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Compressed Sensing for Image Compression: Survey of Algorithms

  • S. K. GunasheelaEmail author
  • H. S. Prasantha
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 906)

Abstract

Compressed sensing (CS) is an image acquisition method, where only few random measurements are taken instead of taking all the necessary samples as suggested by Nyquist sampling theorem. It is one of the most active research areas in the past decade. In this age of digital revolution, where we are dealing with humongous amount of digital data, exploring the concepts of compressed sensing and its applications in the field of image processing is very much relevant and necessary. The paper discusses the basic concepts of compressed sensing and advantages of incorporating CS-based algorithms in image compression. The paper also discusses the drawbacks of CS, and conclusion has been made regarding when the CS-based algorithms are effective and appropriate in image compression applications. As an example, reconstruction of an image acquired in compressed sensing way using \( l_{1} \) minimization, total variation-based augmented Lagrangian method and Bregman method is presented.

Keywords

Compressed sensing Image compression Nyquist sampling theorem 

References

  1. 1.
    Marks, R. J., II. (1991). Introduction to Shannon sampling and interpolation theory. Berlin: Springer.CrossRefGoogle Scholar
  2. 2.
    Lustig, M., Donoho, D., & Pauly, J. (2007). Sparse MRI: The application of compressed sensing for rapid MR imaging. Magnetic Resonance in Medicine, 58(6), 1182–1195.CrossRefGoogle Scholar
  3. 3.
    Donoho, D. L. (2006). Compressed sensing. IEEE Transactions on Information Theory, 52(4), 1289–1306.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Martin, G., Bioucas-Dias, J. M., & Plaza, A. (2015). HYCA: A new technique for hyperspectral compressed sensing. IEEE Transactions on Geoscience and Remote Sensing, 53, 2819–2831.CrossRefGoogle Scholar
  5. 5.
    Takhar, D., Laska, J. N., Wakin, M. B., Duarte, M. F., Baron, D., Sarvotham, S., Kelly, K. F., & Baraniuk, R. G. (2006, January). A new compressed imaging camera architecture using optical-domain compression. Computational Imaging IV, 6065, 43–52.Google Scholar
  6. 6.
    Candès, E., Romberg, J., & Tao, T. (2006). Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 52(2), 489–509.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Donoho, D. (2006). Compressed sensing. IEEE Transactions on Information Theory, 52(4), 1289–1306.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Candès, E., & Tao, T. (2006). Near optimal signal recovery from random projections: Universal encoding strategies? IEEE Transactions on Information Theory, 52(12), 5406–5425.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Black, P. E. (2005, February). “Greedy algorithm” in Dictionary of Algorithms and Data Structures [online], U.S. National Institute of Standards and Technology.Google Scholar
  10. 10.
    Mallat, S. G., & Zhang, Z. (1993). Matching pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing, 41(12), 3397–3415.CrossRefGoogle Scholar
  11. 11.
    Pati, Y. C., Rezaiifar, R., & Krishnaprasad, P. S. (1993). Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition. In Twenty-Seventh Asilomar Conference on Signals, Systems and Computers.Google Scholar
  12. 12.
    Santosa, F., & Symes, W. W. (1986). Linear inversion of band-limited reflection seismograms. SIAM Journal on Scientific and Statistical Computing, 7(4), 1307–1330.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Donoho, D. L., & Stark, P. B. (1989). Uncertainty principles and signal recovery. SIAM Journal on Applied Mathematics, 49, 906–931.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Donoho, D. L., & Logan, B. F. (1992). Signal recovery and the large sieve. SIAM Journal on Applied Mathematics, 52, 577–591.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Donoho, D. L., & Huo, X. (2001). Uncertainty principles and ideal atomic decomposition. IEEE Transactions on Information Theory, 47, 2845–2862.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Donoho, D. L., & Elad, M. (2003). Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization. Proceedings of the National Academy of Sciences of the United States of America, 100, 2197–2202.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Elad, M., & Bruckstein, A. M. (2002). A generalized uncertainty principle and sparse representation in pairs of RN bases. IEEE Transactions on Information Theory, 48, 2558–2567.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Candes, E., & Tao, T. (2006). Near optimal signal recovery from random projections: Universal encoding strategies. IEEE Transactions on Information Theory, 52(12), 5406–5425.