Fresnelets: a new wavelet basis for digital holography

Michael Liebling; Thierry Blu; Michael A. Unser

doi:10.1117/12.449721

5 December 2001 Fresnelets: a new wavelet basis for digital holography

Michael Liebling, Thierry Blu, Michael A. Unser

Proceedings Volume 4478, Wavelets: Applications in Signal and Image Processing IX; (2001) https://doi.org/10.1117/12.449721
Event: International Symposium on Optical Science and Technology, 2001, San Diego, CA, United States

Abstract

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.

Citation Download Citation

Michael Liebling, Thierry Blu, and Michael A. Unser "Fresnelets: a new wavelet basis for digital holography", Proc. SPIE 4478, Wavelets: Applications in Signal and Image Processing IX, (5 December 2001); https://doi.org/10.1117/12.449721

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