1A Convex Optimization Approach to pMRI Reconstruction

Abstract—In parallel magnetic resonance imaging (pMRI) reconstruction without using pre-estimation of coil sensitivity functions, one group of algorithms reconstructs sensitivity encoded images of the coils first followed by the magnitude image reconstruction, e.g. GRAPPA. Another group of algorithms jointly computes the image and sensi-tivity functions by regularized optimization which is a non-convex problem with local only solution. For the magnitude 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 image. As a result, the magnitude image reconstruction for pMRI is formulated into a two-step convex optimization problem, which produces 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 perfor-mance compared with existing algorithms. Index Terms—Medical imaging; Parallel magnetic reso-nance imaging; MRI reconstruction; Convex optimization; Regularized optimization. I