Jo Schlemper^{1}, Jinming Duan^{2}, Cheng Ouyang^{1}, Chen Qin^{1}, Jose Caballero^{1}, Joseph V. Hajnal^{3}, and Daniel Rueckert^{1}

We present simple reconstruction networks for multi-coil data by extending deep cascade of CNN’s and exploiting the data consistency layer. In particular, we propose two variants, where one is inspired by POCSENSE and the other is calibration-less. We show that the proposed approaches are competitive relative to the state of the art both quantitatively and qualitatively.

The proposed networks are direct extensions of deep cascades of CNN (DC-CNN), where the denoising sub-networks and the data consistency layers are interleaved. However, for parallel imaging, the data consistency layer can be extended in two ways, yielding two network variants. In the first approach, sensitivity estimates are required, which can be computed using algorithms such as E-SPIRiT^{[9]}. The input to the CNN is a single, sensitivity-weighted recombined image. At each iteration, the CNN updates an estimate of the combined image. For the data consistency layer, the forward operation is performed, then acquired samples are filled coil-wise as:

$$ s_{\text{rec}} (k) = \begin{cases} \lambda s_\text{cnn}(k) + (1 - \lambda ) s_0(k) \quad \text{ if } k \in \Omega \\ s_\text{cnn}(k) \quad\quad\quad\quad\quad\quad\quad \text{if } k \not \in \Omega \end{cases}\quad \text{(Eq. 1)} $$

where $$$s_\text{cnn}$$$, $$$s_0$$$ are the intermediate CNN reconstruction in $$$k$$$-space and the original $$$k$$$-space data respectively. The result is mapped back to image domain via the adjoint of the encoding matrix. As the operation in the data consistency layer is analogous to the projection step from POCSENSE, the proposed network is termed D(eep)-POCSENSE. The balancing term λ depends on the input noise level, however, this is made trainable as a network parameter. The network is trained using $$$\ell_2$$$ loss:

$$ \ell (\theta) = \sum_{(x_u, x) \in D} \| x - f_\text{cnn} (x_u ; \theta) \|_2^2$$

where $$$x_u$$$ and $$$x$$$ are the initial recombined image and ground truth respectively.

The second approach reconstructs the multiple coil data directly without performing the recombination and the coil images are stacked along the channel-axis and fed into each sub-network. For the data consistency layer, each coil image is Fourier transformed and Eq. 1 is applied individually. As it does not require a sensitivity estimate, the proposed approach is calibration-less. The proposed network, DC-CNN, is trained with the following weighted-$$$\ell_2$$$ loss:

$$\ell (\theta ) = \sum_{(x_u, x) \in D} \sum_{i=1}^{n_\text{coil}} \| C^H_i (x_i - f_{\text{cnn}}(x_{u,1}, \dots, x_{u,n_\text{coil}}; \theta)_i ) \|_2^2$$

where the subscript indexes $$$i$$$-th coil data and $$$C^H_*$$$ is the sensitivity map.

[1] Hammernik, Kerstin, et al. "Learning a variational network for reconstruction of accelerated MRI data." Magnetic Resonance in Medicine 79.6 (2018): 3055-3071.

[2] Mardani, Morteza, et al. "Deep generative adversarial networks for compressed sensing (GANCS) MRI." arXiv:1706.00051 (2017).

[3] Han, Yoseob, and Jong Chul Ye. "k-Space Deep Learning for Accelerated MRI." arXiv1805.03779 (2018).

[4] Cheng, Joseph, et al. “DeepSPIRiT: Generalized Parallel Imaging using Deep Convolutional Neural Networks”, Abstract 0570, in Proceedings of the 26th Annual Meeting of ISMRM, Paris, France, 2018.

[5] Akçakaya, Mehmet, et al. "Scan‐specific robust artificial‐neural‐networks for k‐space interpolation (RAKI) reconstruction: Database‐free deep learning for fast imaging." Magnetic resonance in medicine (2018).

[6] Zhang, P., Wang, F., Xu, W., & Li, Y. (2018, September). Multi-channel Generative Adversarial Network for Parallel Magnetic Resonance Image Reconstruction in K-space. In International Conference on Medical Image Computing and Computer-Assisted Intervention (pp. 180-188). Springer, Cham, 2018.

[7] Schlemper, Jo, et al. "A deep cascade of convolutional neural networks for dynamic MR image reconstruction." IEEE Transactions on Medical Imaging 37.2 (2018): 491-503.

[8] Samsonov, Alexei A., et al. "POCSENSE: POCS‐based reconstruction for sensitivity encoded magnetic resonance imaging." Magnetic Resonance in Medicine: 52.6 (2004): 1397-1406.

[9] Uecker, Martin, et al. "ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA." Magnetic resonance in medicine 71.3 (2014): 990-1001.

[10] Murphy, Mark, et al. "Fast $$$\ell_1 $$$-SPIRiT Compressed Sensing Parallel Imaging MRI: Scalable Parallel Implementation and Clinically Feasible Runtime." IEEE Transactions on Medical Imaging 31.6 (2012): 1250-1262.

The proposed D-POCSENSE architecture. The input to the CNN is a single, sensitivity-weighted recombined image. At each iteration, the CNN updates an estimate of the combined image. The sub-network takes a single recombined image as an input and produces the denoised result as an output. The data consistency is performed by mapping the intermediate output to the raw $$$k$$$-space by applying encoding matrix. The updated image is recombined by the adjoint of the encoding matrix.

The proposed DC-CNN architecture. The network jointly reconstructs each coil data simultaneously. The data consistency opeartion is applied separately for each coil.

The summary of quantitative results.

The reconstruction results from each method for Cartesian undersampling with acceleration factor 4.

The reconstruction results from each method for Cartesian undersampling with acceleration factor 6.