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Structured Sparse Bayesian Modelling for Audio Restoration

  • James Murphy
  • Simon Godsill
Chapter
Part of the Signals and Communication Technology book series (SCT)

Abstract

This chapter shows how sparse solutions can be obtained for a range of problems in a Bayesian setting by using prior models on sparsity structure. As an example, a model to remove impulse and background noise from audio signals via their representation in time-frequency space using Gabor wavelets is presented. A number of prior models for the sparse structure of the signal in this space are introduced, including simple Bernoulli priors on each coefficient, Markov chains linking neighbouring coefficients in time or frequency, and Markov random fields, imposing two dimensional coherence on the coefficients. The effect of each of these priors on the reconstruction of a corrupted audio signal is shown. Impulse removal is also covered, with similar sparsity priors being applied to the location of impulse noise in the audio signal. Inference is performed by sampling from the posterior distribution of the model variables using a Gibbs sampler.

References

  1. 1.
    Balian R (1981) Un principe dincertitude fort en théorie du signal ou en mécanique quantique. CR Acad Sci Paris 292(2):1357–1361Google Scholar
  2. 2.
    Boll S (1979) Suppression of acoustic noise in speech using spectral subtraction. IEEE Trans Acoust Speech Signal Process 27(2):113–120CrossRefGoogle Scholar
  3. 3.
    Candès EJ, Romberg J, Tao T (2006) Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory 52(2):489–509CrossRefzbMATHGoogle Scholar
  4. 4.
    Candès EJ, Tao T (2006) Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans Inf Theory 52(12):5406–5425CrossRefGoogle Scholar
  5. 5.
    Chen SS, Donoho DL, Saunders MA (2001) Atomic decomposition by basis pursuit. SIAM Rev 43(1):129–159MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Donoho DL (2006) Compressed sensing. IEEE Trans Inf Theory 52(4):1289–1306MathSciNetCrossRefGoogle Scholar
  7. 7.
    Erkelens JS, Heusdens R (2008) Tracking of nonstationary noise based on data-driven recursive noise power estimation. IEEE Trans Audio Speech Lang Process 16(6):1112–1123CrossRefGoogle Scholar
  8. 8.
    Feichtinger HG, Strohmer T (1998) Gabor analysis algorithms: theory and applications. Birkhäuser, BostonGoogle Scholar
  9. 9.
    Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans Pattern Anal Mach Intell 6:721–741CrossRefzbMATHGoogle Scholar
  10. 10.
    Gilks WR, Gilks WR, Richardson S, Spiegelhalter DJ (1996) Markov chain Monte Carlo in practice. Chapman & Hall/CRC, LondonGoogle Scholar
  11. 11.
    Godsill SJ , Rayner PJW (1998) Digital audio restoration:a statistical model-based approach. Springer, Berlin (ISBN 3 540 76222 1, Sept 1998)Google Scholar
  12. 12.
    Godsill SJ (2010) The shifted inverse-gamma model for noise floor estimation in archived audio recordings. Appl Signal Process 90.991-999(Special Issue on Preservation of Ethnological Recordings)Google Scholar
  13. 13.
    Gustafsson S, Martin R, Jax P, Vary P (2002) A psychoacoustic approach to combined acoustic echo cancellation and noise reduction. IEEE Trans Speech Audio Process 10(5):245–256CrossRefGoogle Scholar
  14. 14.
    Low F (1985) Complete sets of wave packets. A passion for physics-essays in honor of Geoofrey Chew. World Scientific, Singapore, pp 17–22Google Scholar
  15. 15.
    Mallat SG, Zhang Z (1993) Matching pursuits with time-frequency dictionaries. IEEE Trans sig process 41(12):3397–3415CrossRefzbMATHGoogle Scholar
  16. 16.
    McGrory CA, Titterington DM, Reeves R et al (2009) DM Titterington, R. Reeves, and A.N. Pettitt. Variational Bayes for estimating the parameters of a hidden Potts model. Stat Comput 19(3):329–340Google Scholar
  17. 17.
    Murphy J, Godsill S (2011) Joint Bayesian removal of impulse and background noise. In: Proceedings of the IEEE international conference on acoustics, speech and signal processing (ICASSP), pp 261–264Google Scholar
  18. 18.
    Murphy J, (2013) Sparse audio restoration in hidden states, hidden structures: bayesian learning in time series models, PhD Thesis, Cambridge UniversityGoogle Scholar
  19. 19.
    Niss M (2005) History of the Lenz-Ising model 1920–1950: from ferromagnetic to cooperative phenomena. Arch Hist Exact Sci 59(3):267–318MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Qian S, Chen D ( 1993) Discrete Gabor transform. IEEE Trans Sig Process 41(7):2429–2438Google Scholar
  21. 21.
    Soon IY, Koh SN, Yeo CK (1998) Noisy speech enhancement using discrete cosine transform. Speech Commun 24(3):249–257CrossRefGoogle Scholar
  22. 22.
    Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc Ser B (Methodol) 58: 267–288Google Scholar
  23. 23.
    Wolfe PJ, Godsill SJ, Ng WJ (2004) Bayesian variable selection and regularisation for time-frequency surface estimation. J R Stat Soc Ser B 66(3):575–589 Read paper (with discussion)Google Scholar
  24. 24.
    Wolfe PJ, Godsill SJ (2005) Interpolation of missing data values for audio signal restoration using a Gabor regression model. In: Proceedings of the IEEE international conference on acoustics, speech and signal processing (ICASSP), pp. 517–520Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of EngineeringCambridge UniversityWuppertalCambridge

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