Construction of Hilbert transform pairs of wavelet bases and gabor-like transforms

IEEE Transactions on Signal Processing

Volumn 57, Issue 9, 2009, Pages 3411-3425

  Chaudhury, Kunal Narayan ^a   Unser, Michael ^a  

^a EPFL   (Switzerland)

Author keywords

Analytic signal; B spline multiresolution; Biorthogonal wavelet basis; Directional Hilbert transform; Dual tree complex wavelet transform; Gabor function; Hilbert transform; Time frequency localization

Indexed keywords

ANALYTIC SIGNAL; B-SPLINE MULTIRESOLUTION; BIORTHOGONAL WAVELET BASIS; DIRECTIONAL HILBERT TRANSFORM; DUAL-TREE COMPLEX WAVELET TRANSFORM; GABOR FUNCTION; HILBERT TRANSFORM; TIME-FREQUENCY LOCALIZATION;

FAST FOURIER TRANSFORMS; MULTIRESOLUTION ANALYSIS; PARTIAL DISCHARGES; SPLINES;

MATHEMATICAL TRANSFORMATIONS;

EID: 69349085849     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2009.2020767     Document Type: Article

References (40)

1. N. G. Kingsbury, "Shift invariant properties of the dual-tree complex wavelet transform," in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Mar. 1999, pp. 1221-1224.
2. N. G. Kingsbury, "Complex wavelets for shift invariant analysis and filtering of signals," J. Appl. Comput. Harmon. Anal., vol. 10, no. 3, pp. 234-253, May 2001.
3. P. F. C. de Rivaz and N. G. Kingsbury, "Bayesian image deconvolution and denoising using complex wavelets," in Proc. IEEE Int. Conf. Image Processing (ICIP), 2001, vol. 2, pp. 273-276.
4. L. Sendur and I. W. Selesnick, "Bivariate shrinkage with local variance estimation," IEEE Signal Process. Lett., vol. 9, no. 12, pp. 438-441, 2002.
5. S. Hatipoglu, S. K. Mitra, and N. G. Kingsbury, "Image texture description using complex wavelet transform," in Proc. IEEE Int. Conf. Image Processing (ICIP), 2000, vol. 2, pp. 530-533.
6. I. W. Selesnick, "Hilbert transform pairs of wavelet bases," IEEE Signal Process. Lett., vol. 8, no. 6, pp. 170-173, 2001.
7. I. W. Selesnick, "The design of approximate Hilbert transform pairs of wavelet bases," IEEE Trans. Signal Process., vol. 50, no. 5, pp. 1143-1152, May 2002.
8. R. Yu and H. Ozkaramanli, "Hilbert transform pairs of biorthogonal wavelet bases," IEEE Trans. Signal Process., vol. 54, no. 6, pp. 2119-2125, Jun. 2006.
9. F. C. A. Fernandes, R. L. C. Van Spaendonck, and C. S. Burrus, "A new framework for complex wavelet transforms," IEEE Trans. Signal Process., vol. 51, no. 7, pp. 1825-1837, Jul. 2003, 43.
10. C. Chaux, L. Duval, and J. C. Pesquet, "Image analysis using a dual-tree M-band wavelet transform," IEEE Trans. Image Process., vol. 15, no. 8, pp. 2397-2412, Aug. 2006.
11. D. Gabor, "Theory of communication," J. Inst. Elect. Eng., vol. 93, pp. 429-457, 1946.
12. B. Van der Pol, "The fundamental principles of frequency modulation," J. IEE, vol. 93, pp. 153-158, 1946.
13. J. Ville, "Theorie et application de la notion de signal analytique," Cables Transmiss., vol. 93, no. III, pp. 153-158, 1948.
14. M. S. Pattichis and A. C. Bovik, "Analyzing image structure by multidimensional frequency modulation," IEEE Trans. Pattern Anal. Mach. Intell., vol. 29, no. 5, pp. 753-766, May 2007.
15. M. Unser and T. Blu,"Wavelet theory demystified," IEEE Trans. Signal Process., vol. 51, no. 2, pp. 470-483, Feb. 2003.
16. M. Unser and T. Blu, "Fractional splines and wavelets," SIAM Rev. vol. 42, no. 1, pp. 43-67, Mar. 2000.
