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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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