  The compressive sensing (CS) scheme exploits much fewer measurements than
suggested by the Nyquist-Shannon sampling theorem to accurately reconstruct
images, which has attracted considerable attention in the computational imaging
community. While classic image CS schemes employed sparsity using analytical
transforms or bases, the learning-based approaches have become increasingly
popular in recent years. Such methods can effectively model the structures of
image patches by optimizing their sparse representations or learning deep
neural networks, while preserving the known or modeled sensing process. Beyond
exploiting local image properties, advanced CS schemes adopt nonlocal image
modeling, by extracting similar or highly correlated patches at different
locations of an image to form a group to process jointly. More recent
learning-based CS schemes apply nonlocal structured sparsity prior using group
sparse representation (GSR) and/or low-rank (LR) modeling, which have
demonstrated promising performance in various computational imaging and image
processing applications. This article reviews some recent works in image CS
tasks with a focus on the advanced GSR and LR based methods. Furthermore, we
present a unified framework for incorporating various GSR and LR models and
discuss the relationship between GSR and LR models. Finally, we discuss the
open problems and future directions in the field.
