Multi denoising approximate message passing for optimal recovery with lower computational cost

An emerging issue in large-scale inverse problems is constituted by the interdependency between computational and recovery performance; in particular in practical application, such as medical imaging, it is crucial to provide high quality estimates given bounds on computational time. While most work in this direction has gone down the lines of improving optimisation schemes, in this paper we are proposing and investigating a different approach based on a multi denoising approximate message passing (MultiD-AMP) framework for Compressive Sensing (CS) image reconstruction which exploits an hierarchy of denoisers by starting with a low fidelity model and then using the estimate as starting point for a higher fidelity models through an iterative reconstruction algorithm. MultiD-AMP achieves lower time complexity and same accuracy compared to using the same most accurate denoiser as in D-AMP. The novelty of our approach is based on exploiting the deterministic state evolution of AMP, which means the predictability of the recovery performances, to design a strategy for selecting the denoiser from a set ordered by both computational complexity and statistical efficiency. We apply the MultiD-AMP framework for image reconstruction given noisy Gaussian random linear measurements. Furthermore, we extend and show the applicability of MultiD-AMP for CS to image reconstruction.