# Text in the Dark: Extremely Low-Light Text Image Enhancement

Che-Tsung Lin111These authors contributed equally to this work. Chun Chet
Ng222These authors contributed equally to this work. Zhi Qin Tan  Wan Jun Nah
Xinyu Wang  Jie Long Kew  Pohao Hsu  Shang Hong Lai  Chee Seng Chan
<EMAIL_ADDRESS>Christopher Zach

###### Abstract

Text extraction in extremely low-light images is challenging. Although
existing low-light image enhancement methods can enhance images as pre-
processing before text extraction, they do not focus on scene text. Further
research is also hindered by the lack of extremely low-light text datasets.
Thus, we propose a novel extremely low-light image enhancement framework with
an edge-aware attention module to focus on scene text regions. Our method is
trained with text detection and edge reconstruction losses to emphasize low-
level scene text features. Additionally, we present a Supervised Deep Curve
Estimation model to synthesize extremely low-light images based on the public
ICDAR15 (IC15) dataset. We also labeled texts in the extremely low-light See
In the Dark (SID) and ordinary LOw-Light (LOL) datasets to benchmark extremely
low-light scene text tasks. Extensive experiments prove our model outperforms
state-of-the-art methods on all datasets. Code and dataset will be released
publicly at https://github.com/chunchet-ng/Text-in-the-Dark.

###### keywords:

Extremely Low-Light Image Enhancement, Edge Attention, Text Aware
Augmentation, Scene Text Detection, Scene Text Recognition

††journal: Signal Processing: Image Communication

[chalmers]organization=Chalmers University of Technology, state=Gothenburg,
country=Sweden

[um]organization=Universiti Malaya, state=Kuala Lumpur, country=Malaysia

[nthu]organization=National Tsing Hua University, state=Hsinchu,
country=Taiwan

[adelaide]organization=The University of Adelaide, state=Adelaide,
country=Australia

We present a new method to enhance low-light images, especially scene text
regions.

We developed a novel Supervised-DCE model to synthesize extremely low-light
images.

We create 3 new low-light text datasets SID-Sony-Text, SID-Fuji-Text, and LOL-
Text.

Our new datasets assess enhanced low-light images with scene text extraction
tasks.

Our method achieves the best results on all datasets quantitatively &
qualitatively.

##  1 Introduction

(a)

(b)

(c)

(d)

Figure 1: From left to right: (a) Original images; (b) Enhanced results with
our proposed method; (c-d) Zoomed-in (2x) regions of the blue and green
bounding boxes. Top row: SID-Sony-Text; Middle row: SID-Fuji-Text; Bottom row:
LOL-Text. Extremely low-light images in the SID dataset are significantly
darker than those in the LOL dataset, and our model enhances the images to the
extent that texts are clearly visible with sharp edges.

Scene text understanding involves extracting text information from images
through text detection and recognition, which is a fundamental task in
computer vision. However, performance drops sharply when images are captured
under low-light conditions. The main difficulty in detecting text in low-light
images is that low-level features, such as edges and character strokes, are no
longer prominent or hardly visible. On the other hand, enhancing images
captured in extremely low-light conditions pose a greater challenge than
ordinary low-light images due to the higher noise levels and greater
information loss. For instance, we show the difference in darkness level in
Figure 1, where it is evident that the See In the Dark (SID) datasets [1] are
darker and, in theory, more difficult to enhance than the LOw-Light (LOL)
dataset [2]. Quantitatively, we calculated the PSNR and SSIM values for two
subsets of SID, SID-Sony and SID-Fuji, and LOL by comparing each image against
pure black images in Table 1. Based on each dataset’s average perceptual
lightness (L* in the CIELAB color space), images in SID are at least 15 times
darker than those in LOL. Hence, low-light image enhancement is a necessary
pre-processing step for scene text extraction under such conditions.

Over the years, many general or low-light image enhancement models have been
proposed to improve the interpretability and extraction of information in
images by providing better input for subsequent image content analysis. Early
methods [3, 18, 19] typically attempted to restore the statistical properties
of low-light images to those of long-exposure images from a mathematical
perspective. On the other hand, deep learning-based methods [2, 1, 23, 25] aim
to learn the mapping between low-light images and their corresponding long-
exposure versions via regression. To the best of our knowledge, most existing
low-light image enhancement works have not explicitly addressed the restored
image quality in terms of downstream scene text tasks.

Dataset PSNR $\uparrow$ SSIM $\uparrow$ Avg. L* $\downarrow$ SID-Sony [1]
44.350 0.907 0.009 SID-Fuji [1] 41.987 0.820 0.004 LOL [2] 23.892 0.195 0.142
Pure Black $\infty$ 1.000 0.000

Table 1: The difference between the extremely low-light dataset, SID, and the
ordinary low-light dataset, LOL, is shown in terms of PSNR and SSIM values,
computed by comparing short-exposure images against pure black images. Avg. L*
is the average perceptual lightness in the CIELAB color space, calculated
based on short-exposure images. Scores are averaged across training and test
sets. Higher PSNR and SSIM values, along with lower Avg. L*, indicate darker
images that are more challenging for image enhancement and scene text
extraction.

Recent advancements in visual attention mechanisms have demonstrated their
effectiveness in identifying and boosting salient features in images. Channel-
only attention [11, 12, 13], spatial attention [14, 15] or the subsequent
channel-spatial attention [16, 17] modules were proposed to emphasize the most
informative areas. However, these methods cannot preserve texture details,
especially fine-grained edge information that is intuitively needed to enhance
extremely low-light images with complex textures. To overcome this limitation,
we introduce Edge-Aware Attention (Edge-Att). This novel attention module
simultaneously performs channel and spatial attention-based feature learning
on high-level image and edge features. Our model also considers text
information in the image through a text-aware loss function. This way, our
model can effectively enhance low-light images while preserving fine-grained
edge information, texture details, and legibility of text.

The scarcity of extremely low-light text datasets presents a hurdle for
further research. To address this, we annotated all text instances in both the
training and testing sets of the SID and LOL datasets, creating three new low-
light text datasets: SID-Sony-Text, SID-Fuji-Text, and LOL-Text. We then
proposed a novel Supervised Deep Curve Estimation (Supervised-DCE) model to
synthesize extremely low-light scene text images based on the commonly used
ICDAR15 (IC15) scene text dataset. It allows researchers to easily translate
naive scene text datasets into extremely low-light text datasets. In addition
to the previously published conference version of this work [45], we have made
four significant extensions. Firstly, we propose a novel dual encoder-decoder
framework that can achieve superior performance on low-light scene text tasks
(Section 3.1). Secondly, we introduce a new image synthesis method capable of
generating more realistic extremely low-light text images (Section 4.1).
Thirdly, we have further annotated texts in the Fuji and LOL datasets, thereby
forming the largest low-light scene text datasets to date (Section 5).
Fourthly, comprehensive experiments and analyses are carried out to study the
latest methods along with our proposed methods on all synthetic and real low-
light text datasets. The main contributions of our work are as follows:

  * 1.

We present a novel scene text-aware extremely low-light image enhancement
framework with dual encoders and decoders to enhance extremely low-light
images, especially scene text regions within them. Our proposed method is
equipped with Edge-Aware Attention modules and trained with new Text-Aware
Copy-Paste (Text-CP) augmentation. Our model can restore images in challenging
lighting conditions without losing low-level features.

  * 2.

We developed a Supervised-DCE model to synthesize extremely low-light images.
This allows us to use existing publicly available scene text datasets such as
IC15 to train our model alongside genuine ones for scene text research under
such extreme lighting conditions.

  * 3.

We labeled the texts in the SID-Sony, SID-Fuji, and LOL datasets and named
them SID-Sony-Text, SID-Fuji-Text, and LOL-Text, respectively. This provides a
new perspective for objectively assessing enhanced extremely low-light images
through scene text tasks.

##  2 Related Works

Low-light Image Enhancement. Retinex theory assumes that an image can be
decomposed into illumination and reflectance. Most Retinex-based methods
enhance results by removing the illumination part [3], while others such as
LIME [18] keep a portion of the illumination to preserve naturalness. BIMEF
[19] further designs a dual-exposure fusion framework for accurate contrast
and lightness enhancement. RetinexNet [2] combines deep learning and Retinex
theory, adjusting illumination for enhancement after image decomposition. The
recent successes of generative adversarial networks (GANs) [20] have attracted
attention from low-light image enhancement because GANs have proven successful
in image translation. Pix2pix [21] and CycleGAN [22] have shown good image-
translation results in paired and unpaired image settings, respectively. To
overcome the complexity of CycleGAN, EnlightenGAN [23] proposed an
unsupervised one-path GAN structure. Besides general image translation, [1]
proposed learning-based low-light image enhancement on raw sensor data to
replace much of the traditional image processing pipeline, which tends to
perform poorly on such data. EEMEFN [24] also attempted to enhance images
using multi-exposure raw data that is not always available.

Zero-Reference Deep Curve Estimation (Zero-DCE) [25] designed a light-weight
CNN to estimate pixel-wise high-order curves for dynamic range adjustment of a
given image without needing paired images. [26] designed a novel Self-
Calibrated Illumination (SCI) learning with an unsupervised training loss to
constrain the output at each stage under the effects of a self-calibrated
module. ChebyLighter [27] learns to estimate an optimal pixel-wise adjustment
curve under the paired setting. Recently, the Transformer [28] architecture
has become the de-facto standard for Natural Language Processing (NLP) tasks.
ViT [29] applied the attention mechanism in the vision task by splitting the
image into tokens before sending it into Transformer. Illumination Adaptive
Transformer (IAT) [30] uses attention queries to represent and adjust ISP-
related parameters. Most existing models enhance images in the spatial domain.
Fourier-based Exposure Correction Network (FECNet) [31] presents a new
perspective for exposure correction with spatial-frequency interaction and has
shown that their model can be extended to low-light image enhancement.

Scene Text Extraction. Deep neural networks have been widely used for scene
text detection. CRAFT [32] predicts two heatmaps: the character region score
map and the affinity score map. The region score map localizes individual
characters in the image, while the affinity score map groups each character
into a single word instance. Another notable scene text detection method is
Pixel Aggregation Network (PAN) [33] which is trained to predict text regions,
kernels, and similarity vectors. Both text segmentation models have proven to
work well on commonly used scene text datasets such as IC15 [34] and TotalText
[35]. Inspired by them, we introduced a text detection loss in our proposed
model to focus on scene text regions during extremely low-light image
enhancement. Furthermore, state-of-the-art text recognition methods such as
ASTER [36] and TRBA [37] are known to perform well on images captured in
complex scenarios. ASTER [36] employs a flexible rectification module to
straighten the word images before passing them to a sequence-to-sequence model
with the bi-directional decoder. The experimental results of ASTER showed that
the rectification module could achieve superior performance on multiple scene
text recognition datasets, including the likes of IC15 and many more. Besides,
TRBA [37] provided interesting insights by breaking down the scene text
recognition framework into four main stages: spatial transformation, character
feature extraction, followed by sequence modeling, and the prediction of
character sequences. Given these methods’ robustness on difficult texts, they
are well-suited to recognize texts from enhanced low-light images.

