Bo Zhu^{1,2}, Berkin Bilgic^{1}, Congyu Liao^{1}, Bruce R. Rosen^{1}, and Matthew S Rosen^{1,2}

Automated Transform by Manifold Approximation (AUTOMAP) is a generalized MR image reconstruction framework based on supervised manifold learning and universal function approximation implemented with a deep neural network architecture. Here we investigate the effect of significant sampling trajectory error in spiral acquisitions, where mismatch between training and runtime scanner trajectories may result in unpredictable reconstruction artifacts. We demonstrate through Monte Carlo analysis that the error in AUTOMAP reconstruction increases smoothly as a function of trajectory error, demonstrating reasonable robustness to trajectory deviation. We find these simulation results are consistent with reconstruction performance on real scanner data acquired from human subjects.

We quantified
the effect of reconstruction of data that deviates from the nominal (trained) sampling
trajectory using Monte Carlo simulations of varying amplitudes of trajectory
error (Figure 3a) from a spiral acquisition. To examine a range of potential
errors on realistic trajectories, we measured the actual trajectory during a
spiral acquisition (described below) and computed the difference vectors of
samples between the actual and ideal designed trajectories. These vectors were
scaled to generate a set of variable-amplitude offset sampling trajectories. Trajectories
from zero deviation (perfect match) to 4× the measured
deviation were used to encode 100 T1-weighted ground-truth reference images
from the Human Connectome Project (HCP)^{3} into simulated k-space data, which was reconstructed
with both AUTOMAP and non-uniform FFT.
Spiral k-space
data was then acquired from a healthy volunteer on a Siemens 3T Prisma MRI scanner
using a 32-channel head coil. A constant spiral trajectory was designed to
cover a field of view of 256 x 256 mm^{2} with 2 x 2 mm^{2}
in-plane resolution and 5 mm slice thickness. This was achieved using 10 spiral
interleaves, each having an 8 ms readout duration which included a 1 ms
rewinder, with slew rate 133 mT/m/ms and maximum gradient strength 24 mT/m; TE
= 35 ms, TR = 200 ms, and flip angle = 20°.
Data from the multichannel receiver head coil
was SVD coil-compressed to one channel.
A calibration acquisition measurement was made to measure the actual sampling trajectory^{4}.

The AUTOMAP neural network (Figure 1) is
composed of a series of fully-connected layers and convolutional layers. The fully-connected
layers approximate the between-manifold projection from the sensor to image
domain, representing the spatial decoding domain transform function. The
convolutional layers force the image to be represented sparsely in the convolutional
feature space. The AUTOMAP network
was trained on phase-modulated HCP brain images with non-Cartesian encoding
with the designed spiral trajectory (more
details of the AUTOMAP architecture and training protocol can be found in ^{1}). Reconstructions
were compared with non-uniform FFT regridding reconstruction^{5} with kernel size $$$J_d$$$ = 6
samples, using code available from
http://web.eecs.umich.edu/~fessler/irt/irt/.

RESULTS AND DISCUSSION

We found that the error in the AUTOMAP reconstruction increases smoothly as a function of trajectory error, similar to that in conventional NUFFT reconstruction (Figure 3b), demonstrating reasonable robustness to trajectory deviation. AUTOMAP achieved superior reconstruction accuracy compared with NUFFT out to very large trajectory errors, more than 3.5-fold larger than the measured experimental deviation (Figure 3b). Reconstructions and error maps for 1x, 2x, and 4x Trajectory Errors (based on the same reference image) are shown in Figure 4.

These simulation results are consistent with the reconstruction performance on real scanner data acquired from human subjects. Figure 5 shows AUTOMAP and NUFFT reconstructions of a 10-interleave spiral MR acquisition, where both methods assume the nominal trajectory. Although there is no ground truth with which to calculate reconstruction error, image SNR was measured to be higher in the AUTOMAP output (21.6 vs 17.6). Figure 5c-d displays windowed versions of Figure 5a-b, revealing coherent object-dependent and ringing artifacts in the NUFFT reconstruction (Figure 5d); these are much reduced in the AUTOMAP reconstruction, primarily exhibiting standard Gaussian white noise (Figure 5c).

