Convergence behavior of iterative SENSE reconstruction with non-Cartesian trajectories

In non-Cartesian SENSE reconstruction based on the conjugate gradient (CG) iteration method, the iteration very often exhibits a "semi-convergence" behavior, which can be characterized as initial convergence toward the exact solution and later divergence. This phenomenon causes difficulties in automatic implementation of this reconstruction strategy. In this study, the convergence behavior of the iterative SENSE reconstruction is analyzed based on the mathematical principle of the CG method. It is revealed that the semi-convergence behavior is caused by the ill-conditioning of the underlying generalized encoding matrix (GEM) and the intrinsic regularization effect of CG iteration. From the perspective of regularization, each iteration vector is a regularized solution and the number of iterations plays the role of the regularization parameter. Therefore, the iteration count controls the compromise between the SNR and the residual aliasing artifact. Based on this theory, suggestions with respect to the stopping rule for well-behaved reconstructions are provided. Simulated radial imaging and in vivo spiral imaging are performed to demonstrate the theoretical analysis on the semi-convergence phenomenon and the stopping criterion. The dependence of convergence behavior on the undersampling rate and the noise level in samples is also qualitatively investigated.