Dynamic Multi-Coil Reconstruction using Variational Networks
Kerstin Hammernik1, Matthias Schloegl2, Erich Kobler1, Rudolf Stollberger2,3, and Thomas Pock1

1Institute of Computer Graphics and Vision, Graz University of Technology, Graz, Austria, 2Institute of Medical Engineering, Graz University of Technology, Graz, Austria, 3BioTechMed-Graz, Graz, Austria


In this work, we present a variational network for reconstructing dynamic multi-coil data. Incorporation of parallel imaging increases the acceleration potential due to additional spatial information, but was not considered so far in other learning-based reconstruction approaches for dynamic MRI. We show that variational network reconstructions with learned spatio-temporal regularization lead to further improvements in image quality compared to state-of-the-art Compressed Sensing approaches for different CINE cardiac datasets and acceleration factors with 10-times faster reconstruction time.


Recent developments in deep learning for accelerated MR image reconstruction have shown improved results over Compressed Sensing (CS)-based approaches for both static1 and dynamic2-4 imaging. However, most of current learning-based approaches for dynamic MRI2-4 are limited to single-coil data only, although MR data is always acquired with multiple receiver coils. In parallel imaging (PI), coil-sensitivity information is additionally included in the reconstruction, which increases the spatial information and therefore the acceleration potential. Thus, it is natural to include PI in every reconstruction framework, which becomes already computationally demanding for current iterative CS reconstruction for dynamic MRI and even more for learning-based network architectures that involve many free parameters. In this work, we formulate a variational network (VN)1,5,6 that learns spatio-temporal regularization and dataterm weights from dynamic multi-coil MR data. The highly efficient VN is tested on CINE cardiac data for different contrasts and accelerations.


To obtain a spatio-temporal reconstruction $$$u\in{\mathbb{C}^{N_xN_yN_t}}$$$ of size $$$N_x\times{N_y}\times{N_t}$$$, we define a VN1,5 by an unrolled incremental gradient descent (GD) scheme with $$$J$$$ components and a fixed number of iterations $$$N$$$, illustrated in Figure 1: $$u^{n+1}=u^{n}-\sum\limits_{j=0}^J{\nabla}h_j(u^{n}),\quad0\leq{n}\leq{N-1}.$$ The term $$$\sum\limits_{j=0}^Jh_j(u^{n})$$$ is defined by a generalized CS model for dynamic MR image reconstruction, including a data-fidelity $$h_0(u)=\frac{\lambda}{2}\Vert{Au-f}\Vert_2^2,\quad\lambda>0$$ and a regularization term $$h_j(u)=\langle 1,\rho_j(k_j*u)\rangle,{\quad}j=1,\cdots,J.$$ The data-fidelity $$$h_0(u)$$$ enforces consistency of the reconstruction $$$u$$$ to the acquired rawdata $$$f\in\mathbb{C}^{N_xN_yN_tN_c}$$$ for $$$N_c$$$ coils. The dynamic multi-coil forward operator $$$A$$$, i.e. $$A:u=(u_t)_{t=1,\cdots,N_t}\mapsto(\mathcal{F}_t\left[c_iu_t\right])_{i=1,\cdots,N_c;\,t=1,\cdots,N_t},$$ involves coil-sensitivity profiles $$$c_i$$$ and Fourier transforms with temporally varying sampling masks $$$\mathcal{F}_t$$$. In the regularization part, prior information in terms of convolutions with 2D+t complex filter kernels $$$k_j\in\mathbb{C}^{s{\times}s{\times}s_t}$$$ of size $$$s{\times}s{\times}s_t$$$, followed by non-linear potential functions $$$\rho:\mathbb{C}^{N_xN_yN_t}\mapsto\mathbb{R}^{N_xN_yN_t}$$$ are applied to the reconstruction $$$u$$$. All prior information including filter kernels, activation functions $$$\phi=\rho^\prime$$$ and regularization parameters $$$\lambda$$$ are learned from pairs of undersampled data and corresponding fully-sampled reference data using a mean-squared-error loss1.

