Your browser does not support JavaScript!
http://iet.metastore.ingenta.com
1887

access icon free Cauchy diversity measures: a novel methodology for enhancing sparsity in compressed sensing

© The Institution of Engineering and Technology
Received 03/11/2012, Accepted 15/04/2013, Revised 13/03/2013, Published

As a new enchanting theory, compressed sensing (CS) demonstrates that a sparse signal can be recovered through a surprisingly small number of linear measurements by solving a problem of ℓ 1 norm minimisation (which can be thought as a special case of the signomial diversity measures). However, the traditional CS model with ℓ 1 norm minimisation can not fully exploit the sparsity especially when the degree of sparsity increases or the measurements number reduces. In this study, the Cauchy diversity measures is incorporated into the proposed model to deal with the above difficulties. The simulation results demonstrate that under the same condition, this new model offers a superior reconstruction precision compared with the common used signomial diversity measures.

Inspec keywords: compressed sensing; minimisation; signal reconstruction

Other keywords: linear measurements; CS model; compressed sensing; signomial diversity measures; sparse signal recovery; â„“1 norm minimisation; reconstruction precision; Cauchy diversity measures; enchanting theory; sparsity enhancement

Subjects: Optimisation techniques; Signal processing and detection; Signal processing theory; Optimisation techniques

