These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Rizwan Ahmad

1]Biomedical Engineering, Ohio State University, Columbus, OH 43210, USA

2]Electrical and Computer Engineering, COMSATS University, Islamabad, Pakistan

3]Electrical and Computer Engineering, Ohio State University, Columbus, OH
43210, USA

# MRI Recovery with Self-Calibrated Denoisers without Fully-Sampled Data

Sizhuo Liu<EMAIL_ADDRESS>Muhammad Shafique<EMAIL_ADDRESS>Philip
Schniter<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [  [

###### Abstract

In many MRI applications, acquiring fully sampled training data is
challenging. We present a self-supervised image reconstruction method, termed
ReSiDe, capable of recovering images solely from undersampled data. ReSiDe is
inspired by plug-and-play (PnP) methods and employs a denoiser as a
regularizer. However, unlike traditional PnP approaches that utilize generic
denoisers or train application-specific denoisers using high-quality images or
image patches, ReSiDe directly trains the denoiser on the image or images that
are being reconstructed from the undersampled data. We introduce two
variations of our method: ReSiDe-S and ReSiDe-M. ReSiDe-S is scan-specific and
works with a single set of undersampled measurements, while ReSiDe-M operates
on multiple sets of undersampled measurements. More importantly, the denoisers
trained in ReSiDe-M are stored for faster PnP-based inference without further
training. To improve robustness, the denoising strength in ReSiDe is auto-
tuned using the discrepancy principle. Studies I, II, and III compare ReSiDe-S
and ReSiDe-M against other self-supervised or unsupervised methods using data
from T1- and T2-weighted brain MRI, MRXCAT digital perfusion phantom, and
first-pass cardiac perfusion, respectively. ReSiDe-S and ReSiDe-M outperform
other methods in terms of reconstruction signal-to-noise ratio and structural
similarity index measure for Studies I and II, and in terms of expert scoring
for Study III. In summary, we present a self-supervised image reconstruction
method and validate it in both static and dynamic MRI applications. These
developments can benefit MRI applications where the availability of fully
sampled training data is limited.

###### keywords:

self-supervised, MRI, reconstruction

##  1 Introduction

Magnetic resonance imaging (MRI) is a well-established diagnostic tool that
provides several advantages over other imaging modalities, including exquisite
soft-tissue contrast, high spatial and temporal resolution, multiple contrast
mechanisms, and radiation-free acquisition. MRI is routinely used in a broad
range of clinical applications, including neuro, musculoskeletal, abdominal,
and cardiovascular imaging. However, long scan times remain a challenge in MRI
as they can reduce patient comfort, make acquisition sensitive to motion, and
decrease patient throughout. Consequently, accelerating MRI has become a
highly active area of research, with various acquisition and processing
techniques being explored to reduce scan times [1].

In parallel MRI (pMRI), which is available on all commercial MRI scanners, the
data are acquired in parallel across multiple receive coils [2]. The resulting
multi-coil data, even when subsampled below the Nyquist rate, can be jointly
processed to reconstruct the underlying image. Typically, pMRI can speed up
the acquisition process by a factor of two to three. To achieve further
acceleration, pMRI can be combined with methods that utilize prior information
about the image. For example, compressed sensing (CS) leverages sparsity-based
priors and can be readily paired with pMRI [3]. The combination of pMRI and CS
yields acceleration rates higher than those of pMRI alone, and these
reconstruction methods are becoming increasingly available on commercial
scanners [4].

More recently, deep learning (DL) methods have been developed to reconstruct
images from highly undersampled MRI data. Several studies suggest that DL
methods can outperform sparsity-driven CS methods in terms of image quality
[5]. Most of the DL methods are based on supervised learning, where a
reconstruction network is trained on a large fully sampled training dataset
[6]. Outside of a handful of 2D applications, such training datasets are not
available. For other applications, e.g., cardiac imaging, collecting fully
sampled data may not be feasible [7]. Therefore, self-supervised DL (SSDL)
techniques have recently gained significant interest for MRI reconstruction
[8]. These techniques do not require fully sampled training datasets and
instead leverage the redundant information within the undersampled data to
guide the training process.

Several SSDL methods have been proposed for image denoising, including single-
instance deep generative prior methods such as deep image prior (DIP) and deep
decoder [9, 10]. These methods model an image as the output of a generator
network, with both network parameters and input code vectors trained on an
image-specific basis. Another popular SSDL denoising method is Noise2Noise
[11], which denoises images using two noisy copies of the same image. However,
acquiring multiple copies of an image is not practical for most MRI
applications. To address this issue, other SSDL-based denoising methods have
recently been proposed, including Noise2Void [12], Noise2Self [13], and
Self2Self [14], which operate on a single noisy image. These methods train a
network to predict a pixel from its neighboring pixels or predict one group of
pixels from another. Since noise is assumed to be independent across pixels
and thus difficult to predict, the network denoises the image by implicitly
learning the underlying structure in the image. For SSDL-based denoising, Xu
et al. took a different approach and proposed a method called Noisy-As-Clean
[15]. This method works by adding synthetic noise to the noisy images and
training a denoising network to remove the added noise. The trained network is
then used to denoise the images on which it was trained. In a separate
development, Stein’s unbiased risk estimator (SURE)-based loss was used for
unsupervised training of denoising networks [16].

