  In parallel magnetic resonance imaging (pMRI) reconstruction without using
estimation of coil sensitivity functions, one group of algorithms reconstruct
sensitivity encoded images of the coils first followed by the magnitude only
image reconstruction, e.g. GRAPPA, and another group of algorithms jointly
compute the image and sensitivity functions by regularized optimization which
is a non-convex problem with local only solutions. For the magnitude only image
reconstruction, this paper derives a reconstruction formulation, which is
linear in the magnitude image, and an associated convex hull in the solution
space of the formulated equation containing the magnitude of the image. As a
result, the magnitude only image reconstruction for pMRI is formulated into a
two-step convex optimization problem, which has a globally optimal solution. An
algorithm based on split-bregman and nuclear norm regularized optimizations is
proposed to implement the two-step convex optimization and its applications to
phantom and in-vivo MRI data sets result in superior reconstruction performance
compared with other algorithms.
