2805

Robust Partial Fourier Reconstruction with Zero-shot Deep Untrained Generative Prior
So Hyun Kang*1, Jihoo Kim*1, JaeJin Cho2,3, Clarissa Z. Cooley2,3, Berkin Bilgic2,3,4, and Tae Hyung Kim1
1Department of Computer Engineering, Hongik University, Seoul, Korea, Republic of, 2Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown, MA, United States, 3Department of Radiology, Harvard Medical School, Boston, MA, United States, 4Harvard/MIT Health Sciences and Technology, Cambridge, MA, United States

Synopsis

Keywords: Machine Learning/Artificial Intelligence, Machine Learning/Artificial Intelligence, Image Reconstruction, Partial Fourier, Brain, Multi-echo MRI, Low-field MR

Motivation: We introduce a novel partial Fourier reconstruction method.

Goal(s): The objective is to enhance partial Fourier reconstruction by integrating the traditional phase constraint with the recent zero-shot deep learning approach.

Approach: The proposed method combines the virtual conjugate coils (VCC) phase constraint with zero-shot deep untrained generative prior (ZS-DUGP), assuming MRI can be nonlinearly represented by untrained networks, enabling simultaneous image reconstruction and prior learning without external training data. This approach enables robust partial Fourier reconstruction.

Results: Evaluation across diverse datasets, including the fastMRI, the QALAS multi-echo data, and the low-field MR data, validates its enhanced performance compared to existing techniques.

Impact: We propose a novel partial Fourier reconstruction combining virtual conjugate coils with a zero-shot untrained generative network prior. It provides robust reconstruction without external training dataset, evaluated across various scenarios (parallel imaging, multi-echo/contrast imaging, low-field MR) demonstrating its utility.

Introduction

We present a novel partial Fourier reconstruction method based on zero-shot deep learning. The partial Fourier method1–4 is a widely used accelerated MRI technique, where reconstruction is often performed by employing the phase smoothness constraint or conjugate symmetry in k-space5–12.
Other venues for accelerated imaging include the recently developed scan-specific deep learning reconstruction techniques13–15. Among these, zero-shot MRI reconstruction using a deep untrained generative network (ZS-DUGP)16 was proven to be powerful in multi-contrast imaging. This approach assumes a prior, suggesting that MRI images can be nonlinearly represented by an untrained generative neural network, enabling simultaneous image reconstruction and prior learning without the need for an external training dataset. It demonstrated improved reconstruction performance, especially in multi-echo/contrast scenarios.
In this study, we introduce a novel partial Fourier reconstruction method based on ZS-DUGP. The proposed method is built upon the same prior used in ZS-DUGP and incorporates a phase constraint by exploiting virtual conjugate coils. We evaluated the proposed method on various datasets, including a fastMRI17,18, a QALAS multi-echo dataset19–21, and a low-field dataset22,23, demonstrating improved performance compared to existing methods.
Code: https://anonymous.4open.science/r/Partial_Fourier_Reconstruction_With_Zero-Shot_Deep_Untrained_Generative_Prior-5495/

Theory

We employ the following MRI data acquisition model,$$\mathbf{y}=\mathbf{U}\mathbf{F}\mathbf{S}\mathbf{x}=\mathbf{U}\mathbf{F}\mathbf{S}\mathbf{P}\mathbf{m}$$Here, $$$\mathbf{y}\in\mathbb{C}^m$$$ represents the acquired k-space data, $$$\mathbf{U}$$$ is undersampling, $$$\mathbf{F}$$$ is the Fourier transform, $$$\mathbf{S}$$$ represents coil sensitivity, and $$$\mathbf{x}\in\mathbb{C}^n$$$ represents the underlying image to be reconstructed. We decompose the image $$$\mathbf{x}$$$ as $$$\mathbf{x}=\mathbf{P}\mathbf{m}$$$, where $$$\mathbf{P}\in\mathbb{C}^{n\times n}$$$ is a diagonal matrix for the image phase, and $$$\mathbf{m}\in \mathbb{C}^n$$$ represents the magnitude. Assuming that the phase $$$\mathbf{P}$$$ is estimated in advance, we will employ the virtual conjugate coils (VCC)10 for the phase constraint,$$\tilde{\mathbf{y}}^*=\tilde{\mathbf{U}}\mathbf{F}\mathbf{S}^*\mathbf{P}^*\mathbf{m}$$where $$$\tilde{\color{white}{\cdot}}$$$,$$$\color{white}{\cdot}^*$$$ denote spatial-reversal and complex conjugation, respectively. Additionally, the ZS-DUGP employs the following prior assumption,$$\mathbf{x}\approx f_\theta(\mathbf{z})$$where $$$f_\theta(\cdot)$$$ represents an untrained generative neural network with trainable parameters $$$\theta$$$, and $$$\mathbf{z}$$$ is a fixed random input. The corresponding MAP estimator becomes,$$\hat{\mathbf{m}},\hat{\mathbf{\theta}}=\arg\min_{\mathbf{m},\mathbf{\theta}}\left\|\pmatrix{\mathbf{y}\\\tilde{\mathbf{y}}^*}-\pmatrix{\mathbf{U}\mathbf{F}\mathbf{S}\mathbf{P}\\\tilde{\mathbf{U}}\mathbf{F}\mathbf{S}^*\mathbf{P}^*}\mathbf{m}\right\|_2^2+\lambda\left\|\mathbf{m}-f_\theta(\mathbf{z})\right\|_2^2$$By applying alternating minimization, we obtain,$$\theta^{(i+1)}=\arg\min_{\theta}\left\|\mathbf{m}-f_\theta(\mathbf{z})\right\|_2^2$$$$\mathbf{m}^{(i+1)}=\left(\mathbf{B}^H\mathbf{B}+\mathbf{C}^H\mathbf{C}+\lambda\mathbf{I}\right)^{-1}\left(\mathbf{B}^H\mathbf{y}+\mathbf{C}^H\tilde{\mathbf{y}}^*+\lambda f_{\theta^{(i+1)}}(\mathbf{z})\right)$$where $$$\mathbf{B}=\mathbf{U}\mathbf{F}\mathbf{S}\mathbf{P}$$$, $$$\mathbf{C}=\tilde{\mathbf{U}}\mathbf{F}\mathbf{S}^*\mathbf{P}^*$$$. The first equation represents the loss function for network training, and the second one represents the image update through regularized constrained reconstruction. This iterative procedure allows for simultaneous network training (for prior learning) and image reconstruction without requiring external training data. Fig. 1 illustrates this procedure.
This process assumes that phase can be estimated in advance, often using a fully sampled autocalibration signal (ACS) in conventional methods. In our study, we performed phase estimation without fully sampled ACS. Fig. 2 shows the phase estimation process, where we conducted an initial SENSE reconstruction and extracted the central k-space region that was treated as synthetic ACS. This allowed us to determine the $$$\mathbf{P}$$$ matrix, which is integrated into the iterative process above.

