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