Compressive Sensing by Learning a Gaussian Mixture Model From Measurements

摘要

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 , 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 , 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.