#

###### Abstract

Background: Magnetic resonance imaging (MRI) offers excellent soft tissue
contrast essential for diagnosis and treatment, but its long acquisition times
can cause patient discomfort and motion artifacts.

Purpose: To propose a self-supervised deep learning-based compressed sensing
MRI (DL-based CS-MRI) method named “Adaptive Self-Supervised Consistency
Guided Diffusion Model (ASSCGD)” to accelerate data acquisition without
requiring fully sampled datasets.

Materials and Methods: We used the fastMRI multi-coil brain axial T2-weighted
(T2-w) dataset from 1,376 cases and single-coil brain quantitative
magnetization prepared 2 rapid acquisition gradient echoes (MP2RAGE) T1 maps
from 318 cases to train and test our model. Robustness against domain shift
was evaluated using two out-of-distribution (OOD) datasets: multi-coil brain
axial postcontrast T1-weighted (T1c) dataset from 50 cases and axial
T1-weighted (T1-w) dataset from 50 patients. Data were retrospectively
subsampled at acceleration rates $R\in\\{2\times,4\times,8\times\\}$. ASSCGD
partitions a random sampling pattern into two disjoint sets, ensuring data
consistency during training. We compared our method with ReconFormer
Transformer and SS-MRI, assessing performance using normalized mean squared
error (NMSE), peak signal-to-noise ratio (PSNR), and structural similarity
index (SSIM). Statistical tests included one-way analysis of variance (ANOVA)
and multi-comparison Tukey’s Honesty Significant Difference (HSD) tests.

Results: ASSCGD preserved fine structures and brain abnormalities visually
better than comparative methods at $R=8\times$ for both multi-coil and single-
coil datasets. It achieved the lowest NMSE at $R\in\\{4\times,8\times\\}$, and
the highest PSNR and SSIM values at all acceleration rates for the multi-coil
dataset. Similar trends were observed for the single-coil dataset, though SSIM
values were comparable to ReconFormer at $R\in\\{2\times,8\times\\}$. These
results were further confirmed by the voxel-wise correlation scatter plots.
OOD results showed significant (p $\ll 10^{-5}$) improvements in undersampled
image quality after reconstruction.

Conclusions: ASSCGD successfully reconstructs fully sampled images without
utilizing them in the training step, potentially reducing imaging costs and
enhancing image quality crucial for diagnosis and treatment.

Adaptive Self-Supervised Consistency-Guided Diffusion Model for Accelerated
MRI Reconstruction
Mojtaba Safari1, Zach Eidex1, Shaoyan Pan1, Richard L.J. Qiu1, Xiaofeng Yang,
PhD1,‡
1Department of Radiation Oncology and Winship Cancer Institute, Emory
University, Atlanta, GA 30322, United States of America
‡Corresponding Author

Author to whom correspondence should be addressed. email:
<EMAIL_ADDRESS>

keywords: k-space sampling, fastMRI, accelerated MRI, reconstruction, adaptive
partitioning

##  1 Introduction

Magnetic resonance imaging (MRI) provides excellent soft tissue contrast,
playing a vital role in diagnosis, treatment, and follow-up. However, the
prolonged acquisition times can lead to patient discomfort and motion
artifacts, which compromise image quality. In addition, the recent Lancet
Oncology Commission highlighted severe MRI and other medical imaging device
shortages in low- and middle-income countries [1], potentially resulting in
2.5 million deaths worldwide [2]. Globally, only seven MRI scanners are
installed per million people, primarily due to the high cost of installation,
operation, and maintenance. In addition, the highly sampled k-space required
for high-resolution MRI images prohibitively increases acquisition time. Long
acquisition time reduces imaging throughput and increases patient discomfort
and motion artifacts [3].

The MRI acquisition can be accelerated by reducing the sampled k-space data,
but this is limited by the Nyquist criteria. Compressed sensing (CS) and
parallel imaging (PI) techniques aim to recover fully sampled images from
under-sampled images by exploiting data in a sparse transformed space and
redundant data acquired using uncorrelated radiofrequency coils, respectively
[4, 5]. However, at high acceleration rates, PI and CS methods suffer from
noise amplification [6] and residual artifacts [7], respectively.

Deep learning (DL) algorithms have been extensively used to reconstruct
accelerated high-resolution MRI images. DL-based compressed sensing-MRI (DL-
based CS-MRI) approaches are roughly divided into two categories: data-driven
and physics-guided [8]. The former maps the under-sampled k-space to the
fully-sampled k-space [9, 10, 11]. The latter incorporates the forward
encoding operator knowledge, such as the coil sensitivity and under-sampling
pattern, to solve ill-posed inverse problems. It is mainly divided into two
groups: unroll optimization method and the data consistency (DC) layer. The
former unrolls an iterative reconstruction approach where it alternates
between data consistency and regularization to solve the objective in a fixed
number of iterations [12, 13, 14]. The DC layer is an end-to-end way of
training the unroll models and usually is the latest layer of the networks.
[15, 16, 17].

These methods are typically trained under supervised frameworks where the
reference fully sampled images were utilized to train a model. However,
obtaining the fully sampled images might be impractical in imaging scenarios
such as cardiovascular MRI due to excessive involuntary movements, or
diffusion MRI with echo planar imaging due to quick T2^* signal decay [18].
Additionally, acquiring high-resolution anatomical brain MRI images can be
prohibitively long.

In this study, we propose an adaptive self-supervised consistency-guided
diffusion (ASSCGD) model to reconstruct fully sampled images without requiring
them in the training step. Our proposed ASSCGD model is based on an
adversarial mapper to reconstruct fully sampled MRI images. Our proposed
method was evaluated using both single-coil and multi-coil MRI data, as well
as two out-of-distribution (OOD) datasets. Our method leverages a recently
proposed ReconFormer transformer [19] as a generator and is compared with two
state-of-the-art models. Our contributions are as follows:

  * •

To our knowledge, ASSCGD is the first study proposing a self-supervised method
using an adversarial mapper.

  * •

The proposed method performs the backward diffusion process in smaller steps
that improve sampling efficiency.

  * •

The proposed method’s robustness against domain shift was evaluated at the
test time,

  * •

To our knowledge, It is the first self-supervised method aimed at
reconstructing fully sampled quantitative magnetization prepared 2 rapid
acquisition gradient echoes (MP2RAGE) T1 map,

##  2 Materials and Methods

###  2.1 Compressed sensing MRI

Let $y\in\mathbb{C}^{N}$ represent the observed subsampled k-space measurement
and $x\in\mathbb{C}^{M}$ is the unobserved fully sampled data. The compressed
sensing is formulated as follows:

$\centering y=\mathcal{A}_{\Omega}x+\delta\@add@centering$ (1)

where $\delta\in\mathbb{C}^{N}$ is the additive acquisition noise and
$\mathcal{A}_{\Omega}\in\mathbb{C}^{N\times M}\to\mathbb{C}^{N}$ give $N\ll M$
represents the encoding operator. $\mathcal{A}_{\Omega}$ is composed of a coil
sensitivity map $\mathcal{S}$, a Fourier transform $\mathcal{F}$, and a
sampling map with the specified pattern $\Omega$ controlling the acceleration
rate (R). The mathematical expression for the encoding operator is
$\mathcal{A}_{\Omega}=\Omega\odot\mathcal{F}\odot\mathcal{S}$. The MRI
reconstruction is formulated as an unconstrained optimization problem as
follows [14, 17, 20],

$\centering\underset{x,\varphi}{\arg\min}\parallel
x-f_{\text{ASSCGD}}(y\mid\varphi)\parallel_{2}^{2}+\lambda\parallel
y-\mathcal{A}_{\Omega}x\parallel_{2}^{2}\@add@centering$ (2)

where $f_{\text{ASSCGD}}$ represents our proposed model parametrized by
$\varphi$. The first and second terms represent the regularization and data
consistency, and $\lambda>0$ is a scalar regularization weight that balances
between the data consistency and regularization terms.

