BIOMEDICAL IMAGING GROUP (BIG)
Laboratoire d'imagerie biomédicale (LIB)

Gabor Wavelet Analysis and the Fractional Hilbert Transform

K.N. Chaudhury, M. Unser

Proceedings of the SPIE Optics and Photonics 2009 Conference on Mathematical Methods: Wavelet XIII, San Diego CA, USA, August 2-6, 2009, vol. 7446, pp. 74460T-1–74460T-7.

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. 

@INPROCEEDINGS(http://bigwww.epfl.ch/publications/chaudhury0903.html,
AUTHOR="Chaudhury, K.N. and Unser, M.",
TITLE="{G}abor Wavelet Analysis and the Fractional {H}ilbert Transform",
BOOKTITLE="Proceedings of the {SPIE} Conference on Mathematical Imaging:
	{W}avelet {XIII}",
YEAR="2009",
editor="",
volume="7446",
series="",
pages="74460T-1--74460T-7",
address="San Diego CA, USA",
month="August 2-6,",
organization="",
publisher="",
note="")

© 2009 SPIE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from SPIE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.