Compressive sensing by learning a gaussian mixture model from measurements

Jianbo Yang, Xuejun Liao, Xin Yuan, Patrick Llull, David J. Brady, Guillermo Sapiro, Lawrence Carin

Research output: Contribution to journalArticlepeer-review

148 Scopus citations


Compressive sensing of signals drawn from a Gaussian mixture model (GMM) admits closed-form minimum mean squared error reconstruction from incomplete linear measurements. An accurate GMM signal model is usually not available a priori, because it is difficult to obtain training signals that match the statistics of the signals being sensed. We propose to solve that problem by learning the signal model in situ, based directly on the compressive measurements of the signals, without resorting to other signals to train a model. A key feature of our method is that the signals being sensed are treated as random variables and are integrated out in the likelihood. We derive a maximum marginal likelihood estimator (MMLE) that maximizes the likelihood of the GMM of the underlying signals given only their linear compressive measurements. We extend the MMLE to a GMM with dominantly low-rank covariance matrices, to gain computational speedup. We report extensive experimental results on image inpainting, compressive sensing of high-speed video, and compressive hyperspectral imaging (the latter two based on real compressive cameras). The results demonstrate that the proposed methods outperform state-of-the-art methods by significant margins.

Original languageEnglish (US)
Article number6939730
Pages (from-to)106-119
Number of pages14
JournalIEEE Transactions on Image Processing
Issue number1
StatePublished - Jan 1 2015
Externally publishedYes


  • Compressive sensing
  • Gaussian mixture model (GMM)
  • High-speed video
  • Hyperspectral imaging
  • Inpainting
  • Maximum marginal likelihood estimator (MMLE)
  • Mixture of factor analyzers (MFA)

ASJC Scopus subject areas

  • Software
  • Computer Graphics and Computer-Aided Design


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