  We address the problem of reconstructing high quality images from
undersampled MRI data. This is a challenging task due to the highly ill-posed
nature of the problem. In particular, in dynamic MRI scans, the interaction
between the target structure and the physical motion affects the acquired
measurements leading to blurring artefacts and loss of fine details. In this
work, we propose a framework for dynamic MRI reconstruction framed under a new
multi-task optimisation model called Compressed Sensing Plus Motion (CS+M).
Firstly, we propose a single optimisation problem that simultaneously computes
the MRI reconstruction and the physical motion. Secondly, we show our model can
be efficiently solved by breaking it up into two more computationally tractable
problems. The potentials and generalisation capabilities of our approach are
demonstrated in different clinical applications including cardiac cine, cardiac
perfusion and brain perfusion imaging. We show, through numerical and graphical
experiments, that the proposed scheme reduces blurring artefacts and preserves
the target shape and fine details. We also report the highest quality
reconstruction under highly undersampling rates in comparison to several state
of the art techniques.
