Compressive Sensing via Low-Rank Gaussian Mixture Models

We develop a new compressive sensing (CS) inversion algorithm by utilizing the Gaussian mixture model (GMM). While the compressive sensing is performed globally on the entire image as implemented in our lensless camera, a low-rank GMM is imposed on the local image patches. This low-rank GMM is derived via eigenvalue thresholding of the GMM trained on the projection of the measurement data, thus learned in situ. Extensive results on both simulation data and real data captured by the lensless camera verify the efficacy of the proposed algorithm. Our GMM model degrades to the piecewise linear estimator (PLE) if each patch is represented by a single Gaussian model. Following this, a low-rank PLE algorithm for CS inversion is also developed, which constructs an additional contribution of this paper. Since good results have been obtained via different algorithms when the measurement number is larger (more than 0.1 of the pixel numbers in the image), we hereby spend more efforts on the challenge case with a small number of measurements. Furthermore, we compare the CS reconstruction results using our algorithm with the JPEG compression. Simulation results demonstrate when limited bandwidth is available (a small number of measurements), our algorithm can achieve comparable results as the JPEG.