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Biomedical Imaging Group
Fractional Splines Wavelets

# Wavelet demos

Fractional (α,τ)-spline wavelet demo Screenshot

This applet demonstrates a new wavelet transform that is based on an extended family of fractional splines. The implementation uses a new FFT-based algorithm that can handle IIR filters efficiently. The applet displays the scaling and wavelet basis functions; they can be modified interactively. The transform can be applied to an image in a separable fashion, and then reverted exactly. The parameters are :

• Type of filter : orthonormal, B-spline and dual filters
• Degree α, which can vary continuously between -0.49 and 5.00
• Shift τ, which can vary continuously between -0.50 and 0.50. A shift of zero corresponds to symmetrical filters; for non-zero shifts, the basis functions become asymmetrical; an effect that is more pronounced for small degree.
To the best of our knowledge, these are the first examples ever of a wavelet transform with a fractional order of approximation.
Reference:
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, in press.

Fractional wavelet demo Screenshot

This demo computes a wavelet transform with a continuously-varying order parameter α. The underlying scaling functions and wavelets are symmetric. The 2D wavelet transform is separable and implemented by successive filtering and decimation (using Mallat's algorithm) of the rows and columns of the image. Since the filters are not necessarily compactly supported, the decomposition is implemented in the Fourier domain using FFTs. The parameters are :

• Type of filters : orthonormal, dual and B-spline filters generated by causal, anticausal and symmetric splines.
• Fractional order of filters α
Reference:
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.