Enhanced Partial Fourier MRI With Zero-Shot Deep Untrained Priors

Abstract

We present a novel method for partial Fourier reconstruction using a zero-shot unsupervised deep learning approach. Specifically, this method enhances partial Fourier reconstruction by integrating the traditional phase constraint with recent advancements in zero-shot deep learning techniques. Recently, a zero-shot deep learning method based on an untrained generative prior has been proposed, demonstrating effectiveness in multi-echo/contrast reconstruction. This approach is based on the assumption that MR images can be nonlinearly represented through an untrained artificial neural network, allowing for simultaneous image reconstruction and prior learning without requiring any training data. In this study, we integrate this framework with the virtual conjugate coil (VCC) phase constraint to enable robust partial Fourier reconstruction. We evaluated the proposed method on the fastMRI dataset, the QALAS multi-contrast dataset, and a low-field dataset. Our experiment results confirm that the proposed method yields improved reconstruction quality compared to existing methods, with approximately 1.15x to 2.5x reductions in NRMSE. The proposed technique enables robust partial Fourier reconstruction, proving valuable in numerous applications. Moreover, it adopts a zero-shot approach, eliminating the need for training data, which enhances its utility in scenarios where data collection is challenging.

I. INTRODUCTION

Partial Fourier methods [1], [2], [3], [4], [5], [6] have been widely studied and utilized for accelerating magnetic resonance imaging (MRI). In partial Fourier MRI, certain regions of the k-space are not sampled, and reconstruction is conducted using phase constraints or conjugate symmetry in k-space. Conventional techniques, such as homodyne reconstruction, conjugate synthesis, and projection on convex sets (POCS) [1], [2], [3], [4], [5], [6], have incorporated explicit phase constraints to fill in missing partial Fourier k-space samples. Subsequently, model-based methods like POCSENSE, Phase-constrained SENSE, VCC-SENSE, VCC-GRAPPA, and other related techniques [7], [8], [9], [10], [11], [12], [13] were developed by integrating phase constraints with parallel imaging approaches, such as SENSE [14], [15] and GRAPPA [16], enabling further acceleration with enhanced fidelity. Later advancements in model-based reconstruction [17], [18], [19], [20] and deep learning [21], [22] have further improved their performance and applicability.

Meanwhile, accelerated MRI through image reconstruction have been extensively studied [23], [24], [25]. In particular, constrained model-based reconstruction [23], [24], [25] and the latest deep learning techniques [26], [27], [28] are being widely used in various applications. Model-based reconstruction techniques employ optimization-based approaches grounded in statistical models [25], [29], where data consistency is derived from the likelihood distribution, while regularization is induced by a user-defined prior model. Model-based reconstruction does not require additional training data and relies solely on individual, scan-specific data, making it flexibly applicable to diverse MR acquisition scenarios. However, it potentially suffers from lengthy computation times and limited reconstruction performance in highly ill-posed cases.

Recent deep learning-based methods utilize artificial neural networks (ANNs) trained with substantial amounts of training data [26], [27], [28], [30], [31]. Initially, deep learning techniques adopted a physics-agnostic approach, allowing the model to learn the entire reconstruction process solely from data, without incorporating a forward model [26], [27], [28]. However, these models have the significant drawback of requiring vast amounts of training data. Recent approaches embed MR physics into deep learning models, enabling them to learn only the priors from the dataset [30], [31], [32]. This approach significantly reduces the training burden and the amount of required training data, while still maintaining strong reconstruction performance. Despite the advantages of fast inference and high performance, deep learning-based methods face significant challenges in obtaining large training datasets and struggle with issues of generalization and hallucination, particularly when there is a significant discrepancy between the training and the test data [33].

To address these issues, methods such as scan-specific deep learning and zero-shot deep learning [34], [35], [36], [37], [38], which do not depend on external training data, have been proposed. These approaches train and utilize ANNs without requiring external datasets [39], [40], enabling flexible application across a variety of acquisition settings and demonstrating reliable reconstruction performance. Notably, our preliminary version of this work, the Zero-Shot Deep Untrained Generative Prior (ZS-DUGP) [41] has successfully addressed multi-echo/contrast imaging by exploiting the premise that spatio-temporal MRI data can be nonlinearly compressed using ANNs. This approach uses images generated from untrained neural networks as priors for constrained image reconstruction. The neural network is trained solely on intermediate images generated during the reconstruction process, allowing for simultaneous image reconstruction and network training without any external training data. However, this preliminary version of ZS-DUGP did not effectively address the partial Fourier reconstruction problem.

In this study, we introduce a novel partial Fourier MRI reconstruction technique that integrates a phase constraint with the Zero-Shot Deep Untrained Generative Prior [41]. Specifically, by incorporating the virtual conjugate coil (VCC) [11], [13] phase constraint into the ZS-DUGP [41] framework, we enhance the robustness of partial Fourier reconstruction. The proposed technique has been validated with a fastMRI dataset, a QALAS dataset, and a low-field MR dataset, demonstrating its effectiveness across various MRI experiment scenarios compared to existing methods. The following sections will provide a detailed description of the proposed method and the experimental results.

