# StegoGAN: Leveraging Steganography
for Non-Bijective Image-to-Image Translation

Sidi Wu 1 Yizi Chen 111footnotemark: 1 Samuel Mermet 2 Lorenz Hurni 1
Konrad Schindler 1 Nicolas Gonthier 3,2 Loic Landrieu 4,2
1 ETH Zurich  2 Univ Gustave Eiffel, IGN, ENSG, LASTIG  3 IGN  4 Univ Gustave
Eiffel, CNRS, Ecole des Ponts, LIGM  Equal contribution.

###### Abstract

Most image-to-image translation models postulate that a unique correspondence
exists between the semantic classes of the source and target domains. However,
this assumption does not always hold in real-world scenarios due to divergent
distributions, different class sets, and asymmetrical information
representation. As conventional GANs attempt to generate images that match the
distribution of the target domain, they may hallucinate spurious instances of
classes absent from the source domain, thereby diminishing the usefulness and
reliability of translated images. CycleGAN-based methods are also known to
hide the mismatched information in the generated images to bypass cycle
consistency objectives, a process known as steganography. In response to the
challenge of non-bijective image translation, we introduce StegoGAN, a novel
model that leverages steganography to prevent spurious features in generated
images. Our approach enhances the semantic consistency of the translated
images without requiring additional postprocessing or supervision. Our
experimental evaluations demonstrate that StegoGAN outperforms existing GAN-
based models across various non-bijective image-to-image translation tasks,
both qualitatively and quantitatively. Our code and pretrained models are
accessible at https://github.com/sian-wusidi/StegoGAN.

##  1 Introduction

Image-to-image translation is an active research subject with impactful
applications ranging from changing the style of images [39, 13] to
automatically creating maps from satellite images [39] or changing the
modality of medical images [10]. When the source and target domains exhibit
substantial differences, ensuring the semantic consistency between input
images and their translation becomes particularly challenging [9]. Our work
explores the surprisingly uncharted field of adversarial, non-bijective image-
to-image translation.

|  matchable features unmatchable feature hallucination
---|---

Image from $\mathcal{X}$ Image from $\mathcal{Y}$ (a) Semantically misaligned
domains.

Image from $\mathcal{X}$… translated to $\mathcal{Y}$ (b) GAN models
hallucinate spurious features.

Image from $\mathcal{Y}$… translated to $\mathcal{X}$… and back to
$\mathcal{Y}$ (c) CycleGAN models hide unmatchable features by steganography.

Figure 1: Non-Bijective Translation. When image domains present classes
without equivalence 1(a), GAN models tend to hallucinate spurious features
when translating images 1(b). A related phenomenon is steganography, where
CycleGAN-based models covertly encode features in low-amplitude patterns to
bypass cycle consistency 1(c). Instead of disabling this phenomenon, we
harness steganography to prevent the hallucination of spurious features.

Non-Bijective Image Translation. Existing translation methods assume a one-to-
one correspondence between classes of the source and target domains: horses to
zebras [39], satellite image features to their cartographic representation
[39, 7], or distinct cell types viewed under varying medical imaging
modalities [10]. However, as illustrated in Figure 1(a), this assumption does
not always hold. For instance, when considering a dataset with horses and one
with zebras in their habitat, zebra images may include background elements
with no equivalent in the source domain—like elephants. Similarly, in map
translation toponyms (_i.e_., place names printed onto the map) do not have
counterparts in satellite images [7] . We qualify classes of the target domain
without equivalent in the source domain as _unmatchable_.

As illustrated in Figure 1(b), by trying to reproduce the distribution of the
target domain, GANs may hallucinate _spurious_ features or textures in the
generated images, _i.e_., objects without equivalent in the source image. This
is particularly frequent for unmatchable classes [9]. While this can be
perfectly acceptable for some applications [26], adding nonexistent tumors in
MRI scans or incorrect toponyms in maps can severely degrade the usefulness of
the translation result. Instead of detecting and removing these artifacts in
post-processing, we propose an approach that directly prevents their
generation using steganography.

GAN Steganography. To ensure its semantic consistency, CycleGAN [39] back-
translates the generated image to the source image. However, unmatchable
classes of the source domain cannot be encoded into meaningful features in the
images generated in the target domain. As shown in Figure 1(c), these models
can instead _cheat_ by encoding the necessary information into quasi-invisible
patterns in the generated images [8]. This process, known as steganography,
allows GANs to perform seemingly impossible back-translation. For instance, in
a map-to-satellite task, the model can restore the correct names of towns from
satellite images that appear visually correct. This phenomenon is often viewed
as a quirky optimization flaw, easily fixable by adding noise or blur [12, 29,
21].

StegoGAN. We propose StegoGAN, a model that leverages steganography to detect
and mitigate semantic misalignment between domains. In settings where the
domain mapping is non-bijective, StegoGAN experimentally demonstrates superior
semantic consistency over other GAN-based models both visually and
quantitatively, without requiring detection or inpainting steps. In addition,
we publish three datasets from open-access sources as a benchmark for
evaluating non-bijective image translation models.

##  2 Related Work

GAN-Based Image Translation. GAN-based image translation models transfer the
style of images between domains with an adversarial perceptual loss [14]. When
pairs of aligned images from both domains are available, the translated images
can also be supervised by their fidelity with target images [19]. In practice,
such pairs are not always available or even possible to obtain. In the absence
of explicit equivalence between images, preserving the semantics of the input
in the generated image is a priority. Multiple approaches have been proposed
to address this challenge, such as density-based regularization [36], spatial
mutual information [29, 33, 21], or cycle consistency losses [39, 23, 18].

Asymmetric Image Translation. Translating between domains with different
semantic distributions is challenging. Existing approaches include focusing
the network’s attention on the most discriminative part of the input image
[32], augmenting the consistency loss with geometric transformations [12],
replacing the consistency reconstruction term with a contrastive loss [29,
21], or ensuring that the translation is robust to small perturbations of the
input [20]. However, these methods assume a bijective relationship between the
classes of the source and target domains.

Closest to our work is the model of Li _et al_. [26], which uses an auxiliary
variable to model the information loss from information-rich domains (such as
natural images) to information-poor domains (such as label maps). In turn,
they use this variable to create realistic poor-to-rich domain translations.
Our work differs as we precisely want to avoid the creation of spurious—albeit
realistic—details when translating to a domain with unmatchable classes.

CycleGAN Steganography. Chu _et al_. [8] discovered that, when faced with
unmatchable classes, CycleGAN [39] hides information in low-amplitude and
high-frequency signals. The model uses these visually imperceptible patterns
to recreate the source image and bypass the cycle loss. This contradicts the
intention of the cycle consistency loss and makes the model more vulnerable to
adversarial attacks [8]. Luckily, multiple approaches can prevent
steganography, such as blurring [12], compressing [11], or adding noise [21]
to the generated source images in the back-translation. Alternatively, the
back-translation from poor to rich domains can be omitted in the cycle
consistency loss [30, 38].

While steganography in CycleGAN can be problematic, it also offers an
opportunity to analyse distribution differences. In StegAnomaly [4], a model
is trained to translate healthy brain scans into a low-entropy domain with
cycle consistency. When removing high-frequency components, the model error
reveals anomalous structures. This approach, like ours, harnesses
steganography for insightful domain analysis, albeit with a different goal.

##  3 Methods

---
|
|  |  |
---|---|---

(a) Backward cycle

(b) Forward cycle

(c) Matchability disentanglement module

(d) Inference

Figure 2: Architecture. To avoid spurious generation of unmatchable classes in
non-bijective image translation, we propose to make the steganographic process
explicit and in feature-space. Our model runs the backward cycle first 2(a),
then the forward translation cycle 2(b). Thanks to our matchability
disentanglement module 2(c), we can separate the matchable and unmatchable
information while translating images from domain $\mathcal{Y}$ to
$\mathcal{X}$. We can then produce generated and reconstructed images with and
without unmatchable features. At inference time 2(d), our model operates like
a normal image translation model.

We consider two image domains $\mathcal{X}$ and $\mathcal{Y}$ with respective
semantic class sets $\mathcal{K}_{\mathcal{X}}$ and
$\mathcal{K}_{\mathcal{Y}}$. The domains $\mathcal{X}$ and $\mathcal{Y}$ are
considered bijective if there exists a function $\phi$ from
$\mathcal{K}_{\mathcal{X}}$ to $\mathcal{K}_{\mathcal{Y}}$ such that each
class $k_{\mathcal{X}}$ has a unique and natural semantically equivalent class
$\phi(k_{\mathcal{X}})$ in $\mathcal{Y}$, and vice-versa. A class of
$\mathcal{K}_{\mathcal{Y}}$ is said to be _unmatchable_ if it doesn’t have an
equivalent in $\mathcal{K}_{\mathcal{X}}$. While this notion is somewhat
subjective, many applications have obvious examples: toponyms are unmatchable
in satellite images, and tumors in scans of healthy patients.

Objective. Our goal is to learn a mapping $G:\mathcal{X}\mapsto\mathcal{Y}$
such that the translation $G(x)$ of any image $x\in\mathcal{X}$ aligns
stylistically with images from $\mathcal{Y}$, while preserving the semantic
content of $x$. If $\mathcal{K}_{\mathcal{Y}}$ contains an unmatchable class
$k^{\text{unmatch}}_{\mathcal{Y}}$, images translated from $\mathcal{X}$ to
$\mathcal{Y}$ should not contain any instances of
$k^{\text{unmatch}}_{\mathcal{Y}}$. However, in an attempt to match the
distribution of $\mathcal{Y}$, GAN models often create spurious instances of
unmatchable classes in their translated images. In this paper, we propose a
method that employs steganography to prevent the generation of such spurious
information.

