# CycleQSM: Unsupervised QSM Deep Learning using Physics-Informed CycleGAN

Gyutaek Oh, Hyokyoung Bae, Hyun-Seo Ahn, Sung-Hong Park, and Jong Chul Ye G.
Oh, H. Bae, H-S. Ahn, S-H. Park, and J. C. Ye are with the Department of Bio
and Brain Engineering, Korea Advanced Institute of Science and Technology
(KAIST), Daejeon 34141, Republic of Korea (e-mail: {okt0711, ippuny3, hyunsea,
sunghongpark, jong.ye}@kaist.ac.kr). J.C. Ye is also with the Department of
Mathematical Sciences, KAIST.

###### Abstract

Quantitative susceptibility mapping (QSM) is a useful magnetic resonance
imaging (MRI) technique which provides spatial distribution of magnetic
susceptibility values of tissues. QSMs can be obtained by deconvolving the
dipole kernel from phase images, but the spectral nulls in the dipole kernel
make the inversion ill-posed. In recent times, deep learning approaches have
shown a comparable QSM reconstruction performance as the classic approaches,
despite the fast reconstruction time. Most of the existing deep learning
methods are, however, based on supervised learning, so matched pairs of input
phase images and the ground-truth maps are needed. Moreover, it was reported
that the supervised learning often leads to underestimated QSM values. To
address this, here we propose a novel unsupervised QSM deep learning method
using physics-informed cycleGAN, which is derived from optimal transport
perspective. In contrast to the conventional cycleGAN, our novel cycleGAN has
only one generator and one discriminator thanks to the known dipole kernel.
Experimental results confirm that the proposed method provides more accurate
QSM maps compared to the existing deep learning approaches, and provide
competitive performance to the best classical approaches despite the ultra-
fast reconstruction.

###### Index Terms:

Quantitative susceptibility mapping, unsupervised deep learning, optimal
transport, cycleGAN

##  I Introduction

Quantitative susceptibility mapping (QSM) is a magnetic resonance imaging
(MRI) technique that measures magnetic susceptibility values from MRI phase
images [1, 2]. QSM offers image contrast that is sensitive to iron deposition,
which is useful for the diagnosis of various diseases [3, 4, 5, 6].

From a mathematical point of view, a phase image can be expressed by the
convolution between the magnetic susceptibility distribution and the dipole-
shaped kernel. Thus, a magnetic susceptibility distribution can be obtained by
deconvolution: for example, dividing the phase by the dipole kernel in the
Fourier domain. However, the dipole kernel in the Fourier domain has zeros
along the conical surface, which makes the dipole inversion an ill-posed
inverse problem.

To overcome this limitation, various dipole inversion approaches have been
investigated. Calculation of susceptibility through multiple orientation
sampling (COSMOS) [7] is considered as the gold standard algorithm for the
dipole inversion. COSMOS restores accurate QSM from multiple head orientation
data. Unfortunately, the acquisition of MR data along multiple head
orientations for COSMOS reconstruction takes too much subject’s time and
effort.

Accordingly, algorithms have been developed for dipole inversion from single
head orientation data. For example, filling the area around the conical
surface with a threshold value is one of the method for dipole inversion [8,
9]. Also, the regularization by edge information of the magnitude image can
mitigate the ill-posedness of dipole inversion [10, 11]. In addition, dipole
inversion algorithms based on compressed sensing [12, 13] have been studied.
However, the streaking artifact in the reconstructed QSM and the difficulty of
hyper-parameter tuning are limitations of the above algorithms.

Recently, inspired by the successes of deep learning approaches for medical
image reconstruction [14, 15, 16, 17], deep learning algorithms have also been
extensively studied for QSM reconstruction [18, 19, 20, 21, 22, 23, 24]. These
algorithms show comparable performance as the classical methods despite the
fast computational time. However, most of the existing deep learning methods
for QSM reconstruction are based on supervised learning which requires matched
pairs of phase images and ground-truth QSM labels. Nonetheless, it has been
reported that the reconstructed QSM values are often underestimated [25].

To address this, here we propose a novel physics-informed cycleGAN approach
for unsupervised QSM reconstruction, inspired by our recent theory of optimal
transport driven cycleGAN (OT-cycleGAN) [26, 27, 28]. Since the forward
mapping from the susceptibility map to the phase image is given as a
convolution with the known dipole kernel, our mathematical theory [26] leads
to a simple OT-cycleGAN architecture composed of a single pair of generator
and discriminator. The resulting architecture, dubbed CycleQSM, exhibits
faster and more stable training than the original cycleGAN composed of two
sets of deep learning based generator and discriminators. Furthermore, it
provides better performance than the existing deep learning algorithms, and
results in the comparable performance to the best available classical
approach.

This paper is organized as follows. Section II reviews the classical and deep
learning approaches for QSM reconstruction. Section III then describes the
physics for dipole inversion and proposes physics-informed OT-cycleGAN for QSM
reconstruction. Next, experimental data sets and details of our method are
presented in Section IV. Our experimental results and the discussion are shown
in Section V and Section VI, respectively. Finally, Section VII concludes our
work.