The application of SSDL extends beyond image denoising, and many of these
methods can be applied to solve inverse problems, including image
reconstruction. For instance, DIP can readily solve inverse problems with a
known forward operator. Yoo et al. employed DIP for reconstructing dynamic
MRI, wherein a generative network was trained to map a low-dimensional
manifold to a series of dynamic images [17]. More recently, Bell et al.
introduced a more robust adaptation of DIP by training the network to function
as a denoiser instead of a generator [18]. In a separate study, Hamilton et
al. integrated low-rank constraint with DIP and applied it to free-breathing
cardiac imaging [19]. Scan-specific robust artificial neural network for
k-space interpolation (RAKI) is another self-supervised method proposed for
MRI reconstruction [20]. RAKI is similar to Noise2Void but operates in
k-space. In RAKI, a network is first trained on the fully sampled auto-
calibration signal (ACS) region and then used to predict missing k-space
samples from neighboring measured samples. Both RAKI and its recent extension,
called residual RAKI [21], can be viewed as nonlinear extensions of
traditional GRAPPA [22]. However, due to their scan-specific nature, DIP and
RAKI are computationally slow, which limits their translational potential. In
2020, Yaman et al. proposed self-supervised learning via data undersampled
(SSDU), a self-supervised learning method that resembles Noise2Self but with a
loss function defined in k-space [23]. In SSDU, the acquired undersampled
k-space is divided into two subsets, and an unrolled network is trained to
infer images from the first subset such that those images are consistent with
the second subset. At the inference stage, the trained network in SSDU can
rapidly map an undersampled, aliased image to a fully sampled image. More
recently, Millard and Chiew used the Noisier2Noise [24] framework to further
improve SSDU by ensuring that the sampling masks of the two subsets have
similar distributions [25]. Cole et al. [26] also proposed training a network
to map undersampled, aliased images to fully sampled images but with an
adversarial loss, where the discriminator is fed two unrelated undersampled
images: one from the image reconstruction network output and one from an
independent set of measurements. Finally, SURE-based loss has also been
recently used for unsupervised training of image reconstruction networks [27,
28].

In this work, we propose an SSDL method, called recovery with a self-
calibrated denoiser (ReSiDe), for image reconstruction. Our main contribution
involves integrating the self-supervised denoising scheme of Noisy-As-Clean
[15] with the PnP framework to solve the inverse problem in MRI
reconstruction. Additionally, we employ the discrepancy principle to
iteratively adapt the strength of the denoiser, providing faster convergence
and more control over the quality of the final reconstruction. Finally, we
propose a faster, more robust version of ReSiDe that is applicable to cases
where multiple undersampled sets of measurements are available. Using data
from brain MRI, MRXCAT perfusion phantom, and first-pass perfusion MRI, we
show that ReSiDe outperforms other self-supervised and unsupervised image
reconstruction methods. These developments significantly expand our
preliminary description of ReSiDe [29], which did not include auto-tuning, was
applicable only to a single set of measurements, and used only one T1 and one
T2-weighted image for validation.

##  2 Theory

In MRI, the data are sampled in the spatial frequency domain, called k-space,
and the MRI reconstruction entails estimating the underlying image from noisy
and potentially undersampled k-space measurements. The measured noisy k-space
data are related to the image via

$\boldsymbol{y}=\boldsymbol{A}\boldsymbol{x}+\boldsymbol{\eta},$ (1)

where $\boldsymbol{x}\in{\mathbb{C}}^{N}$ is a vectorized $N$-pixel image,
$\boldsymbol{y}\in{\mathbb{C}}^{M}$ is the pMRI data measured from $C$ receive
coils, $\boldsymbol{\eta}\in{\mathbb{C}}^{M}$ is circularly symmetric zero-
mean white Gaussian noise with variance $\sigma^{2}$, and
$\boldsymbol{A}\in{\mathbb{C}}^{M\times N}$ is a known forward operator that
subsumes coil sensitivity maps, discrete Fourier transform, and k-space
undersampling. In MRI, the value of $\sigma^{2}$ can be reliably estimated
from noise pre-scan, which typically takes a fraction of a second and can be
integrated with any pulse sequence.

To reduce acquisition time, the k-space data are often prospectively
undersampled to achieve an acceleration rate $R\triangleq\frac{CN}{M}>1$. At
high acceleration rates, the problem in Equation (1) becomes ill-posed. In
that case, a common remedy is to inject prior knowledge about $\boldsymbol{x}$
using a regularizer, resulting in the optimization problem of the form

$\hat{\boldsymbol{x}}=\operatorname*{arg\,min}_{\boldsymbol{x}}\frac{1}{\sigma^{2}}\|\boldsymbol{y}-\boldsymbol{A}\boldsymbol{x}\|^{2}_{2}+\mathcal{R}(\boldsymbol{x}),$
(2)

where the first term enforces data consistency and $\mathcal{R}(\cdot)$
represents the regularizer. For CS-based MRI reconstruction, popular choices
for $\mathcal{R}(\boldsymbol{x})$ include total variation [30] and
$\lambda\|\boldsymbol{\Phi}\boldsymbol{x}\|_{1}$, where $\boldsymbol{\Phi}$
represents a linear sparsifying transfrom and $\lambda>0$ controls the
regularization strength [3]. It has been shown that a simple sparsity-based
regularizer may not fully capture the rich structure in medical images [31].
To leverage more complex priors, Venkatakrishnan et al. proposed an
algorithmic framework called plug-and-play (PnP) [32]. In PnP, an off-the-
shelf image denoiser, $\boldsymbol{f}(\cdot)$, is called within an iterative
algorithm, e.g., primal-dual splitting (PDS) [33], for solving Equation (2). A
PDS-based implementation of PnP is given in Algorithm 1. In the subsequent
sections, we will use this algorithm as a starting point to explain ReSiDe.
Note, the implementation of PnP or ReSiDe is not limited to PDS and can be
carried out using other algorithms, including alternating direction method of
multiplier (ADMM) and fast iterative shrinkage and thresholding algorithm
(FISTA) [34].

Algorithm 1 PnP implemented using PDS

1:$\nu>0$, $\sigma$, $\boldsymbol{A}$, $\boldsymbol{y}$, $\boldsymbol{f}$,
$\gamma=\frac{\nu}{\sigma^{2}}\|\boldsymbol{A}\|_{2}^{2}$,
$\boldsymbol{x}_{0}=\boldsymbol{A}{{}^{\textsf{H}}}\boldsymbol{y}$,
$\boldsymbol{z}_{0}=\boldsymbol{A}\boldsymbol{x}_{0}-\boldsymbol{y}$

2:$\hat{\boldsymbol{x}}=\boldsymbol{x}_{T}$ % final reconstructed image

3:for $t=1,2,\dots,T$ do

4:
$\boldsymbol{u}_{t}=\boldsymbol{x}_{t-1}-\frac{\nu}{\sigma^{2}}\boldsymbol{A}^{\textsf{H}}\boldsymbol{z}_{t-1}$
% intermediate image

