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

Background

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.

Methods

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.

Results

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
figure1

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
figure2

(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).

Conclusions

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

Funding

NIH:K99HL111410-01; R01EB008743-01A2.

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Correspondence to Mehmet Akcakaya.

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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 (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

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Akcakaya, M., Basha, T.A., Manning, W.J. et al. Efficient calculation of g-factors for CG-SENSE in high dimensions: noise amplification in random undersampling. J Cardiovasc Magn Reson 16, W28 (2014). https://doi.org/10.1186/1532-429X-16-S1-W28

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Keywords

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