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Fractional Splines, Wavelet Bases, and Applications

M. Unser

Plenary talk, Proceedings of the First IFAC Workshop on Fractional Differentiation and Its Applications (FDA'04), Bordeaux, French Republic, July 19-21, 2004, vol. 2004-1, pp. 31-35.


The purpose of this presentation is to describe a recent family of basis functions—the fractional B-splines—which appear to be intimately connected to fractional calculus. Among other properties, we show that they are the convolution kernels that link the discrete (finite differences) and continuous (derivatives) fractional differentiation operators. We also provide simple closed forms for the fractional derivatives of these splines. The fractional B-splines satisfy a fundamental two-scale relation. Consequently, they can be used as building blocks for constructing a variety of orthogonal and semi-orthogonal wavelet bases of L2; these are indexed by a continuous order parameter γ = α + 1, where α is the (fractional) degree of the spline. We show that the corresponding wavelets behave like multiscale differentiation operators of fractional order γ. This is in contrast with classical wavelets whose differentiation order is constrained to be an integer. We also briefly discuss some recent applications in medical and seismic imaging.

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AUTHOR="Unser, M.",
TITLE="Fractional Splines, Wavelet Bases, and Applications",
BOOKTITLE="Proceedings of the First IFAC Workshop on Fractional
	Differentiation and Its Applications ({FDA'04})",
YEAR="2004",
editor="",
volume="2004-1",
series="",
pages="31--35",
address="Bordeaux, French Republic",
month="July 19-21,",
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note="Plenary talk")
© 2004 ENSEIRB. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from ENSEIRB. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.
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