Joseph Yitan Cheng^{1}, Feiyu Chen^{2}, Marcus T. Alley^{1}, John M. Pauly^{2}, and Shreyas S. Vasanawala^{1}

To increase the flexibility and scalability of deep convolution neural networks in the context of MRI reconstruction, a framework is proposed using bandpass filtering. The introduction of bandpass filtering enables us to leverage imaging physics while ensuring that the final reconstruction is consistent with known measurements to maintain diagnostic accuracy. We demonstrate this architecture for reconstructing subsampled datasets of contrast-enhanced T1-weighted volumetric scans of the abdomen. Additionally, we demonstrate the generality of the framework through the reconstruction of wave-encoded 2D single-shot fast-spin-echo scans of the abdomen. The proposed technique performs comparably with state-of-the-art techniques while offering the ability for simple parallelization and increase computational speed.

We propose to train and apply ConvNets on patches of k-space
data (Figure
1).
More specifically, a bandpass filter is used to isolate the
reconstruction to small localized patches in k-space. With contiguous k-space patches,
the ability to exploit and enforce the data acquisition model^{1,2} is maintained. For each patch,
the inverse problem of estimating missing data samples is solved using an unrolled
ConvNet architecture^{3–6}. Here, we developed a ConvNet based on the iterative soft-shrinkage algorithm (ISTA) as illustrated in Figure 2.
The final network consists of 8 “iterations” of ISTA with trainable step sizes
and convolutional weights. Example input and output for select bands of
k-space are shown in Figure 3. The phase induced by the
off-center bandpass filtering is modeled to simplify the de-noising operation.
A low-resolution image and the location of the k-space patch are included as
additional inputs. The bandpass filtering was designed for 64x64 patches, and
the final image is reconstructed by averaging the results from overlapping
patches.

With IRB approval, gadolinium-contrast-enhanced T1-weighted volumetric abdominal
scans were acquired on GE MR750 3T scanners using intrinsic navigation^{7} and modest subsampling $$$(R=1.2–2)$$$.
Raw data were coil-compressed^{8} to 6 coils, and images were
first reconstructed using soft-gated^{9,10} compressed-sensing-based
parallel imaging with L1-ESPIRiT^{2}. The Cartesian scans were
separated into individual $$$x$$$-slices, and subsampling was performed in the $$$(k_y,k_z)$$$-plane. This dataset consisted of 229,
14, and 58 pediatric subjects (44006, 2688, and 11136 slices) for training,
validation, and testing, respectively. Uniform and variable-density poisson-disc
subsampling were used $$$(R=2–9)$$$.

To demonstrate the generality of the approach, we adapted
the imaging model to consider non-Cartesian trajectories: the imaging model
included sampling along the non-Cartesian trajectories. T2-weighted 2D
wave-encoded^{11–13} single-shot fast-spin-echo 3T
scans were acquired with variable-density subsampling $$$(R=3.2)$$$. Raw data
were coil compressed^{8} to 18 coils. Due to T2 decay
and patient motion, fully sampled datasets cannot be obtained; thus, L1-ESPIRiT
reconstruction was considered as ground truth. This dataset consisted of 104,
8, and 25 subjects (5005, 383, and 1231 slices) for training, validation, and
testing, respectively.

The ISTA-based ConvNet was implemented with and without the
proposed bandpass filtering in TensorFlow^{14}. Comparison L1-ESPIRiT reconstructions
and the generation of subsampling masks were performed using BART^{15}. Images were evaluated with
peak-signal-to-noise-ratio (PSNR), root-mean-square-error normalized by the
norm of the reference (NRMSE), and structural-similarity metric^{16} (SSIM).

The parallelization of conventional reconstruction algorithms is limited by the need to reconstruct the entire k-space together. Here, we demonstrated the ability to truly parallelize the entire reconstruction by separating the process into k-space patches. As a result, the reconstruction can be performed in a hybrid bandpass-filtered space that can exploit unique properties of image sparsity and data consistency.

With the proposed approach, the input data dimensions into the ConvNets are reduced which decreases memory footprint and increases computational speed for training and inference. For a 256x256 image reconstructed with 64x64 patches, the proposed approach offers over 28x less operations for each Fourier transform. Our approach enables the processing of high-dimensional (>256) and multi-dimensional (3+) images. Thus, possible resolutions and data dimensions does not need to be limited by the computation hardware or the acceptable computation duration for high-speed imaging applications.

This highly-scalable ConvNet was demonstrated for accelerated imaging reconstructions of Cartesian and wave-encoded acquisitions. This approach can be easily extended to other trajectories and other imaging models to fully leverage the capabilities of deep learning for MRI.

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