An Improved Reweighted Method for Optimizing the Sensing Matrix of Compressed Sensing

In compressed sensing (CS), coherence is a simple and practical measure of the quality of a sensing matrix. The smaller the coherence of the sensing matrix, the better the reconstruction result. Most strategies for optimizing the coherence of a sensing matrix are only applicable to gray-value matrices and are not to binary matrices with fast computation speed and smaller storage space. Under a certain condition, the coherence can be improved by weighting the sensing matrix to make its condition number equal to 1, i.e., all non-zero singular values become the same. In this paper, we propose a method for reweighting CS models to reduce the condition number of the sensing matrix so as to improve its coherence, prove that the condition number decreases monotonically to 1 as the weighting times approaches infinity, and then obtain a coherence improvement model equivalent to the original CS model. Finally, we give the RwOMP recovery algorithm based on the proposed reweighted method and verify its superiority by using different binary sensing matrices for CS experiments.