On Optimality in ROVir

We recently published an approach named ROVir (Region-Optimized Virtual coils) that uses the beamforming capabilities of a multichannel magnetic resonance imaging (MRI) receiver array to achieve coil compression (reducing an original set of receiver channels into a much smaller number of virtual channels for the purposes of dimensionality reduction), while simultaneously preserving the MRI signal from desired spatial regions and suppressing the MRI signal arising from unwanted spatial regions. The original ROVir procedure is computationally-simple to implement (involving just a single small generalized eigendecomposition), and its signal-suppression capabilities have proven useful in an increasingly wide range of MRI applications. Our original paper made claims about the theoretical optimality of this generalized eigendecomposition procedure, but did not present the details. The purpose of this write-up is to elaborate on these mathematical details, and to introduce a new greedy iterative ROVir algorithm that enjoys certain advantages over the original ROVir calculation approach. This discussion is largely academic, with implications that we suspect will be minor for practical applications -- we have only observed small improvements to ROVir performance in the cases we have tried, and it would have been safe in these cases to still use the simpler original calculation procedure with negligible practical impact on the final imaging results.