##  II Related Works

###  II-A Classical Methods for QSM Reconstruction

Various methods have been developed to solve the ill-posed dipole inversion
problem. For example, thresholded $k$-space division (TKD) [8, 9] is a method
of filling values near the conical surface of the dipole kernel with a
threshold value. Instead of filling the dipole kernel with a specific value,
some methods used additional information for the dipole inversion. Schweser et
al. [29] proposed homogeneity enabled incremental dipole inversion (HEIDI)
using homogeneity information from gradient echo phase images. Morphology
enabled dipole inversion (MEDI) [10, 11] is a method that exploits the
structural consistency between the magnitude image and the reconstructed
quantitative susceptibility map. Li et al. [30] introduced a method based on a
sparse linear equation and least squares algorithm (iLSQR) to remove streaking
artifacts in reconstructed QSM. Compressed sensing [31, 32] based algorithms
were also examined. Annihilating filter-based low-rank Hankel matrix approach
(ALOHA) for QSM [13] exploits the sparsity of the data in a certain transform
domain to interpolate the missing $k$-space data in the dipole spectral null.

Although above algorithms showed high performance, there are limitations of
the above algorithms, such as streaking artifacts, difficulties in optimizing
hyper-parameters, and high computational complexity.

###  II-B Deep Learning Methods for QSM Reconstruction

To overcome the limitations of traditional methods, QSM reconstruction
algorithms based on deep learning have been investigated. Yoon et al. [18] and
Bollman et al. [19] proposed QSMnet and deepQSM, respectively, which are 3D
U-net [33] structures designed for QSM reconstruction. By using of COSMOS
images as ground-truth QSM labels, they showed comparable results to those of
classical approaches. Chen et al. [20] suggested QSMGAN where the adversarial
loss was utilized, and Gao et al. [21] introduced xQSM where the octave
convolutional layers were applied. Polak et al. [24] also introduced nonlinear
dipole inversion with deep learning using a variational neural network, which
combines optimization of nonlinear QSM data model and the data fidelity term.

Although these methods have shown improved image quality compared to
conventional methods, the requirement of matched pairs containing phase images
and susceptibility maps is a limitation of aforementioned supervised methods.
Moreover, in a recent review paper [25], the authors pointed out that one of
the biggest issues of supervised learning approach is the generalization error
that happens when test data have different characteristics to training data.
For example, compared with the conventional QSM, deep learning QSM results
underestimated the susceptibility values when the susceptibility range in the
training data differs from the test data. Moreover, this effect is shown
severe when training with synthetic data, which lacks enough variability and
structure [25].

Recently, some deep learning algorithms based on weakly supervised or
unsupervised learning were introduced. Liu et al. [23] proposed weakly
supervised learning for QSM reconstruction (wTFI). wTFI reconstructs QSM
without background field removal, and enables the restoration of
susceptibility values near the edges of the brain, which can disappear with
background field removal. Liu et al. [22] proposed unsupervised QSM
reconstruction method (uQSM) by taking advantage of nonlinear data consistency
loss and the total variation loss.

###  II-C Contributions

The contributions of our work are as follows.

  * •

Unlike the supervised learning approaches [18, 19, 20, 21, 24], our method
does not require matched QSM labels for the training so that the trained model
is less sensitive to the lack of training data with enough variability and
structures. Accordingly, the underestimation issue of QSM values in the
supervised learning can be largely overcome.

  * •

In contrast to the existing unsupervised learning approaches [23, 22], our
method learns statistical properties of unpaired QSM labels during training,
which makes the training more stable. This leads to the more accurate QSM map
estimation with less outliers.

  * •

Unlike the classical approaches [8, 9, 10, 11, 12, 13] and deep image prior
(DIP)[34], our method is a feed-forward neural network that provides
instantaneous reconstruction once the cycleGAN training is done. This makes
the algorithm very practical.

Figure 1: CycleQSM architecture. Our cycleGAN architecture has only one
generator and one discriminator because the generator for producing phase
images is replaced by the element-wise product with the known dipole kernel in
the Fourier domain. Here, generated susceptibility map is multiplied by the
brain mask before entering the discriminator.

##  III Theory

###  III-A Dipole Inversion

When the tissue is brought into the magnetic field, the tissue becomes
magnetized. Magnetic susceptibility $\chi$ is the quantitative measure of the
degree of magnetization. In MRI, the magnetization of tissues generates the
magnetic perturbation along the main magnetic field. This magnetic
perturbation, or the phase signal can be represented as follows:

$\displaystyle b(\vec{r})=d(\vec{r})*\chi(\vec{r}),\quad r\in{\mathbb{R}}^{3}$
(1)

where $b$ is the phase signal, $d$ is the dipole kernel, and $*$ is the
convolution operation. Here, the dipole kernel $d$ is represented by

$d(\vec{r})=\frac{1}{4\pi}\frac{3\cos^{2}\theta-1}{|\vec{r}|^{3}}$

where $\theta$ is the angle between $\vec{r}$ and the main magnetic field,
whose Fourier domain spectrum is given by

$\displaystyle\hat{d}(\vec{k})=\frac{1}{3}-\frac{k^{2}_{z}}{|\vec{k}|^{2}}$
(2)

where $\vec{k}=[k_{x},k_{y},k_{z}]^{\intercal}$ is a $k$-space vector.

Accordingly, one of the simplest dipole inversions could be achieved by
element-wise division in the Fourier domain.

