Statistical compressed sensing based on Bayesian principal component analysis

Statistical compressed sensing (SCS), a new framework of compressed sensing, efficiently samples and reconstructs a collection of signals by using a mixture of unconstrained Gaussian models, in which each signal is assumed to be drawn from one of them. The theoretical analysis of SCS demonstrates that a Gaussian signal with fast eigenvalue decay (the decay parameter α ≥ 3) can be recovered well by SCS, the obtained performance is comparable to the best M-term linear approximation which can be depicted by a constrained Gaussian model. However, a is rarely more than 3 for natural images. We propose to model the signal data with a collection of the constrained Gaussians to obtain the best M-term linear approximations. To this end, we introduce a latent variable probabilistic model for signals and develop an expectation maximization algorithm similar to the technic of Bayesian PCA. The developed algorithm iteratively reconstructs the signals, updates the parameters and automatically determines the dimensions of the constrained Gaussians under the Bayesian framework. The experimental results show that the reconstruction performance can be improved by the proposed method.