###  2.2 Diffusion model

The diffusion model inspired by nonequilibrium thermodynamics aims to
approximate complex and intractable distributions with a tractable one like
normal Gaussian [21]. It consists of two process: the forward process and the
reverse process.

#### Forward process:

The forward process utilizes a noise scheduler to add Gaussian noise to the
noise-free $y_{0}$ using a first-order Markov process $q(y_{t}|y_{t-1})$ in a
large number of steps $T$, eventually converting $y_{T}$ to a normal
multivariate Gaussian $y_{T}\sim\mathcal{N}(y_{T};\boldsymbol{0},\mathbf{I})$.
This step employs the first-order Markov process to calculate
$q(y_{t}|y_{t-1})$ as follows [22]:

$\displaystyle y_{t}=\sqrt{1-\beta_{t}}y_{t-1}+\sqrt{\beta_{t}}\epsilon$ (3)
$\displaystyle q(y_{t}\mid
y_{t-1})=\mathcal{N}(y_{t};\sqrt{1-\beta_{t}}y_{t-1},\beta_{t}\mathbf{I})$

where $\epsilon\sim\mathcal{N}(\epsilon;\boldsymbol{0},\mathbf{I})$ and
$\beta_{t}\in(0,1)$ with $\beta_{1}=10^{-4}$ is the noise variance. Assuming
additive Gaussian noise, sampling $y_{t}$ in an arbitrary step $t$ can be
calculated in closed form as follows:

$\centering q(y_{t}\mid
y_{0})=\mathcal{N}(y_{t};\sqrt{\bar{\alpha}_{t}}y_{0},(1-\bar{\alpha}_{t})\mathbf{I})\@add@centering$
(4)

where $\alpha_{t}=1-\beta_{t}$ and
$\bar{\alpha}_{t}=\prod_{s=1}^{t}\alpha_{s}$.

#### Reverse process:

The reverse process gradually learns to remove the added noise in $y_{T}$ to
recover noise-free $y_{0}$. This process of training a network $p_{\varphi}$
to generate $y_{0}$ from Gaussian noise $y_{T}$. The reverse process will
follow the forward process trajectories but in the reverse direction for small
$\beta$ values as follows [21, 23, 24]:

$\centering\underset{\varphi}{\arg\min}\sum_{t=2}^{T}\mathbb{E}_{q(y_{t}\mid
y_{0})}\left[D_{\mathbb{KL}}\left(q(y_{t-1}\mid y_{t},y_{0})\parallel
p_{\varphi}(y_{t-1}\mid y_{t})\right)\right]\@add@centering$ (5)

This is a denoising matching term where it learns the desired denoising
transition step $p_{\varphi}(y_{t-1}\mid y_{t})$ as an approximator to
tractable, ground-truth denoising transition step $q(y_{t-1}\mid y_{t},y_{0})$
given in (6), where it is modeled as a Gaussian.

$\displaystyle q(y_{t-1}\mid
y_{t},y_{0})=\mathcal{N}\left(x_{t-1};\mu_{q}(y_{t},y_{0}),\sigma_{q}(t)\mathbf{I}\right)$
(6)
$\displaystyle\mu_{q}(y_{t},y_{0})=\dfrac{\sqrt{\alpha_{t}}(1-\bar{\alpha}_{t-1})y_{t}+\sqrt{\bar{\alpha}_{t-1}}y_{0}}{1-\bar{\alpha}_{t}}$
$\displaystyle\sigma_{q}(t)=\dfrac{(1-{\alpha}_{t})(1-\bar{\alpha}_{t-1})}{1-\bar{\alpha}_{t}}$

Furthermore, all the $\alpha$ terms are frozen at each timestep, it was shown
that the loss function given in (5) becomes as follows:

$\centering\text{\L{}}=\underset{\varphi}{\arg\min}\dfrac{1}{2\sigma_{q}^{2}(t)}\left[\parallel\mu_{q}(y_{t},y_{0})-\mu_{\varphi}(y_{t})\parallel_{2}^{2}\right]\@add@centering$
(7)

where $\mu_{\varphi}(y_{t})$ is the estimated average recovered image as
follows:

$\mu_{\varphi}(y_{t})=\dfrac{\sqrt{\alpha_{t}}(1-\bar{\alpha}_{t-1})y_{t}+\sqrt{\bar{\alpha}_{t-1}}y_{\varphi}(y_{t})}{1-\bar{\alpha}_{t}}$
(8)

where $y_{\varphi}(y_{t})$ is parameterized by our DL model to recover $y_{0}$
from noisy image $y_{t}$ at a given step $t$.

We employed an adversarial mapper to implicitly capture the conditional
distribution for the reverse process steps. The generator $f_{\text{ASSCGD}}$
is used to sample $\hat{y}_{t}\sim p_{\varphi}(y_{t}\mid y_{t+k})$. At the
same time, a discriminator $\mathcal{D}_{\theta}$ differentiates between
samples $\hat{y}_{t}$ and actual sample $y_{t}$ sampled from the true
denoising distribution $q(y_{t}\mid y_{t+k})$. Our discriminator was coupled
with a gradient penalty to improve learning [15, 24]:

$\displaystyle L_{D}=$ $\displaystyle\sum_{t\geq
0}(\mathbb{E}_{q(y_{0},y_{t})}\mathbb{E}_{q(y_{t+k}\mid
y_{t})}\left[-\log\left(\mathcal{D}_{\theta}(y_{t},y_{t+k})\right)\right]$ (9)
$\displaystyle+\mathbb{E}_{q(y_{t+k})}\mathbb{E}_{\mathcal{N}_{\varphi}(\mu_{\varphi},\sigma_{\varphi})}\left[-\log(1-\mathcal{D}_{\theta}(\hat{y}_{t},y_{t+k}))\right]$
$\displaystyle+\mathbb{E}_{q(y_{0},y_{t})}\mathbb{E}_{q(y_{t+k}\mid
y_{t})}\left[\dfrac{1}{2}\parallel\nabla_{y_{t}}\mathcal{D}_{\theta}(\hat{y}_{t},y_{t+k})\parallel_{2}^{2}\right])$

where $\mathcal{N}_{\varphi}(\mu_{\varphi},\sigma_{\varphi})$ is our generator
parameterized by $\varphi$ to reconstruct mean and variance. The generator
loss becomes:

$\centering L_{G}=\sum_{t\geq
0}\mathbb{E}_{q(y_{t+k})}\mathbb{E}_{\mathcal{N}_{\varphi}(\mu_{\varphi},\sigma_{\varphi})}\left[-\log(\mathcal{D}_{\theta}(y_{t},y_{t+k}))\right]\@add@centering$
(10)

###  2.3 Proposed self-supervised framework and training details

Acquiring fully sampled data can be impractical due to constraints such as
voluntary and involuntary motions, lengthy acquisition times, and signal
decay. These constraints hinder the application of supervised DL-based CS-MRI
approaches. Thus, we proposed a self-supervised approach that randomly divided
the sampling pattern $\Omega$ into two sets $\aleph$ and $\Upsilon$ as given
in (11). These two sets have no elements in common except the center of
k-space.

