On the generalized inverse for MRI reconstruction

Recent studies have suggested that the boundary between data-driven deep-learning non-Cartesian magnetic resonance imaging (MRI) reconstruction methods and conventional optimization-based, iterative reconstruction methods is becoming blurred. For instance, the unrolled iterative reconstruction method can be regarded as a trainable neural network. Another example is that the Moore-Penrose pseudoinverse plays a central role in finding the predefined solution to many imaging processes. However, the application of pseudoinverse in MRI reconstruction was obstructed in clinical imaging, mostly due to the excessive storage required for singular vectors. Since the spatial encoding of MRI is fully determined by the known k-space trajectory, the generalized inverse can be 'iteratively learning in a data-free fashion', which leads to surprising but realizable properties. To compare our method with other conventional methods, numerical simulations were performed using in vivo MRI. The proposed method leads to nearly equivalent image quality with a much shorter run-time (only 0.68%) than the conjugate gradient (CG) method. We discuss the potential impact of the generalized inverse as a feasible reconstruction method for non-Cartesian MRI.