# CLIPtone: Unsupervised Learning for Text-based Image Tone Adjustment

Hyeongmin Lee1,∗ Kyoungkook Kang2,∗ Jungseul Ok1,2 Sunghyun Cho1,2
POSTECH 1GSAI & 2CSE
{hmin970922, kkang831, jungseul<EMAIL_ADDRESS>
###### Abstract

Recent image tone adjustment (or enhancement) approaches have predominantly
adopted supervised learning for learning human-centric perceptual assessment.
However, these approaches are constrained by intrinsic challenges of
supervised learning. Primarily, the requirement for expertly-curated or
retouched images escalates the data acquisition expenses. Moreover, their
coverage of target styles is confined to stylistic variants inferred from the
training data. To surmount the above challenges, we propose an unsupervised
learning-based approach for text-based image tone adjustment, CLIPtone, that
extends an existing image enhancement method to accommodate natural language
descriptions. Specifically, we design a hyper-network to adaptively modulate
the pretrained parameters of a backbone model based on a text description. To
assess whether an adjusted image aligns with its text description without a
ground-truth image, we utilize CLIP, which is trained on a vast set of
language-image pairs and thus encompasses the knowledge of human perception.
The major advantages of our approach are threefold: (i) minimal data
collection expenses, (ii) support for a range of adjustments, and (iii) the
ability to handle novel text descriptions unseen in training. The efficacy of
the proposed method is demonstrated through comprehensive experiments
including a user study.

Figure 1:  We present CLIPtone, a text-based image tone adjustment framework
trained in an unsupervised manner. With its superior understanding of natural
languages, CLIPtone is capable of performing successful adjustments across a
range of text descriptions, including those previously deemed challenging.

††∗ Equal contribution

##  1 Introduction

Image tone adjustment aims for the alteration of the tonal properties of an
image, including brightness, contrast, and color balance. It is also termed
image tone enhancement, as it has primarily been exploited to enhance the
image aesthetics. It is essential for various applications ranging from
photography to medical imaging, and the importance of effective tone
adjustment techniques has grown significantly.

Recent learning-based approaches [15, 35, 17, 47] predominantly rely on
pairwise datasets and supervised learning to learn perceptual adjustments
aligned with human perception. Representative datasets like MIT-Adobe 5K [8]
and PPR10K [31], which comprise pairs of an image and its expert-retouched
version, have emerged, and several methods [17, 25, 36, 50] have been proposed
to learn the mappings between such pairs of images. However, such supervised
learning-based methodologies bear several limitations. First, they demand
cost-intensive datasets stemming from expert retouching. Second, they are
confined to limited amounts of adjustments that are inferred from the
datasets. Lastly, they allow for only automatic image enhancement and lack the
capability for diverse adjustments with any guidance, as the datasets are
composed of paired images without specific guidance annotations.

To reduce the burden of collecting paired images, a few studies [19, 11, 22,
26, 52] suggest weakly-supervised learning-based approaches to learn the
mapping from the unpaired images. They predominantly leverage the adversarial
learning of GANs [16] to capture the tonal properties of a target image set.
Although they do use unpaired data, there is still a requirement to manually
define distinct sets for both source and target images. In addition, they also
share the other limitations of the supervised learning-based approaches.

Meanwhile, a handful of works [42, 23] introduce text-based approaches to
enable controllable adjustments utilizing text descriptions. To this end, they
construct datasets consisting of triplets (i.e., a source image, a target
image, and a text description on the adjustment) and train neural networks on
them. However, due to the reliance on a limited range of descriptions in the
datasets, they cannot accommodate natural language descriptions beyond them.

In this paper, we propose a novel text-based image tone adjustment framework,
CLIPtone, that is trained in an unsupervised manner requiring only source
images and any tone-related text descriptions without the need for paired
images. Our core insight is that the assessment of perceptual adjustments can
be achieved with the recent language-image representation model, CLIP [38], as
it is trained on a vast amount of language-image pairs and thus encapsulates
the human perception. Grounded on this insight, we extend an existing image
enhancement model to enable varied adjustments based on text descriptions.
Specifically, we design a hyper-network to adaptively modulate the pretrained
parameters of a backbone model based on a text description. In order for our
network to learn tone adjustment in an unsupervised manner, we further
introduce a CLIP directional loss and regularization losses.

CLIPtone enjoys several unique benefits stemming from introducing CLIP as
criterion of human perception. It necessitates only arbitrary images and tone-
related text descriptions for its training, which can be collected with
minimal costs. It also supports vast amounts of adjustments previously deemed
challenging with text descriptions, as shown in Fig. 1, thanks to CLIP’s
comprehensive understanding of natural language. Lastly, it is capable of
handling novel text descriptions unseen in training.

Recently, in line with remarkable capabilities of contemporary generative
models, numerous studies [34, 14, 12, 7] have been proposed dictating the
attributes of the generated images with text descriptions. Among them, our
work shares similarities with text-based image colorization and text-based
image editing approaches, in the aspect of modifying input images based on
text descriptions. Compared to these methods, CLIPtone stands out for its
remarkably fewer parameters and shorter inference time.

Our contributions are summarized as follows:

  * •

We present, for the first time, an unsupervised image tone adjustment
framework CLIPtone, that captures human perception on tonal properties
utilizing CLIP, enabling a range of adjustments and zero-shot prediction.

  * •

We design a hyper-network to effectively modulate the pretrained parameters of
an existing image enhancement network and suggest training strategies tailored
for unsupervised learning of CLIPtone.

  * •

Through extensive experiments, we demonstrate that our approach is lighter,
faster, and more effective compared to existing methods.

##  2 Related Work

#### Image Tone Adjustment Methods.

To improve the aesthetic quality of images, a range of learning-based
automatic tone adjustment (or enhancement) methods have been developed. They
have explored optimal neural architectures to directly estimate dense pixel
values [21, 35, 36] or the parameters of traditional physical models such as
color transform matrices [49, 15, 9], and Look-Up Tables (LUTs) [17, 25, 52,
47, 50, 51, 53, 32]. Among these, methods exploiting 3D LUTs [52, 47, 50, 51,
53, 32] have garnered attention for their superior capacity and efficiency.
Since Zeng  _et al_. [52] have proposed adaptive 3D LUTs to input images, Wang
_et al_. [47] proposed spatially-aware 3D LUTs for local adjustment, and Yang
_et al_. [50] increased the capacities of 3D LUTs through non-uniform
sampling.

For neural networks to learn adjustments aligned with human perception,
supervised learning-based approaches [15, 35, 17, 47] have predominantly been
attempted. They rely on paired datasets such as MIT-Adobe 5K [8] and PPR10K
[31], which consist of source images and their expert-retouched counterparts.
To alleviate the collection costs associated with expert retouching, weakly-
supervised learning-based approaches [11, 19, 26, 52] have been introduced.
They use unpaired data comprised of separate source and target sets of images.
These methods primarily leverage the adversarial loss from GANs [16] to
capture the tonal properties of target sets, and the cycle-consistency loss
[55] for preserving the structures of source images.

A few methods [42, 23] employ text descriptions as conditional inputs to
enable diverse and controllable adjustment beyond automatic adjustment. They
construct training datasets consisting of triplets: a source image, a target
image, and a text description on the adjustment and train a network on these
datasets. Specifically, Shi  _et al_. [42] annotated requests to each pair of
the MIT-Adobe 5K dataset [8], and Jiang  _et al_. [23] further proposed
dataset augmentation strategies for imbalanced annotations.

Despite the aforementioned efforts, existing tone adjustment methods have
limitations including high data collection costs (even for curating target
sets), confined adjustments to stylistic variations of training datasets. In
contrast, CLIPtone is independent of these issues thanks to leveraging CLIP as
a perceptual criterion.

#### Text-driven Generative Models.

Recently, cutting-edge generative models have found their uses in image-to-
image translation tasks such as image colorization [20, 10] and editing [5,
12, 7]. Early approaches use GANs [16] trained on specific domains to
manipulate real images through GAN inversion techniques [40, 45, 24, 3],
changing their latent vectors [48, 6, 37] or adapting generator networks [30,
14] to different domains according to text descriptions. More recently, with
advances in auto-regressive models [13] and diffusion models [43, 44, 18],
generative models trained on large amounts of text-image pairs have been
showing superior performance in image synthesis [41]. Several studies have
proposed to finetune these models to manipulate images based on text
descriptions [54, 10, 7] benefiting from their superior generative prior.
While these image manipulation methods seem to have similarities with our
methods, they struggle to maintain the structures of original images and
require excessive time for manipulation.

Figure 2:  We apply tone adjustment filters from Adobe Lightroom Classic [1]
to 500 images from the MIT-Adobe 5K dataset [8], and calculate the relative
similarities between the images and filter names in the CLIP space. For all
filters, the filtered images have higher similarity scores than the source
images, implying that CLIP can assess the tonal properties of images in a
manner aligning with human perception.

Figure 3: CLIPtone consists of a text adapter and a tone adjustment network.
From a target text description, the text adapter calculates a directional
vector within the CLIP embedding space from the source to target text
descriptions and estimates the modulation parameter $\Delta\theta$ for the
AdaInt module and the weight predictor of the tone adjustment network. The
modulated tone adjustment network adaptively constructs an image-text adaptive
3D LUT through fusing basis 3D LUTs and non-uniform sampling, ultimately
adjusting the color values of an input image.

