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  • Workshop presentation
  • Open Access

Efficient calculation of g-factors for CG-SENSE in high dimensions: noise amplification in random undersampling

  • 1,
  • 1,
  • 1, 2 and
  • 1
Journal of Cardiovascular Magnetic Resonance201416 (Suppl 1) :W28

  • Published:


  • Coil Sensitivity
  • Noise Amplification
  • Coil Configuration
  • Sample Correlation Matrix
  • Arbitrary Trajectory


SENSE [1, 2] is one of the most used parallel imaging techniques. In [1], uniform undersampling was employed to efficiently reconstruct an unalised image, whereas in [2], a conjugate gradient-based method (CG-SENSE) was used for reconstruction with arbitrary trajectories. SENSE framework allows the calculation of g-factors, characterizing the noise amplification for a given k-space trajectory and coil configuration [1]. However, calculation of g-factors for arbitrary trajectories in high dimensions is time-consuming [3]. Furthermore, noise characteristics of random undersampling, used in compressed sensing, is not well-understood. In this work, we use a Monte-Carlo (MC) method for fast calculation of g-factors for CG-SENSE similar to [4, 5] and apply it to random Cartesian undersampling trajectories. Theory: SENSE involves a pre-whitening step [1, 2], thus without loss of generality, we assume white noise. SENSE reconstruction solves minm ||y - Em||2, where E is the system matrix, and y are the undersampled measurements. The g-factor for the kth voxel is given by gk = √([E*E]-1k,k [E*E]k,k). Inverting E*E is not feasible in high dimensions. Instead we note the gk corresponds to the kth diagonal of the reconstruction noise covariance matrix (for normalized coil sensitivities), where nrecon = (E*E)-1E*nmeas, and nmeas is measurement noise with identity covariance matrix. We calculate the sample correlation matrix using a MC approach (since sample mean goes to 0), as 1/(p-1)∑p nprecon (nprecon)* for p instances of nrecon. Note we only calculate and store the diagonal elements of this matrix, significantly increasing efficiency.


The MC method was first verified in a numerical simulation, where the g-factor was explicitly calculated for a 2D coil configuration, to determine how many MC simulations suffice. Whole-heart imaging was performed with an isotropic resolution of 1.3 mm using a 32-channel coil array. Two 4-fold accelerated acquisitions were performed, one with uniform undersampling (2 × 2 in the ky-kz plane) and one with random undersampling. Coil sensitivity maps were exported. Images were reconstructed using SENSE (for uniform) and CG-SENSE (for both). g-factors were also calculated with the proposed approach.


Figure 1 shows the results of numerical simulations, indicating the method converges in ~50 MC simulations. Figure 2 shows the reconstructions associated with the two undersampling patterns and reconstructions, and the corresponding g-factors respectively. The results exhibit the semi-convergence property for random undersampling but not for uniform. Furthermore, the g-factor for random undersampling is smaller at its convergent point than for uniform.
Figure 1
Figure 1

g-factor maps calculated from numerical simulations using point-by-point analytical evaluation, as well as the described Monte-Carlo method for various number of simulations (#MC). The Monte-Carlo based approach converges after ~50 simulations.

Figure 2
Figure 2

(a) Reconstructions from two 4-fold accelerated acquisitions with uniform and random sampling (zoomed into the heart). CG-SENSE with uniform undersampling converges in 10-15 iterations, whereas CG-SENSE with random undersampling converges in ~5 iterations (also exhibiting the semi-convergence property attributed to CG-SENSE). (b) The corresponding g-factor maps from 50 MC simulations (depicting the whole slice). g-factor maps for uniform undersampling with CG-SENSE converges to the SENSE maps in 10-15 iterations, exhibiting the folding patterns associated with SENSE reconstructions. g-factor maps for random undersampling are more homogenous, amenable to denoising with a fixed threshold (semi-convergence is also exhibited in these maps). g-factor values taken near the ascending aorta are 1.80 for SENSE; 0.55,1.26,1.60,1.80,1.79 for CG-SENSE with uniform undersampling (iterations 1,3,5,10,15 respectively); and 0.46,0.76,1.09,1.85,2.45 for CG-SENSE with random undersampling (iterations 1,3,5,10,15 respectively).


g-factors for random undersampling is better than those for uniform at high k-space dimensions and high acceleration rates.


NIH:K99HL111410-01; R01EB008743-01A2.

Authors’ Affiliations

Medicine, Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, Massachusetts, USA
Radiology, Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, Massachusetts, USA


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  3. Liu : ISMRM. 2008Google Scholar
  4. Thunberg : MRI. 2007Google Scholar
  5. Robson : MRM. 2008Google Scholar


© Akcakaya et al.; licensee BioMed Central Ltd. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver ( applies to the data made available in this article, unless otherwise stated.