Provided with SEMs and the acquisition grids, we directly reconstructed images without explicit determination of the locations encoded ambiguously with the SEMs using the iTDR. shows the convergence behaviors of the PatLoc reconstructions using only multipolar SEMs (M1+M2) and using the combination of multipolar and nearly linear SEMs (M1+M2+L1+L2). Provided with the reference image in the synthetic data, we calculated the percentage error of the reconstruction at each repetition. Over the first 10 iterations, the percentage error dropped over 80% using either M1+M2 or M1+M2+L1+L2 SEMs. The convergence rates for different accelerated data were different: a higher acceleration rate corresponded to a slower convergence. This can be explained by the deteriorated conditioning of the encoding matrix (Eq. 
) in more accelerated acquisitions. We found that the reconstruction converged after 50 iterations. For acquisitions without acceleration (R=1×1), with 2-fold (R=2×1) and 4-fold (R=2×2) accelerations, the final reconstruction has the percentage error less than 5%. Eight-fold acceleration (R=2×4) converged at 5% and 3% percentage error using M1+M2 and M1+M2+L1+L2 SEMs respectively.
Figure 3 The convergence of the iTDR PatLoc reconstructions using M1+M2 SEM (left) and M1+M2+L1+L2 SEM (right) with R=1×1, R=2×1, R=2×2, and R=2×4 accelerations. Convergent reconstructions were found after 50 iterations in general. (more ...)
shows the reconstructed images and the corresponding error images at different iterations without acceleration and with 8-fold (R=2×4) accelerations. For comparison, results using M1+M2 SEMs and M1+M2+L1+L2 SEMs were shown together. Starting from a zero image, the reconstruction started from the center of the FOV during the first few repetitions. These images corroborated with the convergence plots shown in : a higher acceleration rate converged more slowly. The final reconstructed images without acceleration had a percentage error of 1% and with eight-fold accelerations the reconstruction had percentage errors of 5% and 3% using M1+M2 SEMs and M1+M2+L1+L2 SEMs respectively.
Figure 4 Individual reconstructed images and error images using iterative Time-Domain Reconstruction (iTDR). Each sub-plot shows the reconstruction (left) and the residual error image (right) in each repetition. At a higher acceleration rate, the convergence was (more ...)
Details of the final reconstructions were shown in . For comparison, we also showed the reference images and the reconstruction using conventional linear gradients. Without acceleration (, top row), we found that all reconstructions around cortex were satisfactory. Minor reconstruction error (<1 %) was due to the simulated noise. However, at the center of the FOV, the PatLoc reconstruction using only M1+M2 was blurred. Such a loss of spatial resolution at the center of the image (indicated by a yellow arrow head) was consistent with previous studies (4
): SEMs and coil sensitivity maps from a coil array cannot provide sufficient spatial information to reliably resolve images in the FOV center. Note that by using the L1 and L2 SEMs in conjunction with the M1 and M2 SEMs, we improved the loss of the spatial resolution around the FOV center significantly. The reconstruction was found similar to the one from using then conventional linear gradient system over the FOV uniformly.
Figure 5 Reconstructed images using conventional linear gradient coils and PatLoc system with multipolar (M1+M2) SEMs and with multipolar as well as linear (M1+M2+L1+L2) SEMs. Reconstructions using unaccelerated acquisitions (R=1×1), 4-fold acceleration (more ...)