##  3 Extremely Low-Light Text Image Enhancement

###  3.1 Problem Formulations

Let $x\in R^{W\times H\times 3}$ be a short-exposure image of width $W$ and
height $H$. An ideal image enhancement expects that a neural network
$LE(x;\theta)$ parameterized by $\theta$ can restore this image to its
corresponding long-exposure image, $y\in R^{W\times H\times 3}$, i.e.,
$LE(x;\theta)\simeq y$. However, previous works normally pursued the lowest
per-pixel intensity difference, which should not be the goal for image
enhancement because we usually expect that some high-level computer vision
tasks can work reasonably well on those enhanced images. For example, in terms
of text detection, the goal of the neural network can be the lowest detection
bounding boxes discrepancy, i.e., $B(LE(x;\theta))\simeq B(y)$.

Our novel image enhancement model consists of a U-Net accommodating extremely
low-light images and edge maps using two independent encoders. During model
training, instead of channel attention, the encoded edges guide the spatial
attention sub-module in the proposed Edge-Att to attend to edge pixels related
to text representations. Besides the image enhancement losses, our model
incorporates text detection and edge reconstruction losses into the training
process. This integration effectively guides the model’s attention towards
text-related features and regions, facilitating improved image textual content
analysis. As a pre-processing step, we introduced a novel augmentation
technique called Text-CP to increase the presence of non-overlapping and
unique text instances in training images, thereby promoting comprehensive
learning of text representations.

###  3.2 Network Design

Figure 2: Illustration of the architecture of our proposed framework, designed
to enhance extremely low-light images while incorporating scene text
awareness. Figure 3: (a) Visual representation of our edge decoder, wherein A
and B represent the output from the corresponding convolution blocks in Figure
2 and S denotes the scaling of the image. (b) Illustration of the proposed
Edge-Aware Attention module.

Our model was inspired by U-Net[1] with some refinements. Firstly, the network
expects heterogeneous inputs, i.e., extremely low-light images, $x$, and the
corresponding RCF [38] edge maps, $e$. Secondly, input-edge pairs are handled
by two separate encoders with edge-aware attention modules between them. The
attended features are then bridged with the decoder through skip connections.
Finally, our multi-tasking network predicts the enhanced image, $x^{\prime}$,
and the corresponding reconstructed edge, $e^{\prime}$. The overall
architecture of our network can be seen in Figure 2 and modeled as:

$x^{\prime},e^{\prime}=LE(x,e;\theta).$ (1)

###  3.3 Objectives

Our proposed model is trained to optimize four loss functions. The first two,
Smooth L1 loss and multi-scale SSIM loss focus on enhancing the overall image
quality. The third, text detection loss, targets the enhancement of scene text
regions specifically. The fourth, edge reconstruction loss, focuses on crucial
low-level edge features.

Firstly, we employ smooth L1 loss as the reconstruction loss to better enforce
low-frequency correctness [21] between $x^{\prime}$ and $y$ as:

$\displaystyle\mathcal{L}_{recons}=\left\\{\begin{matrix}&0.5\cdot(x^{\prime}-y)^{2}/\delta,&\text{if
}{|x^{\prime}-y|<\delta}\\\
&|x^{\prime}-y|-0.5\cdot\delta,&\text{otherwise}{}\end{matrix}\right.$ (2)

where we empirically found that $\delta=1$ achieved good result. The authors
of Pix2Pix [21] showed that by utilizing L1 loss, the model can achieve better
results as the generated images are less blurry and proved that L1 loss can
better enforce the learning of low-frequency details, which is also essential
for OCR tasks. On the other hand, the L1 norm is less sensitive to outliers
than the L2 norm, thus resulting in a more robust model towards extreme pixel
intensities.

Secondly, the multi-scale SSIM metric was proposed in [39] for reference-based
image quality assessment, focusing on image structure consistency. An
$M$-scale SSIM between the enhanced image $x^{\prime}$ and ground truth image
$y$ is:

$SSIM_{MS}(x^{\prime},y)=[l_{M}(x^{\prime},y)]^{\tau}\cdot\prod\nolimits^{M}_{j=1}[c_{j}(x^{\prime},y)]^{\phi}[s_{j}(x^{\prime},y)]^{\psi},$
(3)

where $l_{M}$ is the luminance at M-scale; $c_{j}$ and $s_{j}$ represent the
contrast and the structure similarity measures at the $j$-th scale; $\tau$,
$\phi$, and $\psi$ are parameters to adjust the importance of the three
components. Inspired by [39], we adopted the $M$-scale SSIM loss function in
our work to enforce the image structure of $x^{\prime}$ to be close to that of
$y$:

$\mathcal{L}_{SSIM_{MS}}=1-{SSIM_{MS}}(x^{\prime},y).$ (4)

Thirdly, a well-enhanced extremely low-light image implies that we could
obtain similar text detection results on both the enhanced and ground truth
images. As such, we propose to employ CRAFT [32] to localize texts in images
through its region score heatmap. To implicitly enforce our model to focus on
scene text regions, we define the text detection loss, $\mathcal{L}_{text}$
as:

$\mathcal{L}_{text}=\|R(x^{\prime})-R(y)\|_{1},$ (5)

where $R(x^{\prime})$ and $R(y)$ denote the region score heatmaps of the
enhanced and ground truth images, respectively.

Fourthly, the edge reconstruction decoder in our model is designed to extract
edges better, which are essential for text pixels. Figure 3(a) shows an
overview of the edge decoder. The loss at pixel $i$ of detected edge, $e_{i}$,
with respect to the ground truth edge, $g_{i}$ is defined as:

$\displaystyle l(e_{i})=\left\\{\begin{matrix}&\alpha\cdot
log(1-P(e_{i})),&\text{if }g_{i}=0\\\ &\beta\cdot logP(e_{i}),&\text{if
}g_{i}=1\end{matrix}\right.$ (6)

where

$\displaystyle\alpha=\lambda\cdot\frac{\left|Y^{+}\right|}{\left|Y^{+}\right|+\left|Y^{-}\right|},$
(7)
$\displaystyle\beta=\frac{\left|Y^{-}\right|}{\left|Y^{+}\right|+\left|Y^{-}\right|},$

$Y^{+}$ and $Y^{-}$ denote the positive and negative sample sets,
respectively. $\lambda$ is set to 1.1 to balance both types of samples. The
ground truth edge is generated using a Canny edge detector [40], and
P($e_{i}$) is the sigmoid function. Then, the overall edge reconstruction loss
can be formulated as:

$\mathcal{L}_{edge}=\sum_{i=1}^{|I|}\sum_{j=1}^{J}l(e^{j}_{i})+l(e^{{}^{\prime}}_{i}),$
(8)

where $l(e_{i}^{j})$ is the predicted edge at pixel $i$ and level $j$. $J=3$
is the number of side edge outputs in our model. $e_{i}^{{}^{\prime}}$ is the
final predicted edge map from the concatenation of side outputs. $|I|$ is the
number of pixels in a cropped image during training.

Finally, the total joint loss function, $\mathcal{L}_{total\\_en}$ of our
proposed model is:

$\mathcal{L}_{total\\_en}=\omega_{recons}\mathcal{L}_{recons}+\omega_{text}\mathcal{L}_{text}+\omega_{SSIM_{MS}}\mathcal{L}_{SSIM_{MS}}+\omega_{edge}\mathcal{L}_{edge},$
(9)

where $\omega_{recons}$, $\omega_{text}$, $\omega_{SSIM_{MS}}$, and
$\omega_{edge}$ are the weights to address the importance of each loss term
during training.

###  3.4 Edge-Aware Attention

Polarized Self-Attention (PSA) [41] is one of the first works to propose an
attention mechanism catered to high-quality pixel-wise regression tasks.
However, we found that the original PSA module that only considers a single
source of feature map for both channel and spatial attention is ineffective
for extremely low-light image enhancement. Under low light conditions, the
details of content such as the edges of the texts are barely discernible which
is less effective in guiding the network to attend to spatial details.
Therefore, we designed our Edge-Aware Attention (Edge-Att) module to take in
feature maps from two encoders and process them differently, i.e., the feature
maps of extremely low-light images from the image encoder are attended
channel-wise, whereas the spatial attention submodule attends to feature maps
from the edge encoder. By doing so, we can ensure that Edge-Att can attend to
rich images and edge features simultaneously. The proposed attention module is
illustrated in Figure 3(b).

Firstly, the feature map from the image encoder, $F$ is fed into the channel
attention, $A^{ch}(F)\in\mathbb{R}^{C\times 1\times 1}$ with calculation as
follows:

$A^{ch}(F)=\sigma_{3}\left[F_{SG}(W_{z}(\sigma_{1}(W_{v}(F))\times
F_{SM}(\sigma_{2}(W_{q}(F)))))\right],$ (10)

where $W_{q}$, $W_{v}$, and $W_{z}$ are 1x1 convolution layers, $\sigma_{1}$,
$\sigma_{2}$ and $\sigma_{3}$ are tensor reshape operators. $F_{SM}(.)$ and
$F_{SG}(.)$ refer to softmax and sigmoid operators. The output of this branch
is $A^{ch}(F)\bigodot^{ch}F\in\mathbb{R}^{C\times H\times W}$, where
$\bigodot^{ch}$ is a channel-wise multiplication operator.

Secondly, given the edge-branch feature map $E$, the edge-aware spatial
attention, $A^{sp}(E)\in\mathbb{R}^{1\times H\times W}$, is defined as:

$A^{sp}(E)=F_{SG}\left[\sigma_{3}(F_{SM}(\sigma_{1}(F_{GP}(W_{q}(E))))\times\sigma_{2}(W_{v}(E)))\right],$
(11)

where $W_{q}$ and $W_{v}$ are 1x1 convolution layers, $\sigma_{1}$,
$\sigma_{2}$, and $\sigma_{3}$ are three tensor reshape operators. $F_{SM}(.)$
is a softmax operator, $F_{GP}(.)$ is a a global pooling operator, and
$F_{SG}(.)$ is a sigmoid operator. The output of this branch is
$A^{sp}(E)\bigodot^{sp}F\in\mathbb{R}^{C\times H\times W}$, where
$\bigodot^{sp}$ is a spatial-wise multiplication operator, and $F$ is the
image enhancement branch’s feature map. Finally, output of the proposed Edge-
Att module is the composition of two submodules:

$Edge{\text{-}}Att(F,E)=A^{ch}(F)\odot^{ch}F+A^{sp}(E)\odot^{sp}F.$ (12)

###  3.5 Text-Aware Copy-Paste Augmentation

Figure 4: Illustration of the Text-Aware Copy-Paste (Text-CP) data
augmentation. Compared with the original Copy-Paste, our method generates
images with non-overlapping text instances that allow the detection of texts
outside their usual context.

