Image reconstruction analysis

Figure 4 shows the RMSEs between the sum-of-squares image in the reference scan and the reconstructed images using K-InI, self-consistent K-InI, and

1-self-consistent K-InI methods with a zero-value vector as the initial value. The RMSE of K-InI was 0.0182. The RMSE of the self-consistent K-InI decreased from 0.0429 to 0.0310 after 20 iterations and to 0.0253 after 160 iterations. The RMSE of the

1-self-consistent K-InI decreased from 0.0428 to 0.0321 after 40 iterations and then diverged. This divergence is likely due to the iterative calculation of Eq. [13] using only minimally acquired InI data and the estimated reconstruction kernel g . Figure 5 shows the RMSEs of the reconstructions using self-consistent K-InI, and ₁-self-consistent K-InI methods using K-InI reconstruction result as the initial value in the iterative calculation. The RMSE of self-consistent K-InI decreased from 0.0184 to 0.0179 after 20 iterations.

Figure 4. The RMSE of the reconstructed images using K-InI, self-consistent K-InI and ₁-self-consistent K-InI methods when the reconstruction was initialized by a zero vector.

Figure 5. The RMSE of the reconstructed images using K-InI, self-consistent K-InI and ₁-self-consistent K-InI methods when the reconstruction was initialized by the K-InI reconstruction.

The top row of Figure 6 shows the trans-axial sum-of-squares image in the reference scan (Ref) and after K-InI reconstruction. Panels A - D show the reconstructed trans-axial images using self-consistent K-InI and ₁-self-consistent K-InI methods using either a zero-vector or K-InI reconstruction as the initial value after 20, 40, 80, and 160 iterations. The corresponding RMSE’s of the reconstructions were also shown in the figure. K-InI reconstructed image was found similar to the sum-of-squares reference image. But images reconstructed by self-consistent K-InI and

1-self-consistent K-InI were blurred. Using K-InI reconstruction as the initial value can improve both self-consistent K-InI and ₁-self-consistent K-InI reconstructions by showing a smaller RMSE. However, ₁-self-consistent K-InI shows strong streak artifacts along the InI accelerated direction after 40 iterations. In summary, considering the convergence and the computational loading, both self-consistent K-InI and

1-self-consistent used 20 iterations with a zero-vector as the initial value in this study.

Figure 6. The sum-of-squares reference image (SoS) and K-InI reconstruction. Panels A and B show the self-consistent K-InI and the ₁-self-consistent K-InI reconstruction after 20, 40, 80, and 160 iterations using a zero-vector as the initial value. Panels C and

D show the self-consistent K-InI and the ₁-self-consistent K-InI reconstruction after 20, 40, 80, and 160 iterations using K-InI reconstruction as the initial value.

3.2 Spatial resolution analysis

Figure 7 and Figure 8 show the reconstructed visual cortex images with simulated SNR = 1, 10, and 100 using K-InI, self-consistent K-InI, and ₁-self-consistent K-InI methods. All of them revealed similar visual cortex distribution with blurring along InI encoding direction. Three methods were found insensitive to the noise at the simulated SNR. The reconstructions by self-consistent K-InI, and ₁-self-consistent K-InI are more spatially homogeneous.

Figure 7. The reconstructed visual cortex images of K-InI, self-consistent K-InI, and

1-self-consistent K-InI at different SNRs. Reconstructions were linearly scaled between 0 and 1 to illustrate the spatial distribution. The simulated visual cortex ROI is shown as the green area.

Figure 8. The reconstructed visual cortex (V) on the inflated left hemisphere. The dark and light gray on the inflated hemisphere represent sulci and gyri of the brain respectively. The transparent green overlay at the top image and the green circles overlaid on the reconstructed images show the simulated visual cortex. Columns A, B, and C correspond to K-InI, self-consistent K-InI, and ₁ -self-consistent K-InI reconstructions.

We calculated the APSF and SHIFT metrics to quantify the localization accuracy of reconstructions. Figure 9A shows the APSF metrics for K-InI, the self-consistent K-InI, and ₁-self-consistent K-InI reconstructions. We found that the self-consistent K-InI reconstructions have the APSF smaller than 14 mm across SNRs. The APSF of

1-self-consistent K-InI was smaller than 18 mm, and the APSF of K-InI was smaller than 17 mm. The spatial dispersion for the self-consistent K-InI and K-InI was comparable, while ₁-self-consistent K-InI has the highest spatial resolution quantified by APSF. Figure 9B shows the SHIFT metric. We found that the self-consistent K-InI and ₁-self-consistent K-InI reconstructions both had smaller SHIFT metrics than K-InI. When the SNR was higher than 5, the localization precision quantified by the SHIFT metric for self-consistent K-InI and ₁-self-consistent K-InI were found to be approximately 2 mm, while K-InI has larger localization error (SHIFT > 5 mm). Overall, self-consistent K-InI and ₁-self-consistent K-InI provide more accurate localization than K-InI.

Figure 9. The APSF and SHIFT metrics for K-InI, self-consistent K-InI, and

1-self-consistent K- InI reconstructions at visual cortex (V) at different SNRs.

Figure 10 and Figure 11 show the reconstructed sensorimotor cortex images with simulated SNR = 1, 10, and 100 using K-InI, self-consistent K-InI, and

1-self-consistent K-InI methods on brain volume and the inflated brain. Like visual cortex reconstruction, three reconstructions blurred along InI encoding direction. In Figure 10, we found that K-InI significantly shift laterally while self-consistent K-InI and ₁-self-consistent K-InI shows mostly blurring.

Figure 10. The reconstructed sensorimotor (SM) cortex images of K-InI, self-consistent K-InI, and ₁-self-consistent K-InI at different SNRs. Reconstructions were linearly scaled between 0 and 1 to illustrate the spatial distribution. The simulated sensorimotor cortex ROI is shown as the green area.

Figure 11. The reconstructed sensorimotor cortex (SM) on the inflated left hemisphere.

The dark and light gray on the inflated hemisphere represent sulci and gyri of the brain respectively. The transparent green overlay at the top image and the green circles overlaid on the reconstructed images show the simulated visual cortex. Columns A, B, and C correspond to K-InI, self-consistent K-InI, and ₁-self-consistent K-InI

reconstructions.

Figure 12 shows the APSF and SHIFT metrics for the simulated source at the sensorimotor cortex (SM). The APSF of K-InI (40 mm) is about three times larger than that of self-consistent K-InI (12 mm) and ₁-self-consistent K-InI reconstruction (15 mm). In average, the spatial resolution quantified by APSF suggested that self-consistent K-InI and ₁-self-consistent K-InI reconstructions have higher spatial resolution. For localization accuracy, the SHIFT metric for K-InI, self-consistent K-InI, and ₁-self-consistent K-InI were 25 mm, 5 mm, and 7 mm respectively. The self-consistent InI and the ₁-self-consistent InI reconstructions have much higher localization accuracy than K-InI.

Figure 12. The APSF and SHIFT metrics for K-InI, self-consistent K-InI, and

1-self-consistent K-InI reconstructions at the sensorimotor cortex (SM) across different

SNRs.