Accelerated reconstructions are also shown in . At 4-fold acceleration (R=2×2), PatLoc reconstruction with multipolar SEMs (M1+M2) still have prominent loss of spatial information at the FOV center (yellow arrow heads). Using linear gradient system or PatLoc with multipolar and nearly linear SEMs generated comparable reconstructions to the reference image (, middle row). The noise level of the reconstruction was found marginally higher in the 4-fold accelerated case (0.9%) than the unaccelerated case (0.8%), potentially due to the 50% reduction on the sample and/or the noise amplification during the reconstruction. Eight-fold acceleration showed clear differences among reconstructions (, bottom row). As this PatLoc system has eight RF coils, the maximal acceleration rate was eight before transforming the signal equation (Eq. 1
) from an over-determined linear system into an under-determined linear system. Using the linear gradient system, clear residual aliasing artifact along the left-right direction was found in the 2×4 acceleration (green arrow heads). PatLoc reconstructions using only multipolar SEMs (M1+M2) showed the loss of spatial information at the FOV center (yellow arrow heads) and noisy reconstruction at the frontal and occipital areas (magenta arrow heads). With M1, M2, L1, and L2 SEMs, the 8-fold reconstructed PatLoc image showed less aliasing artifact than the linear gradient reconstruction, improved spatial resolution in the FOV center, and reduced noise level in the frontal and occipital lobes.
Since conventional MRI readily provides highly linear gradient coils, we wondered how reconstructions change if we replace the L1+L2 SEMs generated by the PatLoc system with the two linear Bz (Linear 1 + Linear 2) generated by the conventional MRI gradient coils. shows such comparison without acceleration and with 4-fold and 8-fold accelerations (R = 2×2, and R = 2×4). We found that reconstructions using either L1+L2 or Linear 1+ Linear 2 are pretty similar at R=1 and R =2×2. At a high acceleration rate R=2×4, the reconstruction noise was more prominent using the linear Bz. This might be due to the difficulty of using the RF sensitivities to interpolate the missing spatial bases generated by linear Bz.
Figure 6 PatLoc imaging reconstructions using the multipolar (M1+M2) SEMs and the nearly linear (L1+L2) SEMs generated by the PatLoc system or the multipolar (M1+M2) SEMs together with the two linear (Linear 1 + Linear 2) Bz fields generated by the conventional (more ...)
shows the reconstructions of different SNR at R=2×2 and R=2×8. We can see that at a fixed acceleration rate, the reconstruction deteriorated as the SNR decreased. Notably, at SNR = 100, R=2×2 shows fairly good reconstruction (residual error = 0.3%). At R=2×4, the reconstruction shows noticeable noises at SNR = 100 (residual error =3.4%). We consider the reconstruction can work satisfactorily at SNR = 100 or higher.
Accelerated (R=2×2 and R=2×4) PatLoc imaging reconstructions using the multipolar (M1+M2) and the nearly linear (L1+L2) SEMs at SNR = 1000, 500, 200, 100, 50, 20, and 10.
The proposed iTDR reconstruction has capability of reconstructing images using arbitrary SEMs, shows an example of using two SEMs generated by randomly weighting the surface gradient elements. In this example, details of images can be restored in lower left corner of the FOV based on sufficient spatial information from the highly nonlinear SEM pairs and RF coil sensitivities.
Figure 8 iTDR can reconstruct images using arbitrary SEMs. Two SEMs generated by randomly weighting the gradient elements are shown at top. A reference image (bottom left) was encoded by these two SEMs and iTDR reconstructs the highly distorted image (bottom right) (more ...)
To evaluate the spatial resolution, point spread functions (PSFs) corresponding to a pixel at the FOV center and the peripheral of the FOV are shown in . The locations of the non-zero image pixel for the PSF input image are surrounded by cyan boxes and indicated by cyan arrow heads. While the PSF at the FOV periphery was very focal, we found that the center of FOV has a spatially blurred PSF when only M1+M2 SEMs were used. Using L1+L2 SEMs and M1+M2 SEMs, the PSF at the FOV periphery can maintain focal, and the PSF at the FOV center can be improved clearly by suppressing side lobes. The bottom panel of plots the profile of the PSF along a vertical line passing through the non-zero pixel in the PSF input image. The location of the non-zero input PSF image is indicated by a gray dashed line. Quantitatively, the FWHMs at FOV periphery using M1+M2 SEMs and M1+M2+L1+L2 SEMs were both 1.0 pixel. The FWHMs at FOV center using M1+M2 SEMs and M1+M2+L1+L2 SEMs were 7.0 pixels and 2.2 pixels respectively. We also found that the peak of the center FOV PSF shifted by 2 pixels when only M1+M2 SEMs were used.