This work aims to enhance extremely low-light images to improve text detection
and recognition. However, the dataset’s limited number of text instances could
hinder the model’s ability. Although Copy-Paste Augmentation [42] can increase
the number of text instances, overlapping texts introduced by random placement
might confuse CRAFT in text detection loss since CRAFT is not trained to
detect such texts. In the commonly used scene text datasets such as ICDAR15
[34], overlapping texts are marked as ”do not care” regions which are excluded
from models’ training and evaluation. Thus, to adhere to ICDAR’s standard and
to address overlapping text issues, we propose a novel approach called Text-
Aware Copy-Paste Augmentation (Text-CP). Text-CP considers each text box’s
location and size by leveraging uniform and Gaussian distributions derived
from the dataset. For a training image $t$ of width $w_{t}$ and height $h_{t}$
to be augmented, we initialize a set of labeled text boxes in the training set
as $C$, which is:

$\mathit{\emph{C}}=\left\\{(u_{1},v_{1},w_{1},h_{1}),...,(u_{\left|\mathit{\emph{C}}\right|},v_{\left|\mathit{\emph{C}}\right|},w_{\left|\mathit{\emph{C}}\right|},h_{\left|\mathit{\emph{C}}\right|}))\right\\},$
(13)

where each tuple represents the top left position of a text located at $u_{k}$
and $v_{k}$ with width, $w_{k}$, and height, $h_{k}$ with $k$ representing the
index of the current text’s box in the set. We then sample a target number of
text instances, $n_{\text{target}}$, from the set of $C$ to form $C_{t}$,
defined as the set of text boxes to be pasted on that training image, $t$. The
next step is to crop and paste the sampled texts without overlapping. For each
$c_{k}\in C_{t}$, we adopt two uniform distributions in modeling the position
of the texts, $\hat{u}_{k}$ and $\hat{v}_{k}$:

$\displaystyle\hat{u}_{k}\sim U(0,w_{t}),$ (14) $\displaystyle\hat{v}_{k}\sim
U(0,h_{t}).$

As for $w_{k}$ and $h_{k}$, they are sampled from Gaussian distribution as:

$\displaystyle\hat{w}_{k}\sim\mathcal{N}(\mu_{W},\sigma_{W}^{2}),$ (15)
$\displaystyle\hat{h}_{k}\sim\mathcal{N}(\mu_{H},\sigma_{H}^{2}),$

where $\mu$ and $\sigma^{2}$ are estimated means and variances of width $W$
and height $H$ from all the labeled texts in the training set. We illustrate
the overall data augmentation process of Text-CP and its augmented results in
Figure 4. The pseudocode of Text-CP is detailed in the supplementary material.

##  4 Extremely Low-Light Image Synthesis

###  4.1 Problem Formulations

To the best of our knowledge, the research community has not extensively
explored extremely low-light image synthesis, mainly due to the limited
availability of datasets designed explicitly for extremely low-light scene
text. While extremely low-light dataset, SID, and low-light dataset, LOL,
exist, they are not primarily collected with scene text in mind. This scarcity
of dedicated datasets for extremely low-light scene text poses challenges for
evaluating the performance of existing image enhancement methods in terms of
image quality and scene text metrics. In order to address this issue, we
define the extremely low-light image synthesis problem as follows:

$\hat{x}=LS(y;\theta_{s}),$ (16)

where given a long-exposure image $y$, a low-light image synthesis neural
network, $LS(y;\theta_{s})$ parameterized by $\theta_{s}$, will synthesize a
set of images $\hat{x}$, such that $B(LS(y;\theta_{s}))\simeq B(x)$. We want
the synthesized extremely low-light images to be as realistic as possible to
genuine low-light images, $x$.

Therefore, we introduce a Supervised-DCE model focusing on synthesizing a set
of realistic extremely low-light images, enabling existing image enhancement
techniques to leverage publicly available scene text datasets. Consequently,
existing low-light image enhancement methods can benefit from training with
synthetic data to the extent that they can perform better on the downstream
scene text detection task, as detailed in Section 6.5.

###  4.2 Network Design

Figure 5: Illustration of the proposed Supervised-DCE model for extremely low-
light image synthesis.

Zero-DCE [25] was originally proposed to perform image enhancement through
curve estimation. However, its network can only adjust brightness slightly
since the per-pixel trainable curve parameter, $\alpha$, in the quadratic
curve limits the pixel variation. The advantage of performing intensity
adjustment in terms of the quadratic curve is that the pixel range can be
better constrained. In this work, we propose a Supervised-DCE model that
learns to provide reconstructable extremely low-light images with paired
short- and long-exposure images. The framework of our image synthesis network,
Supervised-DCE, can be seen in Figure 5. Our goal is to push most values
closer to zero n the context of synthesizing extremely low-light images.
Accordingly, we propose a reformulation of the DCE model as follows:

$\hat{x}=-(H(y)+U(y))y^{2}+(1+H(y))y,$ (17)

where $y$ is the input (i.e., long-exposure image); $\hat{x}$ is the
synthesized low-light image; $H(y)$ and $U(y)$ are the output of Tanh and ReLU
branches, respectively. By introducing the second $U(y)$ branch, we eliminate
the need for iteratively applying the model to produce the desired output, and
drastic intensity adjustment can be done with only a single iteration.

In the original Zero-DCE model, image enhancement is learned by setting the
exposure value to 0.6 in the exposure control loss. However, manually setting
an exposure value to synthesize extremely low-light images is too heuristic
and inefficient. In our proposed synthesis framework, the overall learning is
done by training SID long-exposure images to be similar to their short-
exposure counterparts. Most importantly, these images are translated in the
hope that their text information is somewhat deteriorated as the one in
genuine extremely low-light images and cannot be easily reconstructed. Then,
the trained model can be used to transform scene text datasets in the public
domain to boost the performance of extremely low-light image enhancement in
terms of text detection.

###  4.3 Objectives

During extremely low-light image synthesis, we expect the output to maintain
spatial consistency while reducing the overall proximity loss:

$\mathcal{L}_{prox}=\|\hat{x}-x\|_{1}+\mathcal{L}_{entropy}(\hat{x},x)+\mathcal{L}_{smoothness}(\hat{x},x),$
(18)

where $\hat{x}$ is the synthesized extremely low-light image given the long-
exposure image $y$, and $x$, is the genuine low-light image, i.e., ground
truth for $\hat{x}$. Entropy loss, $\mathcal{L}_{entropy}$, and smoothness
loss, $\mathcal{L}_{smoothness}$ [43], are also used to encourage the
differences to be both sparse and local. With the introduction of
$\mathcal{L}_{prox}$, we removed the color constancy loss of the original
Zero-DCE model since color constancy can be enforced through the supervised
loss.

The spatial consistency loss, $\mathcal{L}_{spa}$ encourages spatial coherence
of the synthesized image by preserving the difference of neighboring regions
between the input image and its synthesized low-light version:

$\mathcal{L}_{spa}=\frac{1}{\mathcal{M}}\sum_{i=1}^{\mathcal{M}}\sum_{j\in\omega(i)}(|\hat{X}_{i}-\hat{X}_{j}|-\alpha_{s}\log_{10}(9|Y_{i}-Y_{j}|+1))^{2},$
(19)

where $\mathcal{M}$ is the number of local regions, and $\omega(i)$ is four
neighboring regions (top, down, left, right) centered at region $i$. $\hat{X}$
and $Y$ are the averaged intensity values of local regions of the synthesized
images and the long-exposure images, respectively. We introduced logarithm
operation and $\alpha_{s}$ parameter to reduce the large spatial difference of
$Y$ where $\alpha_{s}$ is set to 0.05. We set the local region size to
$4\times 4$, following the original setting of Zero-DCE.

Besides spatial consistency, we also expect the monotonicity relation between
neighboring pixels to be preserved. To achieve this, we reused the
illumination smoothness loss:

$\mathcal{L}_{tv_{Z}}=\sum_{\forall
c\in\xi}(|\nabla_{x}Z^{c}|+|\nabla_{y}Z^{c}|)^{2},\xi=\left\\{R,G,B\right\\},$
(20)

where $\nabla_{x}$ and $\nabla_{y}$ are gradient operations on the x-axis and
y-axis, respectively. Illumination smoothness loss, $\mathcal{L}_{tv_{Z}}$, is
applied on both $H(y)$ and $U(y)$, i.e., the curve parameter maps of the two
branches, respectively, by substituting $Z$ with $H$ and $U$, resulting in
$\mathcal{L}_{tv_{H}}$ and $\mathcal{L}_{tv_{U}}$.

In summary, the overall learning objective, $\mathcal{L}_{total\\_syn}$ to
train our extremely low-light image synthesis network is defined as:

$\mathcal{L}_{total\\_syn}=\omega_{prox}\mathcal{L}_{prox}+\omega_{spa}\mathcal{L}_{spa}+\omega_{tv_{H}}\mathcal{L}_{tv_{H}}+\omega_{tv_{U}}\mathcal{L}_{tv_{U}}.$
(21)

##  5 New Low-Light Text Datasets

Dataset Training Set Testing Set GT Img. Leg. Illeg. $\mu_{W}$ $\mu_{H}$
$\sigma_{W}$ $\sigma_{H}$ GT Img. Leg. Illeg. SID-Sony-Text 161 5937 2128
79.270 34.122 123.635 50.920 50 611 359 SID-Fuji-Text 135 6213 4534 128.579
57.787 183.199 68.466 41 1018 1083 LOL-Text 485 613 1423 23.017 14.011 21.105
17.542 15 28 45 IC15 1000 4468 7418 78.410 29.991 55.947 24.183 500 2077 3153

Table 2: Statistics reported based on long-exposure images for all datasets.
GT Img. stands for ground truth image count, where Leg. and Illeg. stand for
legible and illegible text count, respectively.

In this work, we annotated all text instances in the extremely low-light
dataset, SID [1], and the ordinary low-light dataset, LOL [2]. SID has two
subsets: SID-Sony, captured by Sony $\alpha$7S II, and SID-Fuji, captured by
Fujifilm X-T2. For this work, we included 878/810 short-exposure images and
211/176 long-exposure images at a resolution of 4240×2832/6000×4000 from SID-
Sony and SID-Fuji, respectively. The short-exposure time is 1/30, 1/25, and
1/10, while the corresponding reference (long-exposure) images were captured
with 100 to 300 times longer exposure, i.e., 10 to 30 seconds. In our
experiments, we converted short- and long-exposure SID images to RGB format.
The LOL dataset provides low/normal-light image pairs taken from real scenes
by controlling exposure time and ISO. There are 485 and 15 images at a
resolution of 600×400 in the training and test sets, respectively. We closely
annotated text instances in the SID and LOL datasets following the common IC15
standard. We show some samples in Figure 6. The newly annotated datasets are
named SID-Sony-Text, SID-Fuji-Text, and LOL-Text to differentiate them from
their low-light counterparts.

(a) SID-Sony-Text

(b) SID-Fuji-Text

(c) LOL-Text

Figure 6: Green boxes represent legible texts, and blue boxes represent
illegible texts.

IC15 dataset was introduced in the ICDAR 2015 Robust Reading Competition for
incidental scene text detection and recognition. It contains 1500 scene text
images at a resolution of $1280\times 720$. In this study, IC15 is primarily
used to synthesize extremely low-light scene text images. Detailed statistics
of the text annotations for SID-Sony-Text, SID-Fuji-Text, LOL-Text, and IC15
are shown in Table 2, where we included the statistics for long-exposure
images only for the sake of brevity. In this table, we also report relevant
statistics of the mean and standard deviation of labeled texts’ width and
height to be used by the proposed Text-Aware Copy-Paste augmentation. The text
annotations for SID-Sony-Text, SID-Fuji-Text, and LOL-Text datasets will be
released at https://github.com/chunchet-ng/Text-in-the-Dark.

