Ambiguity and regularization in parallel MRI

In this paper, we formulate the parallel magnetic resonance imaging(pMRI) as a multichannel blind deconvolution problem with subsampling. First, the model allows formal characterization of image solutions consistent with data obtained by uniform subsampling of k-space. Second, the model allows analysis of the minimum set of required calibration data. Third, the filter bank formulation provides analysis of the sufficient sizes of interpolation kernels in widely used reconstruction techniques. Fourth, the model suggests principled development of regularization terms to fight ambiguity and ill-conditioning; specifically, subspace regularization is adapted from the blind image super-resolution work of Sroubek et al. [11]. Finally, characterization of the consistent set of image solutions leads to a cautionary comment on L1 regularization for the peculiar class of piece-wise constant images. Thus, it is proposed that the analysis of the subsampled blind deconvolution task provides insight into both the multiply determined nature of the pMRI task and possible design strategies for sampling and reconstruction.