$\centering\Omega=\aleph\vee\Upsilon\@add@centering$ (11)

We used an under-sampling pattern $\aleph$ to train our proposed model and
define our DC layer as follows.

$\centering\hat{Y}_{t}^{\text{ASSCGD}}(k)=\left\\{\begin{array}[]{c}\hat{Y}_{t}^{\text{ASSCGD}}(k),\,\,\,\,\text{if}\,\,\,\,k\in\aleph\\\
X(k),\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\,k\notin\aleph\end{array}\right.\@add@centering$
(12)

where capital letters refer to the Fourier transform of the corresponding
parameters. In other words, our method updates the k-space lines that were
under-sampled and keeps the original k-space lines that were not sampled
during image acquisition.

Finally, our proposed method defined the loss function given in (7) using a
sampling pattern $\Upsilon$ as follows:

$\centering\text{\L{}}^{\Upsilon}=\underset{\varphi}{\arg\min}\dfrac{1}{2\sigma_{q}^{2}(t)}\dfrac{(1-\alpha_{t})^{2}}{(1-\bar{\alpha}_{t})\alpha_{t}}\left[\parallel\epsilon-\hat{\epsilon}_{\varphi}^{\Upsilon}(y_{t})\parallel_{2}^{2}\right]\@add@centering$
(13)

The final loss function is composed of discriminator loss, generator loss, and
reconstruction loss as follows:

$L_{\text{final}}=\text{\L{}}^{\Upsilon}+\lambda(L_{D}+L_{G})$ (14)

where $\lambda=0.1$ was used to scale and control the ratio between the
losses. Our proposed method used the indices specified by $\aleph$ to
reconstruct k-space at indices given by $\Upsilon$. Figure 1 illustrates the
flowchart of our proposed method. Then the loss function was calculated at the
location indicated by the $\Upsilon$ pattern. In other words, our self-
supervised network was trained to decrease the discrepancy between the
predicted images $y^{\aleph}$ and the acquired measurement $y^{\Upsilon}$ that
was not seen in the training. At the inference step, the test unseen under
-sampled data using a sampling pattern $\Omega$ was used to reconstruct the
fully sampled data.

This self-supervised scenario is similar to the cross-validation concept used
to reduce bias and the likelihood of overfitting. Cross-validation partitions
the dataset into at least two sets where one set is used to train a model and
another set is used to evaluate the model. However, unlike cross-validation in
machine learning, which performs partitioning once for all data, our method
performs random partitioning per image slice.

In the inference step, we did not start from complete noise. Instead, we set
the Gaussian noise covariance to 0.1 and then used them directly as initial
inputs. This approach has been previously employed to reconstruct high-
resolution MR images from under-sampled k-space data and remove MRI motion
artifacts [25, 26].

We used the original implementation of the recently proposed transformer,
named ReconFormer111https://github.com/guopengf/ReconFormer, as a generator.
It incorporates the pyramid structures, enabling scale processing at each
pyramid unit, while the globally columnar structure maintains high-resolution
information [19]. The discriminator consisted of four convolution layers with
kernel size $(3\times 3)$ and padding one. Each convolution layer was followed
by a ReLu activation layer and a batch normalization layer.

The networks were trained using Adam optimizer with a learning rate $2\times
10^{-4}$ to minimize the loss function with batch size four over 25 epochs.
All training was performed on our server using PyTorch [27] framework version
2.1.2, and NVIDIA A100 GPUs with CUDA Toolkit 12.2 for all experiments.

Figure 1: The flowchart of our proposed self-supervised adaptive diffusion
model is illustrated. The coil sensitivity $\mathcal{S}$, random sampling mask
$\Omega$, and k-space were illustrated as inputs. Firstly, the random sampling
pattern $\Omega$ is divided into two non-overlapping sampling masks $\aleph$
and $\Upsilon$ that were used in the training path and loss path as given in
(13), respectively. Then, the adversarial mapper was trained as given in (9)
and (10) using the data sampled in step $t+k$. The sample $y_{t+k}$ used to
recover $\hat{y}_{0}$ where was used to calculate $\hat{y}_{t}$ in a given
step $t$ Equation (4).

###  2.4 Sampling masks

We employed a 1d random sampling pattern $\Omega$ where the k-space center was
excluded from sampling.The center fraction excluded from sampling was set to
$4\%$ of the k-space lines in the horizontal direction. Our proposed ASSCGD
divides the acquired sampling mask $\Omega$ into two disjoint sets $\aleph$
and $\Upsilon$ with except four k-space lines at the center of dataset. Figure
2 illustrates $\Omega$, $\aleph$, and $\Upsilon$ sampling masks for varying
$\rho\in{0.3,0.5,0.7}$ at $R=4\times$.

The sampling pattern $\Upsilon$ was randomly sampled for each different slice
to train the model. Therefore, our subsampled data using $\Upsilon$ will be
able to simulate the ghosting that is present in zero-filled datasets
retrospectively subsampled using $\Omega$ masks. We investigated a uniformly
random selection among elements of $\Omega$ to create subsampled patterns
$\aleph$ and $\Upsilon$. The sampling ratio
$\rho=\frac{\mid\aleph\mid}{\mid\Upsilon\mid}\in\\{0.3,0.5,0.7\\}$ was used to
train and test the proposed model, where $\mid\bullet\mid$ is the number of
elements. In other words, $\rho=0.3$ means 30% of $\Omega$ was randomly
assigned to $\aleph$ to train the model and the rest were assigned to
$\Upsilon$ sampling pattern to define loss function as given in Equation (13).

Figure 2: The sampling masks for three different splitting ratios
$\rho=\frac{\mid\aleph\mid}{\mid\Upsilon\mid}\in\\{0.3,0.5,0.7\\}$ at
$R=4\times$ are illustrated.

###  2.5 Dataset

We utilized publicly available single-coil and multi-coil brain MRI datasets.
Both datasets were retrospectively under-sampled using a random sample
provided in the fastMRI database, with acceleration rates (R) equal to 2, 6,
and 8 [28, 29].