17. J. G. Daugman, "Two-dimensional spectral analysis of cortical receptive field profile," Vis. Res., vol. 20, pp. 847-856, 1980.
18. R. Bracewell, The Fourier Transform and Its Applications. New York: McGraw-Hill, 1986.
19. J. J. Benedetto, Harmonic Analysis and Applications. Boca Raton, FL: CRC Press, 1996.
20. E. M. Stein, Singular Integrals and Differentiability Property of Functions. Princeton, NJ: Princeton Univ. Press, 1970.
21. S. G. Mallat, "A theory for multiresolution signal decomposition: The wavelet representation," IEEE Trans. Pattern Anal. Mach. Intell., vol. 11, no. 7, pp. 674-693, 1989.
22. Y. Meyer, Ondelettes et opérateurs I: Ondelettes. Paris, France: Hermann, 1990.
23. T. Blu and M. Unser, "A complete family of scaling functions: The (α;τ)-fractional splines," in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Apr. 6-10, 2003, vol. VI, pp. 421-424.
24. S. Mallat, A Wavelet Tour of Signal Processing. San Diego, CA: Academic, 1998.
25. E. M. Stein and R. Shakarchi, Complex Analysis. Princeton, NJ: Princeton Univ. Press, 2003.
26. T. Blu and M. Unser, "The fractional spline wavelet transform: Definition and implementation," in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Jun. 2000, pp. 512-515.
27. M. J. Bastiaans, "Gabor's expansion of a signal into Gaussian elementary signals," Proc. IEEE, vol. 68, pp. 538-539, Apr. 1980.
28. M. Unser, A. Aldroubi, and M. Eden, "On the asymptotic convergence of B-spline wavelets to Gabor functions," IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 864-872, Mar. 1992.
29. M. Unser and T. Blu, "Construction of fractional spline wavelet bases," in Proc. SPIE Conf. Mathematical Imaging: Wavelet Applications in Signal and Image Processing VII, Jul. 1999, vol. 3813, pp. 422-431.
30. I. W. Selesnick, R. G. Baraniuk, and N. C. Kingsbury, "The dual-tree complex wavelet transform," IEEE Signal Process. Mag., vol. 22, no. 6, pp. 123-151, Nov. 2005.
31. G. H. Granlund and H. Knutsson, Signal Processing for Computer Vision Dordrecht, The Netherlands: Kluwer, 1995, ch. 4.
32. N. G. Kingsbury, "A dual-tree complex wavelet transform with improved orthogonality and symmetry properties," in Proc. IEEE Int. Conf. Image Processing (ICIP), 2000, vol. 2, pp. 375-378.
33. S. L. Hahn, "Multidimensional complex signals with single-orthant spectra," Proc. IEEE, vol. 80, no. 8, pp. 1287-1300, 1992.
34. T. Bülow and G. Sommer, "Hypercomplex signals - A novel extension of the analytic signal to the multidimensional case," IEEE Trans. Signal Process., vol. 49, no. 11, pp. 2844-2852, 2001.
35. M. Felsberg and G. Sommer, "The monogenic signals," IEEE Trans. Signal Process., vol. 49, no. 12, pp. 3136-3144, 2001.
36. K. G. Larkin, D. J. Bone, and M. A. Oldfield, "Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transforms," J. Opt. Soc. Amer. A, vol. 18, no. 8, pp. 1862-1870, 2001.
37. E. M. Stein and G. Weiss, Fourier Analysis on Euclidean Spaces. Princeton, NJ: Princeton Univ. Press, 1971.
38. W. L. Chan, H. Choi, and R. G. Baraniuk, "Coherent multiscale image processing using dual-tree quaternion wavelets," IEEE Trans. Image Process., vol. 17, no. 7, pp. 1069-1082, Jul. 2008.
39. S. C. Olhede and G. Metikas, "The hyperanalytic wavelet transform," Imperial College Statistics Section Tech. Rep. TR-06-02, 2008, pp. 1-49.
40. M. Unser, "Sampling - 50 years after Shannon," Proc. IEEE, vol. 88, no. 4, pp. 569-587, Apr. 2000.