processing, 53(8):3010–3022, 2005.

    Article  MathSciNet  Google Scholar 

  40. F.G. Meyer, A. Averbuch, and J.O. Stromberg, Fast adaptive wavelet packet image compression, IEEE Trans. on Image Processing, 9(5):792–800, 2000.

    Article  Google Scholar 

  41. F.G. Meyer, A.Z. Averbuch, R.R. Coifman, Multilayered image representation: Application to image compression, IEEE Trans. on Image Processing, 11(9):1072–1080, 2002.

    Article  MathSciNet  Google Scholar 

  42. F.G. Meyer and R.R. Coifman, Brushlets: a tool for directional image analysis and image compression, Applied and Computational Harmonic Analysis, 4:147–187, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  43. P. Moulin and J. Liu, Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors, IEEE Trans. on Information Theory, 45(3):909–919, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  44. G. Peyré, Sparse modeling of textures, Journal of Mathematical Imaging and Vision, 34(1):17–31, May 2009.

    Article  MathSciNet  Google Scholar 

  45. Peyré, Manifold models for signals and images, Computer Vision and Image Understanding, 113(2):249–260, February 2009.

    Article  Google Scholar 

  46. J. Portilla, V. Strela, M.J. Wainwright, and E.P. Simoncelli, Image denoising using scale mixtures of gaussians in the wavelet domain, IEEE Trans. on Image Processing, 12(11):1338–1351, 2003.

    Article  MathSciNet  Google Scholar 

  47. L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60:259–268, 1992.

    Article  MATH  Google Scholar 

  48. E.P. Simoncelli and E.H. Adelson, Noise removal via Bayesian wavelet coring, Proceedings of the International Conference on Image Processing, Laussanne, Switzerland. September 1996.

    Google Scholar 

  49. E.P. Simoncelli,W.T. Freeman, E.H. Adelson, and D.J. Heeger, Shiftable multiscale transforms, IEEE Trans. on Information Theory, 38(2):587–607, 1992.

    Article  MathSciNet  Google Scholar 

  50. J.-L. Starck, E.J. Candès, and D.L. Donoho, The curvelet transform for image denoising, IEEE Trans. on Image Processing, 11:670–684, 2002.

    Article  Google Scholar 

  51. J.-L. Starck, M. Elad, and D.L. Donoho, Redundant multiscale transforms and their application for morphological component separation, Advances in Imaging and Electron Physics, 132:287–348, 2004.

    Google Scholar 

  52. J.-L. Starck, M. Elad, and D.L. Donoho, Image decomposition via the combination of sparse representations and a variational approach. IEEE Trans. on Image Processing, 14(10):1570–1582, 2005.

    Article  MathSciNet  Google Scholar 

  53. J.-L. Starck, M.J. Fadili, and F. Murtagh, The undecimated wavelet decomposition and its reconstruction, IEEE Transactions on Image Processing, 16(2):297– 309, 2007.

    Article  MathSciNet  Google Scholar 

  54. D.S. Taubman and M.W. Marcellin. JPEG 2000: Image Compression Fundamentals, Standards and Practice, Kluwer Academic Publishers, Norwell, MA, USA, 2001.

    Google Scholar 

  55. J.A. Tropp and A.A. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit, Submitted for publication, April 2005.

    Google Scholar 

  56. R. Vidal, Y. Ma, and S. Sastry, Generalized principal component analysis (GPCA), IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(12):1945–1959, December 2005.

    Article  Google Scholar 

  57. M. Zibulevsky and B.A. Pearlmutter, Blind source separation by sparse decomposition in a signal dictionary, Neural Computation, 13(4):863–882, 2001.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, The Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Michael Elad

Authors
  1. Michael Elad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Elad .

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Elad, M. (2010). Sparsity-Seeking Methods in Signal Processing. In: Sparse and Redundant Representations. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7011-4_9

Download citation

Publish with us

Policies and ethics

Search

Navigation