Moreover, we synthesized extremely low-light images based on IC15 by using
U-Net and our proposed Supervised-DCE model, respectively. To study the
difference between these two variations of image synthesis methods, we
generated a total of four sets of images by using the aforementioned two
models trained on SID-Sony and SID-Fuji, individually. Naming convention of
such synthetic datasets follows the format of “{Syn-
IC15}-{Sony/Fuji}-{v1/v2}”. “{Sony/Fuji}” is an indication of which dataset
the image synthesis model is trained on, while “{v1/v2}” differentiates the
image synthesis models where v1 is U-Net and v2 is our proposed Supervised-DCE
model. For instance, the synthetic images generated by a U-Net trained on SID-
Sony and SID-Fuji, are named Syn-IC15-Sony-v1 and Syn-IC15-Fuji-v1. And,
synthetic images generated by our proposed Supervised-DCE model are denoted as
Syn-IC15-Sony-v2 and Syn-IC15-Fuji-v2.

##  6 Experimental Results

###  6.1 Experiment Setup

Datasets and Metrics. All low-light image enhancement methods are trained and
tested on the datasets detailed in Section 5. They are then evaluated in terms
of intensity metrics (PSNR, SSIM), perceptual similarity (LPIPS), and text
detection (H-Mean). For the SID-Sony-Text, SID-Fuji-Text, and LOL-Text
datasets, which are annotated with text bounding boxes only, we used well-
known and commonly used scene text detectors (CRAFT [32] and PAN [33]) to
analyze the enhanced images. For IC15, which provides both text detection and
text recognition labels, we conducted a two-stage text spotting experiment
using the aforementioned text detectors (CRAFT, PAN) and two robust text
recognizers (TRBA [37] and ASTER [36]) on the synthesized IC15 images after
enhancement. The metric for text spotting is case-insensitive word accuracy.

Implementation Details. We trained our image enhancement model for 4000 epochs
using the Adam optimizer [44] with a batch size of 2. The initial learning
rate is set to $1e^{-4}$ and decreased to $1e^{-5}$ after 2000 epochs. At each
training iteration, we randomly cropped a $512\times 512$ patch with at least
one labeled text box inside and applied random flipping and image transpose as
data augmentation strategies. The weightings of each loss term, i.e.,
$\omega_{recons}$, $\omega_{text}$, $\omega_{SSIM_{MS}}$, and $\omega_{edge}$,
were empirically set to 0.2125, 0.425, 0.15, and 0.2125 respectively,
following the work of ELIE_STR [45]. For other image enhancement methods, we
re-trained them on all datasets using the best set of hyperparameters
specified in their respective code repositories or papers.

As for the Supervised-DCE model, we used a batch size of 8 and trained for 200
epochs using the Adam optimizer with default parameters and a fixed learning
rate of $1e^{-4}$. It was trained on $256\times 256$ image patches with loss
weightings of $\omega_{prox}$, $\omega_{spa}$, $\omega_{tv_{A}}$ and
$\omega_{tv_{B}}$, set to 1, 20, 10, and 10 respectively.

###  6.2 Results on SID-Sony-Text and SID-Fuji-Text Datasets

Our model’s performance is demonstrated in Table 3, achieving the highest
H-Mean scores on all datasets with CRAFT and PAN. Following [45], we
illustrate the CRAFT text detection results on SID-Sony-Text in Figure 7.
Qualitative results of existing methods on SID-Fuji-Text are presented in the
supplementary material. The effectiveness of our model in enhancing extremely
low-light images to a level where text can be accurately detected is readily
apparent. In Figure 7, only the images enhanced by our proposed model yield
accurate text detection results. On the other hand, existing methods generally
produce noisier images, resulting in inferior text detection results. While
GAN-enhanced images tend to be less noisy, the text regions are blurry, making
text detection challenging. Moreover, our model achieves the highest PSNR and
SSIM scores on both SID-Sony-Text and SID-Fuji-Text datasets, showing that our
enhanced images are the closest to the image quality of ground truth images.
In short, better text detection is achieved on our enhanced images through the
improvement of overall image quality and preservation of fine details within
text regions.

---
(a) Low-Light

---
(b) LIME [18]

---
(c) BIMEF [19]

---
(d) Zero-DCE [25]

---
(e) Zero-DCE++ [46]

---
(f) SCI [26]

---
(g) CycleGAN [22]

---
(h) EnlightenGAN [23]

---
(i) RetinexNet [2]

---
(j) Pix2Pix [21]

---
(k) ChebyLighter [27]

---
(l) FECNet [31]

---
(m) IAT [30]

---
(n) ELIE_STR [45]

---
(o) Ours

---
(p) Ground Truth

Figure 7: Comparison with state-of-the-art methods on the SID-Sony-Text
dataset is shown in the following manner: for each column, the first row
displays enhanced images marked with blue boxes as regions of interest. The
second row displays zoomed-in regions of enhanced images overlaid with red
text detection boxes from CRAFT [32]. Column 7(a) displays the low-light
image. Columns 7(b) to 7(o) show image enhancement results from all related
methods. The last cell displays ground truth images.

Type Method Image Quality H-Mean PSNR $\uparrow$ SSIM $\uparrow$ LPIPS
$\downarrow$ CRAFT $\uparrow$ PAN $\uparrow$ SID-Sony-Text Input - - - 0.057
0.026 TRAD LIME [18] 13.870 0.135 0.873 0.127 0.057 BIMEF [19] 12.870 0.110
0.808 0.136 0.079 ZSL Zero-DCE [25] 10.495 0.080 0.999 0.196 0.157 Zero-DCE++
[46] 12.368 0.076 0.982 0.218 0.162 SCI [26] 11.814 0.100 1.000 0.201 0.151 UL
CycleGAN [22] 15.340 0.453 0.832 0.090 0.053 EnlightenGAN [23] 14.590 0.426
0.793 0.146 0.075 SL RetinexNet [2] 15.490 0.368 0.785 0.115 0.040 Pix2Pix
[21] 21.070 0.662 0.837 0.266 0.190 ChebyLighter [27] 15.418 0.381 0.787 0.260
0.184 FECNet [31] 22.575 0.648 0.788 0.245 0.188 IAT [30] 19.234 0.562 0.778
0.244 0.176 ELIE_STR [45] 25.507 0.716 0.789 0.324 0.266 Ours 25.596 0.751
0.751 0.368 0.298 GT - - - 0.842 0.661 SID-Fuji-Text Input - - - 0.048 0.005
ZSL Zero-DCE [25] 8.992 0.035 1.228 0.249 0.061 Zero-DCE++ [46] 11.539 0.047
1.066 0.262 0.077 SCI [26] 10.301 0.056 1.130 0.300 0.073 UL CycleGAN [22]
17.832 0.565 0.735 0.277 0.191 EnlightenGAN [23] 18.834 0.572 0.822 0.310
0.277 SL Pix2Pix [21] 19.601 0.599 0.803 0.353 0.296 ChebyLighter [27] 20.313
0.616 0.791 0.412 0.318 FECNet [31] 18.863 0.365 0.829 0.382 0.185 IAT [30]
19.647 0.537 0.844 0.445 0.277 ELIE_STR [45] 19.816 0.614 0.801 0.426 0.333
Ours 21.880 0.649 0.788 0.487 0.356 GT - - - 0.775 0.697 LOL-Text Input - - -
0.333 0.133 ZSL Zero-DCE [25] 14.928 0.587 0.328 0.421 0.229 Zero-DCE++ [46]
15.829 0.537 0.408 0.389 0.242 SCI [26] 14.835 0.549 0.335 0.421 0.171 UL
CycleGAN [22] 19.826 0.734 0.288 0.250 0.133 EnlightenGAN [23] 15.800 0.654
0.300 0.343 0.125 SL Pix2Pix [21] 20.581 0.771 0.247 0.353 0.129 ChebyLighter
[27] 19.820 0.769 0.199 0.353 0.176 FECNet [31] 20.432 0.787 0.231 0.378 0.229
IAT [30] 20.437 0.772 0.234 0.421 0.188 ELIE_STR [45] 19.782 0.824 0.167 0.462
0.235 Ours 21.330 0.828 0.163 0.474 0.294 GT - - - 0.439 0.205

Table 3: Quantitative results of PSNR, SSIM, LPIPS, and text detection H-Mean
for low-light image enhancement methods on SID-Sony-Text, SID-Fuji-Text, and
LOL-Text datasets. Please note that TRAD, ZSL, UL, and SL stand for
traditional methods, zero-shot learning, unsupervised learning, and supervised
learning respectively. Scores in bold are the best of all.

###  6.3 Results on LOL-Text Dataset

To demonstrate the effectiveness of our model in enhancing low-light images
with varying levels of darkness, we conducted experiments on the widely used
LOL dataset, which is relatively brighter than the SID dataset, as depicted in
Table 1. Interestingly, we found that our enhanced images achieved the best
detection results on LOL-Text among existing methods, as shown in Table 3.
Surprisingly, despite the lower resolution (600x400) of the images in LOL, our
method’s enhanced images with sharper and crisper low-level details surpassed
the ground truth images’ H-Mean scores. Qualitative results on the LOL-Text
dataset are illustrated in the supplementary material. Although certain
methods yielded output images with acceptable image quality (i.e., bright
images without color shift), their text detection results were inferior to
ours. Furthermore, our superior results on the LOL-Text dataset emphasize our
method’s ability to generalize well on both ordinary and extremely low-light
images, effectively enhancing a broader range of low-light images while making
the text clearly visible.

###  6.4 Effectiveness of the Proposed Supervised-DCE Model

The goal of image synthesis in our work is to translate images captured in
well-illuminated scenarios to extremely low light. In this work, we choose the
commonly used IC15 scene text dataset as our main synthesis target. The
synthesized dataset then serves as additional data to train better scene text-
aware image enhancement models, which are studied in Section 6.5.

Intuitively, realistic synthesized images should possess similar
characteristics to genuine extremely low-light images. To verify the
effectiveness of our synthesis model, we compared our proposed Supervised-DCE
model (v2) with the U-Net proposed in SID [1] (v1). Specifically, we trained
the synthesizing models on the training set and synthesized the images based
on the corresponding test set. Then, we calculated the PSNR and SSIM of the
synthesized images by comparing them with the genuine ones along with the
average perceptual lightness in CIELAB color space. The comparison was made on
two SID datasets, SID-Sony and SID-Fuji.

In Table 4, we show that v2’s PSNR and SSIM are higher than v1’s, indicating
higher similarity between our synthesized and genuine images. Our new method
(v2) also exhibits closer Avg. L* values and H-Mean scores to the genuine
images than v1, indicating darker and more accurate deterioration of fine text
details. In addition, qualitative results for the proposed Supervised-DCE
model and results of synthetic IC15 datasets including Syn-IC15-Sony-v1, Syn-
IC15-Sony-v2, Syn-IC15-Fuji-v1, and Syn-IC15-Fuji-v2 are presented in the
supplementary material for comprehensive analyses.

