EPFL | Biomedical Imaging Group | Fractional Hilbert Transform

The Fractional Hilbert Transform and Dual-Tree Gabor-Like Wavelet Analysis

K.N. Chaudhury, M. Unser

Proceedings of the Thirty-Fourth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'09), Taipei, Taiwan (People's Republic of China), April 19-24, 2009, pp. 3205–3208.

We provide an amplitude-phase representation of the dual-tree complex wavelet transform by extending the fixed quadrature relationship of the dual-tree wavelets to arbitrary phase-shifts using the fractional Hilbert transform (fHT). The fHT is a generalization of the Hilbert transform that extends the quadrature phase-shift action of the latter to arbitrary phase-shifts—a real shift parameter controls this phase-shift action.

Next, based on the proposed representation and the observation that the fHT operator maps well-localized B-spline wavelets (that resemble Gaussian-windowed sinusoids) into B-spline wavelets of the same order but different shift, we relate the corresponding dual-tree scheme to the paradigm of multiresolution windowed Fourier analysis.

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