EPFL
 Biomedical Imaging GroupSTI
EPFL
  Publications
English only   BIG > Publications > Lévy Processes


 CONTENTS
 Home Page
 News & Events
 People
 Publications
 Tutorials and Reviews
 Research
 Demos
 Download Algorithms

 DOWNLOAD
 PDF
 Postscript
 All BibTeX References

MMSE Estimation of Sparse Lévy Processes

U.S. Kamilov, P. Pad, A. Amini, M. Unser

IEEE Transactions on Signal Processing, vol. 61, no. 1, pp. 137-147, January 1, 2013.



We investigate a stochastic signal-processing framework for signals with sparse derivatives, where the samples of a Lévy process are corrupted by noise. The proposed signal model covers the well-known Brownian motion and piecewise-constant Poisson process; moreover, the Lévy family also contains other interesting members exhibiting heavy-tail statistics that fulfill the requirements of compressibility. We characterize the maximum-a-posteriori probability (MAP) and minimum mean-square error (MMSE) estimators for such signals. Interestingly, some of the MAP estimators for the Lévy model coincide with popular signal-denoising algorithms (e.g., total-variation (TV) regularization). We propose a novel non-iterative implementation of the MMSE estimator based on the belief-propagation (BP) algorithm performed in the Fourier domain. Our algorithm takes advantage of the fact that the joint statistics of general Lévy processes are much easier to describe by their characteristic function, as the probability densities do not always admit closed-form expressions. We then use our new estimator as a benchmark to compare the performance of existing algorithms for the optimal recovery of gradient-sparse signals.


@ARTICLE(http://bigwww.epfl.ch/publications/kamilov1206.html,
AUTHOR="Kamilov, U.S. and Pad, P. and Amini, A. and Unser, M.",
TITLE="{MMSE} Estimation of Sparse {L}{\'{e}}vy Processes",
JOURNAL="{IEEE} Transactions on Signal Processing",
YEAR="2013",
volume="61",
number="1",
pages="137--147",
month="January 1,",
note="")

© 2013 IEEE. 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 IEEE.
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.