II. BACKGROUND AND THEORY

In partial Fourier MRI, the following data acquisition model can be utilized,

$$\mathbf{y}=\mathbf{U}\mathbf{F}\mathbf{S}\mathbf{x}=\mathbf{U}\mathbf{F}\mathbf{S}\mathbf{P}\mathbf{m}$$ (1)

where $\mathbf{y}\in {\mathbb{C}}^{cm}$ represents the acquired undersampled data, $\mathbf{U}\in \{0,1{\}}^{cm\times cn}$ denotes undersampling, $\mathbf{F}\in {\mathbb{C}}^{cn\times cn}$ is the Fourier transform, $\mathbf{S}\in {\mathbb{C}}^{cn\times n}$ indicates the coil sensitivity, where $c$ denotes the number of coils, and $\mathbf{x}\in {\mathbb{C}}^n$ is the image to be reconstructed. The MR image $\mathbf{x}$ can be further decomposed into $\mathbf{x}=\mathbf{P}\mathbf{m}$, where $\mathbf{P}\in {\mathbb{C}}^{n\times n}$ represents the diagonal matrix indicating the image phase, and $\mathbf{m}\in {\mathbb{C}}^n$ is the magnitude of the image. Assuming that the image phase $\mathbf{P}$ is known or can be estimated in advance, the magnitude $\mathbf{m}$ should ideally be completely real-valued. Owing to the complex conjugation property of the Fourier transform, the following condition is met for the complex-conjugated multi-channel images ${\left(\mathbf{S}\mathbf{x}\right)}^{\mathrm{*}}$,

$${\tilde{\mathbf{y}}}^{\mathrm{*}}=\tilde{\mathbf{U}}\mathbf{F}{\left(\mathbf{S}\mathbf{x}\right)}^{\mathrm{*}}=\tilde{\mathbf{U}}\mathbf{F}{\mathbf{S}}^{\mathrm{*}}{\mathbf{x}}^{\mathrm{*}}=\tilde{\mathbf{U}}\mathbf{F}{\mathbf{S}}^{\mathrm{*}}{\mathbf{P}}^{\mathrm{*}}{\mathbf{m}}^{\mathrm{*}}$$ (2)

where $~$ represents the spatial-reversal operator that flips the k-space around the origin, and $\mathrm{*}$ denotes the complex conjugation operator. Utilizing the property that $\mathbf{m}$ is ideally real-valued (i.e., ${\mathbf{m}}^{\mathrm{*}}=\mathbf{m}$), we can construct the following virtual conjugate coils [11], [13], [42],

$${\tilde{\mathbf{y}}}^{\mathrm{*}}=\tilde{\mathbf{U}}\mathbf{F}{\mathbf{S}}^{\mathrm{*}}{\mathbf{P}}^{\mathrm{*}}\mathbf{m}$$ (3)

The corresponding phase-constrained data consistency term, using both the original and VCC data, can be expressed as follows,

$$\stackrel{\mathbf{ˆ}}{\mathbf{m}}=\mathrm{a}\mathrm{r}\mathrm{g}\underset{\mathbf{m}}{\mathrm{m}\mathrm{i}\mathrm{n}}{∥\left[\begin{array}{c}\mathbf{y}\\ {\tilde{\mathbf{y}}}^{\mathrm{*}}\end{array}\right]-\left[\begin{array}{c}\mathbf{U}\mathbf{F}\mathbf{S}\mathbf{P}\\ \tilde{\mathbf{U}}\mathbf{F}{\mathbf{S}}^{\mathrm{*}}{\mathbf{P}}^{\mathrm{*}}\end{array}\right]\mathbf{m}∥}_2^2$$ (4)

which is equivalent to VCC-SENSE [11], [13].

Meanwhile, the proposed ZS-DUGP [41] employs the assumption that the MR image $\mathbf{m}$ can be effectively represented through ANNs. By utilizing a neural network $f_{\theta }\left(\mathbf{z}\right)$ with trainable parameters $\theta$, the following prior model is applicable (where $\mathbf{z}$ is a random input vector),

$$\mathbf{m}~𝒩\left(f_{\theta }\left(\mathbf{z}\right),{\sigma }^2\mathbf{I}\right)$$ (5)

Using this prior model, the following MAP estimator can be derived by integrating it with Eq. (4),

$$\stackrel{\mathbf{ˆ}}{\mathbf{m}}=\mathrm{a}\mathrm{r}\mathrm{g}\underset{\mathbf{m}}{\mathrm{m}\mathrm{i}\mathrm{n}}{∥\left[\begin{array}{c}\mathbf{y}\\ {\tilde{\mathbf{y}}}^{\mathrm{*}}\end{array}\right]-\left[\begin{array}{c}\mathbf{U}\mathbf{F}\mathbf{S}\mathbf{P}\\ \tilde{\mathbf{U}}\mathbf{F}{\mathbf{S}}^{\mathrm{*}}{\mathbf{P}}^{\mathrm{*}}\end{array}\right]\mathbf{m}∥}_2^2+\lambda {∥\mathbf{m}-f_{\theta }\left(\mathbf{z}\right)∥}_2^2$$ (6)

