Compressive Sensing by Learning a Gaussian Mixture Model from Measurements
In this project, we aim to solve the compressive sensing (CS) hyperspectral / video image reconstruction problem. The propose algorithm is robust to different initializations. This is useful for CS reconstruction problems where the suitable training datasets are not available.
Compressive sensing of signals governed by a Gaussian mixture model (GMM) admits closed-form minimum mean squared error (MMSE) reconstruction from incomplete linear measurements. However, an accurate signal model requires training signals that match in statistics the signals being sensed, and can therefore be difficult to obtain in practice. We propose to solve this problem by learning the signal model in situ using the measurements of the signals being sensed, without resorting to other signals to train the model. A key feature of our methods 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), which maximizes the likelihood of the GMM of x given only the linear measurements of x. We extends the MMLE to a GMM with dominantly low-rank covariance matrices, to gain computational speedup. We report extensive experimental results on image in-painting, compressive sensing of high-speed video, and compressive hyperspectral imaging. The results demonstrate the proposed methods outperform state-of-the-art methods by significant margins.