  In Fourier-based medical imaging, sampling below the Nyquist rate results in
an underdetermined system, in which linear reconstructions will exhibit
artifacts. Another consequence of under-sampling is lower signal to noise ratio
(SNR) due to fewer acquired measurements. Even if an oracle provided the
information to perfectly disambiguate the underdetermined system, the
reconstructed image could still have lower image quality than a corresponding
fully sampled acquisition because of the reduced measurement time. The effects
of lower SNR and the underdetermined system are coupled during reconstruction,
making it difficult to isolate the impact of lower SNR on image quality. To
this end, we present an image quality prediction process that reconstructs
fully sampled, fully determined data with noise added to simulate the loss of
SNR induced by a given under-sampling pattern. The resulting prediction image
empirically shows the effect of noise in under-sampled image reconstruction
without any effect from an underdetermined system.
  We discuss how our image quality prediction process can simulate the
distribution of noise for a given under-sampling pattern, including variable
density sampling that produces colored noise in the measurement data. An
interesting consequence of our prediction model is that we can show that
recovery from underdetermined non-uniform sampling is equivalent to a weighted
least squares optimization that accounts for heterogeneous noise levels across
measurements.
  Through a series of experiments with synthetic and in vivo datasets, we
demonstrate the efficacy of the image quality prediction process and show that
it provides a better estimation of reconstruction image quality than the
corresponding fully-sampled reference image.