5: $\boldsymbol{x}_{t}=\boldsymbol{f}(\boldsymbol{u}_{t})$ % denoising

6:
$\boldsymbol{z}_{t}=\frac{\gamma}{1+\gamma}\boldsymbol{z}_{t-1}+\frac{1}{1+\gamma}(\boldsymbol{A}(2\boldsymbol{x}_{t}-\boldsymbol{x}_{t-1})-\boldsymbol{y})$

7:end for

###  2.1 ReSiDe-S

ReSiDe-S is a scan-specific technique that operates on a single set of
undersampled measurements, $\boldsymbol{y}$. A PDS-based implementation of
ReSiDe-S is given in Algorithm 2. ReSiDe-S differs from PnP in the way the
denoiser $\boldsymbol{f}(\cdot)$ is trained. The denoiser used in PnP is
either generic or pretrained using additional training dataset. In contrast,
the denoiser in ReSiDe-S is progressively trained from the image being
recovered. This is akin to blind compressed sensing [35], where both the image
and the sparsifying transform are simultaneously learned during the
reconstruction process. Following the work by Xu et al. [15], we propose
training a DL-based denoiser by adding synthetic noise to the image being
recovered. The denoiser training process is described in Line 5 of Algorithm
2. In summary, $\boldsymbol{u}_{t}$ is an intermediate image at iteration $t$,
and $\mathcal{I}_{i}[\boldsymbol{u}_{t}]$ represents the $i^{\text{th}}$ patch
from $\boldsymbol{u}_{t}$. For training purposes,
$\mathcal{I}_{i}[\boldsymbol{u}_{t}]$ and
$\mathcal{I}_{i}[\boldsymbol{u}_{t}]+\mathcal{N}(\boldsymbol{0},s_{t-1}^{2}\boldsymbol{I})$
act as “clean” and “noisy” patches, respectively. Here,
$\mathcal{N}(\boldsymbol{0},s_{t-1}^{2}\boldsymbol{I})$ represents complex-
valued zero-mean white Gaussian noise with variance $s_{t-1}^{2}$. The
denoiser $\boldsymbol{f}(\cdot;\boldsymbol{\theta})$, parameterized by
$\boldsymbol{\theta}$, is trained using $P\geq 1$ patches in a supervised
fashion by minimizing the loss $\mathcal{L}(\cdot,\cdot)$, which measures the
difference between the denoiser output and clean patches. Once the denoiser is
trained, it is then used to denoise the intermediate image
$\boldsymbol{u}_{t}$ (Line 6 in Algorithm 2), but this time without the added
noise. With pseudo-random or random sampling masks, the aliasing artifacts not
only have a noise-like appearance but also change from one image patch to
another. Therefore, the denoiser trained to remove synthetic noise also
removes some of the image artifacts, moving the intermediate image,
$\boldsymbol{x}_{t}$, closer to the true image, $\boldsymbol{x}$. The process
of training and applying the denoiser is repeated in each of $T$ iterations or
until some convergence criterion is reached.

The strength of the denoiser is controlled by $s_{t}^{2}$, which plays a role
similar to that of regularization strength in CS, i.e., larger $s_{t}^{2}$
leads to smoother images and smaller $s_{t}^{2}$ leads to noisier but sharper
images. As evident from our preliminary results [29], the value of $s_{t}^{2}$
also controls the speed of convergence, with larger values preferred in the
earlier iterations to speed up convergence and smaller values preferred in
later iterations to avoid over-smoothing of the recovered images. To adapt the
value of $s_{t}^{2}$ over iterations, we propose using Morozov’s discrepancy
principle [36, 37] (Line 8 and Line 9 in Algorithm 2). The discrepancy
principle is based on the assumption that the squared residual norm,
$\|\boldsymbol{A}\boldsymbol{x}_{t}-\boldsymbol{y}\|_{2}^{2}$, monotonically
increases with $s_{t}^{2}$, i.e., more aggressive denoising takes the image
away from the least-squares solution. By exploiting this monotonic
relationship, the discrepancy principle updates $s_{t}^{2}$ such that the
value of $\|\boldsymbol{A}\boldsymbol{x}_{t}-\boldsymbol{y}\|_{2}^{2}$
approaches $M\sigma^{2}$, which is the expected value of the squared
$\ell_{2}$ of $\boldsymbol{\eta}$. This is accomplished by using
$\frac{M\sigma^{2}}{\|\boldsymbol{A}\boldsymbol{x}_{t}-\boldsymbol{y}\|_{2}^{2}}$
as a multiplicative correction term, $c_{t}$. In each iteration, the value of
$s_{t}^{2}$ is multiplied with $c_{t}$ to promote the ratio
$\frac{M\sigma^{2}}{\|\boldsymbol{A}\boldsymbol{x}_{t}-\boldsymbol{y}\|_{2}^{2}}$
to be one. The value of $\alpha>0$ (Line 8 in Algorithm 2) controls the
contributions of the corrective term, with larger values leading to a more
rapid adjustment of $\sigma^{2}_{t}$. Optionally, a user-defined scalar
$\tau>0$ can be used to provide further control over the final strength of the
denoiser, with smaller values generating noisy but sharper images. Note,
adjusting $s_{t}^{2}$ directly is much more challenging as it needs to be
varied over iterations, while the fixed value of $\tau$ can be selected once
and then kept constant for a given MRI application.

Algorithm 2 ReSiDe-S

1:$\nu>0$, $\tau>0$, $\sigma$, $s_{0}^{2}$, $\boldsymbol{A}$,
$\boldsymbol{y}$, $\boldsymbol{f}$,
$\gamma=\frac{\nu}{\sigma^{2}}\|\boldsymbol{A}\|_{2}^{2}$,
$\boldsymbol{x}_{0}=\boldsymbol{A}^{\textsf{H}}\boldsymbol{y}$,
$\boldsymbol{z}_{0}=\boldsymbol{Ax}_{0}-\boldsymbol{y}$

2:$\hat{\boldsymbol{x}}=\boldsymbol{x}_{T}$ % final reconstructed image

3:for $t=1,2,\dots,T$ do

4:
$\boldsymbol{u}_{t}=\boldsymbol{x}_{t-1}-\frac{\nu}{\sigma^{2}}\boldsymbol{A}^{\textsf{H}}\boldsymbol{z}_{t-1}$
% intermediate image

