Bayesian compressed sensing of a highly impulsive signal in heavy-tailed noise using a multivariate Cauchy prior

Recent studies reveal that if a signal is highly compressible in some orthonormal basis, then an accurate reconstruction can be obtained from random projections using a very small subset of the projection coefficients, and thus, reducing the complexity of the sensing system. A Bayesian framework was introduced recently with respect to the reconstruction of the original (noisy) signal, providing some advantages when compared with reconstruction methods, employing norm-based constrained minimization approaches. These Bayesian methods were designed by using mixtures of Gaussians to approximate the sparsity of the prior distribution of the projection coefficients. However, there are cases in which a signal exhibits a highly impulsive behavior, and thus, resulting in an even sparser coefficient vector. In this paper, we develop a Bayesian approach for estimating the original signal based on a set of compressed-sensing measurements corrupted by heavy-tailed noise. The prior belief that the vector of projection coefficients should be sparse is enforced by fitting its prior distribution by means of a heavy-tailed multivariate Cauchy distribution. The experimental results show that our proposed method achieves an improved reconstruction performance, in terms of a smaller reconstruction error, while increasing the sparsity using less basis functions, compared with the recently introduced Gaussian-based Bayesian implementation.