EPFL | Biomedical Imaging Group | Fractional Brownian

Fractional Brownian Vector Fields

P.D. Tafti, M. Unser

Multiscale Modeling & Simulation, vol. 8, no. 5, pp. 1645–1670, 2010.

This work puts forward an extended definition of vector fractional Brownian motion (fBm) using a distribution theoretic formulation in the spirit of Gel′fand and Vilenkin's stochastic analysis. We introduce random vector fields that share the statistical invariances of standard vector fBm (self-similarity and rotation invariance) but which, in contrast, have dependent vector components in the general case. These random vector fields result from the transformation of white noise by a special operator whose invariance properties the random field inherits. The said operator combines an inverse fractional Laplacian with a Helmholtz-like decomposition and weighted recombination. Classical fBm's can be obtained by balancing the weights of the Helmholtz components. The introduced random fields exhibit several important properties that are discussed in this paper. In addition, the proposed scheme yields a natural extension of the definition to Hurst exponents greater than one.

@ARTICLE(http://bigwww.epfl.ch/publications/tafti1003.html,
AUTHOR="Tafti, P.D. and Unser, M.",
TITLE="Fractional {B}rownian Vector Fields",
JOURNAL="Multiscale Modeling \& Simulation",
YEAR="2010",
volume="8",
number="5",
pages="1645--1670",
month="",
note="")

© 2010 SIAM. 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 SIAM. 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.