A convex optimisation approach to parallel magnetic resonance imaging reconstruction

Abstract

In parallel magnetic resonance imaging (pMRI), the image reconstruction with unknown coil sensitivity functions is known as a non-convex problem in the existing literatures. The analysis of this paper shows that there exists a convex solution region in the space of the magnitude image and sensitivity encoded image functions, which contains the true magnitude image solution. The derivation of the convex solution region resolves the non-convex difficulty and leads to a convex optimisation formulation of the pMRI reconstruction problem. The formulated problem consists of two steps. Each of the steps solves a regularised convex optimisation problem and provides a globally optimal solution, in the sense that the solution optimises the performance index and is independent of the initial conditions. The applications of the proposed two-step optimisation to in vivo and phantom data sets result in superior pMRI reconstruction performance compared with state-of-the-art algorithms.