Sparse approximation, denoising, and large random frames

Alyson K. Fletcher, Sundeep Rangan, Vivek K. Goyal

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

If a signal x is known to have a sparse representation with respect to a frame, the signal can be estimated from a noise-corrupted observation y by finding the best sparse approximation to y. The ability to remove noise in this manner depends on the frame being designed to efficiently represent the signal while it inefficiently represents the noise. This paper analyzes the mean squared error (MSE) of this denoising scheme and the probability that the estimate has the same sparsity pattern as the original signal. Analyses are for dictionaries generated randomly according to a spherically-symmetric distribution. Easily-computed approximations for the probability of selecting the correct dictionary element and the MSE are given. In the limit of large dimension, these approximations have simple forms. The asymptotic expressions reveal a critical input signal-to-noise ratio (SNR) for signal recovery.

Original languageEnglish (US)
Title of host publicationProceedings of SPIE - The International Society for Optical Engineering
EditorsM. Papadakis, A.F. Laine, M.A. Unser
Pages1-10
Number of pages10
Volume5914
DOIs
StatePublished - 2005
EventWavelets XI - San Diego, CA, United States
Duration: Jul 31 2005Aug 3 2005

Other

OtherWavelets XI
CountryUnited States
CitySan Diego, CA
Period7/31/058/3/05

Fingerprint

Glossaries
approximation
dictionaries
Signal to noise ratio
Recovery
signal to noise ratios
recovery
estimates

Keywords

  • Dictionary-based representations
  • Estimation
  • Isotropic random matrices
  • Nonlinear approximation
  • Stable signal recovery
  • Subspace fitting

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Condensed Matter Physics

Cite this

Fletcher, A. K., Rangan, S., & Goyal, V. K. (2005). Sparse approximation, denoising, and large random frames. In M. Papadakis, A. F. Laine, & M. A. Unser (Eds.), Proceedings of SPIE - The International Society for Optical Engineering (Vol. 5914, pp. 1-10). [59140M] https://doi.org/10.1117/12.615772

Sparse approximation, denoising, and large random frames. / Fletcher, Alyson K.; Rangan, Sundeep; Goyal, Vivek K.

Proceedings of SPIE - The International Society for Optical Engineering. ed. / M. Papadakis; A.F. Laine; M.A. Unser. Vol. 5914 2005. p. 1-10 59140M.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Fletcher, AK, Rangan, S & Goyal, VK 2005, Sparse approximation, denoising, and large random frames. in M Papadakis, AF Laine & MA Unser (eds), Proceedings of SPIE - The International Society for Optical Engineering. vol. 5914, 59140M, pp. 1-10, Wavelets XI, San Diego, CA, United States, 7/31/05. https://doi.org/10.1117/12.615772
Fletcher AK, Rangan S, Goyal VK. Sparse approximation, denoising, and large random frames. In Papadakis M, Laine AF, Unser MA, editors, Proceedings of SPIE - The International Society for Optical Engineering. Vol. 5914. 2005. p. 1-10. 59140M https://doi.org/10.1117/12.615772
Fletcher, Alyson K. ; Rangan, Sundeep ; Goyal, Vivek K. / Sparse approximation, denoising, and large random frames. Proceedings of SPIE - The International Society for Optical Engineering. editor / M. Papadakis ; A.F. Laine ; M.A. Unser. Vol. 5914 2005. pp. 1-10
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