##  3 CLIP as a Perceptual Criterion

As mentioned in Sec. 1, we aim to train a text-based tone adjustment network
without paired images by leveraging CLIP [38]. This is based on a hypothesis
that CLIP inherently encapsulates information about human perception learned
from a vast amount of image-language pairs so that it can be employed for
assessing tone adjustment in a manner consistent with human perception.

To verify our hypothesis, we conduct an experiment to examine whether CLIP can
correctly assess images whose tones are already known. To this end, we adopt
tone adjustment filters manually designed by experts. Specifically, we
randomly sample 500 images from the MIT-Adobe 5K dataset [8], and apply nine
filters such as “aged”, “bright”, “cinematic” from Adobe Lightroom Classic
[1], and construct a dataset consisting of source and filtered image pairs for
each filter. In each dataset, the source images have natural tones while the
filtered images have specific tones produced by each filter. We then compute
their similarities to their corresponding filter names in the CLIP embedding
space. Specifically, adopting the idea of CLIP-IQA [46], we compute relative
similarities defined as:

$S(I,T)=\mathrm{softmax}(\textrm{sim}(I,T),\textrm{sim}(I,T^{\prime}))$ (1)

where $I$ and $T$ are an image and a filter name, respectively.
$\textrm{sim}(I,T)$ is the cosine similarity between the CLIP embeddings of
$I$ and $T$. $T^{\prime}$ is a text description “normal photo”. To obtain CLIP
embeddings, we use the CLIP-RN50 model in this experiment.

Fig. 2 illustrates the result where the filtered images have higher similarity
scores than the source images for all filters. This result indicates that CLIP
holds the potential to serve as a differentiable perceptual criterion for tone
adjustment evaluation. Additional results employing other CLIP models are
included in the supplementary material.

##  4 Network Architecture of CLIPtone

The network architecture of CLIPtone comprises a tone adjustment network and a
text adapter as shown in Fig. 3. The tone adjustment network, for which we
adopt an existing image enhancement network, takes an input image and alters
the tonal properties of the image, while the text adapter takes a target text
description as input and modulates the parameters of the tone adjustment
network for text-based tone adjustment. Our design philosophy can be regarded
as modulating a backbone network using a hyper-network, a strategy whose
efficacy has been demonstrated in various prior studies [39, 33, 4, 27]. Here,
the selection of the backbone network and defining an efficient and effective
modulation strategy are crucial, and we describe our choices in the following.

#### Tone Adjustment Network.

For the tone adjustment network, we adopt the image enhancement network of
Yang  _et al_. [50], which is based on 3D LUTs. Here, we take a brief look at
Yang  _et al_.’s method for completeness. Their method takes an input image
and extracts a feature map using a convolutional neural network (CNN). Then,
from the feature map, a weight predictor network estimates the weights $w$ of
learnable basis LUTs. The basis 3D LUTs are then linearly combined using the
weights $w$ to form a fused 3D LUT. Meanwhile, from the feature map extracted
by the CNN, an AdaInt module estimates sampling coordinates, which determine
the source color values of the fused 3D LUT. Specifically, an AdaInt module
estimates sampling coordinates $x_{c}(i)$ where $c\in\\{r,g,b\\}$ and $i$
denote the color axis and an index along each color axis. For example,
$x_{r}(1)=0.2$, $x_{g}(2)=0.3$, and $x_{b}(3)=0.4$ mean that the color value
$(0.2,0.3,0.4)$ of an input image will be converted to the color value of the
fused 3D LUT at $(1,2,3)$. Then, the non-uniform sampling step resamples the
fused 3D LUT according to the estimated sampling coordinates, and obtains a
final image-adaptive 3D LUT. Finally, the color values of the input image
undergo trilinear-interpolated look-up operations to produce a color-enhanced
image. We refer the readers to [50] for more details.

We build our approach on top of the method of Yang  _et al_. [50] because it
adjusts only the tonal properties of an input image using LUTs without
modifying the contents, and provides an efficient and effective architecture
leveraging basis LUTs. In our approach, we use the basis LUTs from the Yang
_et al_.’s model, which are pre-trained for the image enhancement task,
without modification. We modulate only the parameters of the weight predictor
and the AdaInt module with our text adapter to construct an _image-text
adaptive 3D LUT_ as shown in Fig. 3. These two modules are the most crucial
ones as they determine how to combine the basis 3D LUTs, how to resample a
fused 3D LUT, and ultimately how to adjust an image. All the other parameters
including the basis LUTs are kept frozen. This selective modulation enables
the efficient and effective learning of our text adapter.

#### Text Adapter.

The text adapter modulates the parameters for the AdaInt and weight predictor
modules according to a target text description $\hat{T}$. To this end, the
text adapter first embeds the input target text description $\hat{T}$ into the
CLIP embedding space using a CLIP text encoder $E_{T}$. The text adapter also
uses a source text description $T$, which is fixed to “normal photo”, to
represent the tone of the input image. To embed $T$ and $\hat{T}$, we adopt
the CLIP-RN50 text encoder. Then, it computes a directional vector by
computing the difference between the embeddings of the target and source text
descriptions, i.e., $E_{T}(\hat{T})-E_{T}(T)$, and feeds the directional
vector to a two-layered MLP to compute modulation parameter $\Delta\theta$.
Finally, similar to the previous hyper-network-based approach [4], the text
adapter modulates the parameters of the AdaInt and weight predictor modules
as:

$\hat{\theta}=\theta\otimes(1+s\cdot\Delta\theta)$ (2)

where $\theta$ is the original parameter of the AdaInt module and the weight
predictor, and $\hat{\theta}$ is their modulated parameter. $s$ is a scaling
factor that controls the degree of adjustment, and $\otimes$ denotes Hadamard
product.

Regarding the source description, we have explored different schemes: text-
dependent, image-dependent, and neutral schemes. The text-dependent scheme
derives source descriptions from input texts, _e.g_., antonyms of input texts,
while the image-dependent scheme utilizes the CLIP image embedding of an input
image instead of the CLIP text embedding of a source text description. The
neutral scheme utilizes a fixed text description representing neutrality, such
as “normal photo”.

Among these options, we found that the neutral scheme works the best due to
the following reasons. (i) The image-dependent scheme tends to cause unstable
training due to the continuous variation in the source prompt. (ii) The text-
dependent scheme is impractical as it requires finding antonyms for the input
text, which can be challenging. (iii) The neutral scheme fits our scenario the
best, where an image may already be colorful, but a user may still want to
edit the image with the keyword “colorful” for a more enhanced result.
Additionally, we have evaluated different neutral descriptions, _e.g_.,
“ordinary photo” and “photo”, and empirically found that “normal photo” yields
the best results.

##  5 Unsupervised Learning for CLIPtone.

#### Training Strategy.

We train CLIPtone on two datasets: one for text descriptions and the other for
source images. We note that those are neither paired nor labeled, i.e.,
unsupervised learning. Specifically, for text descriptions, we adopt the Color
Names Dataset [2] encompassing 30,138 color names with duplicates to learn
adjustments to various tones. Before we feed the descriptions to the CLIP text
encoder, we append “photo” to each description (e.g., a color name “red” is
transformed into “red photo”). For source images, we use 4,500 source images
from the MIT-Adobe 5K dataset [8]. We augment the images with random-crop and
horizontal-flip, and random adjustments of bright and saturation, following
previous work [52]. For initialization of the text adapter, the weights are
sampled from a uniform distribution $\mathcal{U}$(0, 0.01), and the biases are
set to zero. For training, we use the Adam optimizer [28] with the learning
rate $1e-3$ and the coefficients $\beta_{1}=0.9$ and $\beta_{2}=0.999$.

#### Training Loss.

Given a randomly sampled pair of a target description $\hat{T}$ and a source
image $I$, we train CLIPtone using a loss $\mathcal{L}$ defined as:

$\mathcal{L}=\lambda_{\text{content}}\mathcal{L}_{\text{content}}+\lambda_{\text{CLIP}}\mathcal{L}_{\text{CLIP}}+\lambda_{\text{LUT}}\mathcal{L}_{\text{LUT}}\;,$
(3)

where $\mathcal{L}_{\text{content}}$, $\mathcal{L}_{\text{CLIP}}$, and
$\mathcal{L}_{\text{LUT}}$ represent a content-preserving loss, a CLIP
directional loss, and an LUT regularization loss, respectively. The parameters
$\lambda_{\text{content}}$, $\lambda_{\text{CLIP}}$, and
$\lambda_{\text{LUT}}$ are used to balance these loss terms, and we set each
of them to $1$.