Methods

To evaluate the proposed method, three datasets were prepared. The first dataset was the fastMRI brain dataset17,18, from which we selected a fully-sampled data from the validation set and extracted a single slice. We removed the oversampled and zero-padded regions to generate a 2D dataset with a matrix size 224x320, and the original 16 channels were coil-compressed into 12 channels. For the experiments, we applied 3x uniform undersampling with 5/8 partial Fourier ($$$R_{eff}=4.8\times$$$). We compared the proposed method with SENSE24, VCC-SENSE10, SENSE+TV25, ZS-SSL15.
The second dataset was the 3D-QALAS dataset19–21, comprising 5 multi-echo data obtained through elliptical scanning. The matrix size was 208x208x176, with a resolution of 1.15mmx1.15mmx1.15mm. We selected one slice for reconstruction and applied a 12x variable density mask with 5/8 partial Fourier ($$$R_{eff}=15.5\times$$$). In the experiments, we compared SENSE24, Subspace with locally low rank constraint (Subspace+LLR)26,27, VCC-SENSE10, SENSE with joint TV (SENSE+jTV)28, and the proposed method.
The third dataset was the low-field dataset, acquired from a 73mT, 50 kg portable “Halbach dome” head scanner22,23. The data was acquired using a single coil and prospective partial Fourier sampling with the matrix size of 256x101. We applied the proposed method for reconstruction.

Results

Fig. 3 shows the reconstruction results for the fastMRI dataset. Fig. 4 presents the results for the QALAS dataset. Compared with other methods, the proposed approach enables partial Fourier reconstruction and demonstrates improved performance with lower NRMSE values. Fig. 5 displays the reconstruction results for the low-field data, confirming that the proposed method can recover the missing k-space regions.

Conclusion

In this study, we proposed a novel partial Fourier reconstruction using the zero-shot deep untrained generative prior. This method combines the phase constraint using virtual conjugate coils with an untrained generative neural network, allowing robust partial Fourier reconstruction without the need for external training data. We experimentally validated the proposed method to various scenarios, such as parallel imaging, multi-echo/contrast imaging, and low-field MR, demonstrating its utility.

Acknowledgements

* These authors contributed equally to this work.
This work was supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2022R1F1A1074786), research grants NIH R01 EB028797, U01 EB025162, P41 EB030006, U01 EB026996, R03 EB031175, R01 EB032378, UG3 EB034875 and by Nvidia Corporation for computing support.

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Figures

Fig. 1. An illustration of the proposed iterative procedure. This procedure iteratively updates the image through constrained reconstruction (Regularized VCC-Recon) and simultaneously trains the network using the intermediate images as training data, eliminating the need for an external training dataset.

Fig. 2. Phase estimation without fully-sampled ACS is performed. Initially, SENSE reconstruction is carried out using the undersampled k-space data. Subsequently, the central k-space region is cropped, and a Hamming window is applied to create synthetic ACS. After applying the inverse Fourier transform, the estimated phase is obtained.

Fig. 3. Retrospective fastMRI reconstruction results from the uniformly undersampled partial Fourier mask with an effective acceleration factor of 4.8x. We compared SENSE, VCC-SENSE, SENSE+TV, ZS-SSL, and the proposed method. As expected, SENSE, SENSE+TV, and ZS-SSL struggled to effectively reconstruct missing partial Fourier k-space samples. In contrast, the proposed method not only reconstructed them but also improved image quality with the lowest NRMSE value.

Fig. 4. Retrospective QALAS reconstruction results using variable density sampling with a partial Fourier mask (R=15.5x). Due to space constraints, only the third echo images are displayed. NRMSE values were computed using all five echo images. We compared SENSE, Subspace+LLR, SENSE+jTV, VCC-SENSE, ZS-SSL, and the proposed method. The proposed method not only reconstructed missing partial Fourier k-space samples but also improved image quality with the lowest NRMSE value.

Fig. 5. Prospective Low-field MR reconstruction results from the partial Fourier sampling. The proposed method successfully reconstructed the missing k-space regions. To enhance the comparative visualization of the original and reconstructed images, both datasets were zero-padded to a size of 256x256 in k-space, followed by inverse Fourier transformation and display.

Proc. Intl. Soc. Mag. Reson. Med. 32 (2024)
2805
DOI: https://doi.org/10.58530/2024/2805