Open Access

Denoising by Sparse Approximation: Error Bounds Based on Rate-Distortion Theory

  • Alyson K Fletcher1,
  • Sundeep Rangan2,
  • Vivek K Goyal3 and
  • Kannan Ramchandran4
EURASIP Journal on Advances in Signal Processing20062006:026318

DOI: 10.1155/ASP/2006/26318

Received: 9 September 2004

Accepted: 30 June 2005

Published: 8 March 2006

Abstract

If a signal https://static-content.springer.com/image/art%3A10.1155%2FASP%2F2006%2F26318/MediaObjects/13634_2004_Article_1632_IEq1_HTML.gif is known to have a sparse representation with respect to a frame, it can be estimated from a noise-corrupted observation https://static-content.springer.com/image/art%3A10.1155%2FASP%2F2006%2F26318/MediaObjects/13634_2004_Article_1632_IEq2_HTML.gif by finding the best sparse approximation to https://static-content.springer.com/image/art%3A10.1155%2FASP%2F2006%2F26318/MediaObjects/13634_2004_Article_1632_IEq3_HTML.gif . Removing noise in this manner depends on the frame efficiently representing the signal while it inefficiently represents the noise. The mean-squared error (MSE) of this denoising scheme and the probability that the estimate has the same sparsity pattern as the original signal are analyzed. First an MSE bound that depends on a new bound on approximating a Gaussian signal as a linear combination of elements of an overcomplete dictionary is given. Further analyses are for dictionaries generated randomly according to a spherically-symmetric distribution and signals expressible with single dictionary elements. Easily-computed approximations for the probability of selecting the correct dictionary element and the MSE are given. Asymptotic expressions reveal a critical input signal-to-noise ratio for signal recovery.

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Authors’ Affiliations

(1)
Department of Electrical Engineering and Computer Sciences, University of California
(2)
Flarion Technologies Inc.
(3)
Department of Electrical Engineering and Computer Science and Research Laboratory of Electronics, Massachusetts Institute of Technology
(4)
Department of Electrical Engineering and Computer Sciences, College of Engineering, University of California

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Copyright

© Alyson K. Fletcher et al. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.