Dataset PSNR SSIM Avg. L* CRAFT PAN Syn-SID-Sony-v1 41.095 0.809 0.176 0.294
0.083 Syn-SID-Sony-v2 45.442 0.942 0.003 0.135 0.014 Genuine SID-Sony - -
0.008 0.057 0.026 Syn-SID-Fuji-v1 39.187 0.784 0.172 0.402 0.042 Syn-SID-
Fuji-v2 41.881 0.863 0.002 0.093 0.002 Genuine SID-Fuji - - 0.004 0.048 0.005

Table 4: The difference between genuine extremely low-light dataset, SID, and
synthetic extremely low-light images generated using U-Net (v1) and
Supervised-DCE (v2). Please note that synthetic images’ PSNR and SSIM values
are based on comparison against genuine low-light images in the test set
instead of pure black images calculated in Table 1. Additionally, we can
notice that v2-images are more realistic and darker, similar to genuine
extremely low-light images due to their higher values of PSNR and SSIM, along
with closer Avg. L*.

###  6.5 Results on Training with Mixed Datasets

We trained top-performing models from Section 6.2 using a mixture of genuine
(SID) and synthetic low-light (IC15) datasets to test whether extremely low-
light image enhancement can benefit from synthesized images. The trained
models were evaluated on their respective genuine low-light datasets. Results
in Table 5 showed a significant increase in H-Mean, and we found that both
versions (v1 and v2) can fill the gap caused by the scarcity of genuine low-
light images. This justifies the creation of a synthetic IC15 dataset for such
a purpose. Furthermore, v2-images, i.e., extremely low-light images
synthesized by our proposed Supervised-DCE, further pushed the limit of H-mean
scores on genuine extremely low-light images, and our enhancement model
benefited the most because it could learn more from text instances and
reconstruct necessary details to represent texts. Despite our method’s
success, a noticeable gap exists between our results and the ground truth,
emphasizing the need for further research and development to achieve even more
accurate and reliable scene text extraction in low-light conditions.

Type Method SID-Sony-Text + Syn-IC15-Sony-v1 SID-Sony-Text + Syn-IC15-Sony-v2
SID-Fuji-Text + Syn-IC15-Fuji-v1 SID-Fuji-Text + Syn-IC15-Fuji-v2 CRAFT
$\uparrow$ PAN $\uparrow$ CRAFT $\uparrow$ PAN $\uparrow$ CRAFT $\uparrow$ PAN
$\uparrow$ CRAFT $\uparrow$ PAN $\uparrow$ Input 0.057 0.026 0.057 0.026 0.048
0.005 0.048 0.005 ZSL Zero-DCE++ [46] 0.230 0.159 0.242 0.153 0.274 0.080
0.281 0.076 SCI [26] 0.240 0.154 0.243 0.160 0.307 0.076 0.313 0.084 UL
CycleGAN [22] 0.180 0.071 0.219 0.143 0.297 0.284 0.310 0.277 EnlightenGAN
[23] 0.205 0.146 0.237 0.163 0.329 0.246 0.342 0.282 SL ELIE_STR [45] 0.348
0.278 0.361 0.296 0.444 0.359 0.466 0.375 Ours 0.383 0.311 0.395 0.319 0.515
0.392 0.549 0.416 GT 0.842 0.661 0.842 0.661 0.775 0.697 0.775 0.697

Table 5: Text detection H-Mean on genuine extremely low-light datasets when
trained on a combination of genuine and synthetic datasets. Scores in bold are
the best of all.

###  6.6 Ablation Study of Proposed Modules

Proposed Modules Image Quality H-Mean Text-CP Dual Encoder Edge-Att Edge
Decoder PSNR $\uparrow$ SSIM $\uparrow$ LPIPS $\downarrow$ CRAFT $\uparrow$
PAN $\uparrow$ - - - - 21.847 0.698 0.783 0.283 0.205 ✓ - - - 21.263 0.658
0.771 0.304 0.252 ✓ ✓ - - 20.597 0.655 0.780 0.335 0.261 ✓ ✓ ✓ - 21.440 0.669
0.776 0.342 0.256 ✓ ✓ - ✓ 21.588 0.674 0.779 0.353 0.285 ✓ - ✓ ✓ 23.074 0.712
0.783 0.350 0.281 - ✓ ✓ ✓ 24.192 0.738 0.784 0.356 0.292 ✓ ✓ ✓ ✓ 25.596 0.751
0.751 0.368 0.298

Table 6: Ablation study of proposed modules in terms of PSNR, SSIM, LPIPS, and
text detection H-Mean on the SID-Sony-Text dataset. Scores in bold are the
best of all.

To understand the effect of each component of our model, we conducted several
ablation experiments by either adding or removing them one at a time. Results
are presented in Table 6. The baseline was a plain U-Net without any proposed
modules. We initiated the ablation study by adding Text-CP data augmentation,
which improved CRAFT H-Mean from 0.283 to 0.304, indicating that involving
more text instances during training is relevant to text-aware image
enhancement for models to learn text representation. Moreover, scores
increased steadily by gradually stacking the baseline with more modules. For
instance, with the help of the dual encoder structure and Edge-Att module in
our proposed framework, CRAFT H-Mean increased from 0.304 to 0.342. This shows
that they can extract image features better and attend to edges that shape
texts in enhanced images. The edge reconstruction loss calculated based on
predictions from the edge decoder helped strengthen the learning of edge
features and empowered encoders in our model. Interestingly, we found that
removing one of the two most representative modules (i.e., dual encoder or
Edge-Att module) led to significant differences in H-Mean because these two
modules’ designs allow them to extract and attend to significant image
features independently. We concluded the experiment by showing that combining
all proposed modules led to the highest scores, as each module played an
integral role in our final network. Further analysis of Edge-Att and Text-CP
are included in the supplementary material to study their effectiveness as
compared to the original versions.

##  7 Conclusion

This paper presents a novel scene text-aware extremely low-light image
enhancement framework consisting of a Text-Aware Copy-Paste augmentation
method as a pre-processing step, followed by a new dual-encoder-decoder
architecture armed with Edge-Aware attention modules. With further assistance
from text detection and edge reconstruction losses, our model can enhance
images to the extent that high-level perceptual reasoning tasks can be better
fulfilled. Extremely low-light image synthesis has rarely been discussed over
the years. Thus, we proposed a novel Supervised-DCE model to provide better
synthesized extremely low-light images so that extremely low-light image
enhancement can benefit from publicly available scene text datasets such as
IC15. Furthermore, our proposed extremely low-light image enhancement model
has been rigorously evaluated against various competing methods, including
traditional techniques and deep learning-based approaches, on challenging
datasets such as SID-Sony-Text, SID-Fuji-Text, LOL-Text, and synthetic IC15.
Through extensive experimentation, our findings consistently demonstrate our
model’s superiority in extremely low-light image enhancement and text
extraction tasks.

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# Spectral Turán Type Problems on Cancellative Hypergraphs

Zhenyu Ni Department of Mathematics, Hainan University, Haikou 570228, P.R.
China (995264@hainanu.edu.cn). Lele Liu College of Science, University of
Shanghai for Science and Technology, Shanghai 200093, P.R. China
(ahhylau@outlook.com). This author is supported by the National Natural
Science Foundation of China (No. 12001370).Corresponding author. Liying Kang
Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
(lykang@shu.edu.cn). This author is supported by the National Natural Science
Foundation of China (Nos. 11871329, 11971298)

###### Abstract

Let $G$ be a cancellative $3$-uniform hypergraph in which the symmetric
difference of any two edges is not contained in a third one. Equivalently, a
$3$-uniform hypergraph $G$ is cancellative if and only if $G$ is
$\\{F_{4},F_{5}\\}$-free, where $F_{4}=\\{abc,abd,bcd\\}$ and
$F_{5}=\\{abc,abd,cde\\}$. A classical result in extremal combinatorics stated
that the maximum size of a cancellative hypergraph is achieved by the balanced
complete tripartite $3$-uniform hypergraph, which was firstly proved by
Bollobás and later by Keevash and Mubayi. In this paper, we consider spectral
extremal problems for cancellative hypergraphs. More precisely, we determine
the maximum $p$-spectral radius of cancellative $3$-uniform hypergraphs, and
characterize the extremal hypergraph. As a by-product, we give an alternative
proof of Bollobás’ result from spectral viewpoint.

Keywords: Hypergraph; Spectral radius; Spectral Turán problem.

AMS Classification: 05C35; 05C50; 05C65.

##  1 Introduction

Consider an $r$-uniform hypergraph (or $r$-graph for brevity) $G$ and a family
of $r$-graphs $\mathcal{F}$. We say $G$ is _$\mathcal{F}$ -free_ if $G$ does
not contain any member of $\mathcal{F}$ as a subhypergraph. The _Turán number_
$\operatorname{ex}(n,\mathcal{F})$ is the maximum number of edges of an
$\mathcal{F}$-free hypergraph on $n$ vertices. Determining Turán numbers of
graphs and hypergraphs is one of the central problems in extremal
combinatorics. For graphs, the problem was asymptotically solved for all non-
bipartite graphs by the celebrated Erdős-Stone-Simonovits Theorem. By contrast
with the graph case, there is comparatively little understanding of the
hypergraph Turán number. We refer the reader to the surveys [6, 9, 12].

In this paper we consider spectral analogues of Turán type problems for
$r$-graphs. For $r=2$, the picture is relatively complete, due in large part
to a longstanding project of Nikiforov, see e.g., [13] for details. However,
for $r\geq 3$ there are very few known results. In [10], Keevash-Lenz-Mubayi
determine the maximum $p$-spectral radius of any $3$-graph on $n$ vertices not
containing the Fano plane when $n$ is sufficiently large. They also obtain a
$p$-spectral version of the Erdős-Ko-Rado theorem on $t$-intersecting
$r$-graphs. Recently, Ellingham-Lu-Wang [4] show that the $n$-vertex
outerplanar $3$-graph of maximum spectral radius is the unique 3-graph whose
shadow graph is the join of an isolated vertex and the path $P_{n-1}$. Gao-
Chang-Hou [7] study the extremal problem for $K_{r+1}^{+}$-free $r$-graphs
among linear hypergraphs, where $K_{r+1}^{+}$ is obtained from the complete
graph $K_{r+1}$ by enlarging each edge of $K_{r+1}$ with $r-2$ new vertices
disjoint from $V(K_{r+1})$ such that distinct edges of $K_{r+1}$ are enlarged
by distinct vertices.