5:
$\boldsymbol{\theta}_{t}=\operatorname*{arg\,min}_{\boldsymbol{\theta}}\frac{1}{P}\sum\limits_{i=1}^{P}\mathcal{L}\big{(}\boldsymbol{f}(\mathcal{I}_{i}[\boldsymbol{u}_{t}]+\mathcal{N}(\boldsymbol{0},s_{t-1}^{2}\boldsymbol{I});\boldsymbol{\theta}),\mathcal{I}_{i}[\boldsymbol{u}_{t}]\big{)}$
% denoiser training

6:
$\boldsymbol{x}_{t}=\boldsymbol{f}(\boldsymbol{u}_{t};\boldsymbol{\theta}_{t})$
% denoising

7:
$\boldsymbol{z}_{t}=\frac{\gamma}{1+\gamma}\boldsymbol{z}_{t-1}+\frac{1}{1+\gamma}(\boldsymbol{A}(2\boldsymbol{x}_{t}-\boldsymbol{x}_{t-1})-\boldsymbol{y})$

8: $c_{t}=\left(\frac{\tau
M\sigma^{2}}{\|\boldsymbol{A}\boldsymbol{x}_{t}-\boldsymbol{y}\|^{2}_{2}}\right)^{\alpha}$
% correction term

9: $s_{t}^{2}=c_{t}s_{t-1}^{2}$ % updating denoiser SNR

10:end for

###  2.2 ReSiDe-M

There are two major limitations of ReSiDe-S. First, it requires de novo
training of a denoising network in each iteration, which is time-consuming and
thus unrealistic for clinical deployment. Second, ReSiDe-S is a scan-specific
method and strictly operates on a single set of undersampled measurements;
however, for most MRI applications, more than one set of undersampled
measurements is generally available. To reduce computation time at the time of
inference and to leverage the availability of multiple sets of undersampled
measurements, we propose ReSiDe-M, which is outlined in Algorithm 3. Here, the
tilde annotation on the top of the symbol indicates it is a joint
representation of $K\geq 1$ sets of measurements. For example,
$\tilde{\boldsymbol{A}}$, $\tilde{\boldsymbol{y}}$, and
$\tilde{\boldsymbol{\sigma}}^{2}$, indicate the forward operator, k-space
measurements, and average noise variance, respectively, from $K$ sets of
measurements.

ReSiDe-M is implemented in two stages: training and inference. The training
stage is similar to ReSiDe-S, where both image recovery and denoiser training
happen in tandem. However, in contrast to ReSiDe-S, the denoiser training in
ReSiDe-M is performed using multiple undersampled datasets. For $K=1$, the
training phase of ReSiDe-M is identical to ReSiDe-S. Although the training
stage of ReSiDe-M can reconstruct images, its main objective is to store the
resulting sequence of denoisers, parameterized by
$\\{\boldsymbol{\theta}_{t}\\}_{t=1}^{T}$. Since the denoisers in ReSiDe-M are
trained on a more diverse, larger set of patches from multiple images, they
are expected to generalize better on unseen images without further training.
The second stage in ReSiDe-M is inference, where an unseen undersampled
dataset is reconstructed using a PnP algorithm, with the denoising in the
$t^{\text{th}}$ iteration of PnP (Line 15 in Algorithm 3) performed using the
denoiser $\boldsymbol{f}(\,\cdot\,;\boldsymbol{\theta}_{t})$ trained in the
first stage. The computational complexity of ReSiDe-M at the inference stage
is similar to that of sparsity-based iterative CS methods. A high-level
description of ReSiDe-M can be found in Figure 1.

Algorithm 3 ReSiDe-M

Training

1:$\nu>0$, $\tau>0$, $\widetilde{\sigma}^{2}$, $s_{0}^{2}$,
$\tilde{\boldsymbol{A}}$, $\tilde{\boldsymbol{y}}$, $\boldsymbol{f}$,
$\gamma=\frac{\nu}{\widetilde{\sigma}^{2}}\|\tilde{\boldsymbol{A}}\|_{2}^{2}$,
$\tilde{\boldsymbol{x}}_{0}=\tilde{\boldsymbol{A}}^{\textsf{H}}\tilde{\boldsymbol{y}}$,
$\tilde{\boldsymbol{z}}_{0}=\tilde{\boldsymbol{A}}\tilde{\boldsymbol{x}}_{0}-\tilde{\boldsymbol{y}}$

2:$\\{\boldsymbol{\theta}_{t}\\}_{t=1}^{T}$ % storing denoisers

3:for $t=1,2,\dots,T$ do

4:
$\tilde{\boldsymbol{u}}_{t}=\tilde{\boldsymbol{{x}}}_{t-1}-\frac{\nu}{\sigma^{2}}\tilde{\boldsymbol{A}}^{\textsf{H}}\tilde{\boldsymbol{z}}_{t-1}$
% intermediate image

5:
$\boldsymbol{\theta}_{t}=\operatorname*{arg\,min}_{\boldsymbol{\theta}}\frac{1}{P}\sum\limits_{i=1}^{P}\mathcal{L}(\boldsymbol{f}(\mathcal{I}_{i}[\tilde{\boldsymbol{u}}_{t}]+\mathcal{N}(\boldsymbol{0},s_{t-1}^{2}\boldsymbol{I});\boldsymbol{\theta}),\mathcal{I}_{i}[\tilde{\boldsymbol{u}}_{t}])$
% denoiser training

6:
$\tilde{\boldsymbol{x}}_{t}=\boldsymbol{f}(\tilde{\boldsymbol{{u}}}_{t};\boldsymbol{\theta}_{t})$
% denoising

7:
$\tilde{\boldsymbol{z}}_{t}=\frac{\gamma}{1+\gamma}\tilde{\boldsymbol{{z}}}_{t-1}+\frac{1}{1+\gamma}(\tilde{\boldsymbol{A}}(2\tilde{\boldsymbol{x}}_{t}-\tilde{\boldsymbol{x}}_{t-1})-\tilde{\boldsymbol{y}})$

8: $c_{t}=\left(\frac{\tau
M\widetilde{\sigma}^{2}}{\|\tilde{\boldsymbol{A}}\tilde{\boldsymbol{x}}_{t}-\tilde{\boldsymbol{y}}\|^{2}_{2}}\right)^{\alpha}$
% correction term

