Introduction

Deep learning (DL) methods for MR image reconstruction have seen an increase in popularity over the last few years. Historically, focus has been on the reconstruction of magnitude images while the image phase is often not considered. Recently, complex-valued reconstructions are increasingly used,^1,2 but their complete evaluation is limited by the absence of phase-specific metrics. Validating the accuracy of the reconstructed phase is important for techniques such as susceptibility-weighted and phase-contrast imaging. Common metrics such as mean squared error (MSE) and peak signal-to-noise ratio can be used to assess complex-valued reconstructions,³ but they do not specifically assess phase. Distinguishing between the magnitude and phase performance allows for a more precise understanding of strengths and weaknesses. Understanding these effects is particularly pertinent given that not all MR sequences value magnitude and phase equally. We introduce a new phase-specific metric, absolute phase disparity (APD), and illustrate its use in evaluating complex-valued DL reconstruction.

Methods

The APD is defined as
$$\text{APD} = \frac{ \sum |\textbf{x}|\, |\text{arg} (\textbf{x}_{pred}^* \textbf{x})|}{\sum |\textbf{x}|}$$
where x is the reference image, x_pred is the predicted image, and ^* denotes the complex conjugate. This expression has several favorable characteristics: 1) It accounts for phase wrap because the argument x^*_predx is proportional to the relative phase difference despite potentially different principal arguments; 2) by weighting the APD by |x|, the the contribution of low signal regions where the phase is impacted by noise is reduced.
The APD can be modified to produce a map. For the i^th pixel in the map,
$$\text{APD}_i = \frac{ |\textbf{x}_i|\, |\text{arg} (\textbf{x}_{pred,i}^* \textbf{x}_i)|}{\text{max} (|\textbf{v}|)}$$

The APD is similar in concept to the normalized absolute error (NAE), a magnitude-specific metric, which is defined as
$$ \text{NAE}_i = \frac{ N||\textbf{x}_{pred,i}| - |\textbf{x}_i||}{\sum|\textbf{x}|}$$
where N is the number of pixels, or voxels, in x. A single NAE value per image can be obtained by averaging over the map.

APD and NAE were used to assess reconstructed images in an experiment investigating the effect of coil overlap on deep learning-based reconstruction methods. NAE serves as a magnitude-specific comparison to help interpret findings. Eight-channel head coil data were synthesized in which coil element radii were increased in 1 cm increments from 8 cm to 12 cm. The Calgary-Campinas dataset⁴ was used to derive the synthesized data and contained 47/20/50 train/validation/test volumes from healthy subjects. Reconstruction was applied to 2D axial slices of which the central 100 were used for testing, and retrospective 2D uniform undersampling was used. The DL model architecture was a deep convolutional cascade⁵ adapted for multi-channel data and trained with MSE applied independently to the real and imaginary channels. The DL model was compared to conjugate gradient SENSE⁶ (CG-SENSE) at R = 6 and 8. The performance of each method as a function of increasing coil overlap was analyzed using the non-parametric Friedman test (α = 0.05). As appropriate, this test was followed by post-hoc one-tailed Wilcoxon signed-rank tests (α = 0.05) between adjacent levels of coil overlap. The Holm-Bonferroni method was used to correct for multiple comparisons.

Results

Table 1 summarizes performance results which are visualized in Figure 1. At R = 6, a larger NAE increase was observed in CG-SENSE compared to the DL model with increasing coil radius. Both methods show a similar, small decrease in APD. At R = 8, there was a greater divergence in method performance. CG-SENSE APD (Figure 2) and NAE (Figure 3) maps showed increased noise amplification compared to the DL model, although the DL model still shows phase and magnitude errors at the noise amplification boundaries.

Discussion

The APD and NAE maps show similar patterns suggesting the phase and magnitude assessments are consistent. The patterns suggest the DL model is more reliant on the anatomy being reconstructed than CG-SENSE. The APD map intensity ranges indicate that generally, the accuracy of the reconstructed phase is highly sensitive to spatial aliasing artifacts. As expected, map values are lower in regions with low signal. Despite training the DL model on independent real and imaginary channels, APD values suggest this is an effective strategy for reconstructing phase. As many modern DL models use loss functions applied to only the magnitude image, such as the structural similarity index measure (SSIM), a question is whether this allows for accurate phase reconstruction.

Conclusion

APD is an effective phase- metric that allows for the quantification and interpretation of phase reconstruction performance. APD results were consistent with the magnitude-metric, NAE, suggesting that the metric provides a reasonable phase assessment. APD has the potential to be effective in developing fully complex DL reconstruction methods.

Acknowledgements

ND is supported by a Natural Sciences and Engineering Research Council (NSERC) BRAIN CREATE award, an NSERC Canada Graduate Scholarship (CGS-M), and an Alberta Graduate Excellence Scholarship (AGES). RS and RF thank NSERC for ongoing operating support for this project (RGPIN/2021-02858 and RGPIN/2021-02867). RS also acknowledges operational support from the NSERC Alliance–Alberta Innovates Advance Program.

References

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