To state our results precisely, we need some basic definitions and notations.
A $3$-graph is _tripartite_ or _$3$ -partite_ if it has a vertex partition
into three parts such that every edge has exactly one vertex in each part. Let
$T_{3}(n)$ be the complete $3$-partite $3$-graph on $n$ vertices with part
sizes $\lfloor n/3\rfloor$, $\lfloor(n+1)/3\rfloor$, $\lfloor(n+2)/3\rfloor$,
and $t_{3}(n)$ be the number of edges of $T_{3}(n)$. That is,

$t_{3}(n)=\Big{\lfloor}\frac{n}{3}\Big{\rfloor}\cdot\Big{\lfloor}\frac{n+1}{3}\Big{\rfloor}\cdot\Big{\lfloor}\frac{n+2}{3}\Big{\rfloor}.$

We call an $r$-graph $G$ _cancellative_ if $G$ has the property that for any
edges $A$, $B$, $C$ whenever $A\cup B=A\cup C$, we have $B=C$. Equivalently,
$G$ is cancellative if $G$ has no three distinct triples $A$, $B$, $C$
satisfying $B\triangle C\subset A$, where $\triangle$ is the symmetric
difference. For graphs, the condition is equivalent to saying that $G$ is
triangle-free. Moving on to $3$-graphs, we observe that $B\Delta C\subset A$
can only occur when $|B\cap C|=2$ for $B\neq C$. This leads us to identify the
two non-isomorphic configurations that are forbidden in a cancellative
$3$-graph: $F_{4}=\\{abc,abd,bcd\\}$ and $F_{5}=\\{abc,abd,cde\\}$.

It is well-known that the study of Turán numbers dates back to Mantel’s
theorem, which states that $\operatorname{ex}(n,K_{3})=\lfloor
n^{2}/4\rfloor$. As an extension of the problem to hypergraphs, Katona
conjectured, and Bollobás [1] proved the following result.

######  Theorem 1.1 ([1]).

A cancellative $3$-graph on $n$ vertices has at most $t_{3}(n)$ edges, with
equality only for $T_{3}(n)$.

In [8], Keevash and Mubayi presented a new proof of Bollobás’ result, and
further proved a stability theorem for cancellative hypergraphs. The main
result of this paper is the following $p$-spectral analogues of Bollobás’
result.

######  Theorem 1.2.

Let $p\geq 1$ and $G$ be a cancellative $3$-graph on $n$ vertices.

  1. $(1)$

If $p\geq 3$, then $\lambda^{(p)}(G)\leq\lambda^{(p)}(T_{3}(n))$, with
equality if and only if $G=T_{3}(n)$.

  2. $(2)$

If $p=1$, then $\lambda^{(1)}(G)=1/9$.

##  2 Preliminaries

In this section we introduce definitions and notation that will be used
throughout the paper, and give some preliminary lemmas.

Given an $r$-graph $G=(V(G),E(G))$ and a vertex $v$ of $G$. The _link_
$L_{G}(v)$ is the $(r-1)$-graph consisting of all $S\subset V(G)$ with
$|S|=r-1$ and $S\cup\\{v\\}\in E(G)$. The _degree_ $d_{G}(v)$ of $v$ is the
size of $L_{G}(v)$. As usual, we denote by $N_{G}(v)$ the neighbor of a vertex
$v$, i.e., the set formed by all the vertices which form an edge with $v$. In
the above mentioned notation, we will skip the index $G$ whenever $G$ is
understood from the context.

The _shadow graph_ of $G$, denoted by $\partial(G)$, is the graph with
$V(\partial(G))=V(G)$ and $E(\partial(G))$ consisting of all pairs of vertices
that belong to an edge of $G$, i.e., $E(\partial(G))=\\{e:|e|=2,\,e\subseteq
f\ \text{for some}\ f\in E(G)\\}$. For more definitions and notation from
hypergraph theory, see e.g., [2].

For any real number $p\geq 1$, the $p$-spectral radius was introduced by
Keevash, Lenz and Mubayi [10] and subsequently studied by Nikiforov [14, 15].
Let $G$ be an $r$-graph of order $n$, the polynomial form of $G$ is a multi-
linear function $P_{G}(\bm{x}):\mathbb{R}^{n}\to\mathbb{R}$ defined for any
vector $\bm{x}=(x_{1},x_{2},\ldots,x_{n})^{\mathrm{T}}\in\mathbb{R}^{n}$ as

$P_{G}(\bm{x})=r\sum_{\\{i_{1},i_{2},\ldots,i_{r}\\}\in
E(G)}x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}.$

The _$p$ -spectral radius_111We modified the definition of $p$-spectral radius
by removing a constant factor $(r-1)!$ from [10], so that the $p$-spectral
radius is the same as the one in [3] when $p=r$. This is not essential and
does not affect the results at all. of $G$ is defined as

$\lambda^{(p)}(G):=\max_{\|\bm{x}\|_{p}=1}P_{G}(\bm{x}),$ (2.1)

where $\|\bm{x}\|_{p}:=(|x_{1}|^{p}+\cdots+|x_{n}|^{p})^{1/p}$.

For any real number $p\geq 1$, we denote by $\mathbb{S}_{p,+}^{n-1}$ the set
of all nonnegative real vectors $\bm{x}\in\mathbb{R}^{n}$ with
$\|\bm{x}\|_{p}=1$. If $\bm{x}\in\mathbb{R}^{n}$ is a vector with
$\|\bm{x}\|_{p}=1$ such that $\lambda^{(p)}(G)=P_{G}(\bm{x})$, then $\bm{x}$
is called an _eigenvector_ corresponding to $\lambda^{(p)}(G)$. Note that
$P_{G}(\bm{x})$ can always reach its maximum at some nonnegative vectors. By
Lagrange’s method, we have the _eigenequations_ for $\lambda^{(p)}(G)$ and
$\bm{x}\in\mathbb{S}_{p,+}^{n-1}$ as follows:

$\lambda^{(p)}(G)x_{i}^{p-1}=\sum_{\\{i,i_{2},\ldots,i_{r}\\}\in
E(G)}x_{i_{2}}\cdots x_{i_{r}}~{}~{}\text{for}\ x_{i}>0.$ (2.2)

It is worth mentioning that the $p$-spectral radius $\lambda^{(p)}(G)$ shows
remarkable connections with some hypergraph invariants. For instance,
$\lambda^{(1)}(G)/r$ is the Lagrangian of $G$, $\lambda^{(r)}(G)$ is the usual
spectral radius introduced by Cooper and Dutle [3], and
$\lambda^{(\infty)}(G)/r$ is the number of edges of $G$ (see [14, Proposition
2.10]).

Given two vertices $u$ and $v$, we say that $u$ and $v$ are _equivalent_ in
$G$, in writing $u\sim v$, if transposing $u$ and $v$ and leaving the
remaining vertices intact, we get an automorphism of $G$.

######  Lemma 2.1 ([14]).

Let $G$ be a uniform hypergraph on $n$ vertices and $u\sim v$. If $p>1$ and
$\bm{x}\in\mathbb{S}_{p}^{n-1}$ is an eigenvector to $\lambda^{(p)}(G)$, then
$x_{u}=x_{v}$.

##  3 Cancellative hypergraph of maximum $p$-spectral radius

The aim of this section is to give a proof of Theorem 1.2. We split it into
Theorem 3.1 – Theorem 3.3, which deal with $p=3$, $p>3$ and $p=1$,
respectively.

###  3.1 General properties on cancellative hypergraphs

We start this subsection with a basic fact.

######  Lemma 3.1.

Let $G$ be a cancellative hypergraph, and $u,v$ be adjacent vertices. Then
$L(u)$ and $L(v)$ are edge-disjoint graphs.

###### Proof.

Assume by contradiction that $e\in E(L(u))\cap E(L(v))$. Since $u$ and $v$ are
adjacent in $G$, we have $\\{u,v\\}\subset e_{1}\in E(G)$ for some edge
$e_{1}$. Hence, $e_{2}=e\cup\\{u\\}$, $e_{3}=e\cup\\{v\\}$ and $e_{1}$ are
three edges of $G$ such that $e_{2}\Delta e_{3}\subset e_{1}$, a
contradiction. ∎

Let $G$ be a $3$-graph and $v\in V(G)$. We denote by $E_{v}(G)$ the collection
of edges of $G$ containing $v$, i.e., $E_{v}(G)=\\{e:v\in e\in E(G)\\}$. For a
pair of vertices $u$ and $v$ in $G$, we denote by $T_{v}^{u}(G)$ a new
$3$-graph with $V(T_{v}^{u}(G))=V(G)$ and

$E(T_{v}^{u}(G))=\big{(}E(G)\setminus
E_{v}(G)\big{)}\cup\\{(e\setminus\\{u\\})\cup\\{v\\}:e\in E_{u}(G)\setminus
E_{v}(G)\\}.$

######  Lemma 3.2.

Let $G$ be a cancellative $3$-graph. Then $T_{v}^{u}(G)$ is also cancellative
for any $u,v\in V(G)$.

###### Proof.

Suppose to the contrary that there exist three edges $e_{1},e_{2},e_{3}\in
T_{v}^{u}(G)$ such that $e_{1}\triangle e_{2}\subset e_{3}$. Recalling the
definition of $T_{v}^{u}(G)$, we deduce that $u$, $v$ are non-adjacent in
$T_{v}^{u}(G)$, and $(e\cup\\{u\\})\setminus\\{v\\}\in E(G)$ for any $e\in
E_{v}(T_{v}^{u}(G))$. On the other hand, since $G$ is cancellative, we have
$v\in e_{1}\cup e_{2}\cup e_{3}$. Denote by $\alpha$ the number of edges
$e_{1}$, $e_{2}$, $e_{3}$ containing $v$. It suffices to consider the
following three cases.

Case 1. $\alpha=3$. We have $v\in e_{1}\cap e_{2}\cap e_{3}$. Hence,
$e_{1}^{\prime}=\left(e_{1}\cup\\{u\\}\right)\setminus\\{v\\}$,
$e_{2}^{\prime}=\left(e_{2}\cup\\{u\\}\right)\setminus\\{v\\}$ and
$e_{3}^{\prime}=\left(e_{3}\cup\\{u\\}\right)\setminus\\{v\\}$ are three edges
in $G$ with $e_{1}^{\prime}\triangle e_{2}^{\prime}\subset e_{3}^{\prime}$.
This contradicts the fact that $G$ is cancellative.

Case 2. $\alpha=2$. Without loss of generality, we assume $v\in(e_{1}\cap
e_{2})\setminus e_{3}$ or $v\in(e_{1}\cap e_{3})\setminus e_{2}$. If
$v\in(e_{1}\cap e_{2})\setminus e_{3}$, then $e_{3}\in E(G)$. It follows that
$e_{1}^{\prime}=(e_{1}\cup\\{u\\})\setminus\\{v\\}$,
$e_{2}^{\prime}=(e_{2}\cup\\{u\\})\setminus\\{v\\}$ and $e_{3}$ are three
edges of $G$ with $e_{1}^{\prime}\triangle e_{2}^{\prime}\subset e_{3}$, which
is a contradiction. If $v\in(e_{1}\cap e_{3})\setminus e_{2}$, then $e_{2}\in
E(G)$. It follows that $e_{1}^{\prime}=(e_{1}\cup\\{u\\})\setminus\\{v\\}$,
$e_{2}$ and $e_{3}^{\prime}=(e_{3}\cup\\{u\\})\setminus\\{v\\}$ are three
edges of $G$ with $e_{1}^{\prime}\triangle e_{2}\subset e_{3}^{\prime}$, a
contradiction.