9: $s_{t}^{2}=c_{t}s_{t-1}^{2}$ % updating denoiser SNR

10:end forInference

11:$\nu>0$, $\sigma$, $\boldsymbol{A}$, $\boldsymbol{y}$, $\boldsymbol{f}$,
$\gamma=\frac{\nu}{\sigma^{2}}\|\boldsymbol{A}\|_{2}^{2}$,
$\boldsymbol{x}_{0}=\boldsymbol{A}^{\textsf{H}}\boldsymbol{y}$,
$\boldsymbol{z}_{0}=\boldsymbol{Ax}_{0}-\boldsymbol{y}$,
$\\{\boldsymbol{\theta}_{t}\\}_{t=1}^{T}$

12:$\hat{\boldsymbol{x}}\leftarrow\boldsymbol{x}_{t}$ % final reconstructed
image

13:for $t=1,2,\dots,T$ do

14:
$\boldsymbol{u}_{t}=\boldsymbol{x}_{t-1}-\frac{\nu}{\sigma^{2}}\boldsymbol{A}^{\textsf{H}}\boldsymbol{z}_{t-1}$
% intermediate image

15:
$\boldsymbol{x}_{t}=\boldsymbol{f}(\boldsymbol{u}_{t};\boldsymbol{\theta}_{t})$
% denoising

16:
$\boldsymbol{z}_{t}=\frac{\gamma}{1+\gamma}\boldsymbol{z}_{t-1}+\frac{1}{1+\gamma}(\boldsymbol{A}(2\boldsymbol{x}_{t}-\boldsymbol{x}_{t-1})-\boldsymbol{y})$

17:end for

Figure 1: A high-level description of ReSiDe-M. In the training stage (a), a
convolutional neural network (CNN)-based denoiser is trained on patches from
intermediate images $\tilde{\boldsymbol{u}}_{t}$ (Line 5 in Algorithm 3). The
resulting sequence of denoisers is stored. In the inference stage (b), the
reconstruction is performed using a PnP method, which conceptually alternates
between data consistency and denoising steps. The denoising in (b) is
performed by calling the denoisers from (a), without further training.

##  3 METHODS

###  3.1 Study I–Brain MRI

In this study, ReSiDe-S and ReSiDe-M were evaluated on T1- and T2-weighted
images from the fastMRI dataset [5]. For each contrast, twenty-two sets of
multi-coil measurements were used. All T1 images were cropped to $320\times
320$, and all T2 images were cropped to $384\times 304$. The multi-coil
k-space data were compressed to eight virtual coils [38]. The data were
retrospectively downsampled at $R=4$ using two realistic Cartesian sampling
masks, i.e., a 1D pseudo-random mask with a 32-line wide ACS region (M1) and a
1D random mask with a 32-line wide ACS region (M2). The M1 mask was kept fixed
across training and testing sets of measurements, while a different M2 mask
was randomly drawn for each training and testing instance. The coil
sensitivity maps were estimated using ESPIRiT [39]. Sixteen out of the 22 sets
of measurements were used for network training in ReSiDe-M and SSDU. Five
measurement sets were used for performance evaluation and for comparing
ReSiDe-S and ReSiDe-M with CS, PnP-BM3D, ConvDecoder [40], and SSDU [23]. The
CS reconstruction employed $\ell_{1}$ regularization in the wavelet domain and
was implemented using SigPy [41]. The PnP-BM3D reconstruction was implemented
using Algorithm 1, with $\boldsymbol{f}(\cdot)$ in Line 5 of Algorithm 1
representing a call to the BM3D denoiser [42].

For each method, one set of measurements was used for manual parameter tuning.
These parameters included: regularization strength, $\lambda$, for CS,
denoising level for BM3D in PnP-BM3D, number of iterations and input size for
ConvDecoder, cardinality of the loss mask and number of epochs for SSDU, and
parameters $\alpha$ and $\tau$, number and the size of patches, and total
number of iterations, $T$, for ReSiDe-S and ReSiDe-M. For CS, the
reconstruction process was repeated with the regularization set at $\lambda/2$
and $2\lambda$ to assess the impact of regularization strength on the image
quality.

###  3.2 Study II–MRXCAT Perfusion Phantom

Twenty-two sets of perfusion image series from the MRXCAT digital phantom were
considered in this study [43]. Each set of measurements was cropped to
$112\times 168$ pixels, with $32$ frames and $4$ receive coils. All data were
retrospectively downsampled at $R=4$ using a 1D pseudo-random Cartesian
sampling mask (M3) [44]. Due to the interleaving nature of M3, the ACS region
was not acquired for individual frames, and the fully sampled time-averaged
k-space was used to estimate coil sensitivity maps. To simulate realistic
data, circularly symmetric zero-mean white Gaussian noise was added to k-space
measurements to generate a signal-to-noise ratio of approximately 12 dB.
Sixteen sets of measurements were used to train ReSiDe-M, and five sets of
measurements were used for performance evaluation and for comparing ReSiDe-S
and ReSiDe-M with CS and PnP-BM4D. The CS reconstruction employed $\ell_{1}$
regularization in the spatiotemporal wavelet domain [45]. The PnP-BM4D
reconstruction was implemented using Algorithm 1, with $\boldsymbol{f}(\cdot)$
in Line 5 of Algorithm 1 representing a call to the BM4D denoiser [46]. As
described in the previous study, one set of measurements was used to manually
optimize free parameters in each method.

###  3.3 Study III–First-Pass Perfusion

This study included 22 first-pass perfusion image series from patients
clinically referred for a cardiac MRI exam at our institute. All measurements
were performed on a commercial 1.5T scanner (MAGNETOM Sola, Siemens
Healthcare, Erlangen, Germany) with a fast low angle shot (FLASH) sequence
using echo-planar imaging (EPI) readout. The data were collected in three
different views, i.e., short-axis, two-chamber, and four-chamber views. The
other imaging parameters were: flip angle 25 degrees, temporal footprint
$75.48$ to $99.36$ ms, matrix size $144\times 108$ to $144\times 144$, field
of view $360\times 270$ to $420\times 380$, echo train length $4$, echo
spacing $6.06$ to $6.29$ ms, slice thickness 8 to 10 mm, and a number of
frames 60. The images were prospectively undersampled in the $k_{x}$-$k_{y}$
domain with an acceleration rate of two and uniform undersampling that was
interleaved across time. Sixteen sets of measurements were used to train
ReSiDe-M, and five sets of measurements were used for performance evaluation
and comparison with PnP-BM4D and CS with $\ell_{1}$ regularization in the
spatiotemporal wavelet domain [45]. Again, one set of measurements was used to
manually optimize free parameters in all methods.