The content-preserving loss $\mathcal{L}_{\text{content}}$ prevents the tone-
adjusted output $\hat{I}$ from excessively deviating from the source image
$I$, and is defined as the mean-squared error (MSE) between $I$ and $\hat{I}$.
The CLIP directional loss $\mathcal{L}_{\textrm{CLIP}}$ aligns the difference
from $I$ to $\hat{I}$ with that from $T$ to $\hat{T}$. To this end, we employ
the CLIP directional loss of StyleGAN-NADA [14], which is defined as:

$\displaystyle\mathcal{L}_{\text{CLIP}}(\Delta{I},\Delta{T})$
$\displaystyle=1-\frac{\Delta{I}\cdot\Delta{T}}{\|\Delta{I}\|\|\Delta{T}\|}\;$
(4)

where $\Delta{I}:=E_{I}(\hat{I})-E_{I}(I)$ and
$\Delta{T}:=E_{T}(\hat{T})-E_{T}(T)$.

Lastly, we devise the LUT regularization loss
$\mathcal{L}_{\text{LUT}}=\lambda_{\text{weight}}\mathcal{L}_{\text{weight}}+\lambda_{\text{interval}}\mathcal{L}_{\text{interval}}$
to ensure that the image-text adaptive 3D LUT is locally smooth, thereby
enabling natural and stable adjustments. We set $\lambda_{\text{weight}}$ and
$\lambda_{\text{interval}}$ to $1e-4$ and $0.5$, respectively. The
regularization is given in two-fold: $\mathcal{L}_{\text{weight}}$ and
$\mathcal{L}_{\text{interval}}$ for the weight predictor module and for the
AdaInt module, respectively. We use L2 regularization
$\mathcal{L}_{\text{weight}}:=\|w\|^{2}$ on the weights $w$ of the basis LUTs
to discourage individual weights from becoming disproportionately large
compared to the others.

$\mathcal{L}_{\text{interval}}$ is a sampling interval loss for regularizing
the adaptive sampling coordinates from the AdaInt module, which is defined as:

$\displaystyle\\!\\!\mathcal{L}_{\text{interval}}\\!=\\!\\!\\!\sum_{l,i,j,k,c}$
$\displaystyle\left|\frac{L_{c}^{(l)}{(i+1,j,k)}-L_{c}^{(l)}{(i,j,k)}}{(x_{r}({i+1})-x_{r}({i}))^{\alpha}}\right|^{2}$
$\displaystyle+$
$\displaystyle\left|\frac{L_{c}^{(l)}(i,j+1,k)-L_{c}^{(l)}(i,j,k)}{(x_{g}(j+1)-x_{g}(j))^{\alpha}}\right|^{2}$
$\displaystyle+$
$\displaystyle\left|\frac{L_{c}^{(l)}(i,j,k+1)-L_{c}^{(l)}(i,j,k)}{(x_{b}(k+1)-x_{b}(k))^{\alpha}}\right|^{2}\\!\\!,$
(5)

where $l\in\\{1,\cdots,L\\}$ is an index to a basis LUT, and $L$ is the number
of the basis LUTs, which is $3$ in our framework following Yang _et al_.’s
method [50]. $c\in\\{r,g,b\\}$ is an index to a color channel.
$L_{c}^{(l)}(i,j,k)$ is the color value of the $c$ color channel of the $l$-th
basis 3D LUT at $(i,j,k)$. $x_{c}(i)$ is the $i$-th sampling coordinate for
the $c$ color channel estimated by the AdaInt module. $\alpha$ is a hyper-
parameter to control the regularization strength. Larger $\alpha$ more
strongly penalizes narrow intervals between consecutive sampling coordinates,
leading to more evenly distributed sampling coordinates, so that disrupt color
changes can be avoided. Conversely, smaller $\alpha$ gives a weaker penalty to
narrow intervals, allowing the sampling coordinates to better reflect the
target text description. In our experiments, we set $\alpha$ to $0.7$.
$\mathcal{L}_{\text{interval}}$ ensures that the distances in the estimated
sampling coordinates, i.e., the sampling intervals, are not too small
especially when neighboring LUT entries have large color difference, thereby
preventing abrupt color changes in tone-adjusted images.

Figure 4:  Qualitative comparisons against baselines for modifications of
input images according to given text descriptions. T2ONet [42] induces only
subtle changes and fails to perform appropriate adjustments. While CLIPstyler
[29] and IP2P [7] make appropriate adjustments aligned with the text
descriptions, they fail to preserve the image contents. In contrast, CLIPtone
successfully makes appropriate adjustments while preserving the contents of
the input images.

##  6 Experiments

###  6.1 Comparative Evaluation

#### Baselines.

As CLIPtone is the first unsupervised learning-based approach for text-based
image tone adjustment, we compare it with other state-of-the-art text-based
image manipulation methods: T2ONet [42], IP2P [7], and CLIPstyler [29]. T2ONet
[42] is a request-based image enhancement model, which is trained on text-
based requests and corresponding before-and-after images in a supervised
manner. IP2P [7] is also a request-based image editing model that fine-tunes a
pretrained text-based image generative model to accommodate a source image and
a text-based request. CLIPstyler [29] is a style transfer model, which is
trained to increase the directional CLIP similarity for a single target text
description. As these models use different forms for target descriptions, we
alter input text descriptions as follows: A text description “krypton” is
transformed to “krypton photo” for CLIPtone, “Make a krypton photo.” for
T2ONet [42] and IP2P [7], and “krypton” for CLIPstyler [29]. For the baseline
models, we use the authors’ official implementations.

#### Qualitative Comparisons.

Fig. 4 shows a qualitative comparison. In the figure, T2ONet [42] barely
changes the input images and fails to perform appropriate adjustment. Although
CLIPstyler [29] and IP2P [7] succeed in making adjustment to fit the
descriptions, they fail to preserve the structures of the input images.
CLIPtone outperforms all the other methods, achieving both appropriate
adjustments reflecting the target descriptions and successful preservation of
the structural information in the input images.

Method |  | Grayscale
---
SSIM $\uparrow$
| CLIP image
---
similarity $\uparrow$
| CLIP text-image
---
directional similarity $\uparrow$
T2ONet [42] | 0.936 | 0.994 | 0.021
CLIPstyler [29] | 0.484 | 0.669 | 0.095
IP2P [7] | 0.765 | 0.932 | 0.073
CLIPtone | 0.858 | 0.975 | 0.083
|  |  |

Table 1: Quantitative comparison of different methods on 500 images from the
MIT-Adobe 5K dataset [8] and manually collected 50 tone-related text
descriptions. CLIPtone achieves overall higher scores in both Grayscale SSIM
and CLIP image similarity, which evaluate structural preservation, as well as
in CLIP text-image directional similarity, which assesses the alignment with
text descriptions, compared to existing methods.

#### Quantitative Comparisons.

For quantitative evaluation, we construct a test set consisting of 500 source
images from MIT-Adobe 5K [8], which are not used for training CLIPtone, and 50
manually collected text descriptions related to tonal properties such as
“warm”, “cold”, and “pastel tone”. We provide all the descriptions used in our
evaluation in the supplementary material.

To assess the adjustments by each method, we focus on two primary factors: the
preservation of image structures, and the alignment with text descriptions. To
assess the structure preservation, we use grayscale SSIM and CLIP image
similarity. Grayscale SSIM computes the SSIM index between two images after
grayscale conversion, while CLIP image similarity measures the cosine
similarity between two images in the CLIP embedding space to assess the
similarity between their contents.

However, it is important to note that both metrics are limited in assessing
the structure preservation since assessing the structure preservation between
two images is not a trivial task, which requires isolating the image structure
from tones and colors. Nevertheless, both metrics are less affected by tones
and colors than other metrics such as PSNR since grayscale SSIM discards the
colors and also includes contrast normalization, and CLIP image similarity
focuses on content similarity rather than tones and colors.

To assess alignment with text descriptions, we measure the CLIP text-image
directional similarity. Specifically, given a pair of an input image and a
target text description, we compute the directional vector between the CLIP
embeddings of the input image and its tone-adjusted image, and the directional
vector between the CLIP embeddings of the source and target text descriptions.
Then, we compute the cosine similarity between the directional vectors.
Meanwhile, as T2ONet [42] supports only predefined descriptions for
adjustments, we evaluate the performance of T2ONet only on 21 compatible
descriptions out of 50.

In Tab. 1, we report the quantitative comparison. The CLIP image similarity
and CLIP text-image directional similarity scores in the table are computed
using the CLIP ViT-B/16 model, which is not used for training any methods for
a fair comparison. Other results with other CLIP models are included in the
supplementary material. In the table, T2ONet [42], which induces subtle
modifications in adjustment process, yields high scores in Grayscale SSIM and
CLIP image similarity. Yet, it markedly underperforms in CLIP text-image
directional similarity than the other methods. Since CLIPstyler [29] is
trained solely on target descriptions, it records the highest CLIP text-image
directional similarity score. However, it struggles to preserve image
structures, leading to the lowest scores in Grayscale SSIM and CLIP image
similarity. CLIPtone achieves high scores in all three metrics, demonstrating
that it outperforms existing methods when considering both primary factors.

Figure 5:  Qualitative comparison against pseudo-ground-truth images created
by tone adjustment filters of Adobe Lightroom Classic [1]. Compared to the
baselines, CLIPtone performs more similar adjustment to the Adobe filters
while preserving the original image structure.

