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Fractional Wavelets: Properties and Applications

M. Unser, T. Blu

Proceedings of the First 2002 SIAM Conference on Imaging Science (SIAG-IS'02), Boston MA, USA, March 4-6, 2002, Session MS1, pp. 33.



We introduce the concept of fractional wavelets which extends the conventional theory to non-integer orders. This allows for the construction of new wavelet bases that are indexed by a continuously-varying order parameter, as opposed to an integer. An essential feature of the method is to gain control over the key wavelet properties (regularity, time-frequency localization, etc…). Practically, this translates into the fact that all important wavelet parameters are adjustable in a continuous fashion so that the new basis functions can be fine-tuned for the application at hand. We present some specific examples of wavelets (fractional splines) and investigate the main implications of the fractional order property. In particular, we prove that these wavelets essentially behave like fractional derivative operators which makes them good candidates for the analysis and synthesis of fractal-like processes. We also consider non-separable extensions to quincunx lattices which are well suited for image processing. Finally, we deal with the practical aspect of the evaluation of these transforms and present a fast implementation based on the FFT.


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