$\displaystyle\hat{\chi}(\vec{k})=\frac{\hat{b}(\vec{k})}{\hat{d}(\vec{k})}$
(3)

However, (3) is not stable because $\hat{D}(\vec{k})$ has zero values at the
conical surface ($\vec{k}^{2}=3k^{2}_{z}$).

###  III-B Optimal Transport Driven CycleGAN

In our recent mathematical theory of optimal transport driven cycleGAN (OT-
cycleGAN) [26, 27, 28], we revealed that various forms of cycleGAN
architecture can be obtained from the dual formulation of an optimal transport
problem, in which the transport cost is the sum of the distances in the
measurement and image domains. In particular, if the forward mapping is known
from the imaging physics as in QSM, the resulting OT-cycleGAN architecture can
be significantly simplified. Here, we briefly review the theory to apply it
for the QSM reconstruction.

Consider the following measurement model:

$\displaystyle{y}$ $\displaystyle=$ $\displaystyle\mathcal{H}{x}\ ,$ (4)

where ${y}\in\mathcal{Y}$ and ${x}\in\mathcal{X}$ denote the measurement and
the unknown image, respectively, and
$\mathcal{H}:\mathcal{X}\mapsto\mathcal{Y}$ is the known deterministic imaging
operator. In contrast to the supervised learning where the goal is to learn
the relationship between the image ${x}$ and measurement ${y}$ pairs, in the
unsupervised learning framework there are no matched image-measurement pairs.
Still we could have sets of images and unpaired measurements, so our goal is
to match the probability distributions rather than each individual samples.
This can be done by finding transportation maps that transport the probability
measures between the two spaces.

Specifically, suppose that the target image space $\mathcal{X}$ is equipped
with a probability measure $\mu$, whereas the measurement space $\mathcal{Y}$
is with a probability measure $\nu$. Then, we can see that the mass transport
from $(\mathcal{X},\mu)$ to $(\mathcal{Y},\nu)$ is performed by the forward
operator $\mathcal{H}$, and the mass transportation from the measurement space
$(\mathcal{Y},\nu)$ to the image space $(\mathcal{X},\mu)$ is done by a
generator $G_{\Theta}:\mathcal{Y}\mapsto\mathcal{X}$, parameterized by
$\Theta$. Then, the following transportation cost is proposed for the optimal
transport problem [26]:

$\displaystyle
c({x},{y};\Theta):=\|{y}-\mathcal{H}{x}\|+\|G_{\Theta}({y})-{x}\|$ (5)

which denotes the sum of the distance between a training sample and a
transported sample in each space. Rather than minimizing the sample-wise cost
using (5), the goal of the optimal transport is to minimize the average
transport cost. More specifically, the optimal transport problem is formulated
to find the joint distribution $\pi$ that leads to the minimum average
transport cost:

$\displaystyle\inf\limits_{\pi\in\Pi(\mu,\nu)}\int_{\mathcal{X}\times\mathcal{Y}}c({x},{y};\Theta)d\pi({x},{y})$
(6)

where $\Pi(\mu,\nu)$ is the set of joint measures whose marginal distributions
in $\mathcal{X}$ and $\mathcal{Y}$ are $\mu$ and $\nu$, respectively.

In [26, 27, 28], the geometric meaning of the optimal transport using (5) is
explained. More specifically, if the first term in (5) is only used, the
optimal transport is to find the joint probability that minimizes the distance
in the empirical distribution $\nu$ and the so-called “push-forward measure”
$\nu_{\mathcal{H}}$. Similarly, if the second term in (5) is only used, the OT
problem is to minimizes the distance in the empirical distribution $\mu$ and
the push-forward measure $\mu_{G}$. By using both terms in (5), our OT
formulation finds the joint measure that minimizes the sum of the two
distances in the measurement and the image spaces.

Using the transportation cost in (5), the Kantorovich dual formulation [35] is
given by [26]

$\displaystyle\min_{\Theta}\mathbb{K}(\Theta,\mathcal{H})=\min_{\Theta}\max_{\Phi}\ell(\Theta;\Phi)$
(7)

where

$\displaystyle\ell(\Theta;\Phi)=\gamma\ell_{cycle}(\Theta)+\ell_{WGAN}(\Theta;\Phi)$
(8)

where $\gamma$ is a suitable hyper-parameter, $\ell_{cycle}$ is the cycle-
consistency loss, $\ell_{WGAN}$ is the Wasserstein GAN loss [36]. More
specifically, $\ell_{cycle}$ and $\ell_{WGAN}$ are given by

$\displaystyle\begin{aligned}
\ell_{cycle}(\Theta)=\int_{\mathcal{X}}\|{x}-G_{\Theta}(\mathcal{H}{x})\|d\mu({x})\\\
+\int_{\mathcal{Y}}\|{y}-\mathcal{H}G_{\Theta}({y})\|d\nu({y})\
,\end{aligned}$ (9) $\displaystyle\begin{aligned} &\ell_{WGAN}(\Theta;\Phi)\\\
&=\int_{\mathcal{X}}\varphi_{\Phi}({x})d\mu({x})-\int_{\mathcal{Y}}\varphi_{\Phi}(G_{\Theta}({y}))d\nu({y})\
.\end{aligned}$ (10)

Note that there exists only a single discriminator $\varphi_{\Phi}$, since the
forward operator $\mathcal{H}$ is assumed known, so there is no need to
compete with the forward operator. This makes the cycleGAN architecture
simple, as will be explained later.