#### Comparison using Pseudo-Ground-Truth Images.

We also compare the baseline methods and ours using pseudo-ground-truth
images, which are generated by applying nine tone adjustment filters of Adobe
Lightroom Classic [1] to our test set as done in Sec. 3. Fig. 5 shows a
qualitative comparison. As the figure shows, CLIPtone excels not only in
preserving image structures but also in adjusting tones, producing results
more similar to the results of the filters designed by experts. For
quantitative analysis, we measure the structural similarity between tone-
adjusted images and their corresponding pseudo-ground-truth images using CLIP
image similarity. We also measure the similarity between the adjustments made
to a tone-adjusted image and to its corresponding pseudo-ground-truth image
for a given input image using CLIP directional similarity. Tab. 2 shows that
CLIPtone achieves the highest CLIP directional similarity score and the second
highest CLIP image similarity score, indicating that CLIPtone performs tone
adjustment similar to the filters manually designed by experts.

Figure 6: Summary of the user study with 20 participants evaluating 30
examples. Each graph represents the average score of each of the compared
methods for each of the following questions: (a) How well the structure of the
input image is preserved. (b) How well the modification suits the text
description. (c) The aesthetic quality of the result image.

Methods | T2ONet [42] | CLIPstyler [29] | IP2P [7] | CLIPtone
---|---|---|---|---
| CLIP image
---
similarity
0.956 | 0.704 | 0.891 | 0.948
| CLIP directional
---
similarity
0.359 | 0.297 | 0.310 | 0.467
|  |  |  |

Table 2: Quantitative comparison using pseudo-ground-truth images created by
tone adjustment filters of Adobe Lightroom Classic [1]. Figure 7:  The effect
of the sampling interval loss $\mathcal{L}_{\text{interval}}$. Without the
loss, CLIPtone may perform excessive adjustments solely to align with the text
descriptions, compromising aesthetic quality.

Method | T2ONet [42] | CLIPstyler [29] | IP2P [7] | CLIPtone
---|---|---|---|---
Param (M) | 21.1 | 6.7 | 1016.9 | 111.4
Time (ms) | 45.1 | 26.6 | 7177.4 | 10.3
|  |  |  |

Table 3:  Comparison of the number of parameters (M) and inference times (ms)
for processing a $512\times 512$-sized image using an NVIDIA GeForce RTX 3090
GPU.

Method |  | Grayscale
---
SSIM $\uparrow$
| CLIP Image
---
Similarity $\uparrow$
| CLIP Text-Image
---
Directional Similarity $\uparrow$
| CLIPtone-0
---
(trained w/o target keyword)
0.859 | 0.968 | 0.087
| CLIPtone
---
(trained w/ target keyword)
0.858 | 0.964 | 0.089
|  |  |

Table 4:  Quantitative comparison of CLIPtone-0 trained without target
keywords and CLIPtone trained with target keywords. CLIPtone-0 performs nearly
as well as CLIPtone in all the metrics.  Figure 8:  Qualitative comparison of
CLIPtone-0 trained without target keywords and CLIPtone trained with target
keywords. In both cases, both models make appropriate and similar adjustments
implying that CLIPtone is capable of robustly adjusting even to novel
descriptions not encountered during training.  Figure 9: Controlling the
strength of adjustment by adjusting the modulation scaling factor
$s\in\\{0,1,2\\}$. A result with the default scaling factor $s=1$ is shown in
the middle.

#### User Study.

Given that image tone adjustment is fundamentally a human-centric task, we
conducted a user study to evaluate the baseline methods and ours. For this
study, we recruited 20 participants from our institution. Each participant was
presented 30 examples consisting of an input image and randomly arranged
resulting images from each method, and was asked to score these on a scale
from 1 (poor) to 5 (excellent) based on the following criteria: (a) How well
the structure of the input image is preserved. (b) How well the modification
suits the text description. (c) The aesthetic quality of the result image. The
complete questionnaires are included in the supplementary material. Fig. 6
illustrates the average of the results of the user study. Against all the
other methods, CLIPtone is preferred in all three aspects, demonstrating that
CLIPtone outperforms existing methods from the perspective of human
perception.

#### Model Complexity.

As shown in Tab. 3, CLIPtone has more parameters than T2ONet [42] and
CLIPstyler [29], each designed to address single and restricted text
descriptions, respectively. Nevertheless, CLIPtone exhibits notably shorter
inference times than the others thanks to its efficient backbone network and
our hyper-network-based approach.

###  6.2 Deeper Analysis of CLIPtone

In this section, we conduct a detailed analysis on CLIPtone, including the
validation of the sampling interval loss and a series of experiments that
demonstrate the unique and practical advantages of CLIPtone.

#### Sampling Interval Loss.

Fig. 7 shows that the absence of $\mathcal{L}_{\text{interval}}$ causes the
model to over-adjust, significantly diminishing the aesthetic quality of the
output image. This happens because the estimated sampling coordinates from the
AdaInt module lead to the construction of an image-text adaptive 3D LUTs with
abrupt color transitions, focusing on matching the target text descriptions.
In contrast, CLIPtone with the sampling interval loss constructs an image-text
adaptive 3D LUT with more gradual color transitions, ensuring that the result
image is not only aligned with the target text descriptions but also visually
appealing.

#### Zero-shot Prediction.

Thanks to leveraging CLIP that is pretrained on a vast amount of image-text
pairs, CLIPtone can also make suitable adjustments for text descriptions
unseen during training. Here we verify this zero-shot prediction capability of
CLIPtone. To this end, we construct a test set using 50 target keywords used
in the quantitative evaluation. We then exclude any text descriptions that
include the keywords from our training set. As a result, we exclude 2,890 text
descriptions, and obtain a subset of 27,248 text descriptions. Then, we train
CLIPtone on the subset. We refer to this version of CLIPtone as ‘CLIPtone-0’.
Finally, we compare CLIPtone-0 and CLIPtone on the test set of the sampled
target keywords. Tab. 4 and Fig. 8 present the quantitative and qualitative
comparisons. These comparisons reveal that CLIPtone-0 performs nearly as well
as CLIPtone, suggesting that CLIPtone is capable of handling novel text
descriptions not included in its training set. A more comprehensive collection
of zero-shot prediction results is included in the supplementary material.

#### Scaling the Strength of Adjustment.

CLIPtone also offers controllable adjustment by changing the scaling factor
$s$ in Eq. (2), which scales the weight offsets of the backbone network
estimated by the text adapter. Fig. 9 illustrates an example that shows smooth
transition of the tone according to the scaling factor $s$.

##  7 Conclusion

In this paper, we propose CLIPtone, the first unsupervised learning-based
approach for text-based image tone adjustment. CLIPtone effectively extends a
3D LUT-based image enhancement model to accommodate a range of adjustments
guided by text descriptions. Thanks to utilizing CLIP for perceptual
supervision, CLIPtone requires no paired datasets, and also supports zero-shot
prediction for unseen text descriptions. Through comprehensive experiments, we
have validated the effectiveness and efficiency of CLIPtone.

#### Limitations.

CLIPtone has some limitations. Firstly, the inherent bias in a pretrained CLIP
model may lead CLIPtone to perform adjustments that do not align with human
perception. We include an analysis in the supplementary material. Secondly,
CLIPtone relies global tone adjustment, thus cannot perform tone adjustment
that requires local tone adjustment. Extending CLIPtone to support local
adjustment could be a promising research avenue. Thirdly, as CLIPtone depends
solely on the directional vectors from text descriptions, it cannot support
adjustments for more diverse and specific styles, particularly those difficult
to express with text alone. Leveraging a CLIP image encoder could be an
interesting future direction.

#### Acknowledgements.

This work was supported by the NRF grant (No.2023R1A2C200494611) and IITP
grant (IITP-2021-0-02068, IITP-2019-0-01906) funded by the Korea government
(MSIT).

