Iterative Reconstruction

Parallel MRI techniques utilize the inherent encoding effect of receiver coil sensitivity for complementing gradient-driven Fourier encoding. As a consequence of this hybrid encoding approach, parallel techniques require advanced reconstruction algorithms beyond mere Fourier transform. Generally, taking coil sensitivity into account renders the reconstruction problem more complex and numerically challenging. In the special case of Cartesian k-space sampling, reconstruction can still be accomplished quite efficiently by direct unfolding in the image domain (1, 2). However, non-Cartesian sampling and other complications, such as B0 inhomogeneity, prevent efficient reconstruction with direct methods. In these cases iterative algorithms offer an efficient and effective alternative. The joint action of gradient and sensitivity encoding creates a general linear mapping of the object’s signal density. Consequently, image reconstruction may likewise be viewed as a linear mapping of the sample values, yielding the final image. Let this mapping be represented by the reconstruction matrix F. It has one row for each voxel to be resolved and one column for each sample value acquired. Thus its size is N×(nC nK) for an N×N image matrix, where nC, nK denote the number of receiver coils used and the number of sampling positions in k-space, respectively. The net encoding effect is conversely described by the (nC nK)×N encoding matrix E, given by