###  3.1 CycleGAN

CycleGAN [39] learns unpaired image translation by enforcing the consistency
between the input image and its back-translation from the generated image. It
uses two generators
$G_{\mathcal{X}\mapsto\mathcal{Y}}:\mathcal{X}\mapsto\mathcal{Y}$ and
$G_{\mathcal{Y}\mapsto\mathcal{X}}:\mathcal{Y}\mapsto\mathcal{X}$, and two
domain discriminators $D_{\mathcal{X}}:\mathcal{X}\mapsto\\{0,1\\}$,
$D_{\mathcal{Y}}:\mathcal{Y}\mapsto\\{0,1\\}$ which predict whether a sample
is generated (0) or real (1).

In the following, when considering images $x\in\mathcal{X}$ or
$y\in\mathcal{Y}$, we define the following short-hands:
$x_{\text{gen}}\vcentcolon=G_{\mathcal{Y}\mapsto\mathcal{X}}(y)$ and
$y_{\text{gen}}\vcentcolon=G_{\mathcal{X}\mapsto\mathcal{Y}}(x)$ for the
generated images, and
$x_{\text{rec}}\vcentcolon=G_{\mathcal{Y}\mapsto\mathcal{X}}(y_{\text{gen}})$
and
$y_{\text{rec}}\vcentcolon=G_{\mathcal{X}\mapsto\mathcal{Y}}(x_{\text{gen}})$
for the reconstructed images. We now detail the losses of CycleGAN.

Adversarial loss. The adversarial loss [14] encourages the discriminators
$D_{\mathcal{X}}$ and $D_{\mathcal{Y}}$ to distinguish between authentic and
generated images, while pushing the generators
$G_{\mathcal{X}\mapsto\mathcal{Y}}$ and $G_{\mathcal{Y}\mapsto\mathcal{X}}$ to
create credible images:

$\displaystyle\\!\\!\\!\\!\mathcal{L}_{\text{GAN}}$
$\displaystyle\\!=\\!\mathbb{E}_{y\sim\mathcal{Y}}\log(D_{\mathcal{Y}}(y))\\!+\\!\mathbb{E}_{x\sim\mathcal{X}}\log(1\\!-\\!D_{\mathcal{Y}}(y_{\text{gen}}))$
$\displaystyle\\!+\\!\mathbb{E}_{x\sim\mathcal{X}}\log(D_{\mathcal{X}}(x))\\!+\\!\mathbb{E}_{y\sim\mathcal{Y}}\log(1\\!-\\!D_{\mathcal{X}}(x_{\text{gen}})).$
(1)

Cycle consistency. The cycle consistency loss ensures that the back-
translation of $y_{\text{gen}}$ to domain $\mathcal{X}$ is close to the
original image $x$, and likewise for $x_{\text{gen}}$ and $y$:

$\displaystyle\mathcal{L}_{\text{cyc}}=\mathbb{E}_{x\sim\mathcal{X}}\|x_{\text{rec}}-x\|+\mathbb{E}_{y\sim\mathcal{Y}}\|y_{\text{rec}}-y\|~{},$
(2)

with $\|\cdot\|$ the pixel-wise $L_{1}$ norm.

Identity loss. The identity loss regularizes the generators to be close to
identity, generally improving color composition:

$\displaystyle\\!\\!\\!\mathcal{L}_{\text{id}}\\!=\\!\mathbb{E}_{x\sim\mathcal{X}}\|G_{\mathcal{Y}\mapsto\mathcal{X}}(x)\\!-\\!x\|\\!+\\!\mathbb{E}_{y\sim\mathcal{Y}}\|G_{\mathcal{X}\mapsto\mathcal{Y}}(y)\\!-\\!y\|.$
(3)

Final Loss. The final objectives are:

$\displaystyle\mathcal{L}(G_{\mathcal{X}\mapsto\mathcal{Y}},G_{\mathcal{Y}\mapsto\mathcal{X}})$
$\displaystyle=\mathcal{L}_{GAN}+\lambda_{\text{cyc}}\mathcal{L}_{\text{cyc}}+\lambda_{\text{id}}\mathcal{L}_{\text{id}}~{},$
(4) $\displaystyle\mathcal{L}(D_{\mathcal{X}},D_{\mathcal{Y}})$
$\displaystyle=-\mathcal{L}_{\text{GAN}}~{},$ (5)

with $\lambda_{\text{cyc}}$ and $\lambda_{\text{id}}$ non-negative
hyperparameters.

###  3.2 StegoGAN

We introduce StegoGAN, a novel model building on the CycleGAN framework [39],
designed specifically for scenarios where domains $\mathcal{X}$ and
$\mathcal{Y}$ lack a bijective relationship. When generating images
$y_{\text{gen}}$ in $\mathcal{Y}$, the generator
$G_{\mathcal{X}\mapsto\mathcal{Y}}$ may add spurious instances of an
unmatchable class $k^{\text{unmatch}}_{\mathcal{Y}}$ in order to deceive the
discriminator $D_{\mathcal{Y}}$. Our goal is to prevent the generation of such
hallucinated instances. To achieve this, we leverage steganography to
explicitly disentangle the matchable and unmatchable information in the
backward cycle ($y\mapsto x_{\text{gen}}\mapsto y_{\text{rec}}$) and prevent
the network from hallucinating in the forward translation ($x\mapsto
y_{\text{gen}}\mapsto x_{\text{rec}}$). See Figure 2 for the overall design of
our approach.

Steganography. In order to faithfully reconstruct $y$ with $y_{\text{rec}}$,
the translated image $x_{\text{gen}}$ must somehow contain information about
the instances of the unmatchable class $k^{\text{unmatch}}_{\mathcal{Y}}$.
CycleGAN methods typically achieve this by hiding low-amplitude and high-
frequency patterns in $x_{\text{gen}}$ that will be decoded by
$G_{\mathcal{X}\mapsto\mathcal{Y}}$ and translated back to instances of
$k^{\text{unmatch}}_{\mathcal{Y}}$ in $y_{\text{rec}}$. Adding low-intensity
noise on each pixel is typically sufficient to destroy the hidden information
and prevent steganography [8].

Steganography is often viewed as an optimization flaw that undermines the
consistency loss. However, in the case of non-bijective translation, this is
the only way to let $G_{\mathcal{X}\mapsto\mathcal{Y}}$ reconstruct instances
of $k^{\text{unmatch}}_{\mathcal{Y}}$ in $y_{\text{rec}}$. Instead of
disabling steganography, we propose to use it to our advantage to detect and
prevent spurious generations. We adapt CycleGAN so that steganography takes
place in feature-space instead of pixel-space, and in an explicit manner.

Backward Cycle. We decompose the generators
$G_{\mathcal{X}\mapsto\mathcal{Y}}$ and $G_{\mathcal{Y}\mapsto\mathcal{X}}$
into two components: an encoder and a decoder, such that
$G_{\mathcal{X}\mapsto\mathcal{Y}}=G_{\mathcal{X}\mapsto\mathcal{Y}}^{\text{dec}}\circ
G_{\mathcal{X}\mapsto\mathcal{Y}}^{\text{enc}}$ and
$G_{\mathcal{Y}\mapsto\mathcal{X}}=G_{\mathcal{Y}\mapsto\mathcal{X}}^{\text{dec}}\circ
G_{\mathcal{Y}\mapsto\mathcal{X}}^{\text{enc}}$. The encoders map their inputs
to feature maps of spatial dimension $H\times W$, where each pixel has $C$
channels. In the following, we denote the intermediary representation of $y$
in $G_{\mathcal{Y}\mapsto\mathcal{X}}$ by
$z_{\text{gen}}\vcentcolon=G_{\mathcal{Y}\mapsto\mathcal{X}}^{\text{enc}}(y)$.
The feature map $z_{\text{gen}}$ encodes information about both matchable and
unmatchable classes, which we want to disentangle.