####  2.5.1 Single-coil data:

The brain MPI-Leipzig Mind-Brain-Body dataset [30, 31] was used to train and
test our proposed model. We utilized high-resolution MP2RAGE T1 maps of 318
patients that were split into two non-overlapping trains and tested with 250
and 68 patients, respectively. The sagittal acquisition orientation of
volumetric MP2RAGE T1 maps with 176 slices were acquired with the imaging
parameters: T_R $=5000$ ms , T_E $=2.92$ ms, TI1$=700$ ms, TI2$=2500$ ms, FA_1
= $4^{\circ}$, FA_2 = $5^{\circ}$, pre-scan normalization, echo spacing $=6.9$
ms, bandwidth $=240$ Hz/pixel, FOV $=256$ mm, voxel size $=1$ mm3 isotropic,
GRAPPA acceleration factor 3, slice order = interleaved, duration= 8 min 22 s.

####  2.5.2 Multi-coil data:

The brain datasets used in the preparation of this article were obtained from
the NYU fastMRI Initiative database222fastmri.med.nyu.edu that was approved by
the NYU School of Medicine Institutional Review Board [28, 29]. We utilized
T2-weighted (T2-w) images of 1051 patients’ data for training and 325
patients’ unseen data for testing. In addition, we evaluated the robustness of
our model to domain shift using two out-of-distribution (OOD) unseen datasets,
T1-weighted (T1-w) and postcontrast T1-w (T1c), using 50 patients’ data for
each MRI sequence.

The sensitivity maps $\mathcal{S}$ were generated from $24\times 24$ center of
k-space using ESPIRiT 333https://mrirecon.github.io/bart/ [32] with a kernel
size $6\times 6$ and the calibration matrix and eigenvalue decomposition
threshold 0.02 and 0.95, respectively.

###  2.6 Quantitative analysis

To evaluate the performance of the proposed ASSCGD model, we compared it
against two benchmark models: SS-MRI [33] and ReconFormer [19], which were
trained under our proposed sampling approach.

Three quantitative metrics were employed: normalized mean square error (NMSE),
peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM),
utilizing the PyTorch Image Quality library
444https://piq.readthedocs.io/en/latest/index.html [34]. The NMSE compared
reconstructed $\hat{y}$ with ground truth $y$ as follows:

$\centering\text{NMSE}(y,\hat{y})=\dfrac{\parallel
y-\hat{y}\parallel_{2}^{2}}{\parallel y\parallel_{2}^{2}}\@add@centering$ (15)

where $\parallel\bullet\parallel_{2}$ is the squared Euclidean distance. Lower
NMSE values indicate better image reconstruction. However, it might be in
favor of blurry images [28].

The PSNR defined below utilizes a logarithmic scaling that makes the
quantification results more aligned with human perception [35].

$\centering\text{PSNR}(y,\hat{y})=\dfrac{y_{\max}^{2}}{\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2}}\@add@centering$
(16)

where $y_{\max}$ is the maximum signal intensity of ground truth images.
Higher PSNR indicates better image reconstruction.

The SSIM quantifies the structural similarity between the reconstructed
$\hat{y}$ and ground truth $y$ is defined as follows:

$\centering\text{SSIM}(y,\hat{y})=\dfrac{(2\mu_{y}\mu_{\hat{y}}+c_{1})(2\sigma_{y\hat{y}}+c_{2})}{(\mu_{y}^{2}+\mu_{\hat{y}}^{2}+c_{1})(\sigma_{y}^{2}+\sigma_{\hat{y}}^{2}+c_{2})}\@add@centering$
(17)

where $\mu_{y}$ and $\mu_{\hat{y}}$ are the average voxel values of $y$ and
$\hat{y}$ images, $\sigma_{y}$ $\sigma_{\hat{y}}$ are the variance, and
$\sigma_{y\hat{y}}$ is the covariance between $y$ and $\hat{y}$ images. The
constants $c_{1}$ and $c_{2}$ stabilize the division, we used
$c_{1}=0.01y_{\max}$ and $c_{2}=0.03y_{\max}$. SSIM ranges between -1 and +1,
with the best similarity achieved by an SSIM equal to +1.

###  2.7 Statistical analysis

The quantitative metrics were compared using one-way analysis of variance
(ANOVA) to evaluate the null hypothesis that the mean values of each method
were the same. The differences with p $<0.05$ was considered statistically
significant. Additionally, a multi-comparison Tukey’s honestly test difference
(HSD) was performed to evaluate pairwise differences between the methods, with
p $<0.05$ indicating statistical significance.

We reported the average values of the quantitative metrics. In addition, we
calculated the $95\%$ confidence intervals (CIs) on the average values using
the percentile bootstrap method (with $n=10000$ iterations) with the bias-
adjusted and accelerated bootstrap method [36].

##  3 Results

In this section, we present the comprehensive evaluation of our proposed
ASSCGD model for MRI reconstruction. The results are organized to highlight
the model’s performance across different datasets and under various
conditions.

We assess the within-domain multi-coil T2-w reconstruction and single-coil
MP2RAGE T1 map reconstruction capabilities of ASSCGD using both qualitative
and quantitative metrics. The model’s ability to preserve fine structures and
reduce artifacts is demonstrated through visual comparisons and statistical
analysis. The impact of loss mask partitioning on the performance of the
proposed method is demonstrated.

We also evaluate the robustness of ASSCGD against domain shifts using OOD
datasets. This analysis illustrates the model’s adaptability to new and unseen
data.

###  3.1 Within-domain reconstruction

Our proposed ASSCGD was evaluated for within-domain reconstruction at
$R=2\times$, $4\times$, and $8\times$ with a partitioning rate $\rho=0.5$. We
conducted both qualitative and quantitative comparisons with the ReconFormer
Transformer and SS-MRI models.

####  3.1.1 Qualitative Results

##### Multi-coil dataset:

Figures 3 and 4 present the qualitative results of within-domain T2-w multi-
coil images for two subjects. Figure 3(a) shows results for a healthy
volunteer, highlighting that our method recovers more details, particularly in
the gold boxed regions, compared to the ReconFormer model trained with our
framework. Difference maps in Figure 3(b) illustrate that our method better
preserves gray matter and edges, indicated by the gold and red boxes.

Figure 3: Within-domain axial T2-w image reconstruction at $R=2\times$,
$4\times$, and $8\times$ and $\rho=0.5$ are illustrated. (a) illustrates the
results for a healthy subject and (b) illustrates the difference map between
the reconstructed and ground truth images. Figure 4: Within-domain axial T2-w
image reconstruction at $R=2\times$, $4\times$, and $8\times$ and $\rho=0.5$
are illustrated. (a) illustrates the results for a subject with evident
abnormality shown by a red box and (b) illustrates the difference map between
the reconstructed and ground truth images.

Figure 4(a) displays results for a patient with brain abnormalities. Our
proposed method could recover the abnormality boundary shown by a red box
close to the ground truth. Furthermore, our method reconstructed details
better (gold boxes) compared with the SS-MRI method. Difference maps in Figure
4(b) confirm these observations.

