Multi-Resolution Compressed Sensing Reconstruction Via Approximate Message Passing

In this paper, we consider the problem of multiresolution compressed sensing (MR-CS) reconstruction, which has received little attention in the literature. Instead of always reconstructing the signal at the original high resolution (HR), we enable the reconstruction of a low-resolution (LR) signal when there are insufficient CS samples to recover an HR signal. We propose an approximate message passing (AMP)-based framework dubbed MR-AMP and derive its state evolution, phase transition, and noise sensitivity, which show that, in addition to its reduced complexity, our method can recover an LR signal with bounded noise sensitivity even when the noise sensitivity of the conventional HR reconstruction is unbounded. We then apply the MR-AMP to image reconstruction using either soft-thresholding or a total variation denoiser and develop three pairs of up-/downsampling operators in the transform or spatial domain. The performance of the proposed scheme is demonstrated on both one-dimensional synthetic data and two-dimensional images.