We introduce a network $M$: $\mathbb{R}^{H\times W\times
C}\mapsto[0,1]^{H\times W\times C}$ that assigns an _unmatchability_ score
between $0$ and $1$ to each pixel and channel. Here, a score of $1$ indicates
that the information does not have a counterpart in domain $\mathcal{X}$,
while it does for $0$. This process gives us the unmatchability mask
$M(z_{\text{gen}})$, which we use to split $z_{\text{gen}}$ into its matchable
and unmatchable parts:

$\displaystyle z_{\text{gen}}^{\text{unmatch}}$
$\displaystyle=M(z_{\text{gen}})\odot z$ (6) $\displaystyle
z_{\text{gen}}^{\text{match}}$
$\displaystyle=\left(1-M(z_{\text{gen}})\right)\odot z~{},$ (7)

with $\odot$ the pixel-wise and channel-wise Hadamard product. In our model,
the generated image $x_{\text{gen}}$ is computed using only the matchable part
of the representation:

$\displaystyle
x_{\text{gen}}=G_{\mathcal{Y}\mapsto\mathcal{X}}^{\text{dec}}(z_{\text{gen}}^{\text{match}})~{}.$
(8)

We produce two reconstructions of $y$: $y_{\text{rec}}^{\text{clean}}$, which
is a direct back-translation of $x_{\text{gen}}$ into the $\mathcal{Y}$ domain
:
$y_{\text{rec}}^{\text{clean}}=G_{\mathcal{X}\mapsto\mathcal{Y}}(x_{\text{gen}})$;
and $y_{\text{rec}}$, which is generated by decoding a combination of the
unmatchable part of $z_{\text{gen}}$ and the features extracted by
$G_{\mathcal{X}\mapsto\mathcal{Y}}^{\text{enc}}$ from a noise-perturbed
version of $x_{\text{gen}}$:

$\displaystyle
y_{\text{rec}}=G_{\mathcal{X}\mapsto\mathcal{Y}}^{\text{dec}}\left(G_{\mathcal{X}\mapsto\mathcal{Y}}^{\text{enc}}\left(x_{\text{gen}}+\epsilon\right)+z_{\text{gen}}^{\text{unmatch}}\right)~{},$
(9)

with $\epsilon$ denoting random Gaussian noise of low amplitude applied to
each pixel and channel of $x_{\text{gen}}$. This noise is added to destroy
potential steganographic information in $x_{\text{gen}}$, therefore forcing
$G_{\mathcal{X}\mapsto\mathcal{Y}}$ to rely only on
$z_{\text{gen}}^{\text{unmatch}}$ to reconstruct unmatchable features in
$y_{\text{rec}}$.

The key mechanisms to disentangle matchable and unmatchable information are
twofold: (i) disturbing direct steganography with random noise, and (ii)
explicitly providing unmatchable information to
$G_{\mathcal{X}\mapsto\mathcal{Y}}$ _in feature-space_.

Forward Cycle. In the forward cycle $x\rightarrow y_{\text{gen}}\rightarrow
x_{\text{rec}}$, the generator $G_{\mathcal{X}\mapsto\mathcal{Y}}$ may create
spurious instances of unmatchable classes when translating $x$ to
$\mathcal{Y}$ to fulfill the expectations of the discriminator $D_{y}$. To
address this, we perform two distinct translations of $x$ in $\mathcal{Y}$:
$y_{\text{gen}}$ has explicit access to the steganographic information
$z_{\text{gen}}^{\text{unmatch}}$ extracted from the backward cycle, while
$y_{\text{gen}}^{\text{clean}}$ does not:

$\displaystyle
y_{\text{gen}}=G_{\mathcal{X}\mapsto\mathcal{Y}}^{\text{dec}}\left(G_{\mathcal{X}\mapsto\mathcal{Y}}^{\text{enc}}(x)+z_{\text{gen}}^{\text{unmatch}}\right)$
(10) $\displaystyle
y_{\text{gen}}^{\text{clean}}=G_{\mathcal{X}\mapsto\mathcal{Y}}\left(x\right)~{}.$
(11)

The rationale is that $G_{\mathcal{X}\mapsto\mathcal{Y}}^{\text{dec}}$ has
explicit access to information about the unmatchable classes of $y$, so it is
not incentivized to invent them. For consistency with the backward step, where
the decoder of $G_{\mathcal{Y}\mapsto\mathcal{X}}$ processes only matchable
information as defined in (7), we use the same disentanglement approach for
generating $x_{\text{rec}}$:

$\displaystyle
x_{\text{rec}}=G_{\mathcal{Y}\mapsto\mathcal{X}}^{\text{dec}}\left(\left(1-M\left(z_{\text{rec}}\right)\right)\odot
z_{\text{rec}}\right)~{},$ (12)

where
$z_{\text{rec}}=G_{\mathcal{Y}\mapsto\mathcal{X}}^{\text{enc}}(y_{\text{gen}})$
is the intermediary representation of $y_{\text{gen}}$ in the forward cycle.

Source | Ground Truth | StegoGAN | CycleGAN [39] | DRIT [23] | GcGAN [12] | CUT [29] | SRUNIT [20]
---|---|---|---|---|---|---|---
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
Figure 3: Qualitative Comparison. We report reconstructions from the test sets
of PlanIGN (top two rows), GoogleMap (row 3 and 4), and MRI (last two rows).
Contrary to the other models, StegoGAN does not hallucinate spurious toponyms,
highways (orange roads), or tumors (white areas) and shows better semantic
correspondences during translation.

Mask Regularization. To avoid degenerate behaviors of our explicit
steganography mechanism, we enforce two priors on the unmatchability masks:
(i) given that a well-posed translation problem predominantly involves
matchable features, the masks should be sparse; (ii) to improve the model’s
interpretability, we favor mask values near $0$ or $1$, representing clear
decisions about matchability. To enforce these priors, we regularize the masks
with the non-convex $L_{0.5}$ norm [37]:

$\mathcal{L}_{reg}=\mathbb{E}_{y\sim\mathcal{Y}}\|M(z_{\text{gen}})\|_{0.5}+\mathbb{E}_{x\sim\mathcal{X}}\|M(z_{\text{rec}})\|_{0.5}~{}.$
(13)

Matchable Consistency. The images $y_{\text{gen}}$ and
$y_{\text{gen}}^{\text{clean}}$, as well as $y_{\text{rec}}$ and
$y_{\text{rec}}^{\text{clean}}$, should be identical outside of unmatchable
regions. To enforce this constraint, we design a function $I$ which takes an
unmatchability mask $m$ as input, takes the channel-wise maximum values of all
pixels, flips its value from $[0,1]$ to $[1,0]$, and upsamples the results to
the dimensions of the input images.

$\displaystyle I(m)=\operatorname{upsample}\left(1-\max_{c}m\right)~{}.$ (14)

The obtained consistency masks $I(m)$ have values close to $1$ for pixels with
only matchable content, and close to $0$ otherwise. This enables us to define
a loss for $y_{\text{gen}}^{\text{clean}}$ and $y_{\text{rec}}^{\text{clean}}$
that focuses solely on regions with matchable features:

$\displaystyle\mathcal{L}_{\text{match}}$
$\displaystyle=\mathbb{E}_{y\sim\mathcal{Y}}\|I\left(M(z_{\text{gen}})\right)\odot\left(y_{\text{gen}}-y_{\text{gen}}^{\text{clean}}\right)\|$
$\displaystyle+\mathbb{E}_{x\sim\mathcal{X}}\|I\left(M(z_{\text{rec}})\right)\odot\left(y_{\text{rec}}-y_{\text{rec}}^{\text{clean}}\right)\|~{}.$
(15)

Final Objective. In addition to the standard CycleGAN loss components (4-5),
we integrate $\mathcal{L}_{\text{match}}$ and $\mathcal{L}_{\text{reg}}$ into
the overall loss function
$\mathcal{L}(G_{\mathcal{X}\mapsto\mathcal{Y}},G_{\mathcal{Y}\mapsto\mathcal{X}})$,
weighted by their respective coefficients $\lambda_{\text{reg}}$ and
$\lambda_{\text{match}}$. Crucially, our proposed approach remains
unsupervised, requiring neither aligned images from $\mathcal{X}$ and
$\mathcal{Y}$ nor specific annotations of unmatchable features.

StegoGAN (ours)CycleGAN [39]DRIT [23]GcGAN [12]CUT [29]SRUNIT [20] | $0$$25$$45$$65$$20$$35$$\leftarrow$ RMSE | $0$$25$$45$$65$$20$$45$Acc($\sigma_{1}$) $\rightarrow$ | $0$$25$$45$$65$$40$$70$Acc($\sigma_{2}$) $\rightarrow$
---|---|---|---
$0$$25$$45$$65$020$\leftarrow$ pFPR (‱) | $0$$25$$45$$65$020$\leftarrow$ iFPR (%) | $0$$25$$45$$65$70100$\leftarrow$ FID | $0$$25$$45$$65$410$\leftarrow$ KID
Percentage of images with highways in the target domain.

Figure 4: Results on GoogleMaps. We report the performance of several top-
performing image translation models for different ratios of unmatchable
features in the target domain of the training set. StegoGAN handles higher
ratios better than competing methods.

##  4 Experiments

In this section, we assess the improvements brought by our method for non-
bijective image translation across various datasets and compare them with
existing models, both qualitatively and quantitatively.

Implementation details. We follow the setting of CyleGAN [39] as our baseline
model: the generators are Resnets [16] and the discriminator is based on
PatchGAN [19]. We define the encoders as the first half of the generator’s
layers and the decoders as the second half. The unmatchability mask predictor
$M$ is defined as a small 3-layer convolutional neural network (CNN). We set
$\lambda_{\text{cyc}}=10$, $\lambda_{\text{idt}}=0.5$ as in [39] and
$\lambda_{\text{match}}=1$. The amplitude of the perturbation $\epsilon$ is
$0.01$ as in [8]. The Appendix provides more architecture and training
details.

###  4.1 Datasets

We assess the performance of StegoGAN across several image translation tasks
that feature unmatchable classes. Each dataset follows a consistent structure:
the training set includes images from the source domain $\mathcal{X}$ devoid
of a specific class (_e.g_., tumors, motorways, toponyms) while the target
domain $\mathcal{Y}$ does include that class. The test set comprises _paired_
images from both domains, specifically excluding the unmatchable class. This
setup allows us to quantify the models’ hallucinations: any generated
instances of the unmatchable class are necessarily spurious and due to its
presence in the training set. We release all three curated datasets on the
Zenodo platform and provide details below.