To employ the proposed approach described in Eq. (6), a pre-trained model $f_{\theta }\left(\mathbf{z}\right)$ capable of generating prior images is required. Several studies have explored various MR image reconstructions by exploiting pre-trained networks as priors [37], [43], [44], [45], [46]. However, as previously mentioned, such pre-trained neural networks can introduce potential issues related to data collection, generalization, and hallucination. In this study, we present a method that eliminates the need for any external training data by adopting zero-shot learning. Specifically, the image reconstruction ($\mathbf{m}$) and the prior learning ($\theta$) are performed jointly,

$$\stackrel{\mathbf{ˆ}}{\mathbf{m}},\hat{\theta }=\mathrm{a}\mathrm{r}\mathrm{g}\underset{\mathbf{m},\theta }{\mathrm{m}\mathrm{i}\mathrm{n}}{∥\left[\begin{array}{c}\mathbf{y}\\ {\tilde{\mathbf{y}}}^{\mathrm{*}}\end{array}\right]-\left[\begin{array}{c}\mathbf{U}\mathbf{F}\mathbf{S}\mathbf{P}\\ \tilde{\mathbf{U}}{\mathbf{F}\mathbf{S}}^{\mathrm{*}}{\mathbf{P}}^{\mathrm{*}}\end{array}\right]\mathbf{m}∥}_2^2+\lambda {∥\mathbf{m}-f_{\theta }\left(\mathbf{z}\right)∥}_2^2$$ (7)

While various numerical algorithms can be applied to solve this objective function, we opted for a simple alternating minimization approach. The iterative procedure is as follows (initially, let $f_{{\theta }^{\left(0\right)}}\left(\mathbf{z}\right)=\mathbf{0}$),

$${\mathbf{m}}^{(i+1)}={\left({\mathbf{B}}^H\mathbf{B}+{\mathbf{C}}^H\mathbf{C}+\lambda \mathbf{I}\right)}^{-1}\left({\mathbf{B}}^H\mathbf{y}+{\mathbf{C}}^H{\tilde{\mathbf{y}}}^{\mathrm{*}}+\lambda f_{{\theta }^{\left(i\right)}}(\mathbf{z})\right)$$ (8)

$${\theta }^{(i+1)}=\mathrm{a}\mathrm{r}\mathrm{g}\underset{\theta }{\mathrm{m}\mathrm{i}\mathrm{n}}{∥{\mathbf{m}}^{(i+1)}-f_{\theta }\left(\mathbf{z}\right)∥}_2^2$$ (9)

for $i=0,\dots ,n-1$, with $\mathbf{B}=\mathbf{U}\mathbf{F}\mathbf{S}\mathbf{P},\mathbf{C}=\tilde{\mathbf{U}}\mathbf{F}{\mathbf{S}}^{\mathrm{*}}{\mathbf{P}}^{\mathrm{*}}$. Eq. (8) is the process of constrained image reconstruction to update image $\mathbf{m}$ using the prior obtained from the neural network $f_{{\theta }^{\left(i\right)}}\left(\mathbf{z}\right)$, and Eq. (9) represents the loss function for network training to update the network parameter $\theta$ using the current image ${\mathbf{m}}^{(i+1)}$ as the training data. Through this iterative procedure, both the image reconstruction ($\mathbf{m}$) and prior learning ($\theta$) are performed simultaneously. These procedures (Eq. (8) and Eq. (9)) are iteratively conducted until convergence, or the maximum number of iterations is reached, where we set the maximum number of iterations to $i=15$ because we empirically observed that the results typically do not change significantly beyond this point. The convergence criterion is based on the relative change in the reconstructed image $\mathbf{m}$ between successive iterations, where we halt the process when this change drops below a specified threshold. Notably, the proposed Eq. (7) presents a large-scale non-convex objective function, and finding its optimal solution is both a non-trivial and computationally intensive problem. To address this challenge, our alternating minimization approach, optimizing $\mathbf{m}$ and $\theta$ alternately, can be highly effective in avoiding undesired local minima and accelerating the optimization process [47], [48].

Fig. 1 illustrates the pipeline of the proposed iterative method. Extending to joint multi-echo/contrast reconstruction is also straightforward; one simply needs to modif Eq. (7) to incorporate a multi-contrast MR image $\mathbf{m}\in {\mathbb{C}}^{n\times n_e}$ and a neural network $f_{\theta }\left(\mathbf{z}\right)\in {\mathbb{C}}^{n\times n_e}$ that represents a join multi-contrast prior, where $n_e$ denotes the number of different contrast images.

FIGURE 1.

An illustration of the proposed iterative procedure is presented. This procedure updates the image iteratively through constrained reconstruction (regularized VCC-Recon) using the prior image from the network, while concurrently training the network parameters with intermediate reconstructed images as training data, thus eliminating the need for an external training dataset.