If the sampling trajectory is measured during scan time, AUTOMAP can be trained upon the actual measured trajectory to achieve greater accuracy. The current implementation of AUTOMAP is trained on single trajectories; future work includes exploring strategies to embed trajectory distortion into training to produce more robust reconstruction networks.

[1] Zhu, B., Liu, J. Z., Rosen, B. R., & Rosen, M. S. Image reconstruction by domain transform manifold learning. arXiv preprint arXiv:1704.08841 (2017).

[2] Zhu, B., Liu, J. Z., Rosen, B. R., & Rosen, M. S. Neural Network MR Image Reconstruction with AUTOMAP: Automated Transform by Manifold Approximation, Proc. Intl. Soc. Mag. Reson. Med., 25 p.0640 (2017)

[3] Fan, Q. et al. MGH–USC Human Connectome Project datasets with ultra-high b-value diffusion MRI. NeuroImage 124, 1108–1114 (2016).

[4] Duyn, J. H., Yang, Y., Frank, J. A. & van der Veen, J. W. Simple Correction Method for k-Space Trajectory Deviations in MRI. Journal of Magnetic Resonance 132, 150–153 (1998).

[5] Fessler, J. A. & Sutton, B. P. Nonuniform fast fourier transforms using min-max interpolation. IEEE Trans. Signal Process. 51, 560–574 (2003).

Figure 1: (a)
Schematic representation of AUTOMAP framework. A
mapping between sensor domain and image domain is determined via supervised
learning of sensor (top) and image (bottom) domain pairs. The training process
implicitly learns a low-dimensional joint manifold $$$X×Y$$$ over which the reconstruction function $$$f(x)=ϕ_f∘g∘ϕ_x^{-1}(x)$$$ is conditioned. (b) AUTOMAP is implemented with a deep
neural network architecture composed of fully-connected layers (FC1, FC2)
with hyperbolic tangent activations followed by a convolutional autoencoder
(C1-C3) with rectifier nonlinearity activations.

Figure 2: (a) Magnitude
(b) and phase of T2-weighted raw k-space acquired from a tubo
spin-echo human subject acquisition are properly reconstructed by AUTOMAP (d, e) trained with a simulated forward-encoded
training corpus.

Figure 3: (a) Plot
of the family of k-space
trajectory deviations used in Monte-Carlo reconstruction analysis as described
in the text. (b) Root
mean squared error (RMSE) of AUTOMAP and NUFFT Monte-Carlo reconstructions as a
function of normalized trajectory error, with unity trajectory error
corresponding to the measured experimental trajectory deviation in a 3T
clinical scanner. AUTOMAP reconstruction error increased smoothly as a function
of trajectory error demonstrating reasonable robustness to trajectory
deviation. AUTOMAP achieved superior reconstruction accuracy compared with
NUFFT out to very large trajectory errors (over 3.5× larger than the measured
experimental deviation).

Figure 4: AUTOMAP (a)
and NUFFT (b) reconstructions
of a simulated
spiral acquistion sampled using the measured trajectory of an actual
10-interleave spiral scan, encoded from a reference image from the Human
Connectome Project. Error maps are shown in (c) and (d). As the trajectory error increases to 2x (e,f)
and 4x (g,h), error maps show increasing error for both methods, but more
structural and ringing artifacts present in the NUFFT reconstructions.

Figure 5: AUTOMAP and NUFFT reconstructions of an in-vivo 3T
MR spiral acquisition are shown in (a-b), and mean image SNR (21.6 for
AUTOMAP vs 17.6 for
NUFFT).
The presence of object-dependent reconstruction artifacts is also reduced, as
revealed in the windowed images
(c-d).