Rapid compressed sensing reconstruction: A semi-tensor product approach

In large-scale applications of compressed sensing (CS), the time cost to reconstruct the original signal is too high. To accelerate the reconstruction and reduce the space cost of the measurement matrix, a novel parallel reconstruction approach based on a semi-tensor product (STP) is proposed. A low-dimensional random matrix where the dimensions are 1/4 (or 1/16, 1/64, 1/256, 1/1024 or even 1/4096) that of conventional CS is generated to sample the original data, and then a parallel reconstruction method is proposed to obtain the solution with the iteratively re-weighted least-squares (IRLS) algorithm. The peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and time cost of reconstruction were evaluated and compared with matrices of different dimensions, and comparisons were also conducted with other state-of-the-art methods. Numerical results show that the speed can be effectively improved (10×, or 100×, even 1000×) and the storage space of the matrix can also be remarkably reduced; that is, the matrix can be 1/4096 of conventional CS. Furthermore, the numerical results show that our formulation outperforms conventional CS in speed of reconstruction and in its comparable quality, which is important for real-time and physical implementation of applications.