Case 3. $\alpha=1$. Without loss of generality, we assume $v\in
e_{3}\setminus(e_{1}\cup e_{2})$. Then $e_{1}\in E(G)$ and $e_{2}\in E(G)$. We
immediately obtain that $e_{1}$, $e_{2}$ and
$e_{3}^{\prime}=(e_{3}\cup\\{u\\})\setminus\\{v\\}$ are three edges of $G$
with $e_{1}\triangle e_{2}\subset e_{3}^{\prime}$. This is a contradiction and
proves Lemma 3.2. ∎

######  Lemma 3.3.

Let $p>1$ and $G$ be a complete $3$-partite $3$-graph. Then

$\lambda^{(p)}(G)=\frac{(27\cdot|E(G)|)^{1-1/p}}{9}.$

###### Proof.

Assume that $V_{1}$, $V_{2}$ and $V_{3}$ are the vertex classes of $G$ with
$n_{i}:=|V_{i}|$ and $n_{1}\geq n_{2}\geq n_{3}$. Let
$\bm{x}\in\mathbb{S}_{p,+}^{n-1}$ be an eigenvector corresponding to
$\lambda^{(p)}(G)$. By Lemma 2.1, for $i=1,2,3$ we denote $a_{i}:=x_{v}$ for
$v\in V_{i}$, and set $\lambda:=\lambda^{(p)}(G)$ for short. In light of
eigenequation (2.2), we find that

$\begin{cases}\lambda a_{1}^{p-1}=n_{2}n_{3}a_{2}a_{3},\\\ \lambda
a_{2}^{p-1}=n_{1}n_{3}a_{1}a_{3},\\\ \lambda
a_{3}^{p-1}=n_{1}n_{2}a_{1}a_{2},\end{cases}$

from which we obtain that $a_{i}=(3n_{i})^{-1/p}$, $i=1,2,3$. Therefore,

$\lambda=\frac{(27\cdot
n_{1}n_{2}n_{3})^{1-1/p}}{9}=\frac{(27\cdot|E(G)|)^{1-1/p}}{9}.$

This completes the proof of Lemma 3.3. ∎

###  3.2 Extremal $p$-spectral radius of cancellative hypergraphs

Let $\operatorname{Ex}_{sp}(n,\\{F_{4},F_{5}\\})$ be the set of all $3$-graphs
attaining the maximum $p$-spectral radius among cancellative hypergraphs on
$n$ vertices. Given a vector $\bm{x}\in\mathbb{R}^{n}$ and a set
$S\subset[n]:=\\{1,2,\ldots,n\\}$, we write $\bm{x}(S):=\prod_{i\in S}x_{i}$
for short. The _support set_ $S$ of a vector $\bm{x}$ is the index of non-zero
elements in $\bm{x}$, i.e., $S=\\{i\in[n]:x_{i}\neq 0\\}$. Also, we denote by
$x_{\min}:=\min\\{|x_{i}|:i\in[n]\\}$ and
$x_{\max}:=\max\\{|x_{i}|:i\in[n]\\}$.

######  Lemma 3.4.

Let $p>1$, $G\in\operatorname{Ex}_{sp}(n,\\{F_{4},F_{5}\\})$, and
$\bm{x}\in\mathbb{S}_{p,+}^{n-1}$ be an eigenvector corresponding to
$\lambda^{(p)}(G)$. If $u,v$ are two non-adjacent vertices, then
$x_{u}=x_{v}$.

###### Proof.

Assume $u$ and $v$ are two non-adjacent vertices in $G$. Since $G$ is a
cancellative $3$-graph, we have $T_{u}^{v}(G)$ is also cancellative by Lemma
3.2. It follows from (2.1) and (2.2) that

$\displaystyle\lambda^{(p)}(T_{u}^{v}(G))$ $\displaystyle\geq 3\sum_{e\in
E(G)}\bm{x}(e)-3\sum_{e\in E_{u}(G)}\bm{x}(e)+3\sum_{e\in
E_{v}(G)}\bm{x}(e\setminus\\{v\\})\cdot x_{u}$
$\displaystyle=\lambda^{(p)}(G)-3\lambda^{(p)}(G)x_{u}^{p}+3\lambda^{(p)}(G)x_{v}^{p-1}x_{u}$
$\displaystyle=\lambda^{(p)}(G)+3\lambda^{(p)}(G)(x_{v}^{p-1}-x_{u}^{p-1})\cdot
x_{u},$

which yields that $x_{u}\geq x_{v}$. Likewise, we also have $x_{v}\geq x_{u}$
by considering $T_{v}^{u}(G)$. Hence, $x_{u}=x_{v}$, completing the proof of
Lemma 3.4. ∎

######  Lemma 3.5.

Let $p>1$, $G\in\operatorname{Ex}_{sp}(n,\\{F_{4},F_{5}\\})$, and $u,v$ be two
non-adjacent vertices. Then there exists a cancellative $3$-graph $H$ such
that

$L_{H}(u)=L_{H}(v),~{}~{}\lambda^{(p)}(H)=\lambda^{(p)}(G),\ \text{and}\
d_{H}(w)\leq d_{G}(w),~{}~{}w\in V(G).$ (3.1)

###### Proof.

Assume that $\bm{x}\in\mathbb{S}_{p,+}^{n-1}$ is an eigenvector corresponding
to $\lambda^{(p)}(G)$. By Lemma 3.4, $x_{u}=x_{v}$. Without loss of
generality, we assume $d_{G}(u)\geq d_{G}(v)$. In view of (2.1) and (2.2), we
have

$\displaystyle\lambda^{(p)}(T_{u}^{v}(G))$ $\displaystyle\geq 3\sum_{e\in
E(G)}\bm{x}(e)-3\sum_{e\in E_{u}(G)}\bm{x}(e)+3\sum_{e\in
E_{v}(G)}\bm{x}(e\setminus\\{v\\})\cdot x_{u}$
$\displaystyle=\lambda^{(p)}(G)+3\lambda^{(p)}(G)(x_{v}^{p-1}-x_{u}^{p-1})\cdot
x_{u}$ $\displaystyle=\lambda^{(p)}(G).$

Observe that $T_{u}^{v}(G)$ is a cancellative $3$-graph and
$G\in\operatorname{Ex}_{sp}(n,\\{F_{4},F_{5}\\})$. We immediately obtain that
$\lambda^{(p)}(T_{u}^{v}(G))=\lambda^{(p)}(G)$. It is straightforward to check
that $H:=T_{u}^{v}(G)$ is a cancellative $3$-graph satisfying (3.1), as
desired. ∎

Next, we give an estimation on the entries of eigenvectors corresponding to
$\lambda^{(p)}(G)$.

######  Lemma 3.6.

Let $G\in\operatorname{Ex}_{sp}(n,\\{F_{4},F_{5}\\})$ and
$\bm{x}\in\mathbb{S}_{p,+}^{n-1}$ be an eigenvector corresponding to
$\lambda^{(p)}(G)$. If $1<p\leq 3$, then

$x_{\min}>\Big{(}\frac{3}{4}\Big{)}^{2/(p-1)}\cdot x_{\max}.$

###### Proof.

Suppose to the contrary that
$x_{\min}\leq\big{(}\frac{3}{4}\big{)}^{2/(p-1)}\cdot x_{\max}$. Let $u$ and
$v$ be two vertices such that $x_{u}=x_{\min}$ and $x_{v}=x_{\max}>0$. Then we
have

$\left(1+\frac{x_{u}}{x_{v}}\right)\left(\frac{x_{u}}{x_{v}}\right)^{p-1}\leq\bigg{(}1+\left(\frac{3}{4}\right)^{2/(p-1)}\bigg{)}\left(\frac{3}{4}\right)^{2}\leq\frac{7}{4}\cdot\frac{9}{16}<1,$

which implies that

$x_{v}^{p}-x_{u}^{p}>x_{u}^{p-1}x_{v}.$ (3.2)

On the other hand, by eigenequations we have

$\sum_{e\in E_{v}(G)\setminus
E_{u}(G)}\bm{x}(e)\geq\lambda^{(p)}(G)(x_{v}^{p}-x_{u}^{p}).$ (3.3)

Now, we consider the cancellative $3$-graph $T_{u}^{v}(G)$. In light of (2.1)
and (3.3), we have

$\displaystyle\lambda^{(p)}(T_{u}^{v}(G))$ $\displaystyle\geq 3\sum_{e\in
E(G)}\bm{x}(e)-3\sum_{e\in E_{u}(G)}\bm{x}(e)+3\sum_{e\in E_{v}(G)\setminus
E_{u}(G)}\bm{x}(e\setminus\\{v\\})\cdot x_{u}$
$\displaystyle\geq\lambda^{(p)}(G)-3\lambda^{(p)}(G)x_{u}^{p}+3\lambda^{p}(G)(x_{v}^{p}-x_{u}^{p})\cdot\frac{x_{u}}{x_{v}}$
$\displaystyle>\lambda^{(p)}(G)+3\lambda^{(p)}(G)\Big{(}-x_{u}^{p}+x_{u}^{p-1}x_{v}\cdot\frac{x_{u}}{x_{v}}\Big{)}$
$\displaystyle=\lambda^{(p)}(G),$

where the third inequality is due to (3.2). This contradicts the fact that $G$
has maximum $p$-spectral radius over all cancellative hypergraphs. ∎

Now, we are ready to give a proof of Theorem 1.2 for $p=3$.

######  Theorem 3.1.

Let $G$ be a cancellative $3$-graph on $n$ vertices. Then
$\lambda^{(3)}(G)\leq\lambda^{(3)}(T_{3}(n))$ with equality if and only if
$G=T_{3}(n)$.

###### Proof.

According to Lemma 3.5, we assume that
$G^{*}\in\operatorname{Ex}_{sp}(n,\\{F_{4},F_{5}\\})$ is a $3$-graph such that
$L_{G^{*}}(u)=L_{G^{*}}(v)$ for any non-adjacent vertices $u$ and $v$.

Our first goal is to show $G^{*}=T_{3}(n)$ by Claim 3.1 – Claim 3.3. Assume
that $\bm{x}\in\mathbb{S}_{3,+}^{n-1}$ is an eigenvector corresponding to
$\lambda^{(3)}(G^{*})$; $u_{1}$ is a vertex in $G^{*}$ such that
$x_{u_{1}}=x_{\max}$ and $u_{2}$ is a vertex with $x_{u_{2}}=\max\\{x_{v}:v\in
N_{G^{*}}(u_{1})\\}$. Let $U_{1}:=V(G^{*})\setminus N_{G^{*}}(u_{1})$ and
$U_{2}:=V(G^{*})\setminus N_{G^{*}}(u_{2})$. Since $u_{2}\in V(G^{*})\setminus
U_{1}$, there exists a vertex $u_{3}$ such that $\\{u_{1},u_{2},u_{3}\\}\in
E(G^{*})$. Let $U_{3}=V(G^{*})\setminus N_{G^{*}}(u_{3})$. Recall that for any
non-adjacent vertices $u$ and $v$ we have $L_{G^{*}}(u)=L_{G^{*}}(v)$. Hence,
the sets $U_{1}$, $U_{2}$ and $U_{3}$ are well-defined.

