Fractional Wavelets
Principal 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.
Main Contribution
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…
Collaboration:
Period: 2000-2006
Funding: Grant 200020-101821 from the Swiss National Science Foundation
Major Publications
- , , Fractional Splines and Wavelets, SIAM Review, vol. 42, no. 1, pp. 43–67, March 2000.
- , , 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.
- , , Wavelet Theory Demystified, IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 470–483, February 2003.
- , , Wavelets, Fractals, and Radial Basis Functions, IEEE-SPS best paper award, IEEE Transactions on Signal Processing, vol. 50, no. 3, pp. 543–553, March 2002.
- , , 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.
- , , 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.
- , , , An Orthogonal Family of Quincunx Wavelets with Continuously Adjustable Order, IEEE Transactions on Image Processing, vol. 14, no. 4, pp. 499–510, April 2005.