## References

  * [1] URL https://www.adobe.com/au/products/photoshop-lightroom-classic.html.
  * [2] URL https://github.com/meodai/color-names.
  * Alaluf et al. [2021] Yuval Alaluf, Or Patashnik, and Daniel Cohen-Or.  Restyle: A residual-based stylegan encoder via iterative refinement.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pages 6711–6720, 2021.
  * Alaluf et al. [2022] Yuval Alaluf, Omer Tov, Ron Mokady, Rinon Gal, and Amit Bermano.  Hyperstyle: Stylegan inversion with hypernetworks for real image editing.  In _Proceedings of the IEEE/CVF conference on computer Vision and pattern recognition_ , pages 18511–18521, 2022.
  * Avrahami et al. [2022] Omri Avrahami, Dani Lischinski, and Ohad Fried.  Blended diffusion for text-driven editing of natural images.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 18208–18218, 2022.
  * Bau et al. [2021] David Bau, Alex Andonian, Audrey Cui, YeonHwan Park, Ali Jahanian, Aude Oliva, and Antonio Torralba.  Paint by word.  _arXiv preprint arXiv:2103.10951_ , 2021.
  * Brooks et al. [2023] Tim Brooks, Aleksander Holynski, and Alexei A Efros.  Instructpix2pix: Learning to follow image editing instructions.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 18392–18402, 2023.
  * Bychkovsky et al. [2011] Vladimir Bychkovsky, Sylvain Paris, Eric Chan, and Frédo Durand.  Learning photographic global tonal adjustment with a database of input/output image pairs.  In _CVPR 2011_ , pages 97–104. IEEE, 2011.
  * Chai et al. [2020] Yoav Chai, Raja Giryes, and Lior Wolf.  Supervised and unsupervised learning of parameterized color enhancement.  In _Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision_ , pages 992–1000, 2020.
  * Chang et al. [2023] Zheng Chang, Shuchen Weng, Peixuan Zhang, Yu Li, Si Li, and Boxin Shi.  L-cad: Language-based colorization with any-level descriptions using diffusion priors.  In _Thirty-seventh Conference on Neural Information Processing Systems_ , 2023.
  * Chen et al. [2018] Yu-Sheng Chen, Yu-Ching Wang, Man-Hsin Kao, and Yung-Yu Chuang.  Deep photo enhancer: Unpaired learning for image enhancement from photographs with gans.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 6306–6314, 2018.
  * Crowson et al. [2022] Katherine Crowson, Stella Biderman, Daniel Kornis, Dashiell Stander, Eric Hallahan, Louis Castricato, and Edward Raff.  Vqgan-clip: Open domain image generation and editing with natural language guidance.  In _European Conference on Computer Vision_ , pages 88–105. Springer, 2022.
  * Esser et al. [2021] Patrick Esser, Robin Rombach, and Bjorn Ommer.  Taming transformers for high-resolution image synthesis.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 12873–12883, 2021.
  * Gal et al. [2022] Rinon Gal, Or Patashnik, Haggai Maron, Amit H Bermano, Gal Chechik, and Daniel Cohen-Or.  Stylegan-nada: Clip-guided domain adaptation of image generators.  _ACM Transactions on Graphics (TOG)_ , 41(4):1–13, 2022.
  * Gharbi et al. [2017] Michaël Gharbi, Jiawen Chen, Jonathan T Barron, Samuel W Hasinoff, and Frédo Durand.  Deep bilateral learning for real-time image enhancement.  _ACM Transactions on Graphics (TOG)_ , 36(4):1–12, 2017.
  * Goodfellow et al. [2014] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.  Generative adversarial nets.  _Advances in neural information processing systems_ , 27, 2014.
  * He et al. [2020] Jingwen He, Yihao Liu, Yu Qiao, and Chao Dong.  Conditional sequential modulation for efficient global image retouching.  In _Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XIII 16_ , pages 679–695. Springer, 2020.
  * Ho et al. [2020] Jonathan Ho, Ajay Jain, and Pieter Abbeel.  Denoising diffusion probabilistic models.  _Advances in neural information processing systems_ , 33:6840–6851, 2020.
  * Hu et al. [2018] Yuanming Hu, Hao He, Chenxi Xu, Baoyuan Wang, and Stephen Lin.  Exposure: A white-box photo post-processing framework.  _ACM Transactions on Graphics (TOG)_ , 37(2):1–17, 2018.
  * Huang et al. [2022] Zhitong Huang, Nanxuan Zhao, and Jing Liao.  Unicolor: A unified framework for multi-modal colorization with transformer.  _ACM Transactions on Graphics (TOG)_ , 41(6):1–16, 2022.
  * Ignatov et al. [2017] Andrey Ignatov, Nikolay Kobyshev, Radu Timofte, Kenneth Vanhoey, and Luc Van Gool.  Dslr-quality photos on mobile devices with deep convolutional networks.  In _Proceedings of the IEEE international conference on computer vision_ , pages 3277–3285, 2017.
  * Ignatov et al. [2018] Andrey Ignatov, Nikolay Kobyshev, Radu Timofte, Kenneth Vanhoey, and Luc Van Gool.  Wespe: weakly supervised photo enhancer for digital cameras.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops_ , pages 691–700, 2018.
  * Jiang et al. [2021] Wentao Jiang, Ning Xu, Jiayun Wang, Chen Gao, Jing Shi, Zhe Lin, and Si Liu.  Language-guided global image editing via cross-modal cyclic mechanism.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pages 2115–2124, 2021.
  * Kang et al. [2021] Kyoungkook Kang, Seongtae Kim, and Sunghyun Cho.  Gan inversion for out-of-range images with geometric transformations.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pages 13941–13949, 2021.
  * Kim et al. [2021] Hanul Kim, Su-Min Choi, Chang-Su Kim, and Yeong Jun Koh.  Representative color transform for image enhancement.  In _Proceedings of the IEEE/CVF international conference on computer vision_ , pages 4459–4468, 2021.
  * Kim et al. [2020] Han-Ul Kim, Young Jun Koh, and Chang-Su Kim.  Global and local enhancement networks for paired and unpaired image enhancement.  In _Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XXV 16_ , pages 339–354. Springer, 2020.
  * Kim et al. [2022] Seongtae Kim, Kyoungkook Kang, Geonung Kim, Seung-Hwan Baek, and Sunghyun Cho.  Dynagan: Dynamic few-shot adaptation of gans to multiple domains.  In _SIGGRAPH Asia 2022 Conference Papers_ , pages 1–8, 2022.
  * Kingma and Ba [2014] Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  _arXiv preprint arXiv:1412.6980_ , 2014.
  * Kwon and Ye [2022] Gihyun Kwon and Jong Chul Ye.  Clipstyler: Image style transfer with a single text condition.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 18062–18071, 2022.
  * Li et al. [2020] Bowen Li, Xiaojuan Qi, Thomas Lukasiewicz, and Philip HS Torr.  Manigan: Text-guided image manipulation.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 7880–7889, 2020.
  * Liang et al. [2021] Jie Liang, Hui Zeng, Miaomiao Cui, Xuansong Xie, and Lei Zhang.  Ppr10k: A large-scale portrait photo retouching dataset with human-region mask and group-level consistency.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 653–661, 2021.
  * Liu et al. [2023] Chengxu Liu, Huan Yang, Jianlong Fu, and Xueming Qian.  4d lut: learnable context-aware 4d lookup table for image enhancement.  _IEEE Transactions on Image Processing_ , 32:4742–4756, 2023.
  * Mahabadi et al. [2021] Rabeeh Karimi Mahabadi, Sebastian Ruder, Mostafa Dehghani, and James Henderson.  Parameter-efficient multi-task fine-tuning for transformers via shared hypernetworks.  _arXiv preprint arXiv:2106.04489_ , 2021.
  * Meng et al. [2021] Chenlin Meng, Yutong He, Yang Song, Jiaming Song, Jiajun Wu, Jun-Yan Zhu, and Stefano Ermon.  Sdedit: Guided image synthesis and editing with stochastic differential equations.  _arXiv preprint arXiv:2108.01073_ , 2021.
  * Moran et al. [2020] Sean Moran, Pierre Marza, Steven McDonagh, Sarah Parisot, and Gregory Slabaugh.  Deeplpf: Deep local parametric filters for image enhancement.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 12826–12835, 2020.
  * Moran et al. [2021] Sean Moran, Steven McDonagh, and Gregory Slabaugh.  Curl: Neural curve layers for global image enhancement.  In _2020 25th International Conference on Pattern Recognition (ICPR)_ , pages 9796–9803. IEEE, 2021.
  * Patashnik et al. [2021] Or Patashnik, Zongze Wu, Eli Shechtman, Daniel Cohen-Or, and Dani Lischinski.  Styleclip: Text-driven manipulation of stylegan imagery.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pages 2085–2094, 2021.
  * Radford et al. [2021] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al.  Learning transferable visual models from natural language supervision.  In _International conference on machine learning_ , pages 8748–8763. PMLR, 2021.
  * Ratzlaff and Fuxin [2019] Neale Ratzlaff and Li Fuxin.  Hypergan: A generative model for diverse, performant neural networks.  In _International Conference on Machine Learning_ , pages 5361–5369. PMLR, 2019.
  * Richardson et al. [2021] Elad Richardson, Yuval Alaluf, Or Patashnik, Yotam Nitzan, Yaniv Azar, Stav Shapiro, and Daniel Cohen-Or.  Encoding in style: a stylegan encoder for image-to-image translation.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 2287–2296, 2021.
  * Rombach et al. [2022] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer.  High-resolution image synthesis with latent diffusion models.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 10684–10695, 2022.
  * Shi et al. [2021] Jing Shi, Ning Xu, Yihang Xu, Trung Bui, Franck Dernoncourt, and Chenliang Xu.  Learning by planning: Language-guided global image editing.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 13590–13599, 2021.
  * Sohl-Dickstein et al. [2015] Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli.  Deep unsupervised learning using nonequilibrium thermodynamics.  In _International conference on machine learning_ , pages 2256–2265. PMLR, 2015.
  * Song and Ermon [2020] Yang Song and Stefano Ermon.  Improved techniques for training score-based generative models.  _Advances in neural information processing systems_ , 33:12438–12448, 2020.
  * Tov et al. [2021] Omer Tov, Yuval Alaluf, Yotam Nitzan, Or Patashnik, and Daniel Cohen-Or.  Designing an encoder for stylegan image manipulation.  _ACM Transactions on Graphics (TOG)_ , 40(4):1–14, 2021.
  * Wang et al. [2023] Jianyi Wang, Kelvin CK Chan, and Chen Change Loy.  Exploring clip for assessing the look and feel of images.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , pages 2555–2563, 2023.
  * Wang et al. [2021] Tao Wang, Yong Li, Jingyang Peng, Yipeng Ma, Xian Wang, Fenglong Song, and Youliang Yan.  Real-time image enhancer via learnable spatial-aware 3d lookup tables.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pages 2471–2480, 2021.
  * Xia et al. [2021] Weihao Xia, Yujiu Yang, Jing-Hao Xue, and Baoyuan Wu.  Tedigan: Text-guided diverse face image generation and manipulation.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 2256–2265, 2021.
  * Yan et al. [2016] Zhicheng Yan, Hao Zhang, Baoyuan Wang, Sylvain Paris, and Yizhou Yu.  Automatic photo adjustment using deep neural networks.  _ACM Transactions on Graphics (TOG)_ , 35(2):1–15, 2016.
  * Yang et al. [2022a] Canqian Yang, Meiguang Jin, Xu Jia, Yi Xu, and Ying Chen.  Adaint: Learning adaptive intervals for 3d lookup tables on real-time image enhancement.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 17522–17531, 2022a.
  * Yang et al. [2022b] Canqian Yang, Meiguang Jin, Yi Xu, Rui Zhang, Ying Chen, and Huaida Liu.  Seplut: Separable image-adaptive lookup tables for real-time image enhancement.  In _European Conference on Computer Vision_ , pages 201–217. Springer, 2022b.
  * Zeng et al. [2020] Hui Zeng, Jianrui Cai, Lida Li, Zisheng Cao, and Lei Zhang.  Learning image-adaptive 3d lookup tables for high performance photo enhancement in real-time.  _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 44(4):2058–2073, 2020.
  * Zhang et al. [2022] Fengyi Zhang, Hui Zeng, Tianjun Zhang, and Lin Zhang.  Clut-net: Learning adaptively compressed representations of 3dluts for lightweight image enhancement.  In _Proceedings of the 30th ACM International Conference on Multimedia_ , pages 6493–6501, 2022.
  * Zhang et al. [2023] Lvmin Zhang, Anyi Rao, and Maneesh Agrawala.  Adding conditional control to text-to-image diffusion models.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pages 3836–3847, 2023.
  * Zhu et al. [2017] Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros.  Unpaired image-to-image translation using cycle-consistent adversarial networks.  In _Proceedings of the IEEE international conference on computer vision_ , pages 2223–2232, 2017.


