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Sparse Gabor Time–Frequency Representation Based on \(\ell _{1/2}\)–\(\ell _2\) Regularization

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Abstract

The discrete Gabor transform (DGT) is an important and widely used time–frequency analysis tool in signal processing. In this paper, a sparse time–frequency representation (TFR) method based on the DGT is studied using mixed \(\ell _{1/2}\)–\(\ell _2\) norm regularization which can overcome the over-sparse problem and permit a better sparsity and stability. Because the width of the window function in the DGT directly affects the time–frequency resolution and concentration of the Gabor spectrum, an adaptive optimal window width (AOWW) selection algorithm based on information entropy theory can be applied to search for the optimal width in the DGT. In this paper, the traditional DGT is converted into a constraint optimization problem by minimizing a mixed \(\ell _{1/2}\)–\(\ell _2\) norm sparse constraint of the Gabor coefficients. The numerical experimental results show that the proposed sparsity-based TFR based on the conventional DGT is an effective and powerful tool for analyzing and processing nonstationary signals, obtaining a higher time–frequency concentration and sparsity of the Gabor TFR of a given signal.

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References

  1. A. Akan, L.F. Chaparro, Multi-window Gabor expansion for evolutionary spectral analysis. Signal Process. 63(3), 249–262 (1997)

    Article  MATH  Google Scholar 

  2. J. Akhtar, K.E. Olsen, Sparse frequency extrapolation of spectrograms. in 2017 25th European Signal Processing Conference (EUSIPCO). IEEE (2017), pp. 1135–1139

  3. P. Balazs, M. Dörfler, F. Jaillet, Theory, implementation and applications of nonstationary Gabor frames. J. Comput. Appl. Math. 236(6), 1481–1496 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. M.J. Bastiaans, Gabor’s expansion of a signal into Gaussian elementary signals. Opt. Eng. 68(4), 538–539 (1980)

    Google Scholar 

  5. P. Busch, T. Heinonen, P. Lahti, Heisenberg’s uncertainty principle. Phys. Rep. 452(6), 155–176 (2013)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. E. Cands, J. Romberg, Sparsity and incoherence in compressive sampling. Inverse Probl. 23(3), 969–985(17) (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Cohen, Time–frequency distributions—a review. Proc. IEEE 77(7), 941–981 (1989)

    Article  Google Scholar 

  9. I. Daubechies, R. Devore, M. Fornasier et al., Iteratively reweighted least squares minimization for sparse recovery. Commun. Pure Appl. Math. 63(1), 1–38 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. N. Delprat, Global frequency modulation laws extraction from the Gabor transform of a signal: a first study of the interacting components case. IEEE Trans. Speech Audio Process. 5(1), 64–71 (1997)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Dörfler, E. Matusiak, Nonstationary Gabor frames-existence and construction. Int. J. Wavelets Multiresolut. Inf. Process. 12(3), 1481–1496 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Elad, M.A.T. Figueiredo, Y. Ma, On the role of sparse and redundant representations in image processing. Proc. IEEE 98(6), 972–982 (2010)

    Article  Google Scholar 

  14. E. Erelebi, Speech enhancement based on the discrete Gabor transform and multi-notch adaptive digital filters. Appl. Acoust. 65(8), 739–762 (2004)

    Article  Google Scholar 

  15. J. Florent, Time–frequency jigsaw puzzle: adaptive multiwindow and multilayered Gabor expansions. Int. J. Wavelets Multiresolut. Inf. Process. 5(02), 293–315 (2009)

    MathSciNet  MATH  Google Scholar 

  16. D. Gabor, Theory of communication. J. Inst. Electr. eng. 93(3), 429–457 (1946)

    Google Scholar 

  17. N. Holighaus, Structure of nonstationary Gabor frames and their dual systems. Appl. Comput. Harmonic Anal. 37(3), 442–463 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. L. Hui, Z. Qingfu, X. Zong-Ben, A preference-based multiobjective evolutionary approach for sparse optimization. IEEE Trans. Neural Netw. Learn. Syst. 29(5), 1716–1731 (2018)

    Article  MathSciNet  Google Scholar 

  19. S.D. Li, Discrete multi-Gabor expansions. IEEE Trans. Inf. Theory 45(6), 1954–1967 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Q.F. Lian, J. Gong, M.H. You, Time-domain characterization of multiwindow Gabor systems on discrete periodic sets. Indian J. Pure Appl. Math. 44(1), 47–76 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. T. Liang, H.K. Kwan, Noise reduction for NMR FID signals via oversampled real-valued discrete Gabor transform. IEICE Trans. Inf. Syst. 88–D(7), 1511–1518 (2005)

    Google Scholar 

  22. C. Lu, S. Joshi, J.M. Morris, Parallel lattice structure of block time-recursive generalized Gabor transforms. Signal Process. 57(2), 195–203 (1997)

    Article  MATH  Google Scholar 

  23. B.S. Manjunath, W.Y. Ma, Texture features for browsing and retrieval of image data. IEEE Trans. PAMI 18(8), 837–842 (1996)

    Article  Google Scholar 

  24. J.M. Morris, Y. Lu, Discrete Gabor expansion of discrete-time signals in \(\ell ^2(z)\) via frame theory. Signal Process. 40(2–3), 155–181 (1994)

    Article  MATH  Google Scholar 

  25. I. Orović, S. Stanković, M. Amin, Compressive sensing for sparse time–frequency representation of nonstationary signals in the presence of impulsive noise. in Compressive Sensing II, vol. 8717. International Society for Optics and Photonics (2013), p. 87170A

