Introduction

The fractional splines are an extension of the polynomial splines for all fractional degrees α > -1. Their basic constituents are piecewise power functions of degree α. One constructs the corresponding B-splines through a localization process similar to the classical one, replacing finite differences by fractional differences (c.f definitions). The fractional B-splines share virtually all the properties of the classical B-splines, including the two-scale relation, and can therefore be used to define new wavelet bases with a continuously-varying order parameter. They only lack positivity and compact support.

The fractional splines have the following remarkable properties:

• Generalization: for α integer, they are equivalent to the classical polynomial splines. As can be seen in the animations below, the fractional B-splines interpolate the polynomial ones in very much the same way as the gamma function (included in the definition) interpolates the factorials.
• Regularity: the fractional splines are α-Hölder continuous; their critical Sobolev exponent is α + 1/2.
• Decay: the fractional B-splines decay at least like |x|^-α-2; they are compactly supported for α integer (α odd in the symmetric case).
• Order of approximation: the fractional splines have a fractional order of approximation α + 1; a property that has not been encountered before in wavelet theory.
• Vanishing moments: the fractional spline wavelets have ceil(α)+1 vanishing moments, while the fractional B-splines reproduce the polynomials of degree ceil(α).
• Fractional derivatives: simple formulæ are available for obtaining the fractional derivatives of B-splines. In addition, fractional spline wavelets essentially behave like fractional derivative operators.
• Stretching the bounds of wavelet theory: for -1/2 < α < 0, the fractional B-splines don't have the standard regularity factor (1+z). Yet, they yield perfectly valid wavelet bases with some rather strange characteristics (order of approximation lesser than 1, singularities at the integers, etc...).

Despite their non-conventional properties and lack of compact support, the fractional spline wavelets are perfectly implementable. Try our software demo, which allows you to apply a generalized version of the Battle-Lemarié wavelet transform to some images. You can adjust the parameter α >= 0 which is allowed to vary in a continuous fashion. The transform is orthogonal and fully reversible.

Downlable Matlab code or Java code that implements the various kinds of fractional wavelet transform.

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