Structured dictionary for computations reduction in sparse recovery

One of the main concerns about the signal recovery in compressed sensing (CS) literature is the computational cost which largely depends on the dimensionality of measurement matrix (also named dictionary). In this paper, a simply structured dictionary is designed to reduce the computational cost for the CS signal recovery. The structured dictionary is composed of two parts: pilot part and non-pilot part. By collecting several atoms (i.e., columns of the dictionary) to form a unit and assigning each unit some unique rows of the dictionary, it allows us to exclude certain number of non-supporting atoms by detecting the energy of the corresponding measurements with a theoretically derived threshold. An official CS algorithm is then exploited for more detailed signal recovery with a dictionary of smaller dimensionality. Under some conditions, structured dictionary can even transfer the undetermined CS signal recovery problem to a determined case. Therefore, computational cost can be greatly reduced. The price for the design is a slight sacrifice in recovery accuracy. Based on the existing block orthogonal matching pursuit (BOMP) algorithm, we develop structured BOMP (StBOMP) algorithm. Our numerical study with StBOMP algorithm demonstrates the effectiveness of the designed dictionary in computational cost reduction.