TABLE I: Data sets for experiments

| 2016 QSM Challenge | Cornell | ALOHA-QSM | 2019 QSM Challenge
---|---|---|---|---
Resolution | Voxel pitch (mm3) | 1.06$\times$1.06$\times$1.06 | 0.9375$\times$0.9375$\times$1 | 0.75$\times$0.75$\times$1.1 | 0.64$\times$0.64$\times$0.64
Matrix size | 160$\times$160$\times$160 | 256$\times$256$\times$146 | 240$\times$320$\times$128 | 164$\times$205$\times$205
No. of Cases | Total | 1 | 1 | 5 | 4 (2 SNR / 2 Contrast)
QSM Ground-truth | 1 (COSMOS) | 1 (COSMOS) | 0 | 2 (Simulation)
Training | 1 | 0 | 3 | 4
Test | 0 | 1 | 2 | 0

###  III-C CycleQSM

In this section, we will see how our OT-cycleGAN formulation can be used for
QSM. In the dipole inversion, a phase image $b\in B$ and a susceptibility map
$\chi\in X$ correspond to a noisy measurement and an unobserved image,
respectively. Therefore, the forward model of the dipole inversion can be
formulated as

$\displaystyle b={\mathcal{F}}^{-1}\hat{d}{\mathcal{F}}\chi$ (11)

where ${\mathcal{F}}$ and ${\mathcal{F}}^{-1}$ are 3D Fourier transform and 3D
inverse Fourier transform, respectively. By identifying ${y}:=b,{x}:=\chi$ and
$\mathcal{H}:={\mathcal{F}}^{-1}\hat{d}{\mathcal{F}}$, the cycle-consistency
loss (9) and the GAN loss (10) can be represented by

$\displaystyle\begin{aligned} \ell_{cycle}(\Theta)=\int_{X}\|\chi-
G_{\Theta}({\mathcal{F}}^{-1}\hat{d}{\mathcal{F}}\chi)\|d\mu(\chi)\\\
+\int_{B}\|b-{\mathcal{F}}^{-1}\hat{d}{\mathcal{F}}\ G_{\Theta}(b)\|d\nu(b)\
,\end{aligned}$ (12)

and

$\displaystyle\begin{aligned} &\ell_{WGAN}(\Theta;\Phi)\\\
&=\int_{X}\varphi_{\Phi}(\chi)d\mu(\chi)-\int_{B}\varphi_{\Phi}(G_{\Theta}(b))d\nu(b)\
.\end{aligned}$ (13)

Although these costs can be used directly for the QSM reconstruction, we
provide additional modifications for a better performance. First, we employ
the least squares GAN (LSGAN) [37] instead of WGAN loss for faster and more
stable training. The link between WGAN and LSGAN was also explained in [27].
Next, we add the gradient difference loss ($\ell_{grad}$) proposed in [18] to
maintain the edge information.

$\displaystyle\begin{aligned} \ell_{grad}(\Theta)=\int_{X}\|\nabla\chi-\nabla
G_{\Theta}({\mathcal{F}}^{-1}\hat{d}{\mathcal{F}}\chi)\|d\mu(\chi)\\\
+\int_{B}\|\nabla b-\nabla{\mathcal{F}}^{-1}\hat{d}{\mathcal{F}}\
G_{\Theta}(b)\|d\nu(b)\end{aligned}$ (14)

where $\nabla$ is a gradient operator for the 3D volume. To preserve image
details and remove noise in reconstructed QSM, we also employ the total
variation (TV) loss ($\ell_{TV}$).

$\displaystyle\ell_{TV}(\Theta)=\int_{B}\|\nabla G_{\Theta}(b)\|d\nu(b)\ .$
(15)

Therefore, the final cost function for CycleQSM can be formulated as

$\displaystyle\begin{aligned}
\ell(\Theta;\Phi)=&\gamma\ell_{cycle}(\Theta)+\ell_{LSGAN}(\Theta,\Phi)\\\
&+\eta\ell_{grad}(\Theta)+\rho\ell_{TV}(\Theta)\end{aligned}$ (16)

where $\gamma$, $\eta$, and $\rho$ are appropriate hyper-parameters. Note that
the additional losses in (14) and (15) are average values with respect to the
marginal distributions. Therefore, the addition of these terms does not change
the optimal transport interpretation. In fact, as analyzed in [38], (16) can
be obtained as a dual formulation of the optimal transport problem with the
following transportation cost:

$\displaystyle c_{QSM}(b,\chi;\Theta)$ $\displaystyle:=$
$\displaystyle~{}\|b-{\mathcal{F}}^{-1}\hat{d}{\mathcal{F}}\chi\|+\|G_{\Theta}(b)-\chi\|$
$\displaystyle+\eta\left(\|\nabla\chi-\nabla
G_{\Theta}({\mathcal{F}}^{-1}\hat{d}{\mathcal{F}}\chi)\|+\|\nabla
b-\nabla{\mathcal{F}}^{-1}\hat{d}{\mathcal{F}}\ G_{\Theta}(b)\|\right)$
$\displaystyle+\rho\|\nabla G_{\Theta}(b)\|$

where $b\in\mathcal{Y}$ and $\chi\in\mathcal{X}$ with probability measures
$\nu$ and $\mu$, respectively.