The proposed process assumes that the image phase $\mathbf{P}$ is either already known or can be estimated in advance. In conventional phase-constrained reconstruction methods, the phase is often estimated from a low-resolution prescan or using a fully-sampled autocalibration signal (ACS) [9]. This approach can be effective because MRI phase typically exhibits smooth characteristics [17], and the central k-space region, containing low-frequency information, includes most of the phase information. However, this approach requires an additional burden for prescan or ACS acquisition.

Instead, in this study, we estimated the phase without relying on a prescan or a fully-sampled ACS [49]. We initially performed SENSE reconstruction on the undersampled data. We then selected only the central k-space region of the reconstructed image and utilized it as a synthetic ACS. This approach is potentially effective because the central k-space region generally exhibits high SNR, allowing for relatively accurate reconstruction. To further enhance the phase estimation and reduce sidelobes, we applied a Hamming window of the same size to this synthetic ACS. We then converted the synthetic ACS into the image domain, from which we extracted the low-resolution magnitude and the estimated phase. This approach facilitates the estimation of the image phase $\mathbf{P}$, as illustrated in Fig. 2. Our empirical findings indicate that the size of the synthetic ACS did not significantly affect the final reconstruction results; we used a size of 64 × 64 in our study. Fig. 3 compares three phase images: the ideal phase (as a reference), the estimated phase from fully-sampled ACS (using the central 64 × 64 k-space region), and the phase estimated by our proposed method without fully-sampled ACS (using the undersampled ACS shown in Fig. 4). This comparison demonstrates that our method can estimate a phase comparable to that obtained from fully-sampled ACS, despite using only undersampled data. Notably, the phase estimation presented here depends on the quality of the synthetic ACS, potentially compromising its accuracy. To address this, we treated the image $\mathbf{m}$ to be reconstructed as complex-valued in Eq. (7), allowing us to compensate for the imperfections in the phase estimation process and enabling the restoration of residual phase.

FIGURE 2.

Phase estimation is performed without a fully-sampled ACS. Initially, SENSE reconstruction is conducted from undersampled k-space data. The central k-space region is then cropped, and a hamming window is applied to generate synthetic ACS. Following the application of the inverse fourier transform, the estimated phase is obtained. Our empirical findings suggest that including a larger readout k_x region in the synthetic ACS is usually beneficial.

FIGURE 3.

Comparison of three phase images: the ideal phase (reference), the estimated phase from the central 64 × 64 fully-sampled ACS, and the estimated phase from our proposed method without fully-sampled ACS (using the 4x-undersampled ACS). The data and sampling mask used for our method are shown in Fig. 4.

FIGURE 4.

Experiment results with the fastMRI data are presented. The data was retrospectively undersampled using 1D 4x-uniform undersampling and 5/8 partial Fourier, resulting in an effective acceleration factor of 6.4x. For a performance comparison, SENSE, VCC-SENSE, LORAKS, MoDL, and ZS-SSL, along with the proposed method, were evaluated. The reconstructed k-space, images, and their error images with NRMSE, PSNR, and SSIM values are presented. The proposed method not only successfully reconstructed the missing partial fourier k-space samples but also enhanced image quality, achieving the best NRMSE, PSNR, and SSIM values.

III. METHODS

A. NETWORK ARCHITECTURE

The neural network $f_{\theta }$ used for this method employs a custom convolutional decoder [36], [40] structure. The target image for training this network, as given in Eq. (9), uses ${\mathbf{m}}^{(i+1)}$, splitting this complex-valued tensor into real and imaginary components that are concatenated along the channel dimension. The input tensor $\mathbf{z}$ is randomly generated with a size of 8 × 8 and 24 channels. This network is made up of 6 upsampling blocks. Within each upsampling block, the data passes through one (1st block) or two (2nd to 5th blocks) 2D convolution layers, followed by a 2D transposed convolution, with each layer interconnected by a ReLU activation function. The final (6th) block consists of three 2D convolutions without a transposed convolution. As it passes through each block, the number of channels changes from (input channels = 24) to 48, 96, 96, 96, 96, and then to (output channels = 2c), respectively. In each transposed convolution layer, the feature size doubles in both the horizontal and vertical dimensions, resulting in a final output size of 512 × 512. This output is then cropped to align with the dimensions of the target MR image ${\mathbf{m}}^{(i+1)}$. If the target MR image ${\mathbf{m}}^{(i+1)}$ exceeds 512 × 512 in size, the network can be adjusted by either appending additional upsampling blocks or increasing the size of the input tensor to expand the output size. Because the network consists solely of convolutional layers, there are no restrictions on the input size, and the number of upsampling blocks can also be easily customized to match the output image size. The loss function for training this network employs MSE loss, as presented in Eq. (9), where the network is iteratively trained and fine-tuned with an updated target ${\mathbf{m}}^{(i+1)}$. The network architecture is illustrated in Fig. 1.