PlanIGN. $\mathcal{X}:$ Aerial Photo, $\mathcal{Y}:$ Maps,
$k^{\text{unmatch}}_{\mathcal{Y}}:$ Text. We construct a dataset using open
data from the French National Mapping Agency (IGN), comprising $1900$ aerial
(ortho-)images at $3$m spatial resolution, and two versions of their
corresponding topographic maps: one with and one without toponyms. This
dataset presents a clear unmatchable class: place names. The training set
includes $1000$ maps with toponyms and $1000$ aerial images, while the test
set comprises $900$ map samples without toponyms and their corresponding
aerial images.

GoogleMaps. $\mathcal{X}:$ Aerial Photo, $\mathcal{Y}:$ Maps,
$k^{\text{unmatch}}_{\mathcal{Y}}:$ Highways. The GoogleMaps dataset [19] is a
standard benchmark for image translation tasks [21, 20]. It contains $1096$
map/image pairs for training and $1098$ for testing. To create a controlled
non-bijective scenario, we exclude all satellite images that show highways and
sample the maps of the training set to contain varying proportions of maps
with highways, ranging from $0\%$ to $65\%$, for a fixed total of $548$ maps.
For the test set we selected $898$ pairs without highways.

Brats MRI $\mathcal{X}:$ T1 Scans, $\mathcal{Y}:$ FLAIR,
$k^{\text{unmatch}}_{\mathcal{Y}}:$ Tumors. Lastly, we used a dataset of brain
MRI scans [28] with two modalities: T1 (naive) and FLAIR (T2 Fluid Attenuated
Inversion Recovery) [15]. We adapt the protocol that Cohen _et al_. [9] used
for the Brats2013 datasets [27] to the more recent Brats2018 [3] dataset by
varying the percentage of scans with tumors in the target domain. We selected
transverse slices from the $60^{\circ}$ to $100^{\circ}$ range in the
caudocranial direction [1] for both T1 and FLAIR scans. Each scan was
classified as tumorous if more than $1\%$ of its pixels were labeled as such,
and as healthy if it contained no tumor pixels. The training set contains
$800$ images from each modality, with all source images (T1) being healthy and
the target domain (FLAIR) comprising $60\%$ tumorous scans. The test set
contains $335$ paired scans of healthy brains.

###  4.2 Evaluation metrics

We use a broad range of metrics to evaluate the performance of StegoGAN and
other image translation algorithms in the non-bijective setting:

FID and KID. The Fréchet Inception Distance (FID) [17] and Kernel Inception
Distance (KID) [5] are widely used to quantify the similarity between the
distributions of real and generated images in the target domain.

RMSE, Acc($\sigma_{1}$), and Acc($\sigma_{2}$). As the test sets comprise
paired images from both domains, we can directly compare the Root Mean Square
Error (RMSE) between the real and predicted images in the target domain. We
count a predicted pixel as correctly predicted if it deviate by less than a
fixed threshold in any of the color channels [12]. We use $\sigma_{1}=5$ and
$\sigma_{2}=10$ for the GoogleMap dataset and $\sigma_{1}=2$ and
$\sigma_{2}=5$ for the less colorful PlanIGN dataset.

pFPR and iFPR. In the GoogleMap dataset, highways are always depicted in
orange, allowing us to label pixels where all color channels differ by less
than $20$ units from $(240,160,30)$ as highways. In the Brats MRI dataset, we
use a pretrained tumor detector [6] to find spurious tumors in the generated
images. This allows us to compute the average false positive rate per pixel
(pFPR) and per instance (iFPR) of the generated images.

###  4.3 Results

Table 1: Quantitative Comparison on PlanIGN. Our model shows a remarkably
better performance than other existing models.

Method | RMSE$\downarrow$ | Acc($\sigma_{1}$)$\uparrow$ | Acc($\sigma_{2}$)$\uparrow$ | FID$\downarrow$ | KID$\downarrow$
---|---|---|---|---|---
CUT [29] | 30.5 | 46.7 | 55.8 | 68.4 | 2.8
CycleGAN [39] | 27.0 | 15.1 | 57.6 | 97.5 | 6.6
DRIT [23] | 34.8 | 33.6 | 36.9 | 76.4 | 3.8
GcGAN [12] | 32.7 | 54.5 | 56.9 | 110.8 | 8.2
SRUNIT [20] | 32.3 | 48.8 | 52.8 | 60.2 | 2.2
StegoGAN (ours) | 22.5 | 66.1 | 74.8 | 58.4 | 2.4

Qualitative Results. Figure 3 showcases StegoGAN’s qualitative performance
against other image translation algorithms. Notably, StegoGAN effectively
avoids generating unmatchable classes such as texts, highways, and tumors,
while producing high-quality image translations.

Quantitative Results. On the PlanIGN dataset (Table 1) and the Brats MRI
dataset (Table 2), StegoGAN outperforms others in fidelity, achieving the
lowest RMSE by a margin of $4.5$ on PlanIGN and by $3.5$ for Brats MRI.
Furthermore, it significantly enhances pixel accuracy, with improvements of
$+11.6$ in Acc($\sigma_{1}$) and $+17.2$ in Acc($\sigma_{2}$) on PlanIGN. In
the MRI dataset, StegoGAN dramatically reduces false positive rates—–over
20$\times$ lower than CycleGAN and 10$\times$ less than the next best model
SRUNIT (for pFPR).

On the GoogleMap dataset, as shown in Figure 4, StegoGAN’s performance is on
par with CycleGAN at $0\%$ unmatchable cases and remains stable even as this
ratio increases, unlike other methods that degrade. Remarkably, StegoGAN
maintains a consistent false positive rate of $0$ across all tests, while this
rate increases for all other methods.

Table 2: Quantitative Comparison on Brats MRI Flair $\rightarrow$ T1. Our
model outperforms competing method in terms of both reconstruction accuracy
and consistency.

Method | RMSE$\downarrow$ | pFPR(‱)$\downarrow$ | iFPR$\downarrow$ | FID$\downarrow$ | KID$\downarrow$
---|---|---|---|---|---
CUT [29] | 39.8 | 17.0 | 23.0 | 103.9 | 8.8
CycleGAN [39] | 39.9 | 21.9 | 22.7 | 89.8 | 7.7
DRIT [23] | 53.0 | 18.5 | 41.2 | 123.4 | 11.6
GcGAN [12] | 41.7 | 24.7 | 22.4 | 61.9 | 3.5
SRUNIT [20] | 42.5 | 15.1 | 21.8 | 58.9 | 2.9
StegoGAN (ours) | 36.3 | 1.1 | 4.2 | 58.5 | 2.4

Target | Source | Mask
---|---|---
|  |
|  |
|  |
Figure 5:  Unmatchability Masks. The unmatchability masks predicted in the
backward cycle follow the instances of unmatchable features in the target
domain: toponyms, highways, and tumors.

Unmatchability Masks. In Figure 5, we illustrate the emergent ability of the
unmatchability masks to trace the outline of unmatchable class instances like
toponyms, highways, and tumors. This aspect highlights the versatility of our
approach, which functions without explicit supervision or aligned images,
offering a tool to explore the pairwise semantic differences between arbitrary
datasets.

###  4.4 Ablation study and analysis

We explore the impact of our main design choices, as well as further
capabilities and limitations of our approach. See the Appendix for further
ablations.

Table 3: Ablation Study on Encoder Depth. We evaluate the impact of changing the depth of the encoder on the reconstruction fidelity and the quality of unmatchability masks. Depth= -1 or 8 means no encoder or no decoder, respectively.  Depth | Mask | Prediction
---|---|---
mIOU$\uparrow$ |  Prec.$\uparrow$ |  recall$\uparrow$ |  RMSE$\downarrow$ |  A($\sigma_{1}$)$\uparrow$ |  A($\sigma_{2}$)$\uparrow$ |  FID$\downarrow$ |  KID$\downarrow$
-1 |  26.6 |  27.1 |  81.2 | 22.4 |  64.3 | 74.2 | 58.8 |  2.5
1 |  25.2 |  25.8 |  81.2 | 22.5 |  66.1 | 74.8 | 58.4 |  2.4
3 |  27.1 |  27.9 |  80.4 | 22.8 |  61.6 | 73.4 | 62.9 |  3.0
5 |  30.8 |  33.4 |  69.5 | 24.2 |  52.7 | 70.6 | 62.3 |  2.6
8 |  47.3 |  60.1 |  60.0 | 24.5 |  53.5 | 70.7 | 62.7 |  2.7

Source | Target | Depth = -1 | Depth = 1 | Depth = 3 | Depth = 5 | Depth = 8
---|---|---|---|---|---|---
|  |  |  |  |  |

Figure 6: Impact of Encoder Depth. We visualize the unmatchability mask for
encoders of different depths for the PlanIGN dataset. Shallower encoders
consider more features as unmatchable.

Encoder Ablation. We conducted an ablation study on the definition of the
intermediary representation $z$ by varying the depth at which the ”encoder”
ends and the ”decoder” starts within the generator. Given that our generators
consist of $9$ consecutive convolutional blocks, we experimented with
different configurations: $-1$ (indicating no decoder), $1$ (the configuration
used in our paper), as well as depths of $3$, $5$, and $8$ (implying no
decoder). We report in Table 3 the reconstruction error of these models, as
well as the fidelity of the consistency mask with the toponym text mask. We
observe that shallow encoders have better reconstruction accuracy while the
consistency masks of deeper encoders better approximate the text masks.

