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Fractional Wavelets

Investigators: Michael Unser, Thierry Blu

Summary: Fractional wavelets are a new type of wavelet bases with a continuously varying order parameter. This allows for a fine tuning of their key properties: order of differentiation (which can be fractional), order of approximation, regularity, and time-frequency localization. The most prominent examples are the fractional spline wavelets, which come in a variety of flavors.

The fractional B-splines with a degree greater than 0. These functions interpolate the conventional B-splines which are represented using a thicker line.
The fractional B-splines with a degree greater than 0. These functions interpolate the conventional B-splines which are represented using a thicker line.

Main Contributions

We have defined new classes of fractional wavelet bases with some remarkable properties. The most-interesting feature from a user's point of view is that these new wavelets are completely tunable with a continuously varying order parameter. They also come in various flavors such as orthogonal, B-splines, symmetric, causal. The latter B-spline wavelets are especially attractive because they tend to be optimally localized in sense of Heisenberg's uncertainty principle. As their order increases, they converge to modulated Gaussian; the localization (equivalent standard deviation) can be tuned in a continuous fashion via the fractional-order parameter. All these wavelet transforms have been implemented via a fast FFT-based algorithm which has been made available to other researchers. Additionally, we have designed fractional wavelets on quincunx lattices with a continuously adjustable order parameter.

More info and demos…

Period: 2000-2006

Funding: Grant 200020-101821 from the Swiss National Science Foundation

Major Publications

[1] 

M. Unser, T. Blu, "Fractional Splines and Wavelets," SIAM Review, vol. 42, no. 1, pp. 43-67, March 2000.

[2] 

T. Blu, M. Unser, "A Complete Family of Scaling Functions: The (α, τ)-Fractional Splines," Proceedings of the Twenty-Eighth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'03), Hong Kong SAR, People's Republic of China, April 6-10, 2003, vol. VI, pp. 421-424.

[3] 

M. Unser, T. Blu, "Wavelet Theory Demystified," IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 470-483, February 2003.

[4] 

T. Blu, M. Unser, "Wavelets, Fractals, and Radial Basis Functions," IEEE Transactions on Signal Processing, vol. 50, no. 3, pp. 543-553, March 2002.

[5] 

T. Blu, M. Unser, "The Fractional Spline Wavelet Transform: Definition and Implementation," Proceedings of the Twenty-Fifth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'00), Istanbul, Turkey, June 5-9, 2000, vol. I, pp. 512-515.

[6] 

M. Unser, T. Blu, "Fractional Wavelets, Derivatives, and Besov Spaces," Proceedings of the SPIE Conference on Mathematical Imaging: Wavelet Applications in Signal and Image Processing X, San Diego CA, USA, August 4-8, 2003, vol. 5207, part I, pp. 147-152.

[7] 

M. Feilner, D. Van De Ville, M. Unser, "An Orthogonal Family of Quincunx Wavelets with Continuously Adjustable Order," IEEE Transactions on Image Processing, vol. 14, no. 4, pp. 499-510, April 2005.

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