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Fresnelets—A New Wavelet Basis for Digital Holography

M. Liebling, T. Blu, M. Unser

Proceedings of the SPIE Conference on Mathematical Imaging: Wavelet Applications in Signal and Image Processing IX, San Diego CA, USA, July 29-August 1, 2001, vol. 4478, pp. 347-352.


We present a new class of wavelet bases—Fresnelets—which is obtained by applying the Fresnel transform operator to a wavelet basis of L2. The thus constructed wavelet family exhibits properties that are particularly useful for analyzing and processing optically generated holograms recorded on CCD-arrays.

We first investigate the multiresolution properties (translation, dilation) of the Fresnel transform that are needed to construct our new wavelet. We derive a Heisenberg-like uncertainty relation that links the localization of the Fresnelets with that of the original wavelet basis. We give the explicit expression of orthogonal and semi-orthogonal Fresnelet bases corresponding to polynomial spline wavelets. We conclude that the Fresnel B-splines are particularly well suited for processing holograms because they tend to be well localized in both domains.

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AUTHOR="Liebling, M. and Blu, T. and Unser, M.",
TITLE="Fresnelets---{A} New Wavelet Basis for Digital Holography",
BOOKTITLE="Proceedings of the {SPIE} Conference on Mathematical
	Imaging: {W}avelet Applications in Signal and Image Processing
	{IX}",
YEAR="2001",
editor="",
volume="4478",
series="",
pages="347--352",
address="San Diego CA, USA",
month="July 29-August 1,",
organization="",
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© 2001 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.
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