  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.
