Double overrelaxation thresholding methods for sparse signal reconstruction

We propose double overrelaxation (DORE) and automatic double overrelaxation (ADORE) thresholding methods for sparse signal reconstruction. The measurements follow an underdetermined linear model, where the regression-coefficient vector is a sum of an unknown deterministic sparse signal component and a zero-mean white Gaussian component with an unknown variance. We first introduce an expectation-conditional maximization either (ECME) algorithm for estimating the sparse signal and variance of the random signal component and then describe our DORE thresholding scheme. The DORE scheme interleaves two successive overrelaxation steps and ECME steps, with goal to accelerate the convergence of the ECME algorithm. Both ECME and DORE algorithms aim at finding the maximum likelihood (ML) estimates of the unknown parameters assuming that the signal sparsity level is known. If the signal sparsity level is unknown, we propose an unconstrained sparsity selection (USS) criterion and show that, under certain conditions, maximizing the USS objective function with respect to the signal sparsity level is equivalent to finding the sparsest solution of the underlying underdetermined linear system. Our ADORE scheme demands no prior knowledge about the signal sparsity level and estimates this level by applying a golden-section search to maximize the USS objective function. We employ the proposed methods to reconstruct images from sparse tomographic projections and compare them with existing approaches that are feasible for large-scale data. Our numerical examples show that DORE is significantly faster than the ECME and related iterative hard thresholding (IHT) algorithms.