##### Single-coil dataset:

Figure 5 presents the qualitative results for high-resolution single-coil
MP2RAGE T1 quantitative map for a healthy volunteer at $R=2\times$, $4\times$,
and $8\times$ with a partitioning rate $\rho=0.5$. Our method could preserve
the small structure shown by gold boxes as well as the Putamen and Caudate
nuclei shown by red boxes with better spatial contrast than the comparative
models, as shown in Figure 5(a) and the difference map shown in Figure 5(b).

Figure 5: Within-domain MP2RAGE T1 quantitative map reconstructed at
$R\in\\{2\times,4\times,8\times\\}$ and $\rho=0.5$ are illustrated. (a)
illustrates the results for a subject and (b) illustrates the difference map
between the reconstructed and the ground truth images.

####  3.1.2 Quantitative Results

Tables 1 and 2 summarize the quantitative metrics for multi-coil and single-
coil datasets, respectively, at different acceleration rates. The ANOVA test
indicated statistically significant differences (p $\ll 10^{-5}$) between the
average values of the methods. The results of the Tukey’s HSD multi-comparison
are presented in the text and tables.

##### Multi-coil dataset:

The quantitative results are listed in Table 1 at $R=2\times$, $4\times$, and
$8\times$ and $\rho=0.5$. The ANOVA test indicated p $\ll 10^{-5}$ for all
metrics indicating that there are statistically significant differences
between average values. Our method got the lowest NMSE for all acceleration
rates except for $R=2\times$ and $R=4\times$. The ReconFormer method achieved
the lowest NMSE value at $R=2\times$, nonetheless, it was not statistically
significantly different from our method (p $=0.65$). Although our method
achieved the lowest NMSE value for $R=8\times$, but it was not statistically
significantly different from ReconFormer (p $=0.83$). Our proposed self-
supervised method achieved the highest PSNR and SSIM values for all
acceleration rates, demonstrating the lowest remaining spatial distortion,
like ghosting, and the highest structural similarities between the ground
truth and reconstructed images, respectively.

Table 1: Within-domain performance for multi-coil axial T2-w fastMRI at
$R\in\\{2\times,4\times,8\times\\}$ and $\rho=0.5$ are provided. The arrows
indicate directions of better performance.

|  | Zero filled | Reconformer | SS-MRI | Ours
---|---|---|---|---|---
| 2$\times$ | 14.99 (12.95 - 17.14) | 0.47 (0.45 - 0.50)∗ | 0.52(0.50 - 0.56)∗ | 0.51 (0.48 - 0.54)
| 4$\times$ | 26.38 (24.43 - 28.41) | 1.51 (1.42 - 1.64) | 4.87 (4.76 - 4.98) | 1.26 (1.18 - 1.35)
NMSE (95% CI) [%] $\downarrow$ | 8$\times$ | 30.78 (28.49 - 33.11) | 3.34 (3.14 - 3.63)∗ | 5.16 (5.06 - 5.26) | 3.26 (3.08 - 3.51)
| 2$\times$ | 17.75 (17.64 - 17.86) | 38.80 (38.65 - 38.96) | 39.05 (38.90 - 39.20) | 39.93 (39.78 - 40.09)
| 4$\times$ | 17.64 (17.53 - 17.77) | 34.23 (34.10 - 34.37) | 27.93 (27.79 - 28.06) | 35.44 (35.31 - 35.58)
PSNR (95% CI) [dB] $\uparrow$ | 8$\times$ | 18.75 (18.62 - 18.87) | 27.54 (27.43 - 27.65) | 30.65 (30.52 - 30.76) | 31.67 (31.55 - 31.79)
| 2$\times$ | 81.53(81.06 - 81.98) | 96.10(96.02 - 96.17) | 96.33 (96.15 - 96.49) | 98.14 (98.06 - 98.21)
| 4$\times$ | 70.89(70.44 - 71.36) | 93.36(93.19 - 93.51) | 76.72 (76.32 - 77.15) | 95.55 (95.38 - 95.69)
SSIM (95% CI) [%] $\uparrow$ | 8$\times$ | 65.17(64.65 - 65.68) | 89.72 (89.49 - 89.93) | 77.01 (76.63 - 77.39) | 91.67 (91.45 - 91.89)
∗ indicates p-value $>0.05$ of Tukey’s HSD.

##### Single-coil dataset:

The quantitative metrics for the within-domain single-coil high-resolution
MP2RAGE T1 map are summarized in Table 2. One-way ANOVA tests indicated
significant differences (p $\ll 10^{-5}$) between methods for all metrics.
Tukey’s HSD test confirmed that ASSCGD performed significantly better than
comparative methods for most metrics. Our method achieved the lowest NMSE
values that were statistically different from the comparative methods for all
acceleration rates except with $R=2$ where our method achieved a performance
that did differ statistically significant (p-value $=0.97$) from ReconFormer.
Our method achieved the highest PSNR values for all acceleration rates that
were statistically significantly (p $<0.05$) from comparative methods.
Furthermore, our proposed method achieved the highest SSIM at $R=4\times$ that
differed statistically significantly higher (p $<0.05$) than the other
methods. Nonetheless, our method and ReconFormer performed similarly in terms
of the SSIM index at $R=2\times$ and $8\times$ with p of $0.15$ and $0.61$,
respectively.

Table 2: Within-domain performance for single-coil high resolution MP2RAGE T1
map at $R\in\\{2\times,4\times,8\times\\}$ and $\rho=0.5$ are provided. The
arrows indicate directions of better performance.

|  | Zero filled | Reconformer | SS-MRI | Ours
---|---|---|---|---|---
| 2$\times$ | 15.37 (13.46 - 17.49) | 0.53 (0.52 - 0.55)∗ | 5.18 (4.97 - 5.40) | 0.56 (0.54 - 0.57)
| 4$\times$ | 22.43 (20.65 - 24.33) | 2.04 (1.99 - 2.09) | 8.75 (8.36 - 9.18) | 1.87 (1.82 - 1.93)
NMSE (95% CI) [%] $\downarrow$ | 8$\times$ | 33.17 (31.79 - 34.72) | 3.99 (3.90 - 4.10) | 10.76 (10.42 - 11.13) | 3.65 (3.56 - 3.75)
| 2$\times$ | 15.46 (15.39 - 15.53) | 35.42(35.31 - 35.53) | 23.07(22.98 - 23.16) | 36.15(36.03 - 36.26)
| 4$\times$ | 15.35 (15.28 - 15.43) | 29.46(29.37 - 29.56) | 21.31(21.21 - 21.40) | 30.51(30.42 - 30.61)
PSNR (95% CI) [dB] $\uparrow$ | 8$\times$ | 16.31 (16.23 - 16.40) | 26.54(26.45 - 26.65) | 20.75(20.66 - 20.85) | 27.94(27.84 - 28.04)
| 2$\times$ | 73.83(73.43 - 74.21) | 95.75(95.68 - 95.81)∗ | 85.12(84.57 - 85.65) | 95.31(95.25 - 95.38)
| 4$\times$ | 68.62(68.29 - 68.94) | 89.11(88.96 - 89.26) | 80.47(80.06 - 80.89) | 89.80(89.63 - 89.95)
SSIM (95% CI) [%] $\uparrow$ | 8$\times$ | 66.13(65.83 - 66.42) | 84.55(84.32 - 84.78)∗ | 77.17(76.75 - 77.62) | 84.77(84.53 - 84.99)
∗ indicates p-value $>0.05$ of Tukey’s HSD.