Figure 9 The point spread function (PSF) at the center of the FOV (left column) and the periphery of the FOV (right column) for unaccelerated PatLoc imaging using either multipolar (M1+M2) or multipolar and linear (M1+M2+L1+L2) SEMs. The cyan boxes and cyan arrow (more ...)
PSFs were also evaluated for reconstructions using the accelerated data. plots the profiles of the PSFs for input images with a non-zero pixel at the FOV center ( left) and at the FOV periphery ( right). We found that acceleration modulated the PSF marginally. Using only multipolar SEMs, the PSFs at the FOV center were blurred with FWHMs of 7.0 pixels, 7.4 pixels, and 7.8 pixels for R=1×1, R=2×2, and R=2×4 respectively. The peak of the PSF was also found shifted by 2 pixels for all acquisitions. Using multipolar and linear SEMs, the PSFs at the FOV center was much focal with FWHMs of 2.2 pixels, 2.3 pixels, and 2.4 pixels for R=1×1, R=2×2, and R=2×4 respectively. At the FOV periphery, all reconstructions using either M1+M2 SEMs or M1+M2+L1+L2 SEMs had a PSF of 1.0 pixel for unaccelerated (R=1×1) and accelerated (R=2×2 and R=2×4) acquisitions.
Figure 10 The point spread function (PSF) at the center of the FOV (left) and the periphery of the FOV (right) for unaccelerated (R=1×1) and accelerated (R=2×2 and R=2×4) PatLoc imaging using either multipolar (M1+M2) or multipolar and linear (more ...)
The spatial resolution analysis using PSF corroborates the local k
-space analysis. shows the local k
-space at 7×7 image voxels evenly distributed over the FOV. Using only multipolar SEMs has a low spatial resolution around the FOV center, consistent with previous studies (13
). FOV periphery does not lose spatial resolution. These local k
-space calculations are consistent with the PSF calculations ( and ). Using both multipolar and nearly linear SEMs, the local k
-space shows improved spatial resolution around the FOV center as the result of increased k
-space coverage. From the local k
-space plot, it is evident that the corresponding local k
-space distribution is the sum of local k
-space using individual SEMs, as also described in Eq. 
The local k-space for unaccelerated PatLoc imaging using either multipolar (M1+M2) or multipolar and linear (M1+M2+L1+L2) SEMs. Improved spatial resolution around FOV center was observed using 4 SEMs due to a wider k-space coverage.
Experimental reconstructions using data with linear gradients, multipolar SEMs (M1+M2), and four SEMs together are shown in . As limited by currently available hardware, the linear gradients were generated from the linear gradient coils in the conventional MRI system. Since our calculation () shows little difference between reconstructions using nearly linear SEMs generated by the generalized PatLoc system and the linear gradients generated by the conventional MRI gradient coils, we considered is what can be achieved using L1/L2 and M1/M2 SEMs by the PatLoc system. Compared to reconstruction using only multipolar SEM acquisitions, the loss of spatial resolution around FOV center was reduced when four SEMs were used together. However, we noticed that around the FOV center the spatial resolution was not recovered completely. There were also more prominent residual aliasing artifacts when four SEMs were used. Similar to a recently published study (13
), this sub-optimal performance is likely due to errors in the estimation of multipolar SEMs and RF coil sensitivity maps . For example, the coil sensitivity maps were estimated from a separate reference scan. Even though such a method ensured consistent coil loading such that estimated coil sensitivity is consistent between the reference scan and PatLoc imaging scan, motion between two scans as well as different eddy current effects can lead to inaccurate coil sensitivity estimation and cause imperfect reconstruction.
Reconstructions of experimental images acquired using linear gradients (L1+L2), multipolar PatLoc SEMs (M1+M2), and the combination of four SEMs (L1+L2+M1+M2).