# From evolved Long-Period-Variable stars to the evolution of M31

Maryam Torki1 Mahdieh Navabi1 Atefeh Javadi1 Elham Saremi1 Jacco Th. van Loon2
and Sepideh Ghaziasgar1 1School of Astronomy, Institute for Research in
Fundamental Sciences (IPM),
Tehran,
2Lennard-Jones Labratories, Keele University, ST5 5BG, UK

(2015)

###### Abstract

One of the ways to understand the genesis and evolution of the universe is to
know how galaxies have formed and evolved. In this regard, the study of star
formation history (SFH) plays an important role in the accurate understanding
of galaxies. In this paper, we used long-period variable stars (LPVs) to
estimate the SFH in the Andromeda galaxy (M31). These cool stars reach their
peak luminosity in the final stage of their evolution; their birth mass is
directly related to their luminosity. Therefore, we construct the mass
function and the star formation history using stellar evolution models.

###### keywords:

stars: AGB and post-AGB - stars: luminosity function, mass function -
galaxies: evolution - galaxies: formation - galaxies: individual: M31 -
galaxies: stellar content - galaxies: structure

††volume: xxx††journal: Title of your IAU Symposium††editors: A.C. Editor,
B.D. Editor & C.E. Editor, eds.

##  1 Introduction

Andromeda, as the closest spiral giant galaxy, offers us a unique opportunity
to know how its various components have formed and evolved. M31 is located at
785± 25 kpc, $(m-M)_{o}=24.47$ mag (McConnachie et al. 2004) and has low
foreground reddening E(B-V)=0.06 mag. Our approach to investigating the star
formation history (SFH) is based on employing long-period variable stars
(LPVs), which we have successfully applied in other galaxies in the Local
Group, such as M33 (Javadi et al. 2011a; 2011b; 2017), Magellanic Clouds
(Rezaeikh et al. 2014), NGC147 and NGC185 (Golshan et al. 2011), IC1613
(Hashemi et al. 2019), Andromeda VII (Navabi et al. 2020; 2021) and Andromeda
I (Saremi et al. 2021). Asymptotic giant branch (AGB) and red supergiants
(RSGs) are basic pillars in our research.

##  2 Data and Technique of Star Formation

In this paper, we used the catalogue of LPVs in M31 from Mould et al. (2004)
to determine star formation (Torki et al. 2019). It should be noted that their
results included near-infrared photometry of almost 2000 variables, most of
which were AGBs. We used the Padova stellar evolutionary models (Marigo et al.
2017) to obtain SFH and assuming that the metallicity is constant over time,
we were able to calculate the mass, age, and pulsation duration of the stars.
The SFH is described by the star formation rate (SFR) and conveys the concept
of how much gas has become a star each year in the past. In fact, SFR, $\xi$
(in M⊙ yr-1) is a function of time and estimated by:

$\xi(t)=\frac{\int_{\rm min}^{\rm max}f_{\rm IMF}(m)m\
dm}{\int_{m(t)}^{m(t+dt)}f_{\rm IMF}(m)\ dm}\ \frac{dn^{\prime}(t)}{\delta
t}.$ (1)

Figure 1: The 2MASS image of the M31 galaxy. The green dots indicate the LPVs
from Mould et al (2004). Figure 2: SFH of the M31 galaxy for constant
metallicity of Z = 0.008 based on data in box 1 of Figure 1. The horizontal
“error bars” show the spread in age within each bin and the statistical errors
are presented in the vertical “error bars”.

where the m is mass, $f_{\rm IMF}$ is the Initial Mass Function (IMF) (Kroupa
2001). The minimum and maximum mass in Kroupa’s IMF are considered 0.02 and
200 M⊙ respectively. In the above equation, it is assumed that stars with mass
between $m(t)$ and $m(t+dt)$ (t is look back time) are LPVs at the present
time, and $\delta t$ is the duration of the star evolutionary phase, which the
number of variables observed ($dn^{\prime}(t)$) in intervals $t$ and $t+dt$
depends on.

##  3 RESULTS

The LPVs are scattered in five rectangular boxes (Figure 1). In this paper, we
estimated the SFH of the M31 galaxy during the broad time interval from 11.8
Gyr (log $t$ = 10.07) to 400 Myr (log $t$ = 8.59) ago in box 1, which its
dimensions are 1.49 kpc$\times$13.09 kpc. The SFR as a function of look-back
time in M31 is shown in Figure 2. The horizontal “error bars” correspond to
the start and end time for which the SFR was calculated in that bin. We see a
peak of star formation at 11.8 Gyr ago; after that, the star formation begins
to decrease until the present-day.

## References

  * [Javadi et al. (2017)] Hamedani Golshan R., Javadi A., van Loon J. Th., Khosroshahi H., Saremi E., 2017, MNRAS, 466, 1764
  * [Bernatowicz et al. (2003)] Hashemi S. A., Javadi A., van Loon J. Th., 2019, MNRAS, 483, 4751
  * [Ott (1993)] Javadi A., van Loon J. Th., Mirtorabi M. T., 2011a, MNRAS, 411, 263
  * [Ott (1993)] Javadi A., van Loon J. Th., Mirtorabi M. T., 2011b, MNRAS, 414, 3394
  * [Lodders & Fegley (1998)] Javadi A., van Loon J. Th., Khosroshahi H. G., Tabatabaei F., Hamedani Golshan R., Rashidi M., 2017, MNRAS, 464, 2103
  * [Anders & Zinner (1993)] Marigo P. et al., 2017, ApJ, 835, 19
  * [Busso et al. (1999)] McConnachie A. W., Irwin M. J., Ferguson A. M. N., Ibata R. A., Lewis G. F., Tanvir N., 2004, MNRAS, 350, 243
  * [Ott (1993)] Mould, J., Saha, A., 2004, ApJ, 154, 623
  * [Nittler et al. (1997)] Rezaeikh S., Javadi A., Khosroshahi H., van Loon J. Th., 2014, MNRAS, 445, 2214
  * [Zinner (1998)] Navabi, M., Noori, M., Saremi, E., Javadi, A., van Loon, J., and Khosroshahi, H., 2020, Proc.conf. Stars and their Variability Observed from Space, 383, 385
  * [Croat, Stadermann & Bernatowicz (2005)] Navabi, M. et al., 2021, ApJ, 910, 127
  * [Zinner (2004)] Saremi, E., Javadi, A., Navabi, M., van Loon, J.T., Khosroshahi, H., Bojnordi Arbab, B., McDonald, I., 2021, ApJ ,923 , 164
  * [Ott et al. (2005)] Torki, M., Javadi, A., van Loon, J. Th., 2019 , Proceedings of IAU Symposium, 343, 512


11institutetext:  ENEA - P.le E. Fermi 1, 8055 Portici (NA), Italy††thanks:
E-mail: 11email<EMAIL_ADDRESS>
# Kuramoto oscillators in random networks

A. Funel

###### Abstract

By means of numerical analysis conducted with the aid of the computer, the
collective synchronization of coupled phase oscillators in the Kuramoto model
in the connected regime of random networks of various sizes is studied. The
oscillators synchronize and achieve phase coherence, and this process is not
significantly affected by the level of connectivity of the network. If the
probability that two oscillators are coupled is around the network
connectivity threshold synchronization still occurs, although in a more
attenuated way. If the size of the network is sufficiently large the
oscillators have a phase transition.