Visualizing these masks in Figure 6, we observe that shallow encoders consider
complex features such as highways and rivers as unmatchable features, while
deeper encoders do not. Shallower encoders seem more influenced by the
variation in appearance (_e.g_., rivers being sometimes covered in vegetation
or with varying colors) while deeper encoders focus on high-level semantics.
We argue that both definitions are equally valid, and that varying the depth
of the encoders can provide insights into the nature of the semantic mismatch
between datasets.

Parameterization. In Table 4, we analyze the effects of omitting the
additional terms $\mathcal{L}_{\text{reg}}$ and $\mathcal{L}_{\text{match}}$
from the loss function $\mathcal{L}$ in Equation 4. We show that while
$\mathcal{L}_{\text{match}}$ generally yields modest improvements across all
metrics, $\mathcal{L}_{\text{reg}}$ is pivotal, particularly for learning the
target distribution. This outcome aligns with our expectations, as the absence
of $\mathcal{L}_{\text{reg}}$ allows the network to transmit all information,
matchable or not, to the $G_{\mathcal{Y}\mapsto\mathcal{X}}$ decoder without
repercussions, impeding the training of the
$G_{\mathcal{X}\mapsto\mathcal{Y}}$ encoder.

Limitations. We augment the CycleGAN framework with a module $M$ and two
hyper-parameters $\lambda_{\text{match}}$ and $\lambda_{\text{reg}}$, thereby
adding to the complexity of its training dynamics. Moreover, the concept of
unmatchability, integral to our approach, is inherently subjective. Given
enough semantic detail, any two distinct datasets could be considered
unmatchable. As a result, fine-tuning the hyperparameter
$\lambda_{\text{reg}}$ is essential to balance the elimination of
hallucinations against the retention of necessary details: increasing its
value leads to more conservative masks and improves the visual appearance of
the generated images, at the cost of more spurious features. Visualizing the
consistency mask is often a useful form of guidance. More details on learning
strategies can be found in the Appendix.

Table 4: Impact of Additional Loss Terms. We evaluate on the GoogleMap dataset
the effect of removing our proposed losses. $\mathcal{L}_{\text{match}}$ has a
small impact while $\mathcal{L}_{\text{reg}}$ is pivotal.

Settings | RMSE$\downarrow$ | Acc($\sigma_{1}$)$\uparrow$ | Acc($\sigma_{2}$)$\uparrow$ | FID$\downarrow$ | KID$\downarrow$
---|---|---|---|---|---
$\mathcal{L}_{\text{reg}}$ | $\mathcal{L}_{\text{match}}$
✓ | ✓ | 22.7 | 41.7 | 67.1 | 77.3 | 6.8
✓ | ✗ | 24.1 | 41.5 | 64.7 | 88.0 | 7.5
✗ | ✓ | 26.7 | 26.7 | 58.9 | 271.1 | 26.6
✗ | ✗ | 25.0 | 25.0 | 61.0 | 303.6 | 33.4

We also acknowledge the potential of recent denoising diffusion models for
image-to-image translation tasks [35, 25]. While our method is not confined to
GANs and could be adapted to diffusion models with cycle consistency losses
[34, 31], unpaired image translation with diffusion models is a nascent field
with unique challenges. We plan to explore this area in future research.

##  5 Conclusions

We have introduced StegoGAN, a model built upon the CycleGAN framework, which
leverages the mechanism of steganography to address the challenges of non-
bijective image-to-image translation. Our model demonstrates an improved
capability to handle divergent distributions between domains, as evidenced by
its performance across various datasets, including aerial imagery, topographic
maps, and MRI scans. We hope that our work will inspire further research in
the little-studied area of non-bijective image translation. We find this
research direction inportant to ensure image translation models are
transferable and applicable in real-world scenarios, where datasets rarely
conform to the level of curation typically found in research benchmarks.

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Supplementary Material

In this appendix, we first detail the construction of the three datasets used
in the experiments (Section 1), then provide additional implementation details
(Section 2), ablation experiments (Section 3), and qualitative results
(Section 4).

##  1 Details on Dataset Construction

###  1.1 GoogleMaps

While this paper focuses on non-bijective translation, the official GoogleMaps
dataset was introduced in CycleGAN [39] for bijective translation between
serial photos and maps. Consequently, we propose a protocol to create
controllable non-bijectivity in this dataset.

We select the “highway” class for its prevalence and its distinctiveness on
maps: they are always represented by the same orange hue. This allows us to
easily detect highways on maps by thresholding in color space: a pixel is a
highway if all color channels are closer than $20$ units from $(240,160,30)$.
In total, $356$ image/map pairs of the train set of the GoogleMaps contain
highways, and $740$ do not.

For the training set, the source domain is always defined as $548$ aerial
images that do not contain highways. We define different versions of the
target domain for the test set by fixing the ratio of maps that contains
highways, from $0\%$ to $60\%$, for a fixed total of $548$ images. The test
set is composed of $899$ pairs of aligned aerial photos and maps _that do not
contain the highways class_ from the test set of the GoogleMaps dataset.

###  1.2 Brats MRI

We adapt the protocol of Cohen _et al_. [9] from the Brats2013 datasets [27]
to the more recent, larger, and more diverse Brats2018 dataset [2]. We
consider two MRI modalities: native (T1) and Fluid Attenuated Inversion
Recovery (FLAIR). We selected transverse slices from the $60^{\circ}$ to
$100^{\circ}$ range in the caudocranial direction [1] for both modalities of
scans.

We label each scan as tumorous if more than $1\%$ of its pixels are labelled
as such, and as healthy if it contains no tumor pixels. We only use high-grade
gliomas (HGG) instead of low-grade gliomas (LGG) as the are more easily
observable [24]. In total, we obtain 5035 pathological pairs and 1135 healthy
pairs. The train set is composed of a source domain of $800$ T1 images of
healthy brains, while the target domain set is composed of FLAIR scans of
which $480$ (60%) are tumorous and $320$ healthy. The test set is composed of
$335$ aligned scans of healthy brains in both modalities.

Figure A-1: Spatial Distribution of Samples in PlanIGN.

###  1.3 PlanIGN

We construct the PlanIGN dataset from two open-access sources available on the
French governmental geoportal: aerial orthophotos and Plan IGN cartographic
product, both projected in RGF93-Lambert-93. As the maps are derived directly
from the orthophotos, we ensure the precise spatial alignment between both
modalities.

#### Sampling.

We consider aligned image/map pairs of resolution $256\times 256$ at a scale
of 1:12500 and a graphics resolution of 96 dpi, corresponding to a ground
sampling distance of $3.3$m per pixel. We randomly select samples across the
French territory with a $3$km buffer between images. We removed images that
were blurry, with significant radiometric aberrations, over sensitive areas,
or for which the roads were significantly occluded. In total, we sample $1900$
such pairs, whose spatial distribution is shown in Figure A-1, and whose
semantic distribution is given in Figure A-2.

#### Processing.

We apply the following processing to the maps to make them easier to
translate:

  * •

We remove underground objects and small paths.

  * •

We homogenize the palette: we use the same color for all roads except
highways, and for buildings and hydrology.

  * •

We optionally add the toponyms to the maps. When we do, we also compute a
toponym mask by applying a 4-pixel dilation to the binary difference between
the map with and without toponyms.

#### Dataset.

The training set is composed of $1000$ orthophotos for the source domain and
$1000$ maps with toponyms for the target domain. The test set is composed of
$900$ aligned pairs of orthophotos and maps without toponyms.

TrainTest$0$$0.2$$0.4$$0.6$$0.8$$1$ProportionRoadsBuildingsHydrologyVegetationToponyms
Figure A-2:  Semantic Distribution in PlanIGN.

##  2 Additional Implementation Details

We follow the same architecture and hyperparameters as CycleGAN, including the
ResNet-based generator [16] with $9$ residual blocks, PatchGAN discriminator
[19], and weights in th loss. We train our model for $200$ epochs with a
learning rate of $0.002$ and the ADAM optimizer [22].

The hyperparameters for each dataset are given in Table A-1. Most methods use
similar hyperparameters with two exceptions:

  * •

Due to the high heterogeneity and noisiness of scans across different MRI
machines, we use a larger batch size of $12$.

  * •

We observed better unmatchability masks with shallower encoders for PlanIGN,
whereas it was the contrary for other datasets. Section 4.4 has pointed out
that shallower encoder seems more influenced by the variation in appearance.
Despite that class like hydrology exits both in the source and target domain,
variations in colors or occlusions by vegetation occurring often in the aerial
images challenge the model to establish correct correspondences. This should
also be empirically regarded as mismatch, which shallower encoder performs
better to capture.

Table A-1: Hyperparameters. We report the value for different parameters
across the datasets used in the experiments.

Dataset | $\lambda_{\text{reg}}$ | Encoder_Depth | Batch_Size | $\lambda_{\text{match}}$
---|---|---|---|---
GoogleMaps | 0.3 | 8 | 1 | 1
PlanIGN | 0.25 | 1 | 1 | 1
Brats | 0.3 | 8 | 12 | 1

##  3 Additional Ablation Study

Table A-2: Impact of $\lambda_{\text{reg}}$. We report the performance on the
GoogleMaps dataset of our method for different values of regularization
strength. Values between $0.3$ and $0.5$ give good results, while the
performance rapidly decreases above $0.6$.

