  Novel Magnetic Resonance (MR) imaging modalities can quantify hemodynamics
but require long acquisition times, precluding its widespread use for early
diagnosis of cardiovascular disease. To reduce the acquisition times,
reconstruction methods from undersampled measurements are routinely used, that
leverage representations designed to increase image compressibility.
  Reconstructed anatomical and hemodynamic images may present visual artifacts.
Although some of these artifact are essentially reconstruction errors, and thus
a consequence of undersampling, others may be due to measurement noise or the
random choice of the sampled frequencies. Said otherwise, a reconstructed image
becomes a random variable, and both its bias and its covariance can lead to
visual artifacts; the latter leads to spatial correlations that may be
misconstrued for visual information. Although the nature of the former has been
studied in the literature, the latter has not received as much attention.
  In this study, we investigate the theoretical properties of the random
perturbations arising from the reconstruction process, and perform a number of
numerical experiments on simulated and MR aortic flow. Our results show that
the correlation length remains limited to two to three pixels when a Gaussian
undersampling pattern is combined with recovery algorithms based on
$\ell_1$-norm minimization. However, the correlation length may increase
significantly for other undersampling patterns, higher undersampling factors
(i.e., 8x or 16x compression), and different reconstruction methods.
