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Fractional Splines Wavelets

# Fractional Spline Wavelets properties

 This section provides a short description of the main wavelet properties; it also explains the role of the various transform parameters. It should help you to select the fractional wavelet transform parameters for the software that is provided. The family of fractional spline wavelet transforms is unique in that the basis functions are adjustable in a continuous manner. This is interesting practically because it allows you to fine-tune the wavelet transform which will provide you with a non-redundant, fully reversible signal representation. The free parameters that you can adjust are α, τ and the transform-type "orthogonal, B-spline or dual".
 The degree α
 This parameter controls a number of key wavelet properties: the parametric form of the basis functions, their smoothness, their space-frequency localization, the order and multi-scale differentiability properties of the transform, and finally, the number of vanishing moments.
 1. Parametric form and smoothness By construction, the fractional spline basis functions are made up of atoms of the form xα. The exponent α also gives their Hölder regularity, meaning that their (fractional) derivatives up to order α are bounded everywhere. Their Sobolev smoothness is α + ½ which expresses the fact that these wavelets are differentiable (α + ½) times in the L2-sense (i.e., finite energy).
 2. Localization The parameter α also directly controls the size (i.e. spatial extent) of the basis functions. For instance, for the B-spline family, the basis functions (resp., wavelets) converge to Gaussians (resp., modulated Gaussians or Gabor functions) with a standard deviation (or equivalent widow size) that is proportional to Sqrt(α). This also means that these functions, for α sufficiently large (say, α > 2), will tend to be optimally localized in the sense of the Heisenberg uncertainty principle; in other words, the product of their space and frequency uncertainties will tend to the minimum that is achievable.
 3. Order and multi-scale differentiation The order of a fractional spline transform is α + 1. This means that the wavelet transform behaves like a multi-scale differentiator of order α + 1. In other words, the wavelet coefficients can be interpreted as being the samples of the α + 1 th fractional derivative of a series of reduced resolution versions of the signal. This property may be interesting for the analysis of fractal-like signals, which will tend to be "whitened" in the transform domain. The order also gives the rate of decay of the approximation error for a scale-truncated reconstruction of the signal.
 4. Vanishing moments The wavelet basis functions have ceil(α + 1) vanishing moments. This means that the wavelets will "kill" all polynomials of degree ceil(α), resulting in a sparse representation for piecewise smooth signals. In other words, the wavelet coefficients will tend to be zero in the smooth regions of the signal where it is well represented by its Taylor series. Of course, the price to pay for having a large number of vanishing moments (large α) is that the basis functions will be less localized, which may produce more ringing around edges.
 The shift parameter τ For large enough values of α, the effect of this parameter is to simply shift the basis functions by the corresponding amount. The parameter has also an effect on symmetry: the basis are entirely symmetric for τ=0, and their weight gets redistributed towards the left or the right depending on the value of τ, an effect that is most visible for smaller α's. The parameter τ also allows for a fine tuning of the modulation phase of the wavelet component. Another interesting property is that you can compute the Hilbert transform of a signal by perform a fractional (α, τ) wavelet transform and reconstructing with (α,τ+½).
 The transform-type The applet will display the scaling and wavelet functions that are used to generate the basis functions of the transform; that is, the functions that are used for re-synthesizing the signal. These are the same as the analysis functions in the orthogonal case, but not otherwise. The transform-type options that are available are as follows:
 1. Orthogonal splines If you don't know what type of transform to use, you may start with an "orthogonal" one, which has some mathematical advantages. It is recommended for signal compression since it will preserve the norm in the transform domain. In other words, you can predict the effect of discarding small coefficients on the reconstruction error. The "orthogonality" property is also recommended for denoising applications since it ensures that white noise in the image domain will be converted into white noise in the transform domain. The down side, however, is that the orthogonal basis functions are not so well localized in the space domain.
 2. B-splines The B-splines wavelets span the same subspaces as the orthogonal ones. However, they are only orthogonal with respect to dilation and not translation (semi-orthogonality property). Their main advantage is that they are much better localized in space. In fact, as α increases, the scaling functions converge to Gaussians (which are nearly isotropic in 2D) and the wavelets to modulated Gaussians (Gabor functions) which are optimally localized in space and frequency. Their width is precisely controlled by α, while their phase can be adjusted with τ. The only down side of the decomposition is that the complementary biorthogonal analysis functions (cf. Dual splines) have much poorer localization and tend to get badly conditioned (slow decay) when α gets too large. Indeed, there is no free lunch with wavelets: if you want to be very well localized on one side (synthesis), you usually end up paying on the other side (analysis).
 3. Dual splines This is the reversed situation of the B-spline case. The nice Gabor-like B-spline functions are on the analysis side, while the "nastier" ones—the dual splines—are on the synthesis side. This makes the type of transform most adequate for feature extraction; for example, texture analysis. In essence, it provides a fully-adjustable, multi-scale, non-redundant decomposition of a signal using Gabor-like functions. Even though, the dual basis functions displayed by the applet may look ugly, you don't have to worry about it in the case of a "feature analysis" application, because the wavelet coefficients that you are extracting are obtained by convolution with the "nice" basis functions displayed in the B-spline mode. The transform is also fully reversible as in the other cases.

 References: M. Unser, A. Aldroubi, M. Eden, "A Family of Polynomial Spline Wavelet Transforms," Signal Processing, vol. 30, no. 2, pp. 141-162, January 1993. M. Unser, T. Blu, "Fractional Splines and Wavelets," SIAM Review, vol. 42, no. 1, pp. 43-67, March 2000. 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. 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. M. Unser, T. Blu, "Wavelet Theory Demystified," IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 470-483, February 2003.