A Bayesian Approach for Online Recovery of Streaming Signals From Compressive Measurements

We consider the progressive reconstruction of a streaming signal from compressive, streaming measurements. We develop a progressive reconstruction algorithm based on sliding window processing, where we reconstruct the streaming signal over small overlapping shifting intervals. Since the consecutive intervals share some common sparse signal vectors, the key idea of this work is to utilize the preliminary information from the preceding interval to improve the performance of the signal recovery algorithm. For this purpose, we propose a novel sparse Bayesian learning (SBL) algorithm, which is highly efficient for recovering streaming signals. One major advantage of SBL is that it provides a measure of uncertainty of the reconstructed signal rather than computing only a point estimate. The proposed SBL algorithm utilizes the previous estimates as well as the correlations among the nonzero coefficients to improve the performance of the algorithm. Since the effect of uncertainty of the reconstructed signal from the preceding interval is specifically taken into account in the recovery process of the current interval, the proposed algorithm is more robust to the error propagation compared to the l1-norm minimization counterpart. Further, we propose a warm-start procedure and derive fast update formulae to reduce the computational cost of the SBL algorithm. In addition, we also discuss the properties of the signal and the underlying approximations which enable the progressive reconstruction of the streaming signal from compressive measurements.