##  IV Method

###  IV-A Experimental Data

To train and test our algorithm, we use data sets from different sources.
First data set is in vivo human brain data that was provided for 2016 QSM
challenge [39]. This data set was acquired from a healthy volunteer by 3T
Siemens scanner with 3D gradient echo (GRE) sequence. Acquisition parameters
of 2016 QSM challenge data are as follows: resolution =
1.06$\times$1.06$\times$1.06 mm3, matrix size = 160$\times$160$\times$160,
TR/TE = 25/35 ms. This data set also contains ground-truth QSM which is
acquired by COSMOS with 12 different orientation data.

Next, we use in vivo healthy human data from the Cornell MRI Research Lab.
This data set was collected using multi-echo GRE sequence with 3T GE system.
The acquisition parameters are as follows: resolution =
0.9375$\times$0.9375$\times$1 mm3, matrix size = 256$\times$256$\times$256,
TR/TE1 = 55/5 ms, $\Delta$TE = 5 ms. QSM is also obtained by COSMOS with 5
different head orientation data.

The third data set was obtained from five healthy volunteers using 3T Siemens
system. This data set was used for ALOHA-QSM [13], and acquired with the
following parameters: resolution = 0.75$\times$0.75$\times$1.1 mm3, matrix
size = 240$\times$320$\times$128, TR/TE1 = 43/9.35 ms, $\Delta$TE = 8.94 ms.

Last, we used the data for 2019 QSM challenge [40]. This data set was obtained
through forward simulation based on in vivo human brain data acquired on 7T MR
system. The acquisition parameters for 2019 challenge data are as follows:
resolution = 0.64$\times$0.64$\times$0.64 mm3, matrix size =
164$\times$205$\times$205, TR/TE1 = 50/4 ms, $\Delta$TE = 8 ms. The local
field data in 2019 challenge data set were simulated with two different
contrast levels and two signal-to-noise ratio (SNR) levels, so total four
local field maps (Sim1Snr1, Sim1Snr2, Sim2Snr1, Sim2Snr2) were provided. In
addition, ground-truth QSM data were provided for two different contrast
levels.

Table I shows the name and number of volumes of each data set. The 2016 QSM
challenge and 2019 QSM challenge data in Table I are used to train our network
as well as other deep learning approaches. For unsupervised learning, we do
not need a matched reference data. So three local field maps of the ALOHA-QSM
data set are also used for training and the remaining two local field maps are
used for inference. Furthermore, the Cornell data set is used for the
quantitative evaluation of our algorithm.

For data preprocessing, brain mask extraction using the brain extraction tool
(BET) [41], multi-coil phase combination by Hermitian inner product (HiP)
[42], and multi-echo phase correction by nonlinear frequency map estimation
[43, 44, 45] are performed. Laplacian-based phase unwrapping [46, 47] is then
applied, and then background phase removal by sophisticated harmonic artifact
reduction for phase data with varying spherical kernel (V-SHARP) [12, 48] is
executed.

###  IV-B Network Architectures

Fig. 1 illustrates the architecture of cycleQSM. Since the forward mapping is
described by the deterministic dipole kernel in (2), cycleQSM has only one
pair of the generator and discriminator. As a result, the training of our
network can be more stable and faster than the conventional cycleGAN. To
provide more information to the generator, the magnitude image is concatenated
with the phase image. The dipole kernel for each training step is generated
depending on the resolution and direction of the main magnetic field of each
data. Accordingly, all the data set in Table I with different spatial
resolution can be fully utilized for training cycleQSM.

In Fig. 1, the reconstructed QSM is multiplied by the brain mask before going
through the discriminator. Without the brain mask, we find that the
discriminator distinguishes real and fake QSM by observing the artifact
outside the brain mask region. We find that the multiplying the brain mask can
stabilize the training process and reduce the artifact outside of the brain
region.

Figure 2: (a) The generator and (b) the discriminator architecture.

Fig. 2(a) shows the architecture of our generator. We use a 3D U-Net [33]
structure to construct our generator. Because of the concatenation of the
magnitude and phase images, our generator has two input channels. The
generator contains 3$\times$3$\times$3 convolution, instance normalization
[49], leaky ReLU, and nearest-neighbor upsampling layers. In addition, the
skip connection via channel concatenation is used for the generator. At the
end of the network, the 1$\times$1$\times$1 convolution produces the final
reconstructed QSM.

We use patchGAN discriminator [50, 51] as shown in Fig. 2(b). The
discriminator consists of 4$\times$4$\times$4 convolution, instance
normalization, and leaky ReLU. Inputs of the discriminator are real
susceptibility maps or generated susceptibility maps that are multiplied with
brain masks.

Figure 3: QSM reconstruction results of Cornell data using various methods.

###  IV-C Experimental Details

Since there are only 8 phase images and 3 QSM label data for network training,
we use random patches during training to increase the amount of data. During
one epoch, we extract a total of 3000 phase and unpaired QSM random patches of
size 64$\times$64$\times$64, respectively. In the inference step, a phase
volume is cropped into patches of size 64$\times$64$\times$64 with a stride of
16$\times$16$\times$16, and all patch inference results are combined to
reconstruct the entire QSM volume. Moreover, we apply data augmentation by
flipping with respect to each axis, and rotating in a plane perpendicular to
the direction of the main magnetic field.

