Ali Pour Yazdanpanah^{1}, Onur Afacan^{1}, and Simon K. Warfield^{1}

Fast data acquisition in Magnetic Resonance Imaging (MRI) is vastly in demand and scan time directly depends on the number of acquired k-space samples. Conventional MRI reconstruction methods for fast MRI acquisition mostly relied on different regularizers which represent analytical models of sparsity. However, recent data-driven methods based on deep learning has resulted in promising improvements in image reconstruction algorithms. In this paper, we propose a deep plug-and-play prior framework for parallel MRI reconstruction problems which utilize a deep neural network (DNN) as an advanced denoiser within an iterative method. We demonstrate that a deep plug-and-play prior framework for parallel MRI reconstruction with a regularization that adapts to the data itself results in excellent reconstruction accuracy and outperforms the clinical gold standard GRAPPA method.

The discretized version of MR imaging model given by

$$\text d=\text E \text x + \text n (1)$$

where $$$\text x$$$ is the unknown MR image, and d is the undersampled k-space data. $$$\text E = \text{PFS}$$$ is an encoding matrix, and $$$F$$$ is a Fourier matrix. $$$P$$$ is a mask representing k-space undersampling pattern and $$$\text S$$$ represents the sensitivity information. Assuming that the interchannel noise covariance has been whitened, the reconstruction relies on the regularized least-square approach:

$$\widehat{\text x} =\underset{\text x}{ argmin} \ \|\text d-\text E\text x\|_{2}^{2}+\beta \text R(\text x) (2)$$

where $$$R$$$ is a regularization functional that promotes sparsity in the solution and $$$\beta$$$ controls the intensity of the regularization.

Our iterative deep plug-and-play prior framework for solving the Eq.2 is provided in Figure 1. DNN architecture is Unet-type convolutional network [6] and Loss minimization was performed using ADAM [7] optimizer. Zero-filled reconstruction is used as an initialization to the algorithm. For least-square case, we have

$$prox (\text d,\text S, \widetilde{\text x};\lambda) = \underset{\text z}{ argmin} \ \frac{1}{2}\|\text z-\widetilde{\text x}\|_{2}^{2}+ \frac{\lambda}{2}\|\text {PFS}\text z-\text d\|_{2}^{2} (3)$$

Since the deep network frameworks work on real-valued parameters, inputs, and outputs, in our method complex data are divided into real and imaginary parts and considered as two-channel input and output.

[1] S. V. Venkatakrishnan, C. A. Bouman, and B. Wohlberg, “Plug-and-play priors for model based reconstruction,” in Proc. IEEE Global Conf. Signal Process. and INf. Process. (GlobalSIP), Austin, TX, USA, December 3-5, 2013, pp. 945–948.

[2] S. Sreehari, S. V. Venkatakrishnan, B. Wohlberg, G. T. Buzzard, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Plug-and-play priors for bright field electron tomography and sparse interpolation,” IEEE Trans. Comp. Imag., vol. 2, no. 4, pp. 408–423, December 2016.

[3] S. H. Chan, X. Wang, and O. A. Elgendy, “Plug-and-play ADMM for image restoration: Fixed-point convergence and applications,” IEEE Trans. Comp. Imag., vol. 3, no. 1, pp. 84–98, March 2017.

[4] U. S. Kamilov, H. Mansour, and B. Wohlberg, “A plug-and-play priors approach for solving nonlinear imaging inverse problems,” IEEE Signal. Proc. Let., vol. 24, no. 12, pp. 1872–1876, December 2017.

[5] Sun, Yu, Brendt Wohlberg, and Ulugbek S. Kamilov. "An online plug-and-play algorithm for regularized image reconstruction." arXiv preprint arXiv:1809.04693, 2018.

[6] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computer-assisted intervention, pages 234–241. Springer, 2015.

[7] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.

[8] Kerstin Hammernik, Teresa Klatzer, Erich Kobler, Michael P Recht, Daniel K Sodickson, Thomas Pock, and Florian Knoll. Learning a variational network for reconstruction of accelerated MRI data. Magnetic resonance in medicine, 79(6):3055–3071, 2018.

[9] Martin Uecker, Peng Lai, Mark J. Murphy, Patrick Virtue, Michael Elad, John M. Pauly, Shreyas S. Vasanawala, Michael Lustig. ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA. Magnetic resonance in medicine, 71(3):990–1001, 2014.

[10] David O Walsh, Arthur F Gmitro, and Michael W Marcellin. Adaptive reconstruction of phased array MR imagery. Magnetic Resonance in Medicine, 43(5):682–690, 2000.

Figure 1. Proposed deep plug-and-play prior framework.

Figure 2. The first row (left to right): Gold standard reconstruction result using fully sampled data, zero-filled reconstruction, GRAPPA reconstruction result with undersampling factor of 2x2, and our reconstruction result with undersampling factor of 2x2 for 3D MPRAGE data. The second row, includes error maps correspond to each reconstruction results for comparison.

Figure 3. The first row (left to right): Gold standard reconstruction result using fully sampled data, zero-filled reconstruction, GRAPPA reconstruction result with undersampling factor of 4, and our reconstruction result with undersampling factor of 4 for 2D coronal knee data. The second row, includes error maps correspond to each reconstruction results for comparison.

Table 1. PSNR and SSIM variations on the two test datasets