Reconstruction of Signals from their Blind Compressive Measurements

This paper proposes a method to reconstruct a signal from its Blind Compressive measurements by formulating it as a constrained optimization problem. It considers two objective functions; one function to recover the sparse representation coefficients and the other function to estimate the signal ensuring the consistency with the given compressed measurements. The sparsifying basis is learned from the reconstructed signals using a probability based transform learning algorithm. The reconstruction of the signal, and the learning of the sparsifying basis are performed using an alternating optimization strategy. The high-frequency artifacts on the reconstructed signal are circumvented by applying total variation minimization. The convergence of the proposed algorithm which uniquely reconstructs the signal up to a practically acceptable lower bound on the estimation error is also established.