In our experiments, we use Adam optimizer with $\beta_{1}=0.5$,
$\beta_{2}=0.999$, and learning rate of 0.00001. Also, we choose $\gamma=10$,
$\eta=1$, and $\rho=0.1$ for (16). Our cycleGAN is trained for 50 epochs, and
implemented in Python by TensorFlow.

###  IV-D Comparison Methods

We compare our algorithm with several conventional methods to verify the
performance of our algorithm. First, TKD [8, 9] is used to replace the values
in the conical surfaces of the dipole kernel with a certain threshold value:

$\displaystyle\hat{d}(\vec{k};a)=\begin{cases}d(\vec{k})&\hat{|}d(\vec{k})|>a\\\
a\cdot\mbox{sign}(d(\vec{k}))&\hat{|}d(\vec{k})|\leq a\end{cases}$ (17)

where $a$ is a threshold value. Next, MEDI [10, 11] is also compared with our
algorithm. The QSM reconstruction by MEDI can be formulated as follows:

$\displaystyle\min_{\chi}\|W(b-{\mathcal{F}}^{-1}\hat{d}{\mathcal{F}}\chi)\|^{2}_{2}+\lambda\|M\nabla\chi\|_{1}$
(18)

where $W$ is the structural weight matrix which is derived by the magnitude
image, and $M$ is the binary mask that contains the edge information of the
magnitude image. In addition, we compare cycleQSM with iLSQR [30] and ALOHA-
QSM [13]. In our comparison experiments, we use $a=0.1$ for TKD, $\lambda=600$
for MEDI, and 30 iteration steps for iLSQR, respectively. Also, hyper-
parameters for ALOHA-QSM are set to $\lambda=10^{1.5}$, $\mu=10^{-1.5}$ for
Cornell data, and $\lambda=10^{2.4}$, $\mu=10^{-2.2}$ for ALOHA-QSM data,
respectively.

We also compare our method with other deep learning methods. Supervised
learning using the same U-Net network is compared with our method. The network
in supervised learning is trained using $L_{1}$ loss during 50 epochs with the
learning rate of 0.0001. Next, deep image prior (DIP) [34] is used for dipole
inversion. DIP is optimized for each volume without training, and the
optimization in DIP can be formulated as follows:

$\displaystyle\min_{\chi}\|W(e^{j{\mathcal{F}}^{-1}\hat{d}{\mathcal{F}}\chi}-e^{jb})\|_{1}+\lambda\|\nabla\chi\|_{1}\
,$ (19)

where $W$ is a noise weighting factor that is obtained from the magnitude
image. We use $\lambda$ in (19) as 0.001 for the best performance. The network
architecture in DIP is same as our generator, but we reduce the number of
channels by half due to the GPU memory limitation. In addition, uQSM [22] is
used for comparison as another unsupervised learning method. We use the
network architecture in [22] for uQSM. uQSM minimizes the loss in (19) with
$\lambda=0.001$, but it is trained with training data first, then the trained
network is used for the reconstruction of test data. uQSM is trained during 50
epochs with the learning rate of 0.0001.

To evaluate algorithms, we use the peak signal-to-noise ratio (PSNR) and the
structural similarity index metrics (SSIM) as quantitative metrics. PSNR and
SSIM are measured for 3D volume of Cornell data which has ground-truth QSM.
Also, we use the root mean square error (RMSE) which is calculated as follows:

$\displaystyle
RMSE=\sqrt{\frac{\sum_{i=1}^{N}(\chi_{i}-\tilde{\chi}_{i})^{2}}{N}}\ ,$ (20)

where $N$ is the number of pixels which in the brain mask, and $\chi_{i}$ ans
$\tilde{\chi}_{i}$ are pixel intensities of ground-truth and reconstructed
QSM.

Figure 4: (a) Deep gray matter structures. The susceptibility values in the
green area are extracted for scatter plots and statistical analysis. (b)
Linear regression between the ground-truth susceptibility values and
reconstructed susceptibility values. The black dashed line shows the graph of
$y=x$, and the red dash-dotted line shows the graph of linear regression
results. The equation of the red dash-dotted line, $R^{2}$ value, fitting
error, and correlation coefficient are written in the upper left corner of
each graph. The error (mean$\pm$standard deviation) in the upper left corner
is calculated as follows: $|\chi_{COSMOS}-\chi_{Reconstruction}|$.

##  V Experimental Results

First, we compare various QSM reconstruction methods for Cornell data in Table
I which contains ground-truth QSM data. Fig. 3 shows the phase image, the
ground-truth QSM, and reconstructed QSM by using conventional and deep
learning based QSM reconstruction methods. As shown in Fig. 3, the
reconstructed QSM from TKD has severe streaking artifacts. MEDI reconstructs
susceptibility maps without the streaking artifact, but the output of MEDI is
extremely smoothed. iLSQR and ALOHA-QSM reconstruct more realistic QSM, but
some streaking artifacts still remain in the reconstructed QSMs. Next,
supervised learning shows similar output to ground-truth without streaking
artifacts. The output of DIP is smoothed and some susceptibility values are
not restored. Also, DIP requires very long reconstruction time compared to
other methods because it has to be optimized to each volume. uQSM can
reconstruct QSM without label data, but uQSM generate smoothed output so that
some of structures in the reconstructed QSM are not recognizable. On the other
hand, our method reconstructs QSM that are close to the ground-truth data. In
addition, the proposed method requires similar reconstruction time as
supervised learning.

