Variational Bayesian Compressed Sensing for Sparse and Locally Constant Signals

Abstract

In this paper, we deal with the signal recovery problem in compressed sensing, that is, the problem of estimating the original signal from its linear measurements. Recovery algorithms can be mainly classified into two types, optimization based algorithms and statistical modeling based algorithms. Basis pursuit (BP) or basis pursuit denoising (BPDN) is one of the most widely used optimization based recovery algorithms, that minimizes the $\ell_{1}$ norm of the signal or its coefficients in some basis under the constraint that its linear transform is equal to or close to the observation signal. There are various extensions of those algorithms depending on the problem structure. When the original signal is an image, the objective function is often the sum of the $\ell_{1}$ norm of the coefficients of the signal in some basis and a total variation (TV) of the image. It can be considered that it requires the image to be sparse in both the specific transform domain and finite differences at the same time. In this paper, we propose a statistical model that represents those sparsities and the signal recovery algorithm based on the variational method. One of the advantages of the statistical approach is that we can utilize the posterior information of the original signal and it is known that it can be used to construct the compressed sensing measurements adaptively. The proposed recovery algorithm and adaptive construction of the compressed sensing measurements are validated on numerical experiments.