  MRI signal acquisition with multiple coils in a phased array is nowadays
commonplace. The use of multiple receiver coils increases the signal-to-noise
ratio (SNR) and enables accelerated parallel imaging methods. Some of these
methods, like GRAPPA or SPIRiT, yield individual coil images in the k-space
domain which need to be combined to form a final image. Coil combination is
often the last step of the image reconstruction, where the root sum of squares
(rSOS) is frequently used. This straightforward method works well for coil
images with high SNR, but can yield problems in images with artifacts or low
SNR in all individual coils. We aim to analyze the final coil combination step
in the framework of linear compression, including principal component analysis
(PCA). With two data sets, a simulated and an in-vivo, we use random
projections as a representation of the whole space of orthogonal projections.
This allows us to study the impact of linear compression in the image space
with diverse measures of reconstruction accuracy. In particular, the $L_2$
error, variance, SNR, and visual results serve as performance measures to
describe the final image quality. We study their relationships and observe that
the $L_2$ error and variance strongly correlate, but as expected minimal $L_2$
error does not necessarily correspond to the best visual results. In terms of
visual evaluation and SNR, the compression with PCA outperforms all other
methods, including rSOS on the uncompressed image space data.