The quantitative metric values of various QSM reconstruction methods are shown
in Table II. As shown in Table II, conventional methods show relatively low
quantitative metric values due to artifacts in the reconstructed QSM or over-
smoothed output. Supervised learning shows the highest quantitative results
because it is trained to minimize $L_{1}$ loss between the output and label
images. DIP and uQSM shows similar metric values to conventional methods
because they show blurry output. Meanwhile, our method shows comparable
quantitative results to supervised learning. From quantitative evaluation, we
confirm that our method outperforms the conventional methods and shows
competitive results compared to supervised learning.

TABLE II: Quantitative evaluation of various QSM reconstruction methods. The
values in the table are calculated within the whole volume.

| PSNR (dB) | SSIM | RMSE (%)
---|---|---|---
Conventional methods | TKD [8, 9] | 38.9398 | 0.9625 | 4.7970
MEDI [10, 11] | 42.8732 | 0.9840 | 3.0674
iLSQR [30] | 42.8234 | 0.9833 | 3.0850
ALOHA-QSM [13] | 41.2693 | 0.9778 | 3.6895
Deep learning methods | Supervised | 43.4971 | 0.9899 | 2.8548
DIP [34] | 41.3845 | 0.9817 | 3.6409
uQSM [22] | 42.7811 | 0.9866 | 3.1001
Proposed | 43.4656 | 0.9890 | 2.8652

TABLE III: The slope, $R^{2}$ value, mean error, and correlation coefficient
(Corr) of various QSM reconstruction methods with Cornell data. The
susceptibility values in gray matter structures are extracted for statistical
analysis. The error (mean$\pm$standard deviation) is calculated as follows:
$|\chi_{COSMOS}-\chi_{Reconstruction}|$.

| Slope | $R^{2}$ | Error | Corr
---|---|---|---|---
Conventional methods | TKD [8, 9] | 1.03 | 0.69 | 0.029 | 0.82
MEDI [10, 11] | 0.95 | 0.92 | 0.013 | 0.96
iLSQR [30] | 0.92 | 0.87 | 0.017 | 0.92
ALOHA-QSM [13] | 1.12 | 0.86 | 0.023 | 0.92
Deep learning methods | Supervised | 0.79 | 0.88 | 0.027 | 0.94
DIP [34] | 0.95 | 0.81 | 0.020 | 0.88
uQSM [22] | 0.85 | 0.83 | 0.024 | 0.91
Proposed | 0.93 | 0.85 | 0.017 | 0.91

Figure 5: QSM reconstruction results of ALOHA-QSM data using various methods.

To verify that our method restores accurate susceptibility values in gray
matter structures, we extract susceptibility values in gray matter structures
in Fig. 4(a), and compare them with ground-truth susceptibility values through
linear regression and correlation coefficient. Fig. 4(b) shows the scatter
plots for the ground-truth and the reconstruction QSM value pairs, and Table
III summarizes the statistical analysis results between susceptibility values
of ground-truth QSM and reconstructed QSM. Because the reconstructed QSM by
TKD has severe streaking artifacts, there are many outliers in the regression
graph of TKD. Therefore, the $R^{2}$ value and correlation coefficient in TKD
is low and the error of TKD is higher than other methods. Next, MEDI shows the
best performance in terms of linear regression and correlation coefficient.
The linear regression result of MEDI is close to $y=x$ and has large $R^{2}$
values. Also, the correlation coefficient of MEDI is the highest among QSM
reconstruction methods. However, note that this accuracy is compromised by the
overly smoothed reconstruction results in Fig. 3. The iLSQR shows slightly
lower performance than MEDI, but the error of iLSQR is relatively small
compared to other methods. The slope of ALOHA-QSM is higher than 1, which
indicates there are some overestimated susceptibility values when using ALOHA-
QSM method.

On the other hand, many points in the graph of supervised learning are below
the line of $y=x$. Furthermore, the slope of supervised learning is lower than
1 and it shows large error due to the underestimation, which confirms the
reported findings [25]. Since DIP cannot reconstruct accurate QSM, there exist
many underestimated or overestimated points in the graph. Also, DIP has low
$R^{2}$ value and correlation coefficient compared to other QSM reconstruction
methods. uQSM also underestimate many susceptibility values, so it shows large
error value. Meanwhile, our method reconstructs accurate susceptibility
values, thus it has fewer underestimated points than supervised learning.
Furthermore, the slope of our method is close to 1, and the error is lower
than other methods except MEDI. From these results, we reveal that our method
provides comparable quantitative results with the classical methods and
outperforms other deep learning methods.

