  This work addresses a central topic in Magnetic Resonance Imaging (MRI) which
is the motion-correction problem in a joint reconstruction and registration
framework. From a set of multiple MR acquisitions corrupted by motion, we aim
at - jointly - reconstructing a single motion-free corrected image and
retrieving the physiological dynamics through the deformation maps. To this
purpose, we propose a novel variational model. First, we introduce an $L^2$
fidelity term, which intertwines reconstruction and registration along with the
weighted total variation. Second, we introduce an additional regulariser which
is based on the hyperelasticity principles to allow large and smooth
deformations. We demonstrate through numerical results that this combination
creates synergies in our complex variational approach resulting in higher
quality reconstructions and a good estimate of the breathing dynamics. We also
show that our joint model outperforms in terms of contrast, detail and blurring
artefacts, a sequential approach.
