Matched filter vs. least-squares for multiple-coil MRI

To achieve the best possible performance in parallel MRI, two reconstruction methods have been widely used in practice: (1): least squares error minimization, e.g. the SENSE method, and (2) maximization of the output signal-to-noise ratio (SNR), known as the matched filter or the MR phased array. We discuss precise assumptions under which methods (1) and (2) become equivalent and also give examples to show that, in general, the two performance criteria can not both be optimized at the same time. We also show that the g-function, which is widely used to assess performance with SENSE, is actually a case of the Cramer-Rao bound (CRB). For unbiased linear reconstructions (not necessarily the particular SENSE reconstruction), the Cramer-Rao bound can be used either to give a general floor on reconstruction error or a ceiling on attainable reconstruction SNR. These results are helpful for performing tradeoffs among algorithms, for optimizing coil array design, and for assessing the ultimate achievable algorithm performance.