  Magnetic Resonance Imaging (MRI) stands as a powerful modality in clinical
diagnosis. However, it is known that MRI faces challenges such as long
acquisition time and vulnerability to motion-induced artifacts. Despite the
success of many existing motion correction algorithms, there has been limited
research focused on correcting motion artifacts on the estimated coil
sensitivity maps for fast MRI reconstruction. Existing methods might suffer
from severe performance degradation due to error propagation resulting from the
inaccurate coil sensitivity maps estimation. In this work, we propose to
jointly estimate the motion parameters and coil sensitivity maps for
under-sampled MRI reconstruction, referred to as JSMoCo. However, joint
estimation of motion parameters and coil sensitivities results in a highly
ill-posed inverse problem due to an increased number of unknowns. To address
this, we introduce score-based diffusion models as powerful priors and leverage
the MRI physical principles to efficiently constrain the solution space for
this optimization problem. Specifically, we parameterize the rigid motion as
three trainable variables and model coil sensitivity maps as polynomial
functions. Leveraging the physical knowledge, we then employ Gibbs sampler for
joint estimation, ensuring system consistency between sensitivity maps and
desired images, avoiding error propagation from pre-estimated sensitivity maps
to the reconstructed images. We conduct comprehensive experiments to evaluate
the performance of JSMoCo on the fastMRI dataset. The results show that our
method is capable of reconstructing high-quality MRI images from
sparsely-sampled k-space data, even affected by motion. It achieves this by
accurately estimating both motion parameters and coil sensitivities,
effectively mitigating motion-related challenges during MRI reconstruction.
