Parallel Magnetic Resonance Imaging: New Methods, New Insights

Parallel imaging can be formulated as an inverse problem using a signal model which predicts multi-channel k-space from magnetization image and coil sensitivities (SENSE). Extending this approach to autocalibration leads to the nonlinear inverse problem of jointly estimat-ing image and sensitivities,1 which is similar to blind multi-channel deconvolution. A second approach to parallel imaging uses linear prediction of the missing samples from a local neigh-bourhood in k-space. Using shift-invariance, the reconstruction weights can be learned from correlations in a fully-sampled calibration area (GRAPPA). This can be formulated using a so-called calibration matrix, a concept which also appears in singular spectrum analysis. It has low-rank, i.e. a signal and a noise subspace. This fact can be exploited for calibra-tionless parallel imaging using structured low-rank matrix completion.2 Linking the different approaches yields new insights. Starting from the signal model, it is possible to identify the subspace of physical k-space signals as a reproducing kernel Hilbert space and to derive a reconstruction formula and error bounds in k-space using approximation theory.3 Coming from the other direction, the ESPIRiT method computes the coil sensitivities from the signal subspace of the calibration matrix and uses them in extended SENSE-based reconstruction