References

    1. 1)
      • 22. Gorodnitsky, I.F., George, J.S., Rao, B.D.: ‘Neuromagnetic source imaging with FOCUSS: a recursive weighted minimum norm algorithm’, J. Electroencephalog. Clin. Neurophysiol., 1995, 95, (4), pp. 231–251 (doi: 10.1016/0013-4694(95)00107-A).
    2. 2)
      • 8. Herman, M., Strohmer, T.: ‘High-resolution radar via compressed sensing’, IEEE Trans. Signal Process., 2009, 57, (6), pp. 2275–2284 (doi: 10.1109/TSP.2009.2014277).
    3. 3)
      • 27. Taswell, C.: ‘Satisficing search algorithms for selecting near-best in adaptive tree-structured wavelet trasforms’, IEEE Trans. Signal Process., 1996, 44, (10), pp. 2423–2438 (doi: 10.1109/78.539028).
    4. 4)
      • 32. Lorenz, M.O.: ‘Methods of measuring the concentration of wealth’, Publ. Am. Stat. Assoc., 1905, 9, (70), pp. 20–219.
    5. 5)
      • 10. Preda, R.O., Vizireanu, D.N.: ‘A robust digital watermarking scheme for video copyright protection in the wavelet domain’, Measurement, 2010, 43, (10), pp. 1720–1726 (doi: 10.1016/j.measurement.2010.07.009).
    6. 6)
      • 18. Saab, R., Chartrand, R., Yilmaz, O.: ‘Stable sparse approximations via nonconvex optimization’. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, ICASSP, 2008, pp. 3885–3888.
    7. 7)
      • 16. Chartrand, R.: ‘Exact reconstruction of sparse signals via nonconvex minimization’, IEEE Signal Process. Lett., 2007, 14, (10), pp. 707–710 (doi: 10.1109/LSP.2007.898300).
    8. 8)
      • 13. Tibshirani, R.: ‘Regression shrinkage and selection via the lasso’, J. Royal Stat. Soc. B., 1996, 58, (1), pp. 267–288.
    9. 9)
      • 35. Zonoobi, D., Kassim, A., Venkatesh, Y.: ‘Gini index as sparsity measure for signal reconstruction from compressive samples’, IEEE J. Sel. Topics Signal Process., 2011, 5, (5), pp. 927–932 (doi: 10.1109/JSTSP.2011.2160711).
    10. 10)
      • 5. Donoho, D.L., Tsaig, Y.: ‘Extensions of compressed sensing’, Signal Process., 2006, 86, (3), pp. 533–548 (doi: 10.1016/j.sigpro.2005.05.027).
    11. 11)
      • 2. Candès, E.J., Tao, T.: ‘Near optimal signal recovery from random projections: universal encoding strategies’, IEEE Trans. Inf. Theory, 2006, 52, (12), pp. 5406–5425 (doi: 10.1109/TIT.2006.885507).
    12. 12)
      • 1. Candès, E.J., Romberg, J., Tao, T.: ‘Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information’, IEEE Trans. Inf. Theory, 2006, 52, (2), pp. 489–509 (doi: 10.1109/TIT.2005.862083).
    13. 13)
      • 33. Gastwirth, , Joseph, L.: ‘The estimation of the lorenz curve and gini index’, Rev. Econ. Stat., 1972, 54, (3), pp. 306–316 (doi: 10.2307/1937992).
    14. 14)
      • 4. Candès, E.J., Wakin, M.: ‘An introduction to compressive sampling’, IEEE Signal Process. Mag., 2008, 25, (2), pp. 21–30 (doi: 10.1109/MSP.2007.914731).
    15. 15)
      • 15. Figueiredo, M., Nowak, R., Wright, S.: ‘Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems’, IEEE J. Sel. Topics Signal Process., 2007, 1, (4), pp. 586–597 (doi: 10.1109/JSTSP.2007.910281).
    16. 16)
      • 20. L.M.O., : ‘Methods of measuring the concentration of wealth’, Publ. Am. Stat. Assoc., 1905, 9, (70), pp. 20–219.
    17. 17)
      • 19. Kreutz-Delgado, K., Rao, B.D.: ‘A general approach to sparse basis selection: majorization, concavity, and affine scaling’. Technical Report UCSD-CIE-97-7-1, University of California, San Diego, 1997.
    18. 18)
      • 3. Donoho, D.L.: ‘Compressed sensing’, IEEE Trans. Inf. Theory, 2006, 52, (4), pp. 5406–5245 (doi: 10.1109/TIT.2006.871582).
    19. 19)
      • 26. Carrillo, R., Aysal, T., Barner, K.: ‘Bayesian compressed sensing using generalized cauchy priors’. Proc. Int. Conf. on Accoustics, Speech, and Signal Proc., 2010, pp. 4058–4061.
    20. 20)
      • 25. Chartrand, R., Yin, W.: ‘Iteratively reweighted algorithms for compressive sensing’. Proc. Int. Conf. on Accoustics, Speech, and Signal Proc., 2008, pp. 3869–3872.
    21. 21)
      • 9. Preda, R.O., Vizireanu, D.N.: ‘Quantisation-based video watermarking in the wavelet domain with spatial and temporal redundancy’, Int. J. Electron., 2011, 98, (3), pp. 393–405 (doi: 10.1080/00207217.2010.547810).
    22. 22)
      • 21. Hooper, D.U., Chapin, F.S.III, et al: ‘Effects of biodiversity on ecosystem functioning: a consensus of current knowledge’, Ecological Monogr., 2005, 75, (1), pp. 3–35 (doi: 10.1890/04-0922).
    23. 23)
      • 6. Lustig, M., Donoho, D.L., Santos, J.M., et al: ‘Compressed sensing MRI’, IEEE Signal Process. Mag., 2008, 25, (2), pp. 72–82 (doi: 10.1109/MSP.2007.914728).
    24. 24)
      • 7. Duarte, M., Davenport, M., Takhar, D., et al: ‘Single-pixel imaging via compressive sampling’, IEEE Signal Process. Mag., 2008, 25, (2), pp. 83–91 (doi: 10.1109/MSP.2007.914730).
    25. 25)
      • 36. Baraniuk, R.G., Cevher, V., Wakin, M.B.: ‘Low-dimensional models for dimensionality reduction and signal recovery: a geometric perspective’, Proc. IEEE, 2010, 98, (6), pp. 959–971 (doi: 10.1109/JPROC.2009.2038076).
    26. 26)
      • 34. Hurley, N., Rickard, S.: ‘Comparing measures of sparsity’, IEEE Trans. Inf. Theory, 2009, 55, (10), pp. 4723–4741 (doi: 10.1109/TIT.2009.2027527).
    27. 27)
      • 24. Rao, B.D., Kreutz-Delgado, K.: ‘An affine scaling methodology for best basis selection’, IEEE Trans. Signal Process., 1999, 47, (1), pp. 187–200 (doi: 10.1109/78.738251).
    28. 28)
      • 12. Natarajan, B.K.: ‘Sparse approximate solutions to linear systems’, SIAM J. Sci. Compon., 1995, 24, (2), pp. 227–234 (doi: 10.1137/S0097539792240406).
    29. 29)
      • 14. Chen, S., Donoho, D., Saunders, M.: ‘Atomic decomposition by basis pursuit’, SIAM J. Sci. Compon., 1998, 20, (1), pp. 33–61 (doi: 10.1137/S1064827596304010).
    30. 30)
      • 29. Beaton, A.E., Tukey, J.W.: ‘The fitting of power series, meaning polynomials, illustrated on band spectroscopic data’, Technometrics, 1974, 16, pp. 147–185 (doi: 10.1080/00401706.1974.10489171).
    31. 31)
      • 30. Cane, G.J.: ‘Linear estimation of parameters of the cauchy distribution based on sample quantiles’, J. Am. Stat. Assoc., 1974, 69, (345), pp. 243–245 (doi: 10.1080/01621459.1974.10480163).
    32. 32)
      • 17. Chartrand, R.: ‘Nonconvex compressed sensing and error correction’. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, ICASSP, 2007, vol. 3, pp. 889–892.
    33. 33)
      • 28. Wickerhauser, M.V.: ‘Adapted wavelet analysis from theory to software, in Peters, A.K., ed. (Wellesley, MA, 1994).
    34. 34)
      • 23. Gorodnitsky, I.F., Rao, B.D.: ‘Sparse signal reconstructions from limited data using FOCUSS: A re-weighted minimum norm algorithm’, IEEE Trans. Signal Process., 1997, 45, (3), pp. 600–616 (doi: 10.1109/78.558475).
    35. 35)
      • 11. Preda, R.O., Vizireanu, D.N.: ‘Robust wavelet-based video watermarking scheme for copyright protection using the human visual system’, J. Electron.c Imaging, 2011, 20, (1), pp. 13–22.
    36. 36)
      • 31. Spiegel, M.R.: ‘Theory and problems of probability and statistics’ (McGraw-Hill, New York, 1992).
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-spr.2012.0329
Loading

Related content

content/journals/10.1049/iet-spr.2012.0329
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading
Print this page
This is a required field
Please enter a valid email address