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Deep Plug-and-Play Prior for Parallel MRI Reconstruction
Ali Pour Yazdanpanah1, Onur Afacan1, and Simon K. Warfield1

1Computational Radiology Laboratory, Boston Children's Hospital and Harvard Medical School, Boston, MA, United States

Synopsis

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.

Introduction

Many approaches for reducing MRI scan time work by acquiring a fraction of the measurement required for a high-quality image. Extensive efforts have been devoted to finding the best regularizers with capable optimization method to solve this issue. The plug-and-play prior framework is proposed by Venkatakrishnan et al. [1] with an idea to utilize the denoiser without any regularization objective as the proximal operator in an iterative method for image reconstruction. The method has been successfully used in different imaging inverse problem applications [2-5]. In [2], the authors used the plug-and-play framework for bright field electron tomography. In [3], plug-and-play alternating direction method of multipliers (ADMM) has been used for image restoration applications. In [4], the authors developed the fast-iterative shrinkage/thresholding algorithm (FISTA) variant of plug-and-play prior for model-based nonlinear inverse scattering and proved that the framework is applicable beyond linear inverse problems. In [5], the authors introduced a scalable version of the plug-and-play framework based on iterative shrinkage/thresholding algorithm (ISTA) which utilized a subset of measurement at every iteration in order to parallelize the algorithm. In all the mentioned papers, a fixed denoiser has been used as the proximal operator which its accuracy isn't ideal in different scenarios for the different applications. However, in this paper, we present a learning-based plug-and-play prior framework for parallel MRI reconstruction which extends the framework to its data-adaptive variant and provides an end-to-end reconstruction scheme.

Methods

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.

Results

In our experiments, we have tested our method with two different datasets. The first dataset has been acquired (3D MPRAGE) on six volunteers with a total of 450 brain images used as the training set. For the second dataset, we have used one of the knee datasets (Coronal fat-saturated proton-density (PD)) presented by [8] which includes a total of 200 images (from central slices) from 10 patients as the training set. 10 images from different patients for each dataset have used for testing purposes. The sensitivity maps were computed from a block of size 24x24 for both brain and knee datasets using ESPIRiT [9] method. Full k-space data reconstructed with the adaptive combine method [10] was used as our gold standard for comparison. Figure 2 display the impact of acceleration factor R=2x2 for zero-filled reconstruction, the clinical gold standard GRAPPA, and our proposed method on 3D MPRAGE brain images. We observed that the proposed method reconstructs artifact-free images, which have better quality than GRAPPA reconstruction, and GRAPPA result shows noise amplification compared to our result (PSNR of ours is 52.93 compared to PSNR of 43.91 for GRAPPA). Figure 3 shows the impact of acceleration factor R=4 for zero-filled reconstruction, GRAPPA, and our proposed method on fat-saturated PD knee data. Similar to Figure 2, GRAPPA result for knee data in Figure 3 shows noise amplification compared to our result (PSNR of ours is 40.48 compared to PSNR of 29.39 for GRAPPA). PSNR and SSIM quantitative variations on two test datasets are depicted in Table 1.

Conclusion

This paper proposes a deep plug-and-play prior framework and demonstrates the effectiveness of learning-based plug-and-play prior framework for parallel MRI reconstruction. The experimental results on two MRI datasets show that our proposed method outperforms the clinical gold standard GRAPPA method.

Acknowledgements

This research was supported in part by NIH grants R01 NS079788, R01 EB019483, R42 MH086984, and by a research grant from the Boston Children's Hospital Translational Research Program.

References

[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.

Figures

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

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