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Candes, E., Romberg, J., & Tao, T. (2006). Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 52(2), 489–509.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Donoho, D. (2006). Compressed sensing. IEEE Transactions on Information Theory, 52(4), 1289–1306.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Candes, E., & Tao, T. (2005). Decoding by linear programming. IEEE Transactions on Information Theory, 51(12), 4203–4215.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhang, Y. (2008, July). On theory of compressed sensing via ℓ1-minimization: Simple derivations and extensions. CAAM Technical Report TR08-11, Department of Computational and Applied Mathematics, Rice University.Google Scholar
  23. 23.
    Chen, S. S. (1995). Basis pursuit, Ph.D. thesis, Stanford University, Department of Statistics.Google Scholar
  24. 24.
    Chen, S. S., Donoho, D. L., & Saunders, M. A. (2001). Atomic decomposition by basis pursuit. SIAM Review, 43(1), 129–159.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rudelson, M., & Vershynin, R. (2005). Geometric approach to error-correcting codes and reconstruction of signals. International Mathematics Research Notices, 64, 4019–4041.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Donoho, D., & Tanner, J. (2005). Neighborliness of randomly-projected simplices in high dimensions. Proceedings of the National Academy of Sciences, 102(27), 9452–9457.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Candles, E. J., Wakin, M. B., & Boyd, S. (2008). Enhancing sparsity by reweighted ℓ1 minimization. Journal of Fourier Analysis and Applications, 14(5), 877–905.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tropp, J. (2006). Just relax: Convex programming methods for identifying sparse signals. IEEE Transactions on Information Theory, 51, 1030–1051.MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tsaig, Y., & Donoho, D. (2005). Extensions of compressed sensing. Signal Processing, 86(3), 533–548.CrossRefGoogle Scholar
  30. 30.
    Rudin, L., Osher, S., & Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D, 60, 259–268.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Chambolle, A. (2004). An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 20, 89–97.MathSciNetCrossRefGoogle Scholar
  32. 32.
    Chan, T. F., Esedoglu, S., Park, F., & Yip, A. (2004). Recent developments in total variation image restoration. CAM Report 05-01, Department of Mathematics, UCLA.Google Scholar
  33. 33.
    Geman, D., & Reynolds, G. (1992). Constrained restoration and the recovery of discontinuities. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(3), 367–383.CrossRefGoogle Scholar
  34. 34.
    Geman, D., & Yang, C. (1995). Nonlinear image recovery with half-quadratic regularization. IEEE Transactions on Image Processing, 4(7), 932–946.CrossRefGoogle Scholar
  35. 35.
    Yang, J., Yin, W., Zhang, Y., & Wang, Y. A fast algorithm for edge-preserving variational multichannel image restoration. Technical Report 08-09, CAAM, Rice University, Submitted to SIIMS.Google Scholar
  36. 36.
    Yang, J., Zhang, Y., & Yin, W. An efficient TVL1 algorithm for deblurring of multichannel images corrupted by impulsive noise. TR08-12, CAAM, Rice University, Submitted to SISC.Google Scholar
  37. 37.
    Li, C. (2009). An efficient algorithm for total variation regularization with applications to the single pixel camera and compressed sensing.Google Scholar
  38. 38.
    Bregman, L. (1967). The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex optimization. USSR Computational Mathematics and Mathematical Physics, 7, 200–217.CrossRefGoogle Scholar
  39. 39.
    Yin, W., Osher, S., Goldfarb, D., & Darbon, J. (2008). Bregman iterative algorithms for L1-minimization with applications to compressed sensing. SIAM Journal on Imaging Sciences, 1, 142–168.CrossRefGoogle Scholar
  40. 40.
    Cai, J. F., Osher, S., & Shen, Z. Linearized Bregman iterations for compressed sensing. UCLA CAM Report, 08-06.Google Scholar
  41. 41.
    Goldstein, T., & Osher, S. The Split Bregman method for L1 regularized problems. UCLA CAM Report, 08-29.Google Scholar
  42. 42.
    Yin, W., & Osher, S. (2012). Error forgetting of Bregman iteration. Journal of Scientific Computing, 54(2), 684–698.MathSciNetzbMATHGoogle Scholar
  43. 43.
    Natarajan, B. K. (1995). Sparse approximate solutions to linear systems. SIAM Journal on Computing, 24(2), 227–234.MathSciNetCrossRefGoogle Scholar
  44. 44.
    Zhang, Y. (2009, May). User’s guide for YALL1: Your algorithms for L1 optimization. Technical Report TR09-17, Department of Computational and Applied Mathematics, Rice University.Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of ECENitte Meenakshi Institute of TechnologyBengaluruIndia

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