Some Studies on Multidimensional Fourier Theory for Hilbert Transform, Analytic Signal and AM–FM Representation
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.
KeywordsFourier 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
Authors would like to thank the editors and anonymous reviewers for their constructive and thorough comments and suggestions which improved the presentation of manuscript.
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.
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- 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
- 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
- 8.D. Gabor, Theory of communication. J. IEE 93, 429–457 (1946)Google Scholar
- 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.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.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
- 20.P. Kovesi, Image features from phase congruency. Videre J. Comput. Vis. Res. 1(3), 1–26 (1999)Google Scholar
- 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.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
- 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
- 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.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.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.P. Singh, Studies on Generalized Fourier Representations and Phase Transforms, arXiv:1808.06550 [eess.SP] (2018)
- 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
- 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)
- 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.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.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