We also employ QSM reconstruction algorithms to ALOHA-QSM data which does not
contain ground-truth QSM. Fig. 5 shows the QSM reconstruction results of
ALOHA-QSM data using various QSM reconstruction methods. As shown in Fig. 5,
TKD generates severe streaking artifacts, so the quality of reconstructed QSM
is degraded and it is difficult to distinguish anatomical structures in the
brain. MEDI also shows smoothed reconstruction results, and checkerboard
artifacts are remarkable when MEDI is applied for ALOHA-QSM data. iLSQR and
ALOHA-QSM show more realistic reconstruction than TKD or MEDI, but slight
noises were seen in the reconstructed susceptibility maps. Supervised learning
does not show artifacts, but the results of supervised learning seem blurry.
Next, like the previous experiment using Cornell data, DIP shows inferior
results. Especially, the severe artifacts are appeared in the sagittal plain.
uQSM generates artifacts and blurred region in reconstructed QSM. On the other
hand, the proposed method does not show the artifact or noise in reconstructed
QSM. Moreover, the contrast and brain structures are recovered accurately when
using our method.

Because ALOHA-QSM data does not contain ground-truth QSM data, we compare mean
susceptibility values of each brain structure in reconstructed susceptibility
maps to verify that our method provides accurate susceptibility values. Fig. 6
shows the mean susceptibility values of brain structures in reconstructed QSM.
Deep learning based algorithms show lower susceptibility values than
conventional algorithms on average. Especially, supervised learning shows the
lowest mean susceptibility value in the putamen and caudate nucleus. On the
contrary, the proposed method shows similar mean susceptibility values with
conventional methods. Therefore, we believe that our method provides proper
susceptibility values.

Figure 6: Mean susceptibility value of each deep gray matter structure in the
reconstructed QSMs using various methods with ALOHA-QSM data. The error bars
indicate the standard deviation of susceptibility values.

##  VI Discussion

To verify our cycleGAN structure, we compare with other cycleGAN structures in
[26]. The first structure is the conventional cycleGAN with two deep
generators and two discriminators. Two generators and discriminators in the
conventional cycleGAN have same architecture as ours. Next, we also compare
our method to cycleGAN with linear unknown kernel as proposed in [26] and
[27]. Because the relationship between the phase image and QSM can be
formulated as the convolution operator in the spatial domain, it is possible
to replace the generator for producing the phase image from the QSM to one
linear convolution kernel that can be trained. We set the size of the
convolution kernel for generating the phase image from the QSM as
20$\times$20$\times$20.

Fig. 7(a) and (b) show the reconstruction results using various cycleGAN
structures. As shown in Fig. 7(a), the reconstruction result by the
conventional cycleGAN has different contrast with the ground-truth QSM. Also,
some susceptibility values are overestimated. Since conventional cycleGAN has
four neural networks which have to be optimized simultaneously, it requires
large training data for stable convergence. Therefore, the difficulty of
convergence of conventional cycleGAN leads to the inferior reconstruction
results. Next, cycleGAN with linear kernel is unable to restore appropriate
susceptibility values. Because each data set that were used for training has
different resolution, the dipole kernels of each data set are also different.
Therefore, it is difficult to replace all dipole kernels to single linear
convolution, and it makes inappropriate reconstruction in cycleGAN with linear
kernel. On the other hand, the proposed cycleGAN reconstructes precise
susceptibility values without any artifacts. Because our cycleGAN employs
deterministic dipole kernel that is generated depending on the resolution of
each data, it is possible to train the networks and reconstruct accurate QSM
with only one generator and one discriminator by utilizing all training data
at different spatial resolution. In Fig. 7(b), various cycleGAN structures are
also applied for ALOHA-QSM data. Compared to other cycleGAN structures, our
cycleGAN reconstructs accurate QSM without the artifact and noise.

Figure 7: QSM reconstruction results of (a) Cornell data and (b) ALOHA-QSM
data using various cycleGAN structures. Yellow arrows indicate artifacts or
errors in the results of cycleGAN and cycleGAN with linear kernel.

We also compare quantitative metric values of each cycleGAN structure using
Cornell data set. From Table IV, we demonstrate that quantitative metric
values of conventional cycleGAN and cycleGAN with linear kernel are lower than
those of our method.

TABLE IV: Quantitative evaluation of various cycleGAN structures. The values
in the table are calculated within the whole volume.

| PSNR (dB) | SSIM | RMSE (%)
---|---|---|---
CycleGAN | 40.1006 | 0.9753 | 4.2208
CycleGAN with linear kernel | 41.9220 | 0.9825 | 3.4224
Proposed | 43.4656 | 0.9890 | 2.8652

##  VII Conclusion

In this paper, we developed a novel unsupervised QSM reconstruction method
called cycleQSM using physics-informed cycleGAN which is derived from optimal
transport perspective. Due to the known dipole kernel, our cycleQSM has only a
single pair of generator and discriminator, which makes the training much
stable.

In our experiments, conventional QSM reconstruction methods showed inferior
qualitative results compared to the proposed method due to the streaking
artifacts or smoothing. Furthermore, the reconstruction time of conventional
methods is relatively long compared to deep learning methods due to the high
computational complexity. Supervised learning generated blurry output and many
susceptibility values were underestimated.

Although the proposed method was trained without the matched pair of phase and
QSM, we demonstrated that the proposed method shows comparable quantitative
results compared to supervised learning. Furthermore, we confirmed that our
method provides accurate susceptibility values without underestimation.
Therefore, we believe that cycleQSM can be a powerful framework for QSM
reconstruction.

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