B. DATASETS

For the evaluation of the proposed method, three different datasets were prepared. The first dataset is the fastMRI brain dataset [33], [50], from which a fully-sampled axial FLAIR k-space data was selected from the validation set. From the original $(readout\times PE)=(640\times 320)$ sized data, the oversampled and zero-padded regions were removed, resulting in a 2D dataset with a matrix size of 320 × 224. For convenience, the original 18-channel data was coil-compressed into 12 channels. For the experiment, the data was retrospectively undersampled using a 1D 4x-uniform mask combined with 5/8 partial Fourier, resulting in the effective acceleration factor of 6.4x. The data and sampling mask are shown in Fig. 4. We compared the proposed method with SENSE [14], [15], VCC-SENSE [11], [13], LORAKS [17], [20] (SENSE-LORAKS version with S-matrix), MoDL [31] (trained with 710 slices from the fastMRI validation dataset), and ZS-SSL [38] techniques.

The second dataset is the 3D-QALAS [51], [52], [53] dataset acquired from a Siemens scanner with a 32-channel head coil, with the Approval of the Institutional Review Board at Massachusetts General Hospital. The 3D-QALAS sequence consists of five readouts, with the initial readout occurring after a T₂ preparation module, and the subsequent four readouts taking place after an inversion pulse, which is used to capture T₁ dynamics. These five multi-contrast images enable simultaneous measurement of T₁ and T₂ maps. Acquiring such a large-scale 3D multi-contrast dataset takes a considerable amount of scan time, and accelerating this process could be advantageous. The obtained data have a matrix size 208 × 208 × 176, with a resolution of 1.15mm × 1.15mm × 1.15mm. Elliptical regions were sampled in the $k_y-k_z$ domain, involving corner cutting in the k-space plane. For the experiment, a single 2D slice with size 208 × 176 from the readout dimension was extracted. The data was retrospectively undersampled using a 2D variable density mask combined with partial Fourier, with an effective acceleration factor of 16.3x. The data and sampling masks are illustrated in Figs. 5, 6, 7, and 8. The proposed method was compared with SENSE [14], [15], VCC-SENSE [11], [13], the subspace method with locally low-rank constraint (Subspace+LLR) [54], [55], J-LORAKS [49], MoDL [31] (trained with 1,040 slices of the QALAS dataset), ZS-SSL [38], and ZS-SSL with joint multi-contrast reconstruction constraint (J-ZS-SSL) [56].

FIGURE 5.

Experiment results with the QALAS multi-contrast data are presented. The data was retrospectively undersampled using 2D variable density undersampling with partial fourier, resulting in an effective acceleration factor of 16.3x. Complementary partial fourier masks were applied to five different contrast datasets. For performance comparison, SENSE, VCC-SENSE, the subspace method with locally low-rank constraint (Subspace+ LLR), J-LORAKS, MoDL, ZS-SSL, and J-ZS-SSL, along with the proposed method, were evaluated. The reconstructed k-space, images, and their error images with NRMSE, PSNR, and SSIM values are presented. Due to space constraints, only the fourth contrast images are displayed, and additional results are included in Figs. 6, 7, and 8. The proposed method not only successfully reconstructed the missing partial Fourier k-space samples but also enhanced image quality, achieving the best NRMSE, PSNR, and SSIM values.

FIGURE 6.

An illustration of the QALAS reconstruction results, including the k-space for all five contrast data.

FIGURE 7.

An illustration of the QALAS reconstruction results, including all five contrast images.

FIGURE 8.

An illustration of the QALAS reconstruction results, including all five contrast error images.

The third dataset is a low-field dataset [57], [58] obtained from a 73mT, 50 kg portable “Halbach dome” head scanner, with the Approval of the Institutional Review Board at Massachusetts General Hospital. The matrix size of the data is 256 × 101, acquired with a single coil, and a prospective partial Fourier sampling (1.3x) was applied. The proposed method was applied to this dataset to demonstrate the feasibility of partial Fourier reconstruction and the potential for image quality improvement. The data and sampling mask are shown in Fig. 10.

FIGURE 10.

Experiment results with the low-field MR data are presented. The data was prospectively undersampled using partial Fourier sampling, and then reconstructed using the proposed method. The reconstructed k-space and images are displayed, demonstrating that the proposed method effectively reconstructed the missing k-space regions with improved image resolution. For improved comparative visualization between the original and reconstructed images, both images were zero-padded to a size of 256 × 256 in k-space.

IV. RESULTS

This section will provide the comparative experimental evaluation of the proposed method using the fastMRI 2D dataset, the QALAS multi-contrast dataset, and the low-field dataset.