####  3.1.3 Voxel-wise correlation

We visualized the scatter plot for 80 randomly selected within-domain multi-
coil and single-coil data to visualize and quantify the agreement between
reconstructed images at $R=8\times$ and $\rho=0.5$ with the ground truth
reference images (see Figure 6). For the single-coil dataset, our method
reconstructed images shown in Figure 6(c) with better conformity with the
references markedly better than SS-MRI shown in Figure 6(a) and comparable to
ReconFormer shown in Figure 6(b). The same results were achieved for multi-
coil data where the SS-MRI performed better than itself trained using the
single-coil dataset. Still there is a noticeable gap between our proposed (see
Figure 6(f)) method and SS-MRI (see Figure 6(d)). Furthermore, ground truth
voxel-wise shown in Figure 6(e) and (f) confirms that our method might be able
to generate images that are more similar to the ground truth.

Figure 6: The scatter plots between ground truth reference images and
reconstructed images at $R=8\times$ and $\rho=0.5$ for single-coil high-
resolution MP2RAGE T1 map and multi-coil axial T2-w are illustrated in the
first row and second row, respectively. Single-coil (a) SS-MRI, (b)
ReconFormer, and (c) Ours. Multi-coil (d) SS-MRI, (e) ReconFormer, and (f)
Ours.

####  3.1.4 Effect of Sampling Mask Ratio

Our proposed method split the sampling mask $\Omega$ randomly into two non-
overlapping masks $\aleph$ and $\Upsilon$, which were used in train and loss
paths, respectively (see Figure 1). The ratio
$\rho=\frac{\mid\aleph\mid}{\mid\Upsilon\mid}$ plays a role in the network’s
performance. We trained and tested the network for varying
$\rho\in\\{0.3,0.5,0.7\\}$ (see Figure 2 for the sampling masks). The effect
of network training with different $\rho$s on the quantitative metrics NMSE,
PSNR, and SSIM are shown in Figure 7 at three acceleration rates
$R\in\\{2,4,8\\}$ that were acquired using multi-coil axial T2-w images. Our
method achieved better performance at $\rho=0.5$ in terms of PSNR and SSIM at
$R=2\times$ and $R=4\times$. However, it performed similarly to $\rho=0.7$ at
$R=8\times$. Furthermore, NMSE metrics indicate that our method achieved the
best performance when trained using $\rho=0.5$ than other splitting ratios
$\rho\in\\{0.3,0.7\\}$.

Figure 7: The effect of network training with varying
$\rho=\frac{\aleph}{\Upsilon}$ is illustrated for within-domain multi-coil
axial T2-w fastMRI images.

###  3.2 Out-of-domain reconstruction

We evaluated our model performance in OOD reconstructions where our proposed
method was trained using multi-coil axial T2-w fastMRI images and then tested
using axial T1c and T1-w shown, respectively, in first and second rows of
Figure 8 at $R\in\\{2\times,4\times,8\times\\}$. Our method statistically
significant (p $\ll 10^{-5}$) improved quantitative metrics.

Our method achieved significantly lower NMSE values (p $\ll 10^{-5}$),
indicating high voxel-wise similarity between the reconstructed and ground
truth images for T1-w and T1c illustrated in Figures 8(a) and (d),
respectively. In addition, high PSNR values confirm minimal remaining spatial
distortion in the reconstructed images for T1-w and T1c illustrated in Figures
8(b) and (e), respectively. Finally, The high SSIM values demonstrate a
remarkable resemblance between the reconstructed and ground truth images (see
Figures 8(c) and (f)).

Figure 8: Out-of-distribution reconstruction quantitative results are
illustrated for multi-coil axial T1-w and T1c in the first and second rows,
respectively, for three difference acceleration rates.

##  4 Discussion

This study introduces the Adaptive Self-supervised Consistency-guided
Diffusion (ASSCGD) model, a novel self-supervised deep learning-based
compressed sensing MRI (DL-based CS-MRI) method designed to mitigate the
challenges associated with prolonged MRI acquisition times and the necessity
for fully sampled datasets. Our approach leverages the synergy between CS and
DL to accelerate MRI acquisition without compromising image quality.

The ASSCGD model’s primary innovation lies in its use of an adversarial mapper
within a self-supervised framework, eliminating the dependence on fully
sampled training datasets. This advancement is particularly significant for
clinical scenarios where acquiring fully sampled data is impractical. By
integrating a diffusion model, our method enhances sampling efficiency and
reconstruction quality. The backward diffusion process, executed in smaller
steps, contributes to robust and efficient image sampling. Extensive testing
on OOD datasets demonstrated significant improvements in NMSE, PSNR, and SSIM
metrics. These results highlight the model’s ability to generalize across
various MRI sequences and patient-specific conditions, underscoring its
versatility for clinical use.

The majority of DL-based CS-MRI methods use supervised learning to train
networks [14, 16, 17, 19, 25]. However, acquiring fully sampled can be
challenging in some practical applications due to the long acquisition time,
physiological constraints, and signal decay. For instance, fully sampled high-
resolution brain MP2RAGE T1 maps can take around 24 minutes to acquire, which
is impractical for large-scale studies and may lead to patient discomfort
unless using acceleration approaches. Such a long acquisition time increases
the likelihood of patient movements, which could substantially reduce the
image quality. Thus, being able to use self-supervised DL-based CS MRI
approaches is imperative to broaden their applications where acquiring such
data are challenging. By integrating a diffusion model, our method enhances
reconstruction quality. The backward diffusion process, executed in smaller
steps, contributes to robust and efficient image sampling. Extensive testing
on within domain and OOD datasets demonstrated significant improvements in
NMSE, PSNR, and SSIM metrics. These results highlight the model’s ability to
generalize across various MRI sequences and patient-specific conditions,
underscoring its versatility for clinical use.

Several self-supervised studies have been proposed to train models without
using fully sampled data. For instance, a data-driven method of de-aliasing
was proposed for single-coil data that performed an image-to-image translation
[37]. However, it did not encode the operator and used similar sampling
patterns for training and loss that increased noise during test time.
Alternatively, a self-supervised study was proposed, assuming data acquired
using two different sampling patterns [38]. Furthermore, physics-driven
methods that unroll the training process were also proposed [20]. However, the
unrolling nature of the method might increase the test burden and time.

Our self-supervised method uses a DC layer that was trained end-to-end and
utilizes the sampling pattern that is feasible to implement clinically. Our
experiment using the multi-coil and single-coil datasets enabled us to recover
target ground truth images at markedly lower acquisition times (see Figures 3
\- 5). Specifically, our method could recover fine details highlighted by gold
and red boxes better than the comparative models. The negligible amount of
remaining aliasing in the reconstructed images was confirmed by Figures 3 \-
5. Furthermore, this observation was confirmed by the low NMSE and high PSNR
achieved in Tables 1 and 2.