##  1 Introduction

The phenomenon of collective synchronization of an ensemble of interacting
entities can be observed in different areas. For example in biology, in
processes involving the regulation of the cell cycle [1, 2]; in physics, in
the study of coupled lasers[3, 4, 5] and Josephson junctions[6]; but also in
human [7] and ecological networks [8]. One of the first important works on the
subject was written by A. Winfree [9] in the 1960s who, studying a population
of interacting oscillators, discovered that if the natural frequencies of the
oscillators were very close, there existed a threshold condition for the
coupling constant that, if exceeded, caused a phase transition in which some
oscillators suddenly synchronized. Winfree’s ideas were taken up by Kuramoto
[10] who proposed a model whose analytical solution could be studied in the
mean-field approximation. The equations of dynamics, for a collection of $N$
oscillators, are:

$\dot{\theta}_{i}=\omega_{i}+\frac{K}{N}\sum_{j=1}^{N}\mbox{sin}(\theta_{j}-\theta_{i})\,.$
(1)

The temporal evolution of the phase $\theta_{i}$ of each oscillator depends on
its natural frequency $\omega_{i}$, and on the phases of the other oscillators
with which it interacts. If $\omega_{i}=\omega_{j}\,\,\forall i,j$ the model
is called homogeneous, otherwise inhomogeneous. The magnitude of the
interaction is given by the coupling constant $K\geq 0$, which is the same for
all oscillators. In order to have a geometric representation of the dynamics,
it is convenient to imagine that the motion occurs on a circle of unit radius
so that the phase of an oscillator is represented by the azimuthal angle
defined by its position, see Fig. 1.

[scale=0.30]kuramoto_phases.pdf

Figure 1: Geometric representation of oscillator dynamics in the Kuramoto
model. Motion occurs on a circle of unit radius. The azimuthal angle $\theta$
represents the phase of the oscillator.

Collective motion can be studied by introducing the order parameter

$Re^{i\psi}=\frac{1}{N}\sum_{j=1}^{N}e^{i\theta_{j}}\,.$ (2)

The modulus $0\leq R\leq 1$ measures phase coherence. If the oscillators are
scattered along the entire circle then $R\approx 0$ while if they are in phase
coherence ($\theta_{i}\approx\theta_{j},\forall i,j$) then $R\approx 1$. The
angle $\psi$ is the average phase. If $K=0$ the oscillators decouple and the
phase of each of them evolves according to the equation
$\theta_{i}(t)=\omega_{i}t+\theta_{i}(0)$ and synchronization, that is
$\dot{\theta}_{i}=\dot{\theta}_{j}=\dot{\theta}$ $\forall i,j$, can only occur
if $\omega_{i}=\omega_{j}\forall i,j$. Note that by summing the members of the
Eq.( 1) we obtain $\sum_{i=1}^{N}\dot{\theta}_{i}=\sum_{i=1}^{N}\omega_{i}$,
which implies that if all the oscillators synchronize they will have a
frequency equal to the average of the natural frequencies:
$\dot{\theta}=\bar{\omega}=N^{-1}\sum_{i=1}^{N}\omega_{i}$. In terms of the
order parameter Eq.( 1) can be written as:

$\dot{\theta}_{i}=\omega_{i}+KR\,\mbox{sin}(\psi-\theta_{i})$ (3)

where it is evident how the single oscillator can be thought as interacting
with a mean-field with effective coupling constant equal to $KR$ and phase
$\psi$. The numerical analysis of the Kuramoto model shows the existence of a
threshold value of the coupling constant below which the oscillators evolve
while remaining in a state of phase incoherence; gradually increasing $K$
beyond this threshold the incoherent state becomes unstable, a small nucleus
of oscillators is formed which synchronize at a frequency equal to
$\bar{\omega}$ while the remaining part moves non-uniformly around the unit
circle; finally, if $K$ reaches a critical value $K_{c}$, collective
synchronization begins. Kuramoto proved that $K_{c}=2/\pi g(0)$ in the limit
$N\rightarrow\infty$, where $g(\omega)$ is the distribution of natural
frequencies, see [11] for details.

The Kuramoto model refers to a fully (all-to-all) connected network. However,
it is interesting to study collective synchronization also in the case of
heterogeneous networks, because of this kind are those of particular
importance in nature, social interactions and technological infrastructures.
Considering the network topology, Eq.( 1) becomes:

$\dot{\theta}_{i}=\omega_{i}+\frac{K}{k_{i}}\sum_{j=1}^{N}A_{ij}\mbox{sin}(\theta_{j}-\theta_{i})$
(4)

where $A_{ij}$ is the adjacency matrix ($A_{ij}=1$ if $\theta_{i}$ and
$\theta_{j}$ are coupled, 0 otherwise) and $k_{i}$ is the number of
connections of oscillator $\theta_{i}$. In this paper, only connected and
symmetric networks ($A_{ij}=A_{ji}$) are considered.

As far as we have been able to ascertain, there is no general theory that
predicts for each type of network the value of the critical threshold $K_{c}$
for the onset of collective synchronization and, therefore, it is necessary to
resort to numerical analysis. In some cases the system goes from a state of
phase incoherence to one of coherence as soon as $K>K_{c}$ and it can be said
that it undergoes a phase transition [12].

In this paper we focus on the study of the Kuramoto model for random networks.
This type of networks are extensively studied both for their theoretical
interest and for their statistical properties that make them a basic model in
different application fields. The novelty presented in this paper is the
accurate numerical analysis of the synchronization process in the connectivity
regime of random networks in the inhomogeneous Kuramoto model. In particular,
the numerical results support the conjecture according to which the critical
threshold of the coupling probability of oscillators above which they almost
certainly synchronize is that of the network connectivity. Furthermore, to the
best of our knowledge, this analysis has never been conducted for networks of
the size of the largest network analyzed in this paper.

##  2 Random networks

The structure of a random network $G(N,p)$ of N nodes, also known as Erdős-
Rényi, is determined by the probability $0\leq p\leq 1$ that a link exists
between two randomly chosen nodes. The degree distribution of a random
network, which is the probability that a randomly chosen node has degree $k$,
is a binomial $P(k)=\binom{N-1}{k}p^{k}(1-p)^{N-1-k}$and the average value of
the degree is $\bar{k}=p(N-1)$. For $p<1/N$ (subcritical region) the network
is mainly composed of isolated nodes, tiny clusters and $\bar{k}<1$. The
largest cluster is a tree whose size is $N_{G}\sim\mbox{ln}N$. For $p=1/N$
(critical point) $\bar{k}=1$, the network is composed of many small
disconnected components, mainly trees, and $N_{G}\sim N^{2/3}$. For
$1/N<p\leq\mbox{ln}N/N=p_{c}$ (supercritical region) $\bar{k}>1$ and a giant
component is formed which contains a finite part of the network and grows
rapidly as $p$ increases. However, the network is not yet connected in this
region. The probability $p_{c}$ is called connectivity threshold because the
network is supposed to became connected for $p$ just above $p_{c}$. For
$p>p_{c}$ one has $\bar{k}\geq\mbox{ln}N$ and $N_{G}\sim N$. At the end of the
connected region $p=1$ and the network becomes fully (all-to-all) connected.
See[13] for details.

##  3 Method

In this work, networks of size $N=50,100,500$ and 1000 are considered. For
each type of network analyzed 10 simulations are performed and for each of
them the distribution of natural frequencies is a Gaussian with
$\bar{\omega}=0$ and $\sigma=0.1$. Since isolated oscillators evolve
independently, their dynamics do not affect the collective synchronization of
those connected to the network. Therefore, we consider the region with
$p>p_{c}$. To study the limit $p\rightarrow p_{c}$, we set $p=p_{c}$ in the
random generator of the numerical library used to generate the networks [14],
exclude disconnected configurations and accept only connected ones that
correspond to the case in which the giant component absorbs all the nodes of
the network and has the minimum number of links. The range of values of the
coupling constant examined in this work is $0\leq K\leq 1$ and we choose to
divide it into 100 steps. For each value of $K$ the system of Eq.( 4) is
numerically integrated up to 5000 time iterations of amplitude 0.1 s. The
order parameter $R$ is calculated by averaging the last 1000 iterations in
order to exclude values due to initial transient phase fluctuations. The value
of $K_{c}$ was estimated by finite-size scaling analysis [15, 16].

##  4 Results

To study how the density of links affects synchronization we consider the
probability increment $\Delta p=(1-p_{c})/10$ and calculate the order
parameter $R(K,p)$ for $p=p_{c}+n\Delta p$ for $n=0,...,10$. The results are
shown in Fig. 2.