A. FASTMRI

Fig. 4 displays the results of the fastMRI experiment. The leftmost column illustrates the gold standard k-space and image, derived through coil combination from the fully sampled data, along with the 6.4x-partial Fourier sampling mask utilized in the experiment. The subsequent columns show the reconstruction results, including the reconstructed k-space, images, and error images. The numbers below the error images indicate the corresponding NRMSE values between the original complex-valued gold standard and the reconstructed images, as well as the PSNR and SSIM values between the real-valued gold standard magnitude images and the reconstructed magnitude images. The qualitative visual comparison in k-space reveals that SENSE and ZS-SSL fall short in reconstructing partial Fourier k-space samples effectively. In contrast, VCC-SENSE, LORAKS, MoDL, and the proposed method successfully addressed this issue, either fully or partially, by employing implicit/explicit phase constraints (VCC-SENSE, LORAKS, and the proposed method) or through supervised learning (MoDL). The visual comparison in the image domain indicates that the proposed method effectively reduces artifacts and preserves resolution. In particular, the error images demonstrate that the proposed method presents the least error within the brain region. Lastly, a quantitative comparison using NRMSE, PSNR, and SSIM values confirmed the improved performance of the proposed method relative to other techniques, as it achieved the best values across all metrics.

The same experiment was repeated on 10 different fastMRI datasets with the same sampling mask, and the results are presented in Table 1. Our method consistently outperformed others across quantitative metrics, achieving the best average NRMSE, PSNR, and SSIM values. Notably, our zero-shot partial Fourier method demonstrates improved performance without requiring training data, outperforming both existing zero-shot and supervised approaches.

TABLE 1.

Quantitative comparison of the fastMRI experiment results conducted on 10 different axial FLAIR slices, presenting the average NRMSE, PSNR, and SSIM values.

	SENSE	VCC-SENSE	LORAKS	MoDL	ZS-SSL	Our Method
NRMSE	0.288	0.192	0.145	0.176	0.164	0.114
PSNR	23.685	26.024	29.625	28.883	29.435	31.659
SSIM	0.708	0.761	0.865	0.814	0.840	0.908

B. 3D-QALAS MULTI-PARAMETRIC MRI

The second experiment was conducted using the 3D-QALAS multi-contrast dataset. Fig. 5 displays only the 4th-contrast results for effective visualization, while the results of the other contrast images are shown in Figs. 6, 7, and 8. In Fig. 5, the leftmost column illustrates the gold standard k-space and image, obtained through coil combination from the fully sampled data, alongside the 16.3x-partial Fourier sampling mask used in the experiment. The subsequent columns show the reconstruction results, including the reconstructed k-space, images, and error images. The numbers below the error images indicate the corresponding NRMSE values between the original complex-valued gold standard and the reconstructed images, calculated using all 5 contrast images. PSNR and SSIM values are also presented between the real-valued gold standard magnitude images and the reconstructed magnitude images. These PSNR and SSIM values were calculated for each contrast and then averaged over 5 different contrasts. The qualitative visual comparison in k-space indicates that SENSE and ZS-SSL fall short in reconstructing partial Fourier k-space samples effectively. Conversely, VCC-SENSE, Subspace+LLR, J-LORAKS, MoDL, J-ZS-SSL, and the proposed method addressed this to varying degrees of success by utilizing explicit or implicit phase constraints (VCC-SENSE, J-LORAKS, and the proposed method), supervised learning (MoDL), or complementary partial Fourier sampling (Subspace+LLR, J-LORAKS, J-ZS-SSL, and the proposed method). The visual comparison in the image domain demonstrates that the proposed method effectively reduces artifacts and preserves resolution. Particularly, the error images in Fig. 8 reveal that the proposed method indicates the least error within the brain region. Lastly, a quantitative comparison using NRMSE, PSNR, and SSIM values confirmed that the proposed method outperformed other techniques, achieving the best values across all metrics.

Fig. 9 shows the MR parameter maps measured from the QALAS reconstruction results, including T₁, T₂, and PD maps. The leftmost column displays the gold standard parameter maps obtained from fully-sampled data, while the remaining columns show maps estimated from each reconstruction result. A qualitative comparison of the estimated parameter maps reveals that the proposed method preserves structural and quantitative information effectively and exhibits lower noise levels, demonstrating improved parameter estimation compared to other methods.

FIGURE 9.

MR parameter maps estimated from the QALAS multi-contrast reconstruction images, displaying the T₁, T₂, and proton density (PD) maps.

These results demonstrate that the proposed method achieves highly robust outcomes even in multi-contrast partial Fourier reconstruction scenarios, outperforming both existing zero-shot and supervised approaches.

C. LOW-FIELD MRI

The third experiment involved applying the proposed method to the low-field data, as shown in Fig. 10. This data was obtained through prospective undersampling using a single coil. The leftmost column displays the sampling mask, while the second column shows the acquired k-space data and the corresponding image. The final column presents the results reconstructed with the proposed method. For effective visualization of the reconstructed images, the original matrix size of 256 × 101 was zero-padded to 256 × 256 in k-space, followed by the inverse Fourier transform. The reconstructed k-space confirms the proposed method’s ability to successfully reconstruct partial Fourier k-space samples, and the reconstructed images illustrate its effectiveness in improving resolution and image quality.

V. DISCUSSION

Previous sections described the advantages of the proposed method across various MR applications. This section will discuss several considerations for employing the proposed method.