$\mathcal{L}_{\text{reg}}$ |  RMSE$\downarrow$ |  Acc($\sigma_{1}$)$\uparrow$ |  Acc($\sigma_{2}$)$\uparrow$ |  pFPR(‱)$\downarrow$ |  iFPR$\downarrow$ |  FID$\downarrow$ |  KID$\downarrow$
---|---|---|---|---|---|---|---
0.1 | 26.1 | 11.0 | 60.4 | 0.1 | 7.7 | 254.4 | 25.8
0.2 | 23.3 | 36.5 | 64.0 | 0.0 | 0.0 | 141.2 | 17.4
0.3 | 22.7 | 41.7 | 67.1 | 0.0 | 0.0 | 77.3 | 6.8
0.4 | 23.5 | 41.8 | 65.7 | 0.0 | 0.2 | 84.8 | 8.3
0.5 | 24.4 | 39.5 | 64.6 | 0.0 | 1.1 | 86.1 | 8.1
0.6 | 25.2 | 40.0 | 64.8 | 19.1 | 16.8 | 103.5 | 11.3
0.7 | 25.7 | 36.8 | 62.7 | 18.8 | 22.2 | 110.1 | 11.3

The hyperparameter $\lambda_{\text{reg}}$ is crucial as it enforces the
sparsity of the unmatchability masks. We report its impact in Table A-2. Too
small values may lead to a too-liberal use of the unmatchability masks,
resulting in a loss of details in the clean generation. Too large values will
prevent our model from using the unmatchability masks altogether.

##  4 Additional Qualitative Results

We provide additional results for GoogleMaps in Figure A-3, PlanIGN in Figure
A-4 and Brats MRI in Figure A-5.

Application to Natural Images. We apply our method to natural image datasets;
see Fig. A-6. StegoGAN performs well in this setting and generates faithful
yet realistic images. Compared to CycleGAN, the clean translation
$y^{\text{clean}}_{\text{gen}}$ produces fewer unmatchable features like snow
or color shifts (Summer $\mapsto$ Winter example), or internal structures of
fruits (Apple $\mapsto$ Orange example). However, since such features often
contribute to the realism and visual appeal of the translated images.
Therefore, our method is better suited for domains that value reliability over
aesthetics, such as medical images or cartography.

Source | Ground Truth | StegoGAN |  CycleGAN [39] |  DRIT [23] |  GcGAN [12] |  CUT [29] |  SRUNIT [20]
---|---|---|---|---|---|---|---
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
Figure A-3: Additional qualitative comparison on Google Photo$\rightarrow$Map. Source | Ground Truth | StegoGAN |  CycleGAN [39] |  DRIT [23] |  GcGAN [12] |  CUT [29] |  SRUNIT [20]
---|---|---|---|---|---|---|---
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
Figure A-4: Additional qualitative comparison on PlanIGN. Source | Ground Truth | StegoGAN |  CycleGAN [39] |  DRIT [23] |  GcGAN [12] |  CUT [29] |  SRUNIT [20]
---|---|---|---|---|---|---|---
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
Figure A-5: Additional qualitative comparison on Brats.  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---
|  |  |  |  |  |  |
Input |  $y^{\text{clean}}_{\text{gen}}$ StegoGAN $y_{\text{gen}}$ |  CycleGAN [39] | Input |  $y^{\text{clean}}_{\text{gen}}$ StegoGAN $y_{\text{gen}}$ |  CycleGAN [39]
Figure A-6: Natural Images Translation. We apply our model to the Summer
$\mapsto$ Winter Yosemite (top row) and Apple $\mapsto$ Orange datasets
(bottom row).


# Compact in-vacuum gamma-ray spectrometer for high-repetition rate PW-class
laser-matter interaction

G. Fauvel<EMAIL_ADDRESS>ELI Beamlines Facility|The Extreme Light
Infrastructure ERIC Za Radnicí 835, 252 41 Dolní Břežany, Czech Republic
Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical
Engineering , 115 19, Czech Republic  K. Tangtartharakul  Center for Energy
Research, University of California San Diego, La Jolla, CA 92093, USA  A.
Arefiev  Center for Energy Research, University of California San Diego, La
Jolla, CA 92093, USA  O. Klimo  Czech Technical University in Prague, Faculty
of Nuclear Sciences and Physical Engineering , 115 19, Czech Republic  M.
Manuel  General Atomics, San Diego, CA 92186, USA  P. Rubovic  ELI Beamlines
Facility|The Extreme Light Infrastructure ERIC Za Radnicí 835, 252 41 Dolní
Břežany, Czech Republic  S. Weber  ELI Beamlines Facility|The Extreme Light
Infrastructure ERIC Za Radnicí 835, 252 41 Dolní Břežany, Czech Republic  F.P.
Condamine  ELI Beamlines Facility|The Extreme Light Infrastructure ERIC Za
Radnicí 835, 252 41 Dolní Břežany, Czech Republic

###### Abstract

With the advent of high repetition rate laser facilities, novel diagnostic
tools compatible with these advanced specifications are in demand. This paper
presents the design of an active gamma-ray spectrometer intended for these
high repetition rate experiments, with particular emphasis on functionality
within a PW level laser-plasma interaction chamber’s extreme conditions. The
spectrometer uses stacked scintillators to accommodate a broad range of gamma-
ray energies, demonstrating its adaptability for various experimental setups.
Additionally, it has been engineered to maintain compactness, electromagnetic
pulse resistance, and ISO-5 cleanliness requirements while ensuring high
sensitivity. The paper also outlines the unfolding process to recover the
gamma-ray spectrum from the spectrometer’s captured image thanks to a
calibration using a 60Co radioactive source.

††preprint: AIP/123-QED

##  I Introduction

In recent years, advancements in laser technology across the globe have led to
an increase not only in power levels but also in repetition rates. Presently,
several facilities are capable of delivering PW class laser pulses at
intervals of a few minutes to secondsDoria  _et al._ (2020); Burdonov  _et
al._ (2021); Hakimi  _et al._ (2022); Volpe  _et al._ (2019). For instance,
the L3 laser at ELI Beamlines in Czech Republic can generate 0.5 PW shots at a
frequency of 3.3 Hz. Similarly, the BELLA center in the USA can reach 0.5 PW
shots at a frequency of 10 Hz Hakimi  _et al._ (2022). This surge in
statistical data presents exciting opportunities for the plasma community, but
it also necessitates corresponding advancements in diagnostic tools.

It is crucial that detectors are compatible with the new energy ranges of
produced particles Ridgers  _et al._ (2012); Zhu  _et al._ (2016); Norreys
(2009); Jansen  _et al._ (2018); Higginson  _et al._ (2018); Jirka  _et al._
(2017) at these new intensity levels. In this range, the gamma-ray photons
produced can provide insights into topics like the production of gamma-ray
bursts in astrophysics Piran (2005); Mészáros (2006), radioactive waste
management Galy  _et al._ (2007), and the production of matter through photon-
photon collision as theorized by Breit-Wheeler Breit and Wheeler (1934);
Krajewska and Kamiński (2012); Blackburn and Marklund (2018). There is also
significant industry interest in reliably and well-characterized production of
high energy photons, which can traverse matter more efficiently and be used as
non-destructive probes Ben-Ismaïl  _et al._ (2011); Wu  _et al._ (2018).

While several methods exist to produce high energy photons Giulietti  _et al._
(2008); Galy  _et al._ (2007); Ledingham and Galster (2010); Cole  _et al._
(2018); Nerush  _et al._ (2014); Nerush and Kostyukov (2015), a recent method
known as Direct Laser Acceleration (DLA) method Stark, Toncian, and Arefiev
(2016); Jansen  _et al._ (2018) has promising laser to photon conversion in
the range of several percents with a low divergence of gamma-ray emission.
This method involves accelerating electrons to relativistic speed in a near-
critical density target due to the ponderomotive force. The electrons produce
a quasi-static azimuthal magnetic field due to a return current, and thanks to
their oscillation within the high-intensity laser beam, they undergo
oscillations generating photons by Compton scattering. This leads to a bright,
collimated synchrotron emission-like of high-energy gamma-rays. At PW levels,
gamma-rays up to a few MeV (for 0.5PW lasers) to hundreds of MeV (for 10PW
lasers) can be producedJansen  _et al._ (2018); Li  _et al._ (2017).

In order to conduct these experiments, we need to design a diagnostic tool
suitable for these high energies. Current designs are sufficient and adaptable
for charged particles Nakamura  _et al._ (2008); Faccini  _et al._ (2010).
However, for photons existing designs, such as those for X-ray spectrometers
using diffraction Young  _et al._ (1998); Bragg and Bragg (1913);
renner_challenges_2019; condamine_high-resolution_2019, cannot be directly
applied as no crystals have a small enough interplanar spacing to diffract
these wavelength (0.1 Å). We therefore need a new type of detector. This new
energy range has been under exploration for several years, and corresponding
detectors have been developed Hannasch  _et al._ (2021); Behrens and Ambrosi
(2002); Chen  _et al._ (2008); Albert  _et al._ (2013); Henderson  _et al._
(2014); Horst  _et al._ (2015); Rhee  _et al._ (2016). These detectors utilize
a pattern of alternating absorbers and Imaging Plates (IPs). By adjusting the
thickness and material type of the absorbers, the energy spectrum can be
unfolded. However, these stacking calorimeters have a significant limitation
linked to the passivity of the detector. IPs must be analyzed between shots, a
time-consuming process reducing the advantage of new high-repetition rate
lasers without accumulating over several shots.

