Sparse signal processing using iterative method with adaptive thresholding (IMAT)

Classical sampling theorem states that by using an anti-aliased low-pass filter at the Nyquist rate, one can transmit and retrieve the filtered signal. This approach, which has been used for decades in signal processing, is not good for high quality speech, image and video signals where the actual signals are not low-pass but rather sparse. The traditional sampling theorems do not work for sparse signals. Modern approach, developed by statisticians at Stanford (Donoho and Candes), give some lower bounds for the minimum sampling rate such that a sparse signal can be retrieved with high probability. However, their approach, using a sampling matrix called compressive matrix, has certain drawbacks: Compressive matrices require the knowledge of all the samples, which defeats the whole purpose of compressive sampling! Moreover, for real signals, one does not need a compressive matrix and we shall show in this invited paper that random sampling performs as good as or better than compressive sampling. In addition, we show that greedy methods such as Orthogonal Matching Pursuit (OMP) are too complex with inferior performance compared to IMAT and other iterative methods. Furthermore, we shall compare IMAT to OMP and other reconstruction methods in term of complexity and show the advantages of IMAT. Various applications such as image and speech recovery from random or block losses, salt & pepper noise, OFDM channel estimation, MRI, and finally spectral estimation will be discussed and simulated.