A. HYPERPARAMETERS

The proposed method requires the adjustment of two hyperparameters: $\lambda$, as utilized in Eq. (8), and the early stopping iteration number for network training, as outlined in Eq. (9). Focusing on the latter, Eq. (9) is analogous to the Deep Image Prior (DIP) [39], and employing an overparameterized neural network can result in overfitting to the incomplete image ${\mathbf{m}}^{(i+1)}$, leading to suboptimal outcomes. To mitigate this, DIP introduced early stopping as a tunable hyperparameter. In this study, we have also implemented early stopping (denoted as $t$) and tuned both $\lambda$ and $t$ to minimize the NRMSE. Specifically for $t$, we assessed the reconstruction performance by increasing $t$ from 100 to 1000 in increments of 100, and from 1000 to 10000 in increments of 1000. For each case, we measured the NRMSE of the final reconstruction result. During this experiment, $\lambda$ was fixed at 0.1. Similarly for $\lambda$, we checked the performance by increasing it from 10⁻⁴ to 10³ in factors of 10. For each case, we measured the NRMSE of the final reconstruction result. During this experiment, $t$ was fixed at 1000. The results are shown in Fig. 11. The selected hyperparameters were $t=1000$ and $\lambda =0.1$, which we used consistently across all datasets. As shown in Fig. 11, our empirical findings with the fastMRI experiment indicate that the results remain stable across a broad spectrum of hyperparameter choices. However, if tuning these hyperparameters is practically challenging, automated hyperparameter tuning methods are available (e.g., SURE-based methods [59], [60], training-based methods [38]), or hyperparameters used on a similar dataset can be reused.

FIGURE 11.

An ablation study on hyperparameters related to reconstruction performance was conducted using the fastMRI dataset. (Top) the relation between the early stopping iteration number (t), used in Eq. (9), and NRMSE was assessed, with t varying from 100 to 10,000, while keeping other hyperparameters constant. It was observed that NRMSE values remained stable approximately from t = 200 to 6000. (Middle) NRMSEs were evaluated by adjusting λ, as specified in Eq. (8), from 10⁻⁴ to 10³, with other hyperparameters held constant. The findings indicated that NRMSE values were stable approximately from λ = 10⁻² to λ = 10¹. (Bottom) NRMSEs were measured based on the number of upsampling blocks in the untrained network, as described in Sec. III-A, with other hyperparameters held constant. The experiment results suggest that reconstruction performance is relatively insensitive to the number of upsampling blocks.

Additionally, an analysis was conducted on the number of upsampling blocks of the suggested untrained network presented in Sec. III-A. The NRMSE values were assessed by increasing the number of the upsampling blocks from the original 6 to 9, with other hyperparameters held constant. The results reveal that performance is relatively insensitive to the number of upsampling blocks, as shown in Fig. 11.

B. COMPUTATION

Another consideration is the time and memory usage required for reconstruction. For instance, reconstructing the 2D fastMRI dataset requires approximately 5 minutes of computation time and 0.5GB of GPU memory on an Nvidia Tesla T4 GPU. Similarly, reconstructing a 2D slice of QALAS data (32 channels, 5 echoes) requires 20 minutes and 1.1GB of GPU memory using the same hardware. Reconstructing 3D data in its entirety is expected to demand significantly more memory and time. However, this experiment was a proof-of-concept implementation, with no specific optimization for reconstruction time and memory. Enhancements in hardware, algorithms, and parallelization are expected to lead to substantial improvements in both reconstruction time and memory usage. Another promising approach is to utilize transfer learning [56], [61]. By introducing a pre-trained network, we can perform the network training in Eq. (9) solely through fine-tuning, significantly reducing the overall computation time. Fig. 12 compares the reconstruction results and computation time between the original training method using an untrained network, and transfer learning using a network pre-trained on a similar 2D slice and fine-tuned on the target dataset. While both methods yield nearly identical reconstruction quality, the original training technique requires approximately 5 minutes, whereas transfer learning completes the reconstruction in only about 45 seconds. These potential improvements are worth exploring in future work.

FIGURE 12.

Comparison of reconstruction results and computation time between our original training method (using an untrained network) and transfer learning (using a pre-trained network and fine-tuning). The experiments were conducted using the fastMRI dataset and the 6.4x undersampling mask shown in Fig. 4. Both methods achieve similar reconstruction quality, but transfer learning significantly reduces computation time.

C. CURRENT LIMITATIONS AND FUTURE DIRECTIONS

A potential limitation of the proposed technique is its reliance on accurate phase estimation. While our synthetic ACS approach shows promise, it may encounter challenges in scenarios with extreme undersampling, complex phase characteristics, or rapid phase variations, which could result in inaccurate phase estimation. Future work could focus on developing more robust phase estimation techniques that incorporate advanced reconstruction methods, as well as exploring the joint optimization of phase and magnitude during the iterative reconstruction process. Extending our method to higher-dimensional data (e.g., 3D MRI) also presents challenges, including increased computational complexity, memory requirements, longer acquisition times, and difficulties in maintaining SNR. Future research could address these challenges by developing GPU-efficient high-dimensional implementations, exploring data partitioning and parallel processing techniques, and investigating transfer learning approaches using pre-trained networks. Despite these challenges, the method’s potential for practical implementation includes addressing various challenging MRI scenarios by accelerating scan speeds and improving SNR without the need for a training dataset. These characteristics make the method particularly useful in diverse low-field and portable MRI settings, as well as in motionprone imaging scenarios such as pediatric, musculoskeletal, and cardiovascular imaging. Furthermore, it has the potential to accelerate EPI acquisition for functional MRI and enhance data acquisition in various quantitative MRI applications, thereby improving overall imaging efficiency and quality.

