Sparse block circulant matrices for compressed sensing

An undetermined measurement matrix can capture sparse signals losslessly if the matrix satisfies the restricted isometry property (RIP) in compressed sensing (CS) framework. However, existing measurement matrices suffer from high computational burden because of their completely unstructured nature. In this study, the authors propose to construct a novel measurement matrix with a specific structure, called sparse block circulant matrix (SBCM), to reduce the computational burden. The RIP of the proposed SBCM is also guaranteed with overwhelming probability. The simulation results validate that SBCM reduces the computational burden significantly whereas keeps similar signal recovery accuracy as Gaussian random matrices.