###  3.4 Quality Assessment

In Studies I and II, where the fully sampled reference was available, image
quality was assessed using the structural similarity index (SSIM) and
reconstruction SNR (rSNR) in dB, defined as
$20\log_{10}\left(\|\boldsymbol{x}\|_{2}/\|\boldsymbol{x}-\hat{\boldsymbol{x}}\|_{2}\right)$.
For Study III, where the fully sampled reference was not available, each
perfusion image series was blindly scored by three expert reviewers, including
two cardiologists, each with more than ten years of experience in cardiac MRI.
Each image series, presented as a movie, was scored on a five-point Likert
scale (1: non-diagnostic, 2: poor, 3: adequate, 4: good, 5: excellent) in
terms of overall image quality.

###  3.5 Implementation of ReSiDe

In Study I, randomly positioned $P=576$ patches and $P=2,\\!306$ patches were
used to train the denoiser in ReSiDe-S and ReSiDe-M, respectively. For
ReSiDe-M, the $2,\\!306$ patches were evenly distributed across the 16
training images. The patch size was fixed at $64\times 64$. For Studies II and
III, randomly positioned $P=288$ patches and $P=4,\\!608$ patches were used to
train the denoiser in ReSiDe-S and ReSiDe-M, respectively. For ReSiDe-M, the
$4,\\!608$ patches were again evenly distributed across the 16 training image
series. The patch size was fixed at $64\times 64\times 20$. The mean squared
error was used as a cost function to train the denoiser. The real and
imaginary parts were split into two channels. The architecture of the denoiser
is shown in Figure 2. Each convolutional layer had 128 kernels of size
$3\times 3$ for Study I and size $3\times 3\times 3$ for Studies II and III.
We used Adam optimizer with the learning rate $10^{-3}$ for Study I and
$10^{-4}$ for Studies II and III. The training in ReSiDe-M was performed on an
NVIDIA RTX 2080 Ti for Study I and an NVIDIA RTX 3090 for Studies II and III.
For Study I, the measurement noise variance $\sigma^{2}$ was estimated from
the outer fringes of k-space. For Study II, the noise was synthetically added;
therefore, the value of $\sigma^{2}$ was precisely known. For Study III, the
value of $\sigma^{2}$ was estimated from the noise pre-scan, which was
included in the raw data file. The value of $\tau$ was set at 0.65, 0.9, and
1.15 for Studies I, II, and III, respectively, and the value of $\alpha$ was
set at 0.1 for all studies. The value of $s_{0}^{2}$ was set such that the
initial SNR for the denoiser training was 5 dB. The number of iterations, $T$,
for ReSiDe-S and ReSiDe-M was set at 80 for all three studies. Within each
iteration, the denoiser was trained for a total of ten epochs. The code for
ReSiDe-S and ReSiDe-M is available here https://github.com/OSU-MR/reside

Figure 2: The denoiser architecture used in ReSiDe-S and ReSiDe-M. Here,
“conv” represents 2D convolutional layer for Study I and 3D convolutional
layer for Studies II and III.

##  4 Results

###  4.1 Study I–Brain MRI

Figure 3 shows a representative example where the values of rSNR and the
multiplicative correction term, $c_{t}$, are plotted as a function of the
number of iterations for ReSiDe-S. Figure 4 and Figure 5 display examples of
reconstructed T1- and T2-weighted images using undersampling masks M1 and M2,
respectively. The first row presents the true image obtained from fully
sampled k-space, alongside the images reconstructed by CS, PnP-BM3D,
ConvDecoder, SSDU, ReSiDe-S, and ReSiDe-M. The second row features two
magnified regions from the images in the first row. The arrows highlight areas
where visible artifacts or blurring is present in some of the reconstructed
images. In the third row, the leftmost panel shows the undersampling masks,
while the remaining panels depict error maps from various reconstructions
after a fivefold amplification. The top section of Table 1 summarizes rSNR and
SSIM values averaged over five T1- and T2-weighted images employing M1 and M2
masks. The rSNR/SSIM of CS reduced from $20.74/0.910$ to $20.36/0.894$ and
$20.39/0.899$ when the value of the regularization strength was changed to
$\lambda/2$ and $2\lambda$, respectively, indicating that the value of
$\lambda$ selected in CS was optimal or near-optimal.

Figure 3: Reconstruction SNR (rSNR) and the multiplicative correction term, $c_{t}$, as a function of iteration number, $t$. These representative curves originate from one of the T1-weighted images reconstructed using ReSiDe-S. Image (Smp.) | CS | PnP-BMXD | CD | SSDU | ReSiDe-S | ReSiDe-M
---|---|---|---|---|---|---
Brain T1 (M1) | $22.77/0.901$ | $\mathbf{24.86/0.963}$ | $14.99/0.766$ | $23.99/0.934$ | $24.39/0.952$ | ${24.60/0.958}$
Brain T1 (M2) | $19.65/0.868$ | $20.16/0.921$ | $14.01/0.740$ | $21.57/0.920$ | $\mathbf{23.35}/0.946$ | $23.29/\mathbf{0.950}$
Brain T2 (M1) | $21.60/0.948$ | $23.26/0.977$ | $14.13/0.804$ | $21.87/0.954$ | $25.45/0.981$ | $\mathbf{25.59/0.981}$
Brain T2 (M2) | $18.92/0.925$ | $19.69/0.947$ | $13.58/0.790$ | $19.83/0.941$ | $\mathbf{24.37}/0.978$ | $24.20/\mathbf{0.978}$
Brain Avg | $20.74/0.910$ | $21.99/0.952$ | $14.18/0.775$ | $21.81/0.937$ | $24.39/0.964$ | $\mathbf{24.42/0.967}$
MRXCAT (M3) | $22.56/0.877$ | $24.21/0.900$ | – | – | $25.89/0.918$ | $\mathbf{26.03/0.921}$
Table 1: Image quality metrics for Study I (top) and Study II (bottom). In
each cell, the first number represents rSNR in dB and the second number
represents SSIM, both averaged over five test samples. The best value in each
row is highlighted in bold font. The “Brain Avg” row represents the average of
the preceding four rows. CD represents ConvDecoder. BMXD represents BM3D for
Study I and BM4D for Study II. Figure 4: An example showing reconstruction of
a T1-weighted image with sampling mask M1. To highlight differences, the
second row magnifies two areas in the brain. The arrows point to features
where some of the methods show artifacts or blurring. The third row shows the
sampling mask (left) and the absolute error maps after fivefold amplification.
Figure 5: An example showing reconstruction of a T2-weighted image with
sampling mask M2. To highlight differences, the second row magnifies two areas
in the brain. The arrows point to features where some of the methods show
artifacts or blurring. The third row shows the sampling mask (left) and the
absolute error maps after fivefold amplification.

