Iterative Recovery of Dense Signals from Incomplete Measurements

Within the framework of compressed sensing, we consider dense signals, which contain both discrete as well as continuous-amplitude components. We demonstrate by a comprehensive numerical study–to the best of our knowledge the first of its kind in the literature–that dense signals can be recovered from noisy, incomplete linear measurements by simple iterative algorithms that are inspired by or are implementations of approximate message passing. Those iterative algorithms are shown to significantly outperform all other algorithms presented so far, when they use a novel noise-adaptive thresholding function that is proposed in this contribution.