Bayesian compressive sensing for approximately sparse signals and application to structural health monitoring signals for data loss recovery

The theory and application of compressive sensing (CS) have received a lot of interest in recent years. The basic idea in CS is to use a specially-designed sensor to sample signals that are sparse in some basis (e.g. wavelet basis) directly in a compressed form, and then to reconstruct (decompress) these signals accurately using some inversion algorithm after transmission to a central processing unit. However, many signals in reality are only approximately sparse, where only a relatively small number of the signal coefficients in some basis are significant and the remaining basis coefficients are relatively small but they are not all zero. In this case, perfect reconstruction from compressed measurements is not expected. In this paper, a Bayesian CS algorithm is proposed for the first time to reconstruct approximately sparse signals. A robust treatment of the uncertain parameters is explored, including integration over the prediction-error precision parameter to remove it as a “nuisance” parameter, and introduction of a successive relaxation procedure for the required optimization of the basis coefficient hyper-parameters. The performance of the algorithm is investigated using compressed data from synthetic signals and real signals from structural health monitoring systems installed on a space-frame structure and on a cable-stayed bridge. Compared with other state-of-the-art CS methods, including our previously-published Bayesian method, the new CS algorithm shows superior performance in reconstruction robustness and posterior uncertainty quantification, for approximately sparse signals. Furthermore, our method can be utilized for recovery of lost data during wireless transmission, even if the level of sparseness in the signal is low.