Figure 2: Order parameter $R$ as a function of the coupling constant $K$ for
random networks of size $N=50,100,500,1000$. For each value of $K$ the system
of Eq.( 4) is numerically integrated up to 5000 time iterations of amplitude
0.1 s. $R$ is calculated by averaging the last 1000 iterations. The figure
shows $R(K)$ for different values of the probability $p$ that determines the
structure of the network in the connected regime. For
$p=p_{c}\simeq\mbox{ln}N/N$ the largest connected component absorbs all the
nodes and has the minimum number of links. This is the configuration for which
the network has the minimum level of connectivity. For $p=1$ the maximum level
of connectivity is achieved and every node is connected to all the others. The
function $R(K)$ behaves roughly the same for $p>p_{c}$ while for $p=p_{c}$ it
grows more slowly as $K$ increases. The figure shows in more detail the
regions where the onset of synchronisation occurs and those where the
oscillators are in phase coherence. For $p>p_{c}$ the value of the critical
threshold of the coupling constant for which synchronization occurs is
$K_{c}\sim 0.16\div 0.19$, while for $p=p_{c}$ it is $K_{c}\sim 0.21$.

What is observed is that the oscillators synchronize and reach the state of
phase coherence for all values of $p$ examined in the connected region. The
behavior of $R(K)$ is very similar for all values of $p>p_{c}$. For $p=p_{c}$,
the system reaches phase coherence, although more slowly. As the size of the
network grows, for $K>K_{c}$ the transition from state $R\simeq 0$ to $R\simeq
1$ is more and more sudden, as the shape of $R(K)$ shows, which means that the
system has a phase transition. This also happens when $p=p_{c}$, as shown in
more detail in Fig. 3, where the derivative $dR/dK$ is also shown. It can be
seen that as $N$ increases the maximum value of $dR/dK$ increases, while its
variation around the maximum shrinks.

[scale=0.50]r_vs_k_pc_random_connected_region.pdf

Figure 3: The figure shows $R(K)$ when $p=p_{c}$ for all analyzed networks. As
the size $N$ of the network increases the transition from $R\simeq 0$ to
$R\simeq 1$ becomes abrupt and the oscillators have a phase transition. This
is also evidenced by the inner figures which show that as $N$ increases the
derivative $dR/dK$ has a larger maximum and the variation around it is
narrower.

Figure 4: The figure shows $R(N,K)$ for all values of $p$. For $K<K_{c}$, R
decays towards a residual of $O(1/\sqrt{N})$. For $K>K_{c}$, R saturates
towards a value $R_{\infty}<1$ with fluctuations of $O(1/\sqrt{N})$. For
$K=K_{c}$ the variation of $R(N,K_{c})$ is minimal. The best estimate is
$K_{c}\sim 0.16\div 0.19$ for $p>p_{c}$, and $K_{c}\sim 0.21$ for $p=p_{c}$.

To estimate $K_{c}$ we consider that for $K<K_{c}$ the oscillators do not
synchronize and $R$ decays around a value of $O(1/\sqrt{N})$. For $K>K_{c}$
$R$ increases and saturates at a value $R_{\infty}<1$, although with
fluctuations of $O(1/\sqrt{N})$. Therefore $K_{c}$ can be estimated
numerically considering that at the onset of synchronization $R(N,K_{c})$ has
the smallest variation with $N$. Fig. 4 shows $R(N,K)$ for all values of $p$
examined in this work. For $p>p_{c}$ we have that $K_{c}\sim 0.16\div 0.19$.
This suggests that the onset of synchronization does not depend too much on
the level of connectivity of the network itself. The theoretical value
predicted by Kuramoto is $K_{c}\sim 0.16$. For $p=p_{c}$ the best estimate is
$K_{c}\sim 0.21$, which confirms that for this network configuration the onset
of synchronization is slower.

##  5 Discussion

The conjecture that the probability threshold above which synchronization is
highly probable is $p_{c}=\mbox{ln}N/N$ has been considered by several
authors. In [17], in the context of optimization of nonconvex cost functions,
authors proved, for the homogeneous Kuramoto model, that if
$p\gg\mbox{ln}N/N^{1/3}$ the oscillators synchronize with high probability for
$N\rightarrow\infty$. In [18], with arguments from spectral graph theory,
authors proved that the same occurs if $p\gg\mbox{ln}^{2}N/N$. The issue has
also been addressed in the context of graphon theory [19, 20, 21, 22, 23]. In
[24] authors proved, still for the case of the homogeneous model and for
$N\rightarrow\infty$, that the oscillators reach phase synchronization with
high probability if $p>p_{c}$. They performed numerical simulations for
networks of size $N\leq 100$. In this work the above conjecture has been
proved numerically even in the more general case of inhomogeneous Kuramoto
model and for larger network sizes.

##  6 Conclusion

We have numerically studied the collective synchronization of coupled phase
oscillators in the Kuramoto model in the connected regime of random networks
of sizes $N=50,100,500$ and 1000. The oscillators synchronize and reach phase
coherence for all connectivity levels of the networks. The behavior of the
order parameter $R$ as a function of the coupling constant $K$ does not show
significant discrepancies between different levels of link density. For the
limiting case in which oscillators are coupled with probability around the
connectivity threshold of the network, synchronization still occurs, although
in a more attenuated way. As the size of the network increases, the transition
from the incoherent to the coherent phase state occurs more and more abruptly
and the oscillators have a phase transition. The critical threshold value for
the onset of synchronization varies in the range $K_{c}\sim 0.16\div 0.19$ for
all the connectivity levels of the networks. In the limiting case when the
network goes from disconnection to connection $K_{c}\sim 0.21$. A random
network of a certain size can be obtained by perturbing a complete network of
the same size by randomly eliminating links. Therefore, from this analysis it
can be deduced that a random perturbation of a complete network of oscillators
coupled according to the Kuramoto model does not affect the synchronization as
long as the perturbation does not disconnect the network.

###### Acknowledgements.

The computing resources and the related technical support used for this work
have been provided by CRESCO/ENEAGRID High Performance Computing
infrastructure and its staff [25]. CRESCO/ENEAGRID High Performance Computing
infrastructure is funded by ENEA, the Italian National Agency for New
Technologies, Energy and Sustainable Economic Development and by Italian and
European research programmes, see http://www.cresco.enea.it/english for
information

## References

  * [1] Banfalvi G. Methods Mol Biol.76120111.
  * [2] Manukyan A., Abraham L., Dungrawala H. Schneider B. L. Methods Mol Biol.7612011173.
  * [3] Ziping J. McCall M. J. Opt. Soc. Am.101993155.
  * [4] Kourtchatov S. Yu, Likhanskii V. V., Napartovich A. P., Arecchi F. T. Lapucci A. Phys. Rev. A5219954089.
  * [5] Takemura, N., Takata, K., Takiguchi, M. Notomi M. Sci. Rep.1120218587.
  * [6] Wiesenfeld K., Colet P. Strogatz S. H. Phys. Rev. E.57199815631.
  * [7] Shahal, S., Wurzberg, A., Sibony, I., Duadi H., Shniderman E., Weymouth D., Davidson N. Fridman M. Nat. Commun.1120203854.
  * [8] Vandermeer J., Hajian-Forooshani Z., Medina N. Perfecto I. R. Soc. Open Sci.82021210122.
  * [9] Winfree A. T. J. Theoret. Biol.16196715.
  * [10] Kuramoto Y. Chemical Oscillations, Waves and Turbulence Springer 19 Berlin 1984\.
  * [11] Strogatz S. H. Physica D: Nonlinear Phenomena14320001.
  * [12] English L. Q. European Journal of Physics292007143.
  * [13] Barabási A. Network science Cambridge University Press 2016\.
  * [14] Hagberg A. A., Schult D. A. Swart P. J. Proceedings of the 7th Python in Science Conference (SciPy2008)200811-15.
  * [15] Marro J. Dickman R. Nonequilibrium phase transitions in lattice models Cambridge University Press 1999\.
  * [16] Moreno Y. Pacheco A. F. Europhysics Letters682004603.
  * [17] Ling S., Xu R. Bandeira A. S. SIAM J. Optim.2920191879.
  * [18] Kassabov M., Strogatz S. H. Townsend A. Chaos: An Interdisciplinary Journal of Nonlinear Science322022093119.
  * [19] Lovász L. American Mathematical Soc.602012.
  * [20] Borgs C., Chayes J., Cohn H. Zhao Y. Transactions of the American Mathematical Society37220193019-3062.
  * [21] Medvedev G. S. SIAM Journal on Mathematical Analysis4620142743-2766.
  * [22] Medvedev G. S. Archive for Rational Mechanics and Analysis2122014781-803.
  * [23] Medvedev G. S. Communications in Mathematical Sciences172019883-898.
  * [24] Nagpal S. V., Nair G. G., Strogatz S. H. Parise F. https://doi.org/10.48550/arXiv.2403.13998 2024
  * [25] Iannone F., Ambrosino F., Bracco G., De Rosa M., Funel A., Guarnieri G., Migliori S., Palombi F., Ponti G., Santomauro G. Procacci, P. Proceedings of the 2019 International Conference on High Performance Computing Simulation (HPCS)20191051-1052.