Compressive signal reconstruction with noise pre-filtering in compressed domain

Compressive Sensing (CS) enables us to exactly reconstruct a signal from a small number of observations if it has a sparse representation in a known basis. But in the situation where the sparsity is not strictly satisfied as the signal contains Gaussian noise, the general practice tries to directly recover the noisy signal, and then filters out the noise from the recovered signal. Because the existence of noise may reduce the reconstruction accuracy, to overcome this drawback, we propose a noise reduction method which preprocesses the signal in compressed domain based on minimum mean square error (MMSE). The method can efficiently suppress the noise but still keep plenty of energy of the desired signal. Both analyses and simulations indicate that the modified method can improve the performance of reconstruction.