  We propose a novel deformation corrected compressed sensing (DC-CS) framework
to recover dynamic magnetic resonance images from undersampled measurements. We
introduce a generalized formulation that is capable of handling a wide class of
sparsity/compactness priors on the deformation corrected dynamic signal. In
this work, we consider example compactness priors such as sparsity in temporal
Fourier domain, sparsity in temporal finite difference domain, and nuclear norm
penalty to exploit low rank structure. Using variable splitting, we decouple
the complex optimization problem to simpler and well understood sub problems;
the resulting algorithm alternates between simple steps of shrinkage based
denoising, deformable registration, and a quadratic optimization step.
Additionally, we employ efficient continuation strategies to minimize the risk
of convergence to local minima. The proposed formulation contrasts with
existing DC-CS schemes that are customized for free breathing cardiac cine
applications, and other schemes that rely on fully sampled reference frames or
navigator signals to estimate the deformation parameters. The efficient
decoupling enabled by the proposed scheme allows its application to a wide
range of applications including contrast enhanced dynamic MRI. Through
experiments on numerical phantom and in vivo myocardial perfusion MRI datasets,
we demonstrate the utility of the proposed DC-CS scheme in providing robust
reconstructions with reduced motion artifacts over classical compressed sensing
schemes that utilize the compact priors on the original deformation
un-corrected signal.