In response to this challenge, several research groups have developed active
gamma-ray spectrometer designs Stránský  _et al._ (2021); Istokskaia  _et al._
(2021). By operating as both absorbers and detectors simultaneously,
scintillators uphold the principle of energy deposition variation while also
ensuring compactness and sensitivity. These spectrometers are usually located
outside the interaction chamber, with the stack of scintillators placed
directly behind a Beryllium window or a thick aluminum flange. However this
additional obstacle can emit supplementary radiation or particles creating
difficulties in retrieving the original spectrum. By positioning the
spectrometer inside the chamber, we can strongly reduce this parasitic signal.

Besides, as interaction chambers continue to increase in size, centimeter-
scale scintillators fail to cover the entire opening angle of the gamma rays,
which is typically from 30° ($\sim$ 500 mrad) for low energy photons to few
degrees ($\sim$150 mrad) for the highest energy photons, as predicted for
gamma flash experimentsStark, Toncian, and Arefiev (2016) . Although
increasing the size of the scintillators is an option, it is impractical to
create very large windows due to pressure and costs constraints. Additionally,
this reduces the spectrometer’s adaptability to position as the angular
position will be only defined by the position of flanges and/or windows around
the interaction chamber. As a result, we decided to focus on designing a
spectrometer that can fit inside the experimental chamber. But, operating
within the confines of an interaction chamber presents its own set of unique
challenges, most notably the reduced available space. The chamber often houses
multiple components such as large optics which can significantly limit the
available real space for additional devices like detectors. To address this,
we introduce a highly compact gamma-ray spectrometer in this paper. The
spectrometer has been engineered with dimensions of $20\times 20\times
25\,\text{cm}^{3}$ , making it particularly well-suited for high-repetition-
rate experiments and adaptable to the rigorous environmental conditions
characteristics of a laser-plasma interaction chamber (EMPs, cleanliness,
vacuum pressure, etc.).

This manuscript describes the design of the high repetition rate spectrometer
suited for the extreme constraints of the PW-level interaction chamber in
Section II. Section III elaborates on the methodology employed for unfolding
the gamma-ray spectrum from the acquired spectrometer images. This process is
crucial for translating the raw data into meaningful energy spectrum. The
paper follows with a discussion of the calibration process, outlining the
strategies employed to align the experimental setup with the theoretical
models.

##  II Design of a gamma-ray spectrometer with stacking scintillators

In this manuscript, gamma rays are classified as photons with energy > 100
keV. To accommodate these high energies, the fluorescence properties of
various materials can be used. Gamma-ray photons deposit some of their energy
within these materials, thereby exciting them. The materials subsequently
relax, emitting fluorescence photons generally within the optical energy
range. This allows the use of an optical Charge-Coupled Device (CCD) to
collect them. By arranging scintillators in a stack configuration in the laser
axis, the first layers act as detectors for lower energy gamma rays and at the
same serve as a filter allowing for the detection of only (ideally) higher
energy gamma-rays in the deeper scintillators .

Energy deposition is primarily dependent on three factors: the energy of the
incident photon, the material of the scintillator, and its thickness. As we
aim to determine the photon, we have two adjustable parameters: the material
and the thickness of the scintillator. Different types of scintillators offer
different absorption coefficients for varying gamma-ray energies. The
spectrometer designed in this study is intended to be used in different
experimental setups covering different energy ranges. Consequently, we employ
various types of scintillators to widen the energy range. The stack uses three
different types of scintillators with various thicknesses related to their
absorption coefficients. Plastic scintillators possess a low absorption
coefficient, facilitating the detection of photons below 100 keVNational
Institute of Standards and Technology (2023a). YAG:Ce scintillators are more
suitable for detection in the MeV rangeAdvatech UK Ltd (2023), and finally,
CsI:Tl scintillators are ideal for energies above 5 to 10 MeVNational
Institute of Standards and Technology (2023b). The limited space within most
chambers led us to use a limited number of scintillators, in a layered
configuration of scintillators adapted to the expected gamma-ray spectrum.
Here the configuration includes two 2 cm thick vinyltoluene (plastic)
scintillators, ten 5 mm thick YAG:Ce scintillators, and five 5 mm thick CsI:Tl
scintillators, all furnished by Advatech UK Ltd. The arrangement of the
scintillators is illustrated in Fig. 1. Note that this modular design allows
for the individual scintillators to be readily interchanged, offering
flexibility to accommodate a range of energy spectra as necessitated by
different experimental requirements. The design prioritizes compactness to
facilitate integration with various experimental setups. Nevertheless, this
compactness comes at the cost of a fixed configuration’s versatility. To
mitigate this, one could envisage a spectrometer with a larger number of
scintillators.

Figure 1: Scheme of the scintillator at the top and physical stack at the
bottom. Each scintillator has the same transversal and vertical dimensions of
5cm each, only the thickness varies. TeflonTM tape is wrapped around each
scintillator to increase photon collection, avoid the overlap of signals
(optical photons going out through an adjacent scintillator) and allow some
spatial separation for the CCD aquisition.

A critical factor in the spectrometer’s performance is the decay time of the
scintillators, particularly the CsI:Tl, which has the longest decay time of
900 ns. This characteristic time can limit maximum repetition rate that can be
used without signal overlap. Even if the scintillator hardware is far above
the required characteristics, the camera becomes the limiting factor for the
repetition rate achievable. As an example the camera used here from Allied
Vision, model "Manta Camera 507-B", can only reach up to 23 fps but can be
easily fixed with a high speed camera. With the current setup operating at a
repetition rate of 10 Hz, the system is within the limits, ensuring clear
temporal separation of detection events.

The fluorescence photons from the scintillators are collected using an optical
CCD combined with an f/1.8 objective to have a large field of view with a
short focal distance as the spectrometer height is only 20 cm. Each
scintillator is wrapped in TeflonTM tape, leaving one polished face exposed,
to enhance photon collection towards the CCD while maintaining sufficient
spatial spacing for scintillator differentiation. To shield the CCD from
Electromagnetic Pulses (EMPs) emitted during laser-plasma interaction, a 5 mm
copper Faraday cage is installed around it. To protect the CCD from the plasma
self-emission (and any other parasitic light), a box composed of 2 mm
aluminium sheets holded by 90° clamps is positioned around the spectrometer.
Using appropriate opto-mechanical elements, we meet cleanliness requirements,
achieving a pressure of 10${}^{-6}\pm$10-7 mbar inside the interaction chamber
and passing Residual Gas Analysis (RGA) tests. The spectrometer’s assembly,
suited for installation within the chamber, is presented in Fig.2.

Figure 2: Spectrometer’s assembly suited for installation inside a PW-class
laser interaction chamber. One side and top panels have been removed for
clarity. Stack configuration is presented in Fig. 1

.

Having underlined the hardware components of the detection system, attention
must now turn to the process of data interpretation. The raw data captured by
the CCD requires a transformation to yield a meaningful gamma-ray spectrum.
This transformation process, commonly referred to as "unfolding," is essential
to convert the digitized image data into a physically interpretable format.
Unfolding is a pivotal step in the experimental pipeline as the quality of the
algorithm directly impacts the accuracy and resolution of the resultant gamma-
ray spectrum, thereby influencing the interpretation of the underlying
physical phenomena. Given the complexity of the data and the risk for various
sources of noise and distortion, the algorithm applied must be robust and
meticulously validated to ensure reliable results. The method is presented in
the next section.

##  III Methodology for Unfolding

In order to accurately reconstruct the energy spectrum of the particle source
under investigation, several computational and data analysis techniques must
be employed on the data captured by the spectrometer. One of the primary
outputs from the spectrometer is the image, which is acquired using a CCD
camera. This scintillation image serves as a raw source of information and is
directly correlated to the energy deposited within various scintillation
materials. It should be noted that the energy deposition is a function of not
only the particles’ inherent energy and type but is also modulated by a set of
factors collectively termed as the "relative light yield."

The concept of relative light yield (RLY) is a composite parameter that
includes a variety of conversion factors, each contributing to the overall
signal transformation process from the FLUKA output to a CCD pixel value
equivalent. These factors are the intrinsic light yield of the scintillation
materials utilized directly dependant on the material but can also vary
depending on the manufacturing of the scintillator crystal. The angular
dependencies affecting photon collection efficiencies for example
scintillators on the left of the image will have less photons collected by the
objective than the scintillators in the center (for a same initial emission).
The quantification efficiency of the CCD camera, which is responsible for
converting the collected photons into a digitized pixel array. This overall
factor allows a simple linear relationship between the FLUKA output and the
CCD value. This parameter, unique for each scintillator, is defined through
the calibration process, detailed in Section IV, thereby enabling the
conversion of the scintillation image to meaningful particle energy. Equation
1 is a simple equation for this RLY parameter, using $\eta_{i}$ as all the
possible conversion which contribute to the final pixel value. The RLY is
defined as $\eta_{RLY}$, the CCD pixel value is noted $E_{CCD}$ and the FLUKA
output is noted $E_{FL}$.

