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Deep Plug-and-Play Prior for Parallel MRI Reconstruction1Computational 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
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