Biomedical Imaging GroupSTI
English only   BIG > Publications > Hilbert Wavelets

 Home Page
 News & Events
 Tutorials and Reviews
 Download Algorithms

 All BibTeX References

On the Hilbert Transform of Wavelets

K.N. Chaudhury, M. Unser

IEEE Transactions on Signal Processing, vol. 59, no. 4, pp. 1890-1894, April 2011.

A wavelet is a localized function having a prescribed number of vanishing moments. In this correspondence, we provide precise arguments as to why the Hilbert transform of a wavelet is again a wavelet. In particular, we provide sharp estimates of the localization, vanishing moments, and smoothness of the transformed wavelet. We work in the general setting of non-compactly supported wavelets. Our main result is that, in the presence of some minimal smoothness and decay, the Hilbert transform of a wavelet is again as smooth and oscillating as the original wavelet, whereas its localization is controlled by the number of vanishing moments of the original wavelet. We motivate our results using concrete examples.

AUTHOR="Chaudhury, K.N. and Unser, M.",
TITLE="On the {H}ilbert Transform of Wavelets",
JOURNAL="{IEEE} Transactions on Signal Processing",

© 2011 IEEE. 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 IEEE.
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.