1 Introduction
Recently, several deep learning approaches have been proposed for accelerated parallel MR image reconstruction
[4, 6, 5, 2, 1, 11]. In this work, we present simple reconstruction networks for multicoil data by extending deep cascade of CNN’s[9]. In particular, we propose two approaches, where one is inspired by POCSENSE[8] and the other is calibrationless. The method is evaluated using a public knee dataset containing 100 subjects[4]. We show that the proposed approaches are competitive relative to the state of the art both quantitatively and qualitatively. ^{†}^{†}Presented at ISMRM 27th Annual Meeting & Exhibition (Abstract #4663)2 Methods
The proposed network architectures. (left) DPOCSENSE architecture. The input to the CNN is a single, sensitivityweighted recombined image. At each iteration, the CNN updates an estimate of the combined image. The subnetwork 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
space by applying encoding matrix. The updated image is recombined by the adjoint of the encoding matrix. (right) The proposed DCCNN architecture. The network jointly reconstructs each coil data simultaneously. The data consistency operation is applied separately for each coil.The proposed networks are direct extensions of deep cascades of CNN (DCCNN), where the denoising subnetworks 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 ESPIRiT[10]. The input to the CNN is a single, sensitivityweighted 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 coilwise as:
(1) 
where , are the th coilweighted image for the intermediate CNN reconstruction in space and the original 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 depends on the input noise level, however, this is made trainable as a network parameter. The network is trained using loss:
(2) 
where and 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 channelaxis and fed into each subnetwork. 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 calibrationless. The proposed network, DCCNN, is trained with the following weighted loss:(3) 
where the subscript indexes th coil data and is the sensitivity map. The proposed architectures are shown in Fig. 3.
3 Evaluation
We used the public knee dataset provided by Hammernik et al.[4]^{1}^{1}1Available at mridata.org.. The dataset contains 100 patients, 20 subjects per acquisition protocol. For each approach, one network was trained to reconstruct all acquisition protocols simultaneously. We used 15 for training and 5 for testing per protocol. The proposed approach was compared with SPIRiT[7] and Variational Network (VN)[4]. We used Cartesian undersampling with acceleration factor (AF) , sampling 24 central region, which was also used as the calibration region for estimating the sensitivity maps. In this work, DPOCSENSE and DCCNN were trained with ,,[9] and convolution kernels with dilation factor 2. The network was trained using Adam with
for 200 epoch with batch size 4. The default parameters were used for both
SPIRiT and VN. We used PSNR and SSIM for the metric.4 Results
Quantitative results are summarised in Table 1 for each acquisition protocol. The proposed methods both outperformed the compressed sensing approach on average. DPOCSENSE achieved the performance close to VN for AF=4, whereas DCCNN was slightly worse. All methods provided similar SSIM. For AF=6, VN achieved the highest PSNR. The sample reconstructions are shown in Fig. 6 for AF=4 and AF=6 respectively. For Axial image, DPOCSENSE gave the most homogeneous image, whereas DCCNN and VN often failed to remove aliasing. For AF=4, all methods generated sharp images. For AF=6, DCCNN performed worse that DPOCSENSE and VN and the residual aliasing is prominent.
5 Discussion and Conclusion
In this work, we proposed simple extensions to DCCNN for parallel imaging. When comparing the two approaches so far explored, DPOCSENSE outperformed DCCNN overall, which leads to the conclusion that incorporating the sensitivity estimate is advantageous. We speculate that this is because it allows intermediate subnetworks to directly operate in the output space as well as directly optimising the loss with respect to the final output. Nevertheless, DCCNN achieved the highest SSIM in some regimes, which shows that a novel way of combining the raw data could lead to improved algorithms. The proposed methods achieved comparable performance to stateoftheart algorithms, however, we note that the variational network produced the best result overall.
6 Note
We observed that training DPOCSENSE and DCCNN networks longer can further remove the residual aliasing present in the reconstruction to eventually reach similar performances. The presented work is now extended to variablesplitting network [3].
7 Acknowledgements
Jo Schlemper is partially funded by EPSRC Grant (EP/P001009/1).
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