VI. CONCLUSION

In this study, we introduce a novel zero-shot partial Fourier reconstruction method using deep untrained generative prior. The proposed method employs VCC phase-constraint for data consistency and integrates regularization based on the concept that MR images can be nonlinearly compressed and represented using ANNs. This strategy facilitates simultaneous prior learning and image reconstruction, eliminating the need for an extra dataset for the network training. The effectiveness of the proposed technique was demonstrated through experiments on the fastMRI, QALAS, and low-field datasets, with results showing improved performance compared to existing methods. The proposed method can be readily adapted to various acquisition conditions, including different sampling trajectories (including non-uniform ones), pulse sequences, and coil geometries, without the requirement for additional data collection for network training. We believe this method is expected to be useful in various MR applications utilizing partial Fourier acquisition, such as echo planar imaging (EPI), quantitative imaging, and low-field MR.

Biographies

SO HYUN KANG received the B.S. degree in computer engineering from Hongik University, Seoul, Republic of Korea, where she is currently pursuing the M.S. degree. Her research interests include computer vision and deep learning security.

JIHOO KIM is currently pursuing the B.S. degree in computer engineering with Hongik University, Seoul, Republic of Korea. His research interest includes computer vision.

JAEJIN CHO received the B.S. degree in electrical engineering from Korea University, Seoul, South Korea, in 2012, and the M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea, in 2014 and 2019, respectively. From 2018 to 2024, he was a Postdoctoral Researcher with the Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, USA. He is currently an Assistant Professor with the Department of Artificial Intelligence and Robotics, Sejong University, Seoul. He leads the Medical Imaging Research and Artificial Intelligence (MIRA) Laboratory, where his research focuses on developing innovative medical image reconstruction techniques through the application of advanced AI technologies.

CLARISSA ZIMMERMAN COOLEY received the B.S., M.Eng., and Ph.D. degrees in electrical engineering from MIT. She is currently an Assistant Professor of radiology with the Harvard Medical School and a member of the MR Physics and Instrumentation Group, Athinoula A. Martinos Center for Biomedical Imaging, MGH. Her research centers on portable MRI system development to improve access to imaging. She received the 2014 ISMRM Young Investigator Award for her contributions to this field.

AMIR HEYDARI received the B.S. degree in industrial engineering from Yazd University, Yazd, Iran, in 2012, and the M.S. degree in systems optimization from the Amirkabir University of Technology, Tehran, Iran, in 2015, where he is currently pursuing the Ph.D. degree in industrial engineering and management systems. He is a Graduate Visiting Researcher with the Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, USA. His research interests include data acquisition and image reconstruction techniques for efficient magnetic resonance imaging.

BERKIN BILGIC received the B.S. degree in electrical and electronics engineering and physics from Bogazici University, in 2008, and the M.S. and Ph.D. degrees in electrical engineering and computer science from Massachusetts Institute of Technology (MIT), in 2010 and 2013, respectively. He completed his postdoctoral training with the Martinos Center for Biomedical Imaging and became an Instructor in radiology with Massachusetts General Hospital (MGH) and Harvard Medical School (HMS), in 2016. In 2019, he was subsequently promoted to an Assistant and an Associate Professor with MGH/HMS, in 2023. He has been an affiliated faculty with the Harvard-MIT Department of Health Sciences and Technology, since 2018. He leads the Bilgic Reconstruction Acquisition for Imaging Neuroscience (BRAIN) Laboratory which develops data acquisition and image reconstruction techniques for efficient magnetic resonance imaging.

TAE HYUNG KIM (Member, IEEE) received the B.S. degree in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea, in 2011, the M.S. degree in electrical engineering-systems and mathematics from the University of Michigan, Ann Arbor, MI, USA, in 2014, and the Ph.D. degree in electrical engineering from the University of Southern California (USC), Los Angeles, CA, USA, in 2020. Before joining Hongik University, he was with the Athinoula A. Martinos Center for Biomedical Imaging, Harvard Medical School, Charlestown, MA, USA, from 2020 to 2022, as a Research Fellow. He is currently an Assistant Professor with the Department of Computer Engineering, Hongik University, Seoul, South Korea. He was a recipient of the Kwanjeong Educational Foundation Scholarship, in 2014, the USC Annenberg Graduate Fellowship, in 2014, and the USC MHI Ph.D. Scholar Award, in 2019.

DATA AVAILABILITY STATEMENT

Sample code and data are available at https://github.com/G-who0212/Partial_Fourier_Reconstruction_With_Zero-Shot_Deep_Untrained_Generative_Prior.