Experimental Setup

Fully sampled retrospectively gated CINE cardiac data were acquired from four healthy volunteers in breathhold using a 3T Siemens Magnetom Skyra and a 26-30-channel spine-/body-coil. Each of these datasets consists of one two-chamber, four-chamber, LVOT-view and four short-axis views, resulting in a total number of 28 datasets. Two image contrasts with the common acquisition parameters, matrix-size $$$192\times192$$$, voxel-size $$$1.8\times1.8\times6$$$mm$$$^3$$$, were used: (1) FLASH TR/TE/FA=5.8ms/3.16ms/12$$$^{\circ}$$$, (2) bSSFP TR/TE/FA=3.9ms/1.72ms/40$$$^{\circ}$$$. Time-frames were cropped to 17 frames with similar temporal-resolution of $$$\Delta$$$t$$$\sim$$$50ms. We trained individual VNs for each contrast using a variable density sampling pattern with 8 reference-lines, also used for coil-sensitivity estimation7, and acceleration factors $$$R\in\{8,12,16\}$$$ on 14 datasets. Sampling patterns were chosen randomly out of a pool of 100 masks in each training iteration to increase the variability of undersampling artifacts. The VN consists of $$$T=20$$$ GD steps, where $$$N_k=36$$$ filter kernels of size $$$7{\times}7{\times}5$$$ are learned along with their corresponding activation functions $$$\phi$$$. Testing was performed on the remaining 14 datasets. The VN reconstructions were compared to the CS-PI method Infimal-Convolution-Total-Generalized-Variation (ICTGV)8 qualitatively and in terms of root-mean-squared-error (RMSE) and structural similarity index (SSIM).


Figure 2 shows two-chamber, four-chamber and short-axis views for $$$R=12$$$. A comparison for investigated acceleration factors is depicted in Figure 3. For bSSFP, the VN results appear differently textured than the ICTGV reconstructions. The VN reconstructions for FLASH appear sharper and with more details and preserved texture compared to ICTGV (see detailed view in Figure 4). The quantitative results in Figure 5 support our observations. While the RMSE values for ICTGV and VN are similar, the SSIM results for VN are improved. Besides the improvements in image quality, VN results can be reconstructed in ~6 sec, hence, $$$10\times$$$ faster than GPU-powered ICTGV reconstructions.

Discussion and Conclusion

In this work, we showed the efficiency of VNs for dynamic multi-coil data, exemplified for CINE cardiac MRI. The proposed VNs allow us to reconstruct images $$$10\times$$$ faster than current PI-CS approaches. The additional information in the temporal domain allows for higher accelerations as in the static case1. The surprisingly small differences between the reconstructions for different accelerations suggest to push the acceleration further in future work. In general, CINE cardiac imaging is an important modality in dynamic multi-coil MR image reconstruction. However, it poses an interesting challenge for learning-based approaches to generate high-quality reference data due to limitations in breathhold capabilities and signal preparation. For bSSFP data, typical acquisition artifacts such as banding seem to impact the quality of learning, reflected in the results. In contrast, FLASH contains less acquisition artifacts, but is characterized by a poorer SNR. This level of SNR leads to an unnatural behaviour of ICTGV, while the VN results appear more natural and sharper.


We acknowledge grant support from the Austrian Science Fund (FWF) under the START project BIVISION, No. Y729, and ERC starting grant ”HOMOVIS”, No. 640156.


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Variational network1 for 2D+t multi-coil image reconstruction, defined as a sequence of GD steps. Prior information such as 2D+t filter kernels $$$k$$$, non-linear activation functions $$$\phi$$$ and dataterm weights $$$\lambda$$$ are learned from pairs of undersampled multi-coil rawdata $$$f$$$ and fully sampled reference images in an end-to-end manner. $$$u^0$$$ is defined by the initial zero-filled solution $$$u^0=A^*f$$$.

Comparison of zero filling, ICTGV, VN and fully-sampled reference reconstructions for selected CINE cardiac test datasets in two-chamber (first and fourth row), four-chamber (second and fifth row) and short-axis (third and sixth row) view with FLASH and bSSFP contrast and acceleration factor of $$$R=12$$$.

Comparison of zero filling, ICTGV, VN and fully-sampled reference reconstructions for selected short-axis CINE cardiac test datasets with FLASH and bSSFP contrast and different acceleration factors $$$R\in\{8,12,16\}$$$.

Detailed comparison between ICTGV and VN reconstructions against fully-sampled reference for a selected time-frame of one FLASH and bSSFB test dataset and acceleration factor $$$R=12$$$.

Quantitative comparison in terms of RMSE and SSIM for different acceleration factors and CINE test datasets. While the pixel-wise RMSE measure performs similarly for PI-CS ICTGV and VN, the patch-based SSIM values are improved for the VN results in all cases.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)