Although our method could achieve the best performance for $\rho=0.5$, Figure
7 suggests that our method might be resilient to variations in the hyper-
parameter $\rho$ around $\rho=0.5$. In addition, out-of-domain reconstruction
indicated in Figure 8 with significantly better performance (p $\ll 10^{-5}$)
indicates reasonable robustness against domain shifts. However, the results
indicate that our method performed better on T1-w (first row in Figure 8)
images than when tested on T1c (second row in Figure 8). That might be due to
contrast agent enhancement in T1c in the image regions that are not evident in
T2-w images.

Despite the promising outcomes, several limitations need to be addressed. Our
model was not tested on prospectively undersampled raw k-space datasets
acquired under parallel imaging frameworks. Future work should explore the
application of ASSCGD to these datasets to validate its clinical utility
further. In addition, we did not train our method using raw multi-coil high-
resolution 3D MRI, such as T1 magnetization-prepared rapid acquisition
gradient echoes MRI images because they are not readily available to the end-
user. . Expanding the training dataset to include such data could enhance the
model’s performance.

##  5 Conclusions

The Adaptive Self-supervised Consistency-guided Diffusion (ASSCGD) model
significantly improves the quality of reconstructed MRI images from
undersampled data without requiring fully sampled training datasets, offering
a promising solution for accelerating MRI acquisition. The proposed method has
the potential to enhance clinical MRI practices by reducing scan times and
improving image quality, which is crucial for accurate diagnosis and treatment
planning. Our method is particularly useful in radiation oncology by reducing
the imaging time and the likelihood of motion artifacts, specifically for MRI-
linear accelerators.

## Conflicts of interest

There are no conflicts of interest declared by the authors.

## Acknowledgement

This research is supported in part by the National Institutes of Health under
Award Number R56EB033332, R01EB032680, and R01CA272991.

## Data availability

Brain MRI data were obtained from the New York University fastMRI initiative
database (https://fastmri.med.nyu.edu/) and the OpenNeuro MPI-Leipzig Mind-
Brain-Body dataset (https://openneuro.org/datasets/ds000221/versions/00002).
The datasets were acquired with the relevant institutional review board
approvals.

