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On the Hilbert Transform of Wavelets

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Apr 2011Embargo end date: 01 Jan 2011Publisher:Institute of Electrical and Electronics Engineers (IEEE)Journal:IEEE Transactions on Signal Processing, volume 59, pages 1,890-1,894 (issn: 1053-587X, eissn: 1941-0476,

Authors: Kunal N. Chaudhury; Michael Unser;

doi: 10.1109/tsp.2010.2103072 , 10.48550/arxiv.1107.4619

arXiv: http://arxiv.org/abs/1107.4619

On the Hilbert Transform of Wavelets

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Abstract

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.

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

Related Organizations

École Normale Supérieure
France

Keywords

Mathematics - Functional Analysis, FOS: Computer and information sciences, Computer Vision and Pattern Recognition (cs.CV), Computer Science - Computer Vision and Pattern Recognition, FOS: Mathematics, Functional Analysis (math.FA)

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SNSF| Spline methods for the continuous/discrete processing of images