A cascade of preconditioned conjugate gradient networks for accelerated magnetic resonance imaging. (October 2022) - Kim, Moogyeong; Chung, Wonzoo
Highlights: A system of equations for direct adoption of transform sparsity to deep learning is derived. A cascade of unfolding networks of PCG is proposed to solve the system of equations. A primal-dual form for effective training of the sparsifying transform is developed. Both the primal and dual variables are updated simultaneously instead of alternating. Numerical simulations on the fastMRI dataset indicate improved performance compared with competing unfolding based methods, and similar performance with a recently proposed method while using significantly fewer network parameters. Abstract: Background and objective: Recent unfolding based compressed sensing magnetic resonance imaging (CS-MRI) methods only reinterpret conventional CS-MRI optimization algorithms and, consequently, inherit the weaknesses of the alternating optimization strategy. In order to avoid the structural complexity of the alternating optimization strategy and achieve better reconstruction performance, we propose to directly optimize the ℓ 1 regularized convex optimization problem using a deep learning approach. Method: In order to achieve direct optimization, a system of equations solving the ℓ 1 regularized optimization problem is constructed from the optimality conditions of a novel primal-dual form proposed for the effective training of the sparsifying transform. The optimal solution is obtained by a cascade of unfolding networks of the preconditioned conjugate gradient (PCG) algorithm trained to minimize the mean element-wise absolute difference ( ℓ 1 loss) between the terminal output and ground truth image in an end-to-end manner. The performance of the proposed method was compared with that of U-Net, PD-Net, ISTA-Net+, and the recently proposed projection-based cascaded U-Net, using single-coil knee MR images of the fastMRI dataset. Results: In our experiment, the proposed network outperformed existing unfolding-based networks and the complex version of U-Net in several subsampling scenarios. In particular, when using the random Cartesian subsampling mask with 25 % sampling rate, the proposed model outperformed PD-Net by 0.76 dB, ISTA-Net+ by 0.43 dB, and U-Net by 1.21 dB on the positron density without suppression (PD) dataset in term of peak signal to noise ratio. In comparison with the projection-based cascade U-Net, the proposed algorithm achieved approximately the same performance when the sampling rate was 25 % with only 1.62 % number of network parameters at the cost of a longer reconstruction time (approximately twice). Conclusion: A cascade of unfolding networks of the PCG algorithm was proposed to directly optimize the ℓ 1 regularized CS-MRI optimization problem. The proposed network achieved improved reconstruction performance compared with U-Net, PD-Net, and ISTA-Net+, and achieved approximately the same performance as the projection-based cascaded U-Net while using significantly fewer network parameters.