Toeplitz block matrices in compressed sensing and their applications in imaging

Recent work in compressed sensing theory shows that sensing matrices whose entries are drawn independently from certain probability distributions guarantee exact recovery of a sparse signal from a small number of measurements with high probability. In most medical imaging systems, the encoding matrices cannot take that form. Instead, they are Toeplitz block matrix. Motivated by this fact, we consider Toeplitz block matrices as the sensing matrices. We show that the probability of perfect reconstruction from a smaller number of filter outputs is also high if the filter coefficients are independently and identically-distributed random variable. Their applications in medical imaging is discussed. Simulation results are also shown to validate the theorem.