  26. M. Öztürk, A. Akan, Y. Çekiç, A robust image watermarking in the joint time–frequency domain. Indian J. Pure Appl. Math. 2010(1), 1–9 (2010)

    Google Scholar 

  27. S.C. Pei, S.G. Huang, STFT with adaptive window width based on the chirp rate. IEEE Trans. Signal Process. 60(8), 4065–4080 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. S. Qian, D. Chen, Discrete Gabor transforms. IEEE Trans. Signal Process. 41(7), 2429–2438 (1993)

    Article  MATH  Google Scholar 

  29. S. Qiu, H.G. Feichtinger, Discrete Gabor structures and optimal representation. IEEE Trans. Signal Process. 43(10), 2258–2268 (1995)

    Article  Google Scholar 

  30. E. Sejdić, I. Djurović, J. Jiang, Time–frequency feature representation using energy concentration: an overview of recent advances. Digit. Signal Process. 19(1), 153–183 (2009)

    Article  Google Scholar 

  31. E. Sejdić, I. Orović, S. Stanković, Compressive sensing meets time–frequency: an overview of recent advances in time–frequency processing of sparse signals. Digit. Signal Process. 77, 22–35 (2018)

    Article  MathSciNet  Google Scholar 

  32. L. Stanković, A measure of some time–frequency distributions concentration. Signal Process. 81(3), 621–631 (2001)

    Article  MATH  Google Scholar 

  33. D.F. Stewart, L.C. Potter, Computationally attractive real Gabor transforms. IEEE Trans. Signal Process. 43(1), 77–84 (1995)

    Article  Google Scholar 

  34. T. Strohmer, Numerical algorithms for discrete Gabor expansions, Gabor Analysis and Algorithms (Springer, Berlin, 1998), pp. 267–294

    Chapter  MATH  Google Scholar 

  35. M. Szyper, New time domain window. Electron. Lett. 31(9), 707–708 (1995)

    Article  Google Scholar 

  36. L. Tao, G.H. Hu, H.K. Kwan, Multiwindow real-valued discrete Gabor transform and its fast algorithms. IEEE Trans. Signal Process. 63(20), 5513–5524 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. D.F. Walnut, Continuity properties of the Gabor frame operator. J. Math. Anal. Appl. 165(2), 479–504 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. J. Wexler, S. Raz, Discrete Gabor expansions. Signal Process. 21(3), 207–220 (1990)

    Article  Google Scholar 

  39. W.J. Williams, A.O. Hero, Uncertainty, information, and time–frequency distributions. Proc. SPIE—Int. Soc. Opt. Eng. 1566, 144–156 (1991)

    Google Scholar 

  40. F.C. Xing, Investigation on solutions of cubic equations with one unknown. J. Cent. Univ. Natl. 12(3), 207–218 (2003)

    Google Scholar 

  41. Z.B. Xu, X.Y. Chang, F.M. Xu, H. Zhang, \(\ell _{1/2}\) regularization: a thresholding representation theory and a fast solver. IEEE Trans. Neural Netw. Learn. Syst. 23(7), 1013–1027 (2012)

    Article  Google Scholar 

  42. Z.B. Xu, H. Liang, Y. Wang, H. Zhang, Representative of \(\ell _{1/2}\) regularization among \(\ell _q (0 < q < 1)\) regularizations: an experimental study based on phase diagram. Acta Automat. Sin. 38(7), 1225–1228 (2012)

    Article  Google Scholar 

  43. L. Zhu, Y. Hao, Y. Song, \({L_{1/2}}\) norm and spatial continuity regularized low-rank approximation for moving object detection in dynamic background. IEEE Signal Process. Lett. 25(1), 15–19 (2018)

    Article  Google Scholar 

  44. M. Zibulski, Y.Y. Zeevi, Gabor representation with oversampling. Proc. SPIE—Int. Soc. Opt. Eng. 1818, 976–984 (1992)

    Google Scholar 

  45. M. Zibulski, Y.Y. Zeevi, Oversampling in the Gabor scheme. IEEE Trans. Signal Process. 41(8), 2679–2687 (1993)

    Article  MATH  Google Scholar 

  46. M. Zibulski, Y.Y. Zeevi, Discrete multiwindow Gabor-type transforms. IEEE Trans. Signal Process. 45(6), 1428–1442 (1997)

    Article  MATH  Google Scholar 

  47. T.P. Zielinski, Joint time–frequency resolution of signal analysis using Gabor transform. IEEE Trans. Instrum. Meas. 50(5), 1436–1444 (2001)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to express our sincere gratitude to the anonymous referees for their constructive comments. This work was supported in part by the National Natural Science Foundation of China under Grant No. 61301295, the Anhui Provincial Natural Science Foundation under Grant No. 1708085MF151, the Natural Science Foundation for the Higher Education Institutions of Anhui Province under Grant No. KJ2018A0018, and the Natural Science Foundation of Anhui Science and Technology University under Grant No. XWYJ201802.

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  1. College of Information and Network Engineering, Anhui Science and Technology University, No. 1501, Huangshan Road, Bengbu, 233030, Anhui, China

    Rui Li

  2. School of Computer Science and Technology, Anhui University, No. 111, Jiulong Road, Hefei, 230601, Anhui, China

    Jian Zhou

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Li, R., Zhou, J. Sparse Gabor Time–Frequency Representation Based on \(\ell _{1/2}\)–\(\ell _2\) Regularization. Circuits Syst Signal Process 38, 4700–4722 (2019). https://doi.org/10.1007/s00034-019-01077-2

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