  In this paper, we consider the problem of multi-resolution 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 not enough CS samples to recover a 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 reduced complexity, our method can recover a 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 total variation denoiser, and develop three
pairs of up-/down-sampling operators in transform or spatial domain. The
performance of the proposed scheme is demonstrated by both 1D synthetic data
and 2D images.