## References

  * [1] Hedvig Hricak, May Abdel-Wahab, Rifat Atun, Miriam Mikhail Lette, Diana Paez, James A Brink, Lluís Donoso-Bach, Guy Frija, Monika Hierath, Ola Holmberg, et al.  Medical imaging and nuclear medicine: a lancet oncology commission.  The Lancet Oncology, 22(4):e136–e172, 2021.
  * [2] Yilong Liu, Alex TL Leong, Yujiao Zhao, Linfang Xiao, Henry KF Mak, Anderson Chun On Tsang, Gary KK Lau, Gilberto KK Leung, and Ed X Wu.  A low-cost and shielding-free ultra-low-field brain mri scanner.  Nature communications, 12(1):7238, 2021.
  * [3] Mojtaba Safari, Xiaofeng Yang, Chih-Wei Chang, Richard LJ Qiu, Ali Fatemi, and Louis Archambault.  Unsupervised mri motion artifact disentanglement: introducing maudgan.  Physics in Medicine and Biology, 2024.
  * [4] Justin P Haldar, Diego Hernando, and Zhi-Pei Liang.  Compressed-sensing mri with random encoding.  IEEE transactions on Medical Imaging, 30(4):893–903, 2010.
  * [5] Anagha Deshmane, Vikas Gulani, Mark A Griswold, and Nicole Seiberlich.  Parallel mr imaging.  Journal of Magnetic Resonance Imaging, 36(1):55–72, 2012.
  * [6] Ricardo Otazo, Daniel Kim, Leon Axel, and Daniel K Sodickson.  Combination of compressed sensing and parallel imaging for highly accelerated first-pass cardiac perfusion mri.  Magnetic resonance in medicine, 64(3):767–776, 2010.
  * [7] Yuchou Chang, Dong Liang, and Leslie Ying.  Nonlinear grappa: A kernel approach to parallel mri reconstruction.  Magnetic Resonance in Medicine, 68(3):730–740, 2012.
  * [8] Mojtaba Safari, Zach Eidex, Chih-Wei Chang, Richard LJ Qiu, and Xiaofeng Yang.  Fast mri reconstruction using deep learning-based compressed sensing: A systematic review.  arXiv preprint arXiv:2405.00241, 2024.
  * [9] Zhaoyang Jin and Qing-San Xiang.  Improving accelerated mri by deep learning with sparsified complex data.  Magnetic Resonance in Medicine, 89(5):1825–1838, 2023.
  * [10] Chentao Cao, Zhuo-Xu Cui, Yue Wang, Shaonan Liu, Taijin Chen, Hairong Zheng, Dong Liang, and Yanjie Zhu.  High-frequency space diffusion model for accelerated mri.  IEEE Transactions on Medical Imaging, 2024.
  * [11] Dominik Narnhofer, Kerstin Hammernik, Florian Knoll, and Thomas Pock.  Inverse gans for accelerated mri reconstruction.  In Wavelets and Sparsity XVIII, volume 11138, pages 381–392. SPIE, 2019.
  * [12] Andreas Kofler, Marie-Christine Pali, Tobias Schaeffter, and Christoph Kolbitsch.  Deep supervised dictionary learning by algorithm unrolling—application to fast 2d dynamic mr image reconstruction.  Medical Physics, 50(5):2939–2960, 2023.
  * [13] Biao Qu, Jialue Zhang, Taishan Kang, Jianzhong Lin, Meijin Lin, Huajun She, Qingxia Wu, Meiyun Wang, and Gaofeng Zheng.  Radial magnetic resonance image reconstruction with a deep unrolled projected fast iterative soft-thresholding network.  Computers in Biology and Medicine, 168:107707, 2024.
  * [14] Mojtaba Safari, Xiaofeng Yang, and Ali Fatemi.  Mri data consistency guided conditional diffusion probabilistic model for mr imaging acceleration.  In Medical Imaging 2024: Clinical and Biomedical Imaging, volume 12930, pages 202–205. SPIE, 2024.
  * [15] Alper Güngör, Salman UH Dar, Şaban Öztürk, Yilmaz Korkmaz, Hasan A Bedel, Gokberk Elmas, Muzaffer Ozbey, and Tolga Çukur.  Adaptive diffusion priors for accelerated mri reconstruction.  Medical Image Analysis, 88:102872, 2023.
  * [16] Maarten L Terpstra, Matteo Maspero, Joost JC Verhoeff, and Cornelis AT van den Berg.  Accelerated respiratory-resolved 4d-mri with separable spatio-temporal neural networks.  Medical physics, 50(9):5331–5342, 2023.
  * [17] Jo Schlemper, Jose Caballero, Joseph V Hajnal, Anthony N Price, and Daniel Rueckert.  A deep cascade of convolutional neural networks for dynamic mr image reconstruction.  IEEE transactions on Medical Imaging, 37(2):491–503, 2017.
  * [18] Kawin Setsompop, R Kimmlingen, E Eberlein, Thomas Witzel, Julien Cohen-Adad, Jennifer A McNab, Boris Keil, M Dylan Tisdall, P Hoecht, Peter Dietz, et al.  Pushing the limits of in vivo diffusion mri for the human connectome project.  Neuroimage, 80:220–233, 2013.
  * [19] Pengfei Guo, Yiqun Mei, Jinyuan Zhou, Shanshan Jiang, and Vishal M Patel.  Reconformer: Accelerated mri reconstruction using recurrent transformer.  IEEE transactions on medical imaging, 2023.
  * [20] Burhaneddin Yaman, Seyed Amir Hossein Hosseini, Steen Moeller, Jutta Ellermann, Kâmil Uğurbil, and Mehmet Akçakaya.  Self-supervised learning of physics-guided reconstruction neural networks without fully sampled reference data.  Magnetic resonance in medicine, 84(6):3172–3191, 2020.
  * [21] Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli.  Deep unsupervised learning using nonequilibrium thermodynamics.  In International conference on machine learning, pages 2256–2265. PMLR, 2015.
  * [22] Jonathan Ho, Ajay Jain, and Pieter Abbeel.  Denoising diffusion probabilistic models.  Advances in neural information processing systems, 33:6840–6851, 2020.
  * [23] Calvin Luo.  Understanding diffusion models: A unified perspective.  arXiv preprint arXiv:2208.11970, 2022.
  * [24] Stanley H Chan.  Tutorial on diffusion models for imaging and vision.  arXiv preprint arXiv:2403.18103, 2024.
  * [25] Jiahao Huang, Angelica I Aviles-Rivero, Carola-Bibiane Schönlieb, and Guang Yang.  Cdiffmr: Can we replace the gaussian noise with k-space undersampling for fast mri?  In International Conference on Medical Image Computing and Computer-Assisted Intervention, pages 3–12. Springer, 2023.
  * [26] Mojtaba Safari, Xiaofeng Yang, Ali Fatemi, and Louis Archambault.  Mri motion artifact reduction using a conditional diffusion probabilistic model (mar-cdpm).  Medical Physics, 51(4):2598–2610, 2024.
  * [27] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al.  Pytorch: An imperative style, high-performance deep learning library.  Advances in neural information processing systems, 32, 2019.
  * [28] Jure Zbontar, Florian Knoll, Anuroop Sriram, Tullie Murrell, Zhengnan Huang, Matthew J. Muckley, Aaron Defazio, Ruben Stern, Patricia Johnson, Mary Bruno, Marc Parente, Krzysztof J. Geras, Joe Katsnelson, Hersh Chandarana, Zizhao Zhang, Michal Drozdzal, Adriana Romero, Michael Rabbat, Pascal Vincent, Nafissa Yakubova, James Pinkerton, Duo Wang, Erich Owens, C. Lawrence Zitnick, Michael P. Recht, Daniel K. Sodickson, and Yvonne W. Lui.  fastMRI: An open dataset and benchmarks for accelerated MRI, 2018\.
  * [29] Florian Knoll, Jure Zbontar, Anuroop Sriram, Matthew J Muckley, Mary Bruno, Aaron Defazio, Marc Parente, Krzysztof J Geras, Joe Katsnelson, Hersh Chandarana, et al.  fastmri: A publicly available raw k-space and dicom dataset of knee images for accelerated mr image reconstruction using machine learning.  Radiology: Artificial Intelligence, 2(1):e190007, 2020.
  * [30] Anahit Babayan, Blazeij Baczkowski, Roberto Cozatl, Maria Dreyer, Haakon Engen, Miray Erbey, Marcel Falkiewicz, Nicolas Farrugia, Michael Gaebler, Johannes Golchert, Laura Golz, Krzysztof Gorgolewski, Philipp Haueis, Julia Huntenburg, Rebecca Jost, Yelyzaveta Kramarenko, Sarah Krause, Deniz Kumral, Mark Lauckner, Daniel S. Margulies, Natacha Mendes, Katharina Ohrnberger, Sabine Oligschläger, Anastasia Osoianu, Jared Pool, Janis Reichelt, Andrea Reiter, Josefin Röbbig, Lina Schaare, Jonathan Smallwood, and Arno Villringer.  ”mpi-leipzig-mind-brain-body”, 2018.
  * [31] Anahit Babayan, Miray Erbey, Deniz Kumral, Janis D Reinelt, Andrea MF Reiter, Josefin Röbbig, H Lina Schaare, Marie Uhlig, Alfred Anwander, Pierre-Louis Bazin, et al.  A mind-brain-body dataset of mri, eeg, cognition, emotion, and peripheral physiology in young and old adults.  Scientific data, 6(1):1–21, 2019.
  * [32] Martin Uecker, Peng Lai, Mark J Murphy, Patrick Virtue, Michael Elad, John M Pauly, Shreyas S Vasanawala, and Michael Lustig.  Espirit—an eigenvalue approach to autocalibrating parallel mri: where sense meets grappa.  Magnetic resonance in medicine, 71(3):990–1001, 2014.
  * [33] Chen Hu, Cheng Li, Haifeng Wang, Qiegen Liu, Hairong Zheng, and Shanshan Wang.  Self-supervised learning for mri reconstruction with a parallel network training framework.  In Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part VI 24, pages 382–391. Springer, 2021.
  * [34] Sergey Kastryulin, Jamil Zakirov, Denis Prokopenko, and Dmitry V. Dylov.  Pytorch image quality: Metrics for image quality assessment, 2022.
  * [35] Zach Eidex, Jing Wang, Mojtaba Safari, Eric Elder, Jacob Wynne, Tonghe Wang, Hui-Kuo Shu, Hui Mao, and Xiaofeng Yang.  High-resolution 3t to 7t adc map synthesis with a hybrid cnn-transformer model.  Medical Physics, 2024.
  * [36] Bradley Efron and Robert J Tibshirani.  Confidence intervals based on bootstrap percentiles.  In An introduction to the bootstrap, pages 168–177. Springer, 1993\.
  * [37] Ortal Senouf, Sanketh Vedula, Tomer Weiss, Alex Bronstein, Oleg Michailovich, and Michael Zibulevsky.  Self-supervised learning of inverse problem solvers in medical imaging.  In Domain Adaptation and Representation Transfer and Medical Image Learning with Less Labels and Imperfect Data: First MICCAI Workshop, DART 2019, and First International Workshop, MIL3ID 2019, Shenzhen, Held in Conjunction with MICCAI 2019, Shenzhen, China, October 13 and 17, 2019, Proceedings 1, pages 111–119. Springer, 2019.
  * [38] Jiaming Liu, Yu Sun, Cihat Eldeniz, Weijie Gan, Hongyu An, and Ulugbek S Kamilov.  Rare: Image reconstruction using deep priors learned without groundtruth.  IEEE Journal of Selected Topics in Signal Processing, 14(6):1088–1099, 2020.