Advertisement

Circuits, Systems, and Signal Processing

, Volume 38, Issue 12, pp 5623–5650 | Cite as

Some Studies on Multidimensional Fourier Theory for Hilbert Transform, Analytic Signal and AM–FM Representation

  • Pushpendra SinghEmail author
  • Shiv Dutt Joshi
Article

Abstract

In this paper, we propose the notion of Fourier frequency vector (FFV) which is inherently associated with the multidimensional (MD) Fourier representation (FR) of a signal. The proposed FFV provides physical meaning to the so-called negative frequencies in the MD-FR that in turn yields MD spatial and MD space-time series analysis. The one-dimensional Hilbert transform (1D-HT) and associated 1D analytic signal (1D-AS) of an 1D signal are well established; however, their true generalization to an MD signal, which possess all the properties of 1D case, are not available in the literature. To achieve this, we observe that in MD-FR the complex exponential representation of a sinusoidal function always yields two frequencies, namely negative frequency corresponding to positive frequency and vice versa. Thus, using the MD-FR, we propose MD-HT and associated MD analytic signal (AS) as a true generalization of the 1D-HT and 1D-AS, respectively, and obtain an explicit expression for the analytic image computation by 2D discrete Fourier transform (2D-DFT). We also extend the Fourier decomposition method for 2D signals that decomposes an image into a set of amplitude-modulated and frequency-modulated (AM–FM) image components. We finally propose a single-orthant Fourier transform (FT) of real MD signals which computes FT in the first orthant, and values in rest of the orthants are obtained by simple conjugation defined in this study.

Keywords

Fourier representation (FR) and Fourier frequency vector Hilbert transform (HT) and analytic signal (AS) Single-orthant Fourier transform (SOFT) Fourier decomposition method (FDM) Linearly independent non-orthogonal yet energy-preserving (LINOEP) vectors 

Notes

Acknowledgements

Authors would like to thank the editors and anonymous reviewers for their constructive and thorough comments and suggestions which improved the presentation of manuscript.

Author Contributions

P. Singh conceived and designed the study, carried out the simulation work, participated in data analysis, and drafted the manuscript; S. D. Joshi discussed and checked the mathematical analyses, coordinated the study and helped in drafting the manuscript. All authors commented and gave final approval for publication.

Funding

There is no funding to support this research.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

Permission to Carry out Fieldwork

No permissions were required prior to conducting this research.