$\begin{array}[]{c}\text{CCD pixel value}=\text{FLUKA output
}\times\sum\text{Conversion factors}\\\ \\\
E_{CCD}=E_{FL}\times\sum_{i}\eta_{i}=E_{FL}\times\eta_{RLY}\\\ \\\
\text{Relative light yield (RLY)}=\sum\text{Conversion factors}\end{array}$
$\displaystyle\eta_{RLY}=\sum_{i}\eta_{i}$ (1)

To calculate the energy deposition in the scintillators, we employ a Monte
Carlo simulation approach using the FLUKA code Ahdida  _et al._ (2022);
Battistoni  _et al._ (2015). This is complemented by the FLAIR software
Vlachoudis (2009) a visual interface used that serves as an interface for
effectively configuring and visualizing the FLUKA simulations. Together, they
provide a robust framework for simulating the experimental setup in a virtual
environment, thereby allowing us to account for material-specific
interactions, scattering, and other complex phenomena that influence the
observed data.

Once the simulation environment is established, the energy spectrum of the
particle source is iteratively adjusted. The aim is to minimize the divergence
between the simulated and experimental data, subject to the modulation imposed
by the RLY of the scintillators used. It is critical to note that while this
is a widely used approach, alternative methodologies exist that might offer
advantages in specific applications. For instance, machine learning techniques
like neural networks or statistical methods such as Bayesian unfolding
algorithms can also be employed to decode the energy spectrum. Although these
advanced techniques are beyond the scope of this work, they represent
important directions for future research. A typical equation used to simulate
the photon spectra is a synchrotron distribution

$S(x)=x\int^{+\infty}_{x}K_{\frac{5}{3}}(\chi)d\chi$ (2)

, with $x=\frac{E}{E_{c}}$ and $E_{c}$ the critical energy and $K_{\alpha}(x)$
the modified Bessel function of order $\alpha$ at the point $x$. When
considering the electron acceleration we obtain the equation :

$S(x)=x^{2}\int_{-1}^{1}\Big{(}\frac{1}{(1-\mu^{2})^{\frac{3}{2}}}K_{\frac{2}{3}}(x(1-\mu^{2})^{-\frac{7}{4}})^{2}d\mu\Big{)}$
(3)

To further account for Bremsstrahlung-like emission, an exponential spectrum
can be used as follows:

$S(E)=\sqrt{E}e^{\frac{-E}{k_{b}T}}$ (4)

Here, $k_{B}T$ represents the effective temperature of the photon distribution
in eV.

The validation of the unfolding algorithm’s accuracy is contingent upon the
availability of empirical data. The algorithm’s consistency is evaluated
through a calibration process that determines the conversion factor between
the energy deposited inside the scintillators and the CCD value. This
calibration not only serves to fine-tune the system’s response but also
provides a basis for assessing the algorithm’s precision. By employing this
real-world data, one can benchmark the algorithm’s performance, ensuring that
the derived gamma-ray spectrum accurately reflects the physical event. The
calibration data thus plays a dual role: it helps in the optimization of the
light yield measurements and furnishes a metric against which the unfolding
algorithm’s accuracy is gauged.

##  IV Calibration using a 60Co radioactive source

The task of calibrating a stacked gamma-ray spectrometer is a complex
challenge that necessitates meticulous planning and resource allocation. One
of the most critical requirements for calibration is the selection of an
appropriate radiation source. The ideal source should possess a well-defined
and accurate energy spectrum, with photon energies exceeding the 500 keV range
and a sufficiently high photon flux to ensure statistical reliability.

In our specific case, we opted to use sources with activities in the TBq
range. After careful consideration, a ${}^{60}\text{Co}$ source was selected
for its suitability for our experiment. This source has an activity of 0.185
TBq and emits photons with two dominant photopeaks at energies of 1.173 MeV
and 1.332 MeV. The experimental setup was configured at a distance of 1 m from
the source which has an opening angle of 11° ($\sim$ 200 mrad).

Data acquisition was executed under varying conditions of gain and exposure
time to optimize the sensitivity and dynamic range of the CCD camera. A
typical scintillation image obtained from this setup is shown in Fig. 3.

Figure 3: Observed image using a 60-Co calibration source at 1m of the
detector. 40 Gain and 0.5s exposure time. Three different type of
scintillators are used, explaining the abrupt changes in scintillation along
the propagation axis. Radiation comes from the left with the source at 1m with
an opening angle of 11°. Contrast was enhanced for better visibility

Upon the successful completion of the data acquisition phase, we proceeded to
start the comparative analysis between our experimentally derived data and the
simulations generated from FLUKA. This meticulous comparison was critical for
the extraction of the RLY key metric defined in Equation 1.

To further refine the data and mitigate inconsistencies or inhomogeneities, we
applied a statistical treatment to the acquired pixel values. Specifically,
the mean pixel value was calculated for each individual scintillator in the
stacked array. This process effectively smoothen the data, making it easier to
directly compare with simulation outputs. The processed mean pixel values are
plotted against the FLUKA simulation results, as illustrated in Fig. 4.

Figure 4: Comparison between CCD output image and FLUKA simulation for a 0.185
TBq 60Co calibration source. Both FLUKA and CCD outputs are normalized.

It is now possible to compute the RLY, normalized and averaged across
specimens of each scintillator type, are presented as follows: for the plastic
scintillators, $(2.3\pm 0.3)\times 10^{-2}$; for YAG:Ce, $(1.87\pm 0.1)\times
10^{-1}$; and for CsI:Tl, $(1\pm 0.1)$ in units of GeV. These quantifications,
which represent the photon-to-light conversion efficiency, are crucial for the
accurate interpretation of the energy spectrum deduced from the observed data.
This means that every FLUKA output will be multiplied by the relative light
yield specific to each scintillator before trying to compare it to the
experimental data.

In order to use this unfolding algorithm it is necessary to simulate a fixed
spectrum, look at the FLUKA output and compare to the experimental data. To
simulate the spectrum it is necesserary to create it using an equation.
However equations might not always represent the actual signal measured
(Double population of photons, different particles, etc.). In order to check
the impact of the simulated spectrum by an equation, we compare it the actual
spectrum generated by the source. A Gaussian shape aimed to represent the
photopeaks of the radioactive source are simulated. The best fit corresponding
to a gaussian shape centered at 1.7 MeV with a width at $\frac{1}{e^{2}}$ of
0.5 MeV is presented in Fig. 5.

Figure 5: Comparison between CCD output image obtained from a 0.185 TBq 60Co
calibration source and FLUKA simulation from a gaussian distribution of
photons centered at 1.7 MeV of width 0.5 MeV. Both FLUKA and CCD outputs are
normalized.

Thus the analysis reveals that the unfolding algorithm is unable to accurately
resolve the dual photopeak structure, and exhibits a systematic shift in the
energy spectrum. This discrepancy arises primarily from two factors. Firstly,
the inherent resolution of the spectrometer constrains its ability to
distinguish between peaks that are in close proximity, such as the observed
1.173 MeV and 1.332 MeV peaks. The spectrometer’s design does not cater to the
high precision required to differentiate between energy levels that are
closely spaced.

Secondly, the source itself contributes to the deviation. The emission
includes not only the desired photons but also electrons and lower-energy
photons. These additional particles interact with the scintillator material,
depositing energy and thereby introducing extraneous signals into the measured
spectrum. Using the entirety photon spectrum emitted by the radioactive source
which includes lower energy photon photons (Bremsstrahlung, etc.) we are able
to almost perfectly match the data as presented in Fig. 6. Consequently, it
becomes imperative to have a comprehensive understanding of the spectrum of
other particles that could potentially interact with the scintillators and the
generating process of the gamma-ray emission to orientate fitting simulations.
Note that the match is not perfect as the radioactive source also emits low
energy electrons which are not included here.

Figure 6: Comparison between CCD output image obtained from a 0.185 TBq 60Co
calibration source and FLUKA simulation from theoretical photon emission
spectrum of a 60Co source. Both FLUKA and CCD outputs are normalized.

##  V Discussion

To conclude, in this manuscript an active gamma-ray spectrometer suited for
PW-class laser interaction chamber is presented for gamma-rays detection from
100 keV to 10 MeV. This spectrometer offers several advantages in its design.
First by using a stack of different scintillators it can reach high repetition
rate of up to 10 Hz which can be easily increased by upgrading the camera
model to a faster camera. This design is modular meaning each scintillator can
be easily replaced or swapped to meet the needs of energy analysis, precision
or maintenance.

The spectrometer’s compatibility with the rigorous demands of a PW-class laser
interaction chamber, such as electromagnetic pulse (EMP) resistance, vacuum
pressure tolerance, and maintenance of cleanliness, is ensured through a
careful selection of materials and design.

It is mandatory to couple this spectrometer with an electron spectrometer. The
latter serves as a magnetic field source to deflect electrons and also
provides more accurate data regarding the contribution of electrons to the
overall signal.

This spectrometer will be implemented in experiments for gamma-ray studies at
ELI-Beamlines in the Experimental Hall E3. The robust design is believed to be
able to withstand the 10 PW-class laser L4f and PW-class laser L3 of
respective repetition rate 0.016 Hz and 10 Hz. The detection of high energy
gamma-rays at such high repetition rate will allow the plasma community to
have insightful data and statistics to study unresolved phenomena like gamma-
ray bursts in astrophysics or understand efficient gamma-ray generation
parameters for example in DLA.

###### Acknowledgements.

We would like to thank V. Istokskaia , L. Giuffrida and B. Lefebvre, R.
Versaci, C. Lacoste, for fruitful conversation regarding the design and
unfolding method. We would like to also thank The National Radiation
Protection Institute (NRPI) in Prague, Cz for their help in the calibration
process. We wish to acknowledge the support of the National Science Foundation
(NSF) and Grantová agentura České republiky (GAČR) for funding on project
number No. 22-42890L.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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