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Statistical compressed sensing of Gaussian mixture models

Publication ,  Journal Article
Yu, G; Sapiro, G
Published in: IEEE Transactions on Signal Processing
December 1, 2011

A novel framework of compressed sensing, namely statistical compressed sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution, and achieving accurate reconstruction on average, is introduced. SCS based on Gaussian models is investigated in depth. For signals that follow a single Gaussian model, with Gaussian or Bernoulli sensing matrices of ${\cal O}(k)$ measurements, considerably smaller than the ${\cal O}(k \log(N/k))$ required by conventional CS based on sparse models, where $N$ is the signal dimension, and with an optimal decoder implemented via linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is shown tightly upper bounded by a constant times the best $k$-term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional sparsity-oriented CS. Stronger yet simpler results further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the best $k$-term approximation with probability one, and the bound constant can be efficiently calculated. For Gaussian mixture models (GMMs), that assume multiple Gaussian distributions and that each signal follows one of them with an unknown index, a piecewise linear estimator is introduced to decode SCS. The accuracy of model selection, at the heart of the piecewise linear decoder, is analyzed in terms of the properties of the Gaussian distributions and the number of sensing measurements. A maximization-maximization (Max-Max) algorithm that iteratively estimates the Gaussian models parameters, the signals model selection, and decodes the signals, is presented for GMM-based SCS. In real image sensing applications, GMM-based SCS is shown to lead to improved results compared to conventional CS, at a considerably lower computational cost. © 2011 IEEE.

Duke Scholars

Published In

IEEE Transactions on Signal Processing

DOI

ISSN

1053-587X

Publication Date

December 1, 2011

Volume

59

Issue

12

Start / End Page

5842 / 5858

Related Subject Headings

  • Networking & Telecommunications
 

Citation

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Yu, G., & Sapiro, G. (2011). Statistical compressed sensing of Gaussian mixture models. IEEE Transactions on Signal Processing, 59(12), 5842–5858. https://doi.org/10.1109/TSP.2011.2168521
Yu, G., and G. Sapiro. “Statistical compressed sensing of Gaussian mixture models.” IEEE Transactions on Signal Processing 59, no. 12 (December 1, 2011): 5842–58. https://doi.org/10.1109/TSP.2011.2168521.
Yu G, Sapiro G. Statistical compressed sensing of Gaussian mixture models. IEEE Transactions on Signal Processing. 2011 Dec 1;59(12):5842–58.
Yu, G., and G. Sapiro. “Statistical compressed sensing of Gaussian mixture models.” IEEE Transactions on Signal Processing, vol. 59, no. 12, Dec. 2011, pp. 5842–58. Scopus, doi:10.1109/TSP.2011.2168521.
Yu G, Sapiro G. Statistical compressed sensing of Gaussian mixture models. IEEE Transactions on Signal Processing. 2011 Dec 1;59(12):5842–5858.

Published In

IEEE Transactions on Signal Processing

DOI

ISSN

1053-587X

Publication Date

December 1, 2011

Volume

59

Issue

12

Start / End Page

5842 / 5858

Related Subject Headings

  • Networking & Telecommunications