Supplementary material

References

  1. 1.
    S. Acton, P. Soliz, S. Russell, M. Pattichis, Content based image retrieval: the foundation for future case-based and evidence-based ophthalmology, in Proceedings of IEEE International Conference on Multimedia and Expo (2008), pp. 541–544Google Scholar
  2. 2.
    S.T. Acton, D.P. Mukherjee, J.P. Havlicek, A.C. Bovik, Oriented texture completion by AM–FM reaction–diffusion. IEEE Trans. Image Process. 10(6), 885–896 (2001)CrossRefGoogle Scholar
  3. 3.
    M. Bahri, E.S.M. Hitzer, A. Hayashi, R. Ashino, An uncertainty principle for quaternion Fourier transform. Comput. Math. Appl. 56, 2398–2410 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Bernstein, J.L. Bouchot, M. Reinhardt, B. Heise, Quaternion and Clifford Fourier Transforms and Wavelets Trends in Mathematics (Birkhäuser, Basel, 2013), pp. 221–246CrossRefGoogle Scholar
  5. 5.
    T. Bulow, G. Sommer, Multi-dimensional signal processing using an algebraically extended signal representation, in ed by. Sommer, G. AFPAC (Springer, Heidelberg, 1997). LNCS, 1315, pp. 148–163Google Scholar
  6. 6.
    M. Felsberg, G. Sommer, The monogenic signal. IEEE Trans. Signal Process. 49(12), 3136–3144 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    L. Fortuna, P. Arena, D. Balya, A. Zarandy, Cellular neural networks: a paradigm for nonlinear spatio-temporal processing. IEEE Circuits Syst. Mag. 1(4), 6–21 (2001)CrossRefGoogle Scholar
  8. 8.
    D. Gabor, Theory of communication. J. IEE 93, 429–457 (1946)Google Scholar
  9. 9.
    G.H. Granlund, H. Knutsson, Signal Processing for Computer Vision (Kluwer, Dordrecht, 1995)CrossRefGoogle Scholar
  10. 10.
    A. Gupta, S.D. Joshi, P. Singh, On the approximate discrete KLT of fractional Brownian motion and applications. J. Frankl. Inst. 355(17), 8989–9016 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Gupta, P. Singh, M. Karlekar, A novel signal modeling approach for classification of seizure and seizure-free EEG signals. IEEE Trans. Neural Syst. Rehabil. Eng. 26(5), 925–935 (2018)CrossRefGoogle Scholar
  12. 12.
    S.L. Hahn, Multidimensional complex signals with single-orthant spectra. Proc. IEEE 80(8), 1287–1300 (1992)CrossRefGoogle Scholar
  13. 13.
    J.J. Havlicek, J. Tang, S. Acton, R. Antonucci, F. Quandji, Modulation domain texture retrieval for CBIR in digital libraries, in 37th IEEE Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA (2003)Google Scholar
  14. 14.
    J. Havlicek, P. Tay, A. Bovik, AM–FM image models: fundamental techniques and emerging trends, in Handbook of Image and Video Processing, pp. 377–395. Elsevier Academic Press (2005)Google Scholar
  15. 15.
    J.P. Havlicek, J.W. Havlicek, N.D. Mamuya, A.C. Bovik, Skewed 2D Hilbert transforms and AM–FM models, in ICIP 98. Proc. International Conference on Image Processing, October 4–7 (1998), 1, pp. 602–606Google Scholar
  16. 16.
    N.E. Huang, Z. Shen, S. Long, M. Wu, H. Shih, Q. Zheng, N. Yen, C. Tung, H. Liu, The empirical mode decomposition and Hilbert spectrum for non-linear and non-stationary time series analysis. Proc. R. Soc. A 454, 903–995 (1988)CrossRefGoogle Scholar
  17. 17.
    K. Kohlmann, Corner detection in natural images based on the 2D Hilbert transform. Signal Proc. 48, 225–234 (1996)CrossRefGoogle Scholar
  18. 18.
    I. Kokkinos, G. Evangelopoulos, P. Maragos, Texture analysis and segmentation using modulation features, generative models, and weighted curve evolution. IEEE Trans. Pattern Anal. Mach. Intell. 31(1), 142–157 (2009)CrossRefGoogle Scholar
  19. 19.
    K.I. Kou, M.S. Liu, J.P. Morais, C. Zou, Envelope detection using generalized analytic signal in 2D QLCT domains. Multidimens. Syst. Signal Process. 28(4), 1343–1366 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    P. Kovesi, Image features from phase congruency. Videre J. Comput. Vis. Res. 1(3), 1–26 (1999)Google Scholar
  21. 21.
    J.V. Lorenzo-Ginori, An approach to the 2D Hilbert transform for image processing applications, in 4th International Conference, ICIAR (2007), Montreal, Canada, August 22–24, proceedingsGoogle Scholar
  22. 22.
    V. Murray, P. Rodriguez, M.S. Pattichis, Robust multiscale AM-FM demodulation of digital images. IEEE Int. Conf. Image Process. 1, 465–468 (2007)Google Scholar
  23. 23.
    V. Murray, M.S. Pattichis, E.S. Barriga, P. Soliz, Recent multiscale AM–FM methods in emerging applications in medical imaging. EURASIP J. Adv. Signal Process. 2012, 23 (2012).  https://doi.org/10.1186/1687-6180-2012-23 CrossRefGoogle Scholar
  24. 24.
    N. Mould, C. Nguyen, J. Havlicek, Infrared target tracking with AM–FM consistency checks, in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation SSIAI 2008, pp. 5–8 (2008)Google Scholar
  25. 25.
    M. Pattichis, G. Panayi, A. Bovik, H. Shun-Pin, Fingerprint classification using an AM–FM model. IEEE Trans. Image Process. 10(6), 951–954 (2001)CrossRefGoogle Scholar
  26. 26.
    M. Pattichis, C. Pattichis, M. Avraam, A. Bovik, K. Kyriakou, AM–FM texture segmentation in electron microscopic muscle imaging. IEEE Trans. Med. Imaging 19(12), 1253–1258 (2000)CrossRefGoogle Scholar
  27. 27.
    S.C. Pei, J.J. Ding, The generalized radial Hilbert transform and its applications to 2-D edge detection (any direction or specified directions), in ICASSP’03. Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, April 6–10 (2003) 3, p. 357–360Google Scholar
  28. 28.
    R. Prakash, R. Aravind, Modulation-domain particle filter for template tracking, in Proceedings of the 19th International Conference on Pattern Recognition ICPR 2008, pp. 1–4 (2008)Google Scholar
  29. 29.
    R.S. Prakash, R. Aravind, Invariance properties of AM–FM image features with application to template tracking, in Proceedings of the Sixth Indian Conference on Computer Vision, Graphics and Image Processing ICVGIP’ 08, pp. 614–620 (2008)Google Scholar
  30. 30.
    P. Singh, Studies on Generalized Fourier Representations and Phase Transforms, arXiv:1808.06550 [eess.SP] (2018)
  31. 31.
    P. Singh, S.D. Joshi, R.K. Patney, K. Saha, The Fourier decomposition method for nonlinear and non-stationary time series analysis. Proc. R. Soc. A 473(2199) (2017)Google Scholar
  32. 32.
    P. Singh, R.K. Patney, S.D. Joshi, K. Saha, Some studies on nonpolynomial interpolation and error analysis. Appl. Math. Comput. 244, 809–821 (2014)MathSciNetzbMATHGoogle Scholar
  33. 33.
    P. Singh, R.K. Patney, S.D. Joshi, K. Saha, The Hilbert spectrum and the energy preserving empirical mode decomposition, arXiv:1504.04104v1 [cs.IT] (2015)
  34. 34.
    P. Singh, S.D. Joshi, R.K. Patney, K. Saha, Fourier-based feature extraction for classification of EEG signals using EEG rhythms. Circuits Syst. Signal Process. 35(10), 3700–3715 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    P. Singh, Novel Fourier quadrature transforms and analytic signal representations for nonlinear and non-stationary time-series analysis. R Soc Open Sci 5(11), 1–26 (2018).  https://doi.org/10.1098/rsos.181131 CrossRefGoogle Scholar
  36. 36.
    P. Singh, Some studies on a generalized Fourier expansion for nonlinear and nonstationary time series analysis, Ph.D. dissertation, Department of Electrical Engineering, IIT Delhi (2016)Google Scholar
  37. 37.
    R.A. Sivley, J.P. Havlicek, Perfect reconstruction AM–FM image models, in IEEE International Conference on Image Processing, pp. 2125–2128 (2006)Google Scholar
  38. 38.
    P. Tay, AM–FM image analysis using the Hilbert–Huang transform, in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation SSIAI 2008, p. 13–16 (2008)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of ECE, School of Engineering and Applied SciencesBennett UniversityGreater NoidaIndia
  2. 2.Department of Electrical EngineeringIndian Institute of Technology DelhiNew DelhiIndia

Personalised recommendations