  Purpose: To characterize the noise distributions in 3D-MRI accelerated
acquisitions reconstructed with GRAPPA using an exact noise propagation
analysis that operates directly in k--space.
  Theory and Methods: We exploit the extensive symmetries and separability in
the reconstruction steps to account for the correlation between all the
acquired k-space samples. Monte Carlo simulations and multi-repetition phantom
experiments were conducted to test both the accuracy and feasibility of the
proposed method; an in-vivo experiment was performed to assess the
applicability of our method to clinical scenarios.
  Results: Our theoretical derivation shows that the direct k-space analysis
renders an exact noise characterization under the assumptions of stationarity
and uncorrelation in the original k-space. Simulations and phantom experiments
provide empirical support to the theoretical proof. Finally, the in-vivo
experiment demonstrates the ability of the proposed method to assess the impact
of the sub-sampling pattern on the overall noise behavior.
  Conclusions: By operating directly in the k-space, the proposed method is
able to provide an exact characterization of noise for any Cartesian pattern
sub-sampled along the two phase-encoding directions. Exploitation of the
symmetries and separability into independent blocks through the image
reconstruction procedure allows us to overcome the computational challenges
related to the very large size of the covariance matrices involved.
