Gradient-based algorithm for designing sensing matrix considering real mutual coherence for compressed sensing systems

This study deals with the issue of designing the sensing matrix for a compressed sensing (CS) system assuming that the dictionary is given. Traditionally, the measurement of small mutual coherence is considered to design the optimal sensing matrix so that the Gram of the equivalent dictionary is as close to the target Gram as possible, where the equivalent dictionary is not normalised. In other words, these algorithms are designed to solve the CS problem using an optimisation stage followed by normalisation. To achieve a global solution, a novel strategy of the sensing matrix design is proposed by using a gradient-based method, in which the measure of real mutual coherence for the equivalent dictionary is considered. According to this approach, a minimised objective function based on alternating minimisation is also developed through searching the target Gram within a set of relaxed equiangular tight frames. Some experiments are done to compare the performance of the newly designed sensing matrix with the existing ones under the condition that the dictionary is fixed. For the simulations of synthetic data and real image, the proposed approach provides better signal reconstruction accuracy.