A likelihood based method for compressive signal recovery under Gaussian and saturation noise

Practical sensing systems have a limited dynamic range, which gives rise to saturation noise in signal acquisi-tion i f the signal attains values outside this range at certain locations. Despite rapid advances in compressed sensing theory and systems over the years, the effect of saturation on compressed sensing measurements, especially in the presence of additive Gaussian noise, has often been overlooked in many existing approaches. Some methods for compressive recovery under saturation noise simply discard saturated measurements, while others impose some ad hoc hard constraints to ensure consistenc y with the known saturation threshold. In this paper, we propose a novel maximum likelihood based estimator for compressive reconstruction in the presence of both Gaussian and saturation noise. Our proposed method ensures probabilistic consistency between the estimated signal and the saturated measurements and helps account for both the saturation effect and the potentially unbounded Gaussian noise in the measurements. We also derive upper bounds on the reconstruction error for our estimator, and argue that they follow intuitive trends. We analyze an important curvatu r e term in these bounds, and show that it is superior to methods such as saturation rejection, indicating better stabilit y and robustness. Furthermore, we present extensive simulation results to demonstrate the effectiveness of ou r method over several other existing methods in reconstructing synthetic signals and images from compressive measurements with saturation and Gaussian noise. We also show simulation results in audio signal de-clipping.