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

Kunal Narayan Chaudhury, Michaël Unser
Ieee Service Center, 445 Hoes Lane, Po Box 1331, Piscataway, Nj 08855-1331 Usa, 2009
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Kunal Narayan Chaudhury, Michaël Unser
Ieee Service Center, 445 Hoes Lane, Po Box 1331, Piscataway, Nj 08855-1331 Usa, 2009
Article de conférence

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https://infoscience.epfl.ch/record/159526?ln=fr

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thumb|Ondelette de Daubechies d'ordre 2. Une ondelette est une fonction à la base de la décomposition en ondelettes, décomposition similaire à la transformée de Fourier à court terme, utilisée dans l

thumb|Portrait de Joseph Fourier. En mathématiques, plus précisément en analyse, la transformation de Fourier est une extension, pour les fonctions non périodiques, du développement en série de Fou

En mathématiques et en traitement du signal, la transformation de Hilbert, ici notée \mathcal{H}, d'une fonction de la variable réelle est une transformation linéaire qui permet d'étendr

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

Kunal Narayan Chaudhury, Michaël Unser

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.

IEEE2009

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On the Shiftability of Dual-Tree Complex Wavelet Transforms

Kunal Narayan Chaudhury, Michaël Unser

The dual-tree complex wavelet transform (DT-ℂWT) is known to exhibit better shift-invariance than the conventional discrete wavelet transform. We propose an amplitude-phase representation of the DT-ℂWT which, among other things, offers a direct explanation for the improvement in the shift-invariance. The representation is based on the shifting action of the group of fractional Hilbert transform (fHT) operators, which extends the notion of arbitrary phase-shifts from sinusoids to finite-energy signals (wavelets in particular). In particular, we characterize the shiftability of the DT-ℂWT in terms of the shifting property of the fHTs. At the heart of the representation are certain fundamental invariances of the fHT group, namely that of translation, dilation, and norm, which play a decisive role in establishing the key properties of the transform. It turns out that these fundamental invariances are exclusive to this group. Next, by introducing a generalization of the Bedrosian theorem for the fHT operator, we derive an explicitly understanding of the shifting action of the fHT for the particular family of wavelets obtained through the modulation of lowpass functions (e.g., the Shannon and Gabor wavelet). This, in effect, links the corresponding dual-tree transform with the framework of windowed-Fourier analysis. Finally, we extend these ideas to the multidimensional setting by introducing a directional extension of the fHT, the fractional directional Hilbert transform. In particular, we derive a signal representation involving the superposition of direction-selective wavelets with appropriate phase-shifts, which helps explain the improved shift-invariance of the transform along certain preferential directions.

IEEE2010

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Gabor Wavelet Analysis and the Fractional Hilbert Transform

Kunal Narayan Chaudhury, Michaël Unser

We propose an amplitude-phase representation of the dual-tree complex wavelet transform (DT-ℂWT) which provides an intuitive interpretation of the associated complex wavelet coefficients. The representation, in particular, is based on the shifting action of the group of fractional Hilbert transforms (fHT) which allow us to extend the notion of arbitrary phase-shifts beyond pure sinusoids. We explicitly characterize this shifting action for a particular family of Gabor-like wavelets which, in effect, links the corresponding dual-tree transform with the framework of windowed-Fourier analysis. We then extend these ideas to the bivariate DT-ℂWT based on certain directional extensions of the fHT. In particular, we derive a signal representation involving the superposition of direction-selective wavelets affected with appropriate phase-shifts.

SPIE2009

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En mathématiques et en traitement du signal, la transformation de Hilbert, ici notée \mathcal{H}, d'une fonction de la variable réelle est une transformation linéaire qui permet d'étendr

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