###  4.2 Study II–MRXCAT Perfusion Phantom

Figure 6 presents a representative frame from reconstructions of an MRXCAT
perfusion phantom. The first row displays the true image derived from fully
sampled k-space, as well as images reconstructed using CS, PnP-BM4D, ReSiDe-S,
and ReSiDe-M methods. The middle row shows a magnification of two selected
regions. The arrows emphasize details that are partially or entirely lost in
some of the reconstructed images. In the bottom row, the leftmost panel shows
the undersampling masks in phase encoding (vertical) and temporal (horizontal)
dimensions. The readout dimension, which is not shown, is fully sampled. The
remaining panels in the third row depict error maps after a fivefold
amplification. The last row in Table 1 summarizes rSNR and SSIM values
averaged over five MRXCAT datasets with M3 mask for CS, PnP-BM4D, ReSiDe-S,
and ReSiDe-M.

Figure 6: A representative frame from MRXCAT perfusion phantom reconstruction. The second row magnifies two areas of the phantom. The arrows point to features where some of the methods show blurring. The third row shows the sampling mask (left) in the phase-encoding (vertical) and temporal (horizontal) dimensions and the absolute error maps after fivefold amplification. |  CS |  PnP-BM4D |  ReSiDe-S |  ReSiDe-M
---|---|---|---|---
E1 |  $3.2$ |  $4.0$ |  $4.4$ |  $\mathbf{5.0}$
E2 |  $2.0$ |  $3.0$ |  $4.0$ |  $\mathbf{4.6}$
E3 |  $2.0$ |  $3.4$ |  $3.6$ |  $\mathbf{3.6}$
Avg |  $2.4$ |  $3.5$ |  $4.0$ |  $\mathbf{4.4}$
Table 2: Image quality scores from three expert reviewers (E1, E2, and E3) on
a five-point Likert scale (5: best, 1: worst), averaged over five perfusion
image series.

###  4.3 Study III–First-Pass Perfusion

Figure 7 and Figure 8 show representative frames from two different first-pass
perfusion image series. Reconstructions from CS, PnP-BM4D, ReSiDe-S, and
ReSiDe-M are shown. The top row shows the entire frame, and the bottom row
displays two magnified regions from the images in the first row. The arrows
highlight the details that are partially lost in some of the reconstructed
images. In Figure 7, the cyan arrows point to the leaflets of the mitral
valve. In Figure 8, the red arrows point to the papillary muscles in the left
ventricle. For CS, PnP-BM4D, ReSiDe-S, and ReSiDe-M, Table 2 provide the image
quality scores averaged over five image series from three cardiac MRI experts,
including two cardiologists.

Figure 7: A representative two-chamber frame from one of the first-pass
perfusion image series. The first row shows the entire frame, while the second
shows two magnified areas from the frame. The visible loss of detail in CS is
highlighted with arrows. Figure 8: A representative short-axis frame from
another first-pass perfusion image series. The first row shows the entire
frame, while the second shows two magnified areas from the frame. The visible
loss of detail in CS is highlighted with arrows.

##  5 DISCUSSION

In this work, we present an SSDL method, called ReSiDe, for MRI
reconstruction. Like PnP methods, ReSiDe integrates a denoiser into the
reconstruction process. However, PnP methods use pretrained denoisers, while
ReSiDe iteratively trains the denoiser on the images being recovered. We
present two variations of ReSiDe, i.e., ReSiDe-S and ReSiDe-M. ReSiDe-S is
scan-specific and utilizes only a single set of undersampled measurements. The
necessity to train the network in each iteration makes ReSiDe-S
computationally slow. In contrast, ReSiDe-M operates on multiple sets of
undersampled measurements and trains the denoiser on patches from multiple
images. More importantly, due to its training on multiple sets of images, the
denoiser in ReSiDe-M generalizes to unseen images. The denoisers trained in
ReSiDe-M are stored and then utilized in a PnP algorithm without further
training. The computation burden of ReSiDe-M after the training stage is
comparable to CS-based iterative methods.

Validation of ReSiDe-S and ReSiDe-M is carried out in three datasets, i.e.,
T1- and T2-weighted images from fastMRI (Study I), digital perfusion image
series from MRXCAT (Study II), and first-pass perfusion data collected from
patients (Study III). In Study I, we compare ReSiDe-S and ReSiDe-M with other
methods that do not require fully sampled data, including CS, PnP-BM3D,
ConvDecoder, and SSDU. As summarized in Table 1, ReSiDe-M and ReSiDe-S
outperform competing methods in terms of rSNR and SSIM, with more than 2 dB
improvement over PnP-BM3D and SSDU. All methods perform better with the M1
mask compared to the M2 mask. This is not surprising because M1 is pseudo-
random and thus avoids large sampling gaps in k-space. Also, M1 is kept
consistent between training and testing, while a different M2 mask is used for
each training and testing instance. Nonetheless, the smaller performance gap
between M1 and M2 for ReSiDe highlights its ability to generalize when the
sampling patterns between training and testing are different. Two examples of
reconstructed images are shown in Figure 4 and Figure 5. Compared to PnP-BM3D,
ConvDecoder, and SSDU, both ReSiDe-S and ReSiDe-M exhibit fewer artifacts
while preserving fine details. The artifacts are particularly pronounced in
ConvDecoder. Despite our best efforts to optimize the code provided by the
original authors here [47], we were unable to improve the performance of
ConvDecoder. The performance of CS was inferior to ReSiDe-S and ReSiDe-M by
more than 3 dB. To explore if this performance gap is due to the suboptimal
selection of the regularization strength in CS, we repeated the CS
reconstruction at lower ($\lambda/2$) and higher ($2\lambda$) regularization
strengths. However, the performance of CS degraded for both those choices,
indicating that the inferior performance of CS cannot be attributed to
suboptimal selection of $\lambda$.

The rSNR and SSIM numbers from Study II followed a trend similar to that of
Study I, with both ReSiDe-S and ReSiDe-M outperforming other methods. Figure 6
shows a representative frame. As shown in the error map, ReSiDe-S and ReSiDe-M
preserve edge information more effectively, whereas CS and PnP-BM4D show
structured errors around the edges. The performance gap is more than 1.5 dB
between ReSiDe and PnP-BM4D and more than 3 dB between ReSiDe and CS.

In Study III, where the image series were subjectively evaluated by three
expert readers, ReSiDe-M consistently outperformed other methods, with
ReSiDe-S being the second best. The example images provided in Figure 7 and
Figure 8 illustrate that the ReSiDe methods are more effective in preserving
small details, e.g., mitral valve leaflets or papillary muscles, especially
compared to CS.

The performance difference between ReSiDe-S and ReSiDe-M, in terms of rSNR and
SSIM, was marginal in Studies I and II. However, we observed that ReSiDe-M
images have more texture and appear sharper than those from ReSiDe-S. We
conjecture that this behavior is related to more expansive training of the
denoiser in ReSiDe-M. In addition to the superior image quality, ReSiDe-M also
offers a computation advantage over ReSiDe-S and many other SSDL methods. For
example, in Study I, ReSiDe-S and ReSiDe-M trainings took 36 minutes and 140
minutes, respectively. However, at the inference stage, the reconstruction
from ReSiDe-M took only 11 seconds per image. In comparison, the training and
inference from SSDU took 127 minutes and 3.3 seconds, respectively. The
inference time for ReSiDe-M for Studies II and III was 25 and 37 seconds,
respectively, per image series. In comparison, PnP-BM4D took 40 minutes and
110 minutes for each image series in Studies II and III, respectively.

The performance of ReSiDe depends on $s_{t}^{2}$, which controls the training
SNR of the denoiser. We employ the discrepancy principle to adapt the value of
$s_{t}^{2}$. Figure 3 shows representative curves for rSNR and $c_{t}$ for one
the T1-weighted images. As expected, the value of the corrective term,
$c_{t}$, converges to one after approximately 80 iterations, which implies
that $s_{t}^{2}$ also converges to a fixed value. Likewise, the rSNR reaches
it maximum value at around 60 iteration mark, with marginal or no drop-off in
rSNR for iterations beyond 60. Similar tends were observed for other datasets
in Studies I, II, and III. The value of $\alpha$ had a marginal impact on the
final rSNR but it did affect the rate of convergence. In particular, larger
values of $\alpha$ led to faster convergence but with rSNR and $c_{t}$ curves
showing more oscillations, especially in the earlier iterations. Therefore,
the value of $\alpha$ was conservatively set at 0.1.

This study has several limitations. First, our current implementation requires
saving a denoiser in each iteration of the training process. Saving a large
number of denoisers can be memory intensive, especially if larger networks are
employed. Future work could explore saving the denoisers less frequently,
e.g., after every tenth iteration. Second, we have used a denoiser
architecture that is based on the residual learning approach proposed in 2017
[48]. It is possible that other network architectures can further improve the
performance of ReSiDe. Third, although using the discrepancy principle
eliminates the need to precisely schedule the denoiser strength [29], both
ReSiDe-S and ReSiDe-M still require selecting an appropriate value for $\tau$.
For the studies presented, we manually selected one value of $\tau$ for each
application. It is not clear whether this value of $\tau$ will remain
reasonable if imaging parameters, e.g., spatial resolution or measurement SNR,
change significantly within each application. However, ReSiDe shares this
limitation with CS and PnP methods, which also require selecting the
regularization or denoising strength. Fourth, we observed ReSiDe-S and
ReSiDe-M to converge to a fixed point after approximately 80 iterations for
all three represented studies. However, a more formal convergence analysis for
ReSiDe is currently missing and beyond the scope of this proof-of-concept
work.

##  6 CONCLUSION

We have presented two self-supervised methods for MRI reconstruction: ReSiDe-S
and ReSiDe-M. ReSiDe-S offers a scan-specific implementation where a single
set of undersampled measurements is used for denoiser training and image
recovery. In contrast, ReSiDe-M trains a denoiser from multiple undersampled
measurements and utilizes that denoiser during inference without further
training. Our validation studies, which used data from brain MRI, perfusion
phantom, and first-pass perfusion, demonstrate that ReSiDe-S and ReSiDe-M
outperform other self-supervised or unsupervised methods in terms of both
qualitative and quantitative metrics. Compared to ReSiDe-S, ReSiDe-M also
offers slightly better image quality and much faster inference.

Acknowledgments This work was funded by NIH grants R01-EB029957, R01-HL151697,
and R01-HL135489.

## Declarations

Ethics approval For the human subject data, approval was granted by the
Institutional Review Board at The Ohio State University (2019H0076).

Consent to participate Informed consent was obtained from all individual
participants included in the study.

Competing interests The authors declare no competing interests.

Code availability The code for ReSiDe-S and ReSiDe-M can downloaded from here
https://github.com/OSU-MR/reside

Competing interests The authors declare no competing interests.

Authors’ contributions S. Liu coded ReSiDe and drafted the first version of
the manuscript. M. Shafique optimized ReSiDe and created the final figures and
tables. P. Schniter contributed by providing valuable feedback for optimizing
ReSiDe and assisted with manuscript preparation. R. Ahmad supported the
project by assisting with data acquisition, troubleshooting ReSiDe, study
design, and manuscript preparation.

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