######  Claim 3.1.

The following statements hold:

  1. $(1)$

$d_{G^{*}}(u_{1})>n(n-1)/9$;

  2. $(2)$

$d_{G^{*}}(u_{2})>n(n-1)/12$;

  3. $(3)$

$d_{G^{*}}(v)>n(n-1)/16$, $v\in V(G^{*})$.

Proof of Claim 3.1. Since $T_{3}(n)$ is a cancellative $3$-graph, it follows
from Lemma 3.3 that

$\lambda^{(3)}(G^{*})\geq\lambda^{(3)}(T_{3}(n))=\frac{\big{(}27\cdot
t_{3}(n)\big{)}^{2/3}}{9}.$

By simple algebra we see

$\lambda^{(3)}(G^{*})\geq\frac{\big{(}(n-2)(n+1)^{2}\big{)}^{2/3}}{9}>\frac{n(n-1)}{9}.$
(3.4)

(1). By eigenequation with respect to $u_{1}$, we have

$\lambda^{(3)}(G^{*})x_{u_{1}}^{2}=\sum_{\\{u_{1},i,j\\}\in
E(G^{*})}x_{i}x_{j}\leq d_{G^{*}}(u_{1})x_{u_{1}}^{2}.$

Combining with (3.4), we get

$d_{G^{*}}(u_{1})\geq\lambda^{(3)}(G^{*})>\frac{n(n-1)}{9}.$ (3.5)

(2). Observe that the definition of $U_{1}$, and $L_{G^{*}}(u)=L_{G^{*}}(v)$
for any pair $u,v\in U_{1}$. We immediately obtain that
$|(e\setminus\\{u_{2}\\})\cap U_{1}|\leq 1$ for each $e\in E_{u_{2}}(G^{*})$.
It follows from $x_{u_{2}}=\max\\{x_{v}:v\in V(G^{*})\setminus U_{1}\\}$ that

$\lambda^{(3)}(G^{*})x_{u_{2}}^{2}=\sum_{\\{u_{2},i,j\\}\in
E(G^{*})}x_{i}x_{j}\leq d_{G^{*}}(u_{2})x_{u_{1}}x_{u_{2}},$

which, together with Lemma 3.6 for $p=3$, gives

$\displaystyle d_{G^{*}}(u_{2})$
$\displaystyle\geq\frac{x_{u_{2}}}{x_{u_{1}}}\cdot\lambda^{(3)}(G^{*})$
$\displaystyle\geq\frac{3}{4}\cdot\lambda^{(3)}(G^{*})$
$\displaystyle>\frac{1}{12}n(n-1).$

The last inequality is due to (3.4).

(3). Let $v$ be an arbitrary vertex in $V(G^{*})$. Then

$\lambda^{(3)}(G^{*})x_{v}^{2}=\sum_{\\{v,i,j\\}\in E(G^{*})}x_{i}x_{j}\leq
d_{G^{*}}(v)x_{u_{1}}^{2}.$

Hence, by Lemma 3.6 and (3.4) we have

$d_{G^{*}}(v)\geq\Big{(}\frac{x_{v}}{x_{u_{1}}}\Big{)}^{2}\cdot\lambda^{(3)}(G^{*})>\frac{1}{16}n(n-1),$

as desired. $\Box$

Next, we consider the graph $H=L_{G^{*}}(u_{1})\cup L_{G^{*}}(u_{2})\cup
L_{G^{*}}(u_{3})$. Let $\phi:E(H)\to[3]$ be a mapping such that $\phi(f)=i$ if
$f\in L_{G^{*}}(u_{i})$, $i\in[3]$. By Lemma 3.1, $\phi$ is an edge coloring
of $H$. For convenience, we denote $L:=V(G^{*})\setminus(U_{1}\cup U_{2}\cup
U_{3})$.

######  Claim 3.2.

If $L\neq\emptyset$, then there is no rainbow star $K_{1,3}$ in the induced
subgraph $H[L]$ with the coloring $\phi$.

Proof of Claim 3.2. Suppose to the contrary that there exist
$v_{1},v_{2},v_{3},v_{4}\in L$ with $\phi(v_{1}v_{2})=1$, $\phi(v_{1}v_{3})=2$
and $\phi(v_{1}v_{4})=3$. We first show that $\\{v_{1},v_{2},v_{3},v_{4}\\}$
induced a clique in $\partial(G^{*})$ by contradiction. Without loss of
generality, we assume $v_{2}v_{3}\notin E(\partial(G^{*}))$. Then
$L_{G^{*}}(v_{2})=L_{G^{*}}(v_{3})$. Since $\phi(v_{1}v_{2})=1$ and
$\phi(v_{1}v_{3})=2$, we have $\\{u_{1},v_{1},v_{2}\\}\in E(G^{*})$ and
$\\{u_{2},v_{1},v_{3}\\}\in E(G^{*})$. This implies that
$e_{1}=\\{u_{1},u_{2},u_{3}\\}$, $e_{2}=\\{u_{1},v_{1},v_{2}\\}$ and
$e_{3}=\\{u_{2},v_{1},v_{2}\\}$ are three edges in $G^{*}$ with
$e_{2}\triangle e_{3}\subset e_{1}$, which is impossible.

On the other hand, since $L=V(G^{*})\setminus(U_{1}\cup U_{2}\cup U_{3})$, we
have $v_{i}u_{j}\in E(\partial(G^{*}))$ for any $i\in[4]$, $j\in[3]$.
Therefore, every pair of vertices in
$\\{v_{1},v_{2},v_{3},v_{4},u_{1},u_{2},u_{3}\\}$ is contained in an edge of
$G^{*}$. Consider the graph

$H^{\prime}:=\bigg{(}\bigcup_{i=1}^{3}L_{G^{*}}(u_{i})\bigg{)}\bigcup\bigg{(}\bigcup_{i=1}^{4}L_{G^{*}}(v_{i})\bigg{)}.$

By Claim 3.1, we have

$\displaystyle|E(H^{\prime})|$ $\displaystyle=\sum_{1\leq i\leq
3}d_{G^{*}}(u_{i})+\sum_{1\leq j\leq 4}d_{G^{*}}(v_{j})$
$\displaystyle>\left(1+\frac{3}{4}+5\times\frac{9}{16}\right)\cdot\frac{1}{9}n(n-1)$
$\displaystyle=\frac{73}{144}n(n-1)$ $\displaystyle>\binom{n}{2},$

a contradiction completing the proof of Claim 3.2. $\Box$

######  Claim 3.3.

$L=\emptyset$.

Proof of Claim 3.3. Suppose to the contrary that $L\neq\emptyset$. For
$i=1,2,3$, let $L_{i}$ be the set of vertices in $L$ which is not contained in
an edge with coloring $i$. By Claim 3.2, we have $L=L_{1}\cup L_{2}\cup
L_{3}$. Without loss of generality, we assume $L_{1}\neq\emptyset$. Let $w$ be
a vertex in $L_{1}$. Then there exists an edge $f$ in $G^{*}$ such that
$f=\\{u_{1},w,w^{\prime}\\}$, where $w^{\prime}\in U_{2}\cup U_{3}$. If
$w^{\prime}\in U_{2}$, then $f^{\prime}=\\{u_{1},u_{3},w^{\prime}\\}\in
E(G^{*})$. Since $G^{*}$ is cancellative, $w$ is not a neighbor of $u_{3}$ in
$G^{*}$. This implies that $w\in U_{3}$, a contradiction to $w\in L$.
Similarly, if $w^{\prime}\in U_{3}$, then $w\in U_{2}$, which is also a
contradiction. $\Box$

Now, we continue our proof. By Claim 3.3, we immediately obtain that $G^{*}$
is a complete $3$-partite $3$-graph with vertex classes $U_{1}$, $U_{2}$ and
$U_{3}$. Hence, $G^{*}=T_{3}(n)$ by Lemma 3.3.

Finally, it is enough to show that $G=T_{3}(n)$ for any
$G\in\operatorname{Ex}_{sp}(n,\\{F_{4},F_{5}\\})$. According to Lemma 3.5 and
Claim 3.3, we can transfer $G$ to the complete $3$-partite $3$-graph
$T_{3}(n)$ by a sequence of switchings $T_{u}^{v}(\,\cdot\,)$ that keeping the
spectral radius unchanged. Let $T_{1},\ldots,T_{s}$ be such a sequence of
switchings $T_{u}^{v}(\,\cdot\,)$ which turn $G$ into $T_{3}(n)$. Consider the
$3$-graphs $G=G_{0},G_{1},\ldots,G_{s}=T_{3}(n)$ in which $G_{i}$ is obtained
from $G_{i-1}$ by applying $T_{i}$. Let $\bm{z}\in\mathbb{S}_{3,+}^{n-1}$ be
an eigenvector corresponding to $\lambda^{(3)}(G_{s-1})$ and
$T_{u}^{v}(G_{s-1})=T_{3}(n)$, and denote

$A:=V(G_{s-1})\setminus\big{(}N_{G_{s-1}}(v)\cup\\{u\\}\cup\\{v\\}\big{)}.$

Hence, we have $L_{G_{s-1}}(w)=L_{G_{s-1}}(v)=L_{T_{3}(n)}(v)$ for each $w\in
A$. In what follows, we shall prove $L_{G_{s-1}}(u)=L_{G_{s-1}}(v)$, and
therefore $G_{s-1}=T_{3}(n)$. If $L_{G_{s-1}}(u)\neq L_{G_{s-1}}(v)$, there
exists an edge $e=v_{1}v_{2}\in L_{G_{s-1}}(u)\setminus L_{G_{s-1}}(v)$ since
$z_{u}=z_{v}$ by Lemma 3.4. Let $M_{1}$ and $M_{2}$ be two subsets of
$V(G_{s-1})$ such that $M_{1}\cup M_{2}=N_{G_{s-1}}(v)$ and
$L_{G_{s-1}}(v)=K_{|M_{1}|,|M_{2}|}$. If $\\{v_{1},v_{2}\\}\subset
N_{G_{s-1}}(v)$, then $\\{v_{1},v_{2}\\}\subset M_{1}$ or
$\\{v_{1},v_{2}\\}\subset M_{2}$. It follows that there exists a vertex $w\in
N_{G_{s-1}}(v)$ such that $f_{1}:=\\{v,w,v_{1}\\}\in E(G_{s-1})$ and
$f_{2}:=\\{v,w,v_{2}\\}\in E(G_{s-1})$. However, $f_{1}\Delta
f_{2}\subset\\{u,v_{1},v_{2}\\}\in E(G_{s-1})$, a contradiction. So we obtain
$\\{v_{1},v_{2}\\}\cap A\neq\emptyset$. Without loss of generality, we assume
$v_{1}\in A$. Then $L_{G_{s-1}}(v_{1})=L_{G_{s-1}}(v)$, i.e., $uv_{2}\in
L_{G_{s-1}}(v)$. Thus, $u\in N_{G_{s-1}}(v)$, a contradiction. This implies
that $G_{s-1}=T_{3}(n)$. Likewise, $G_{i-1}=G_{i}$ for each $i\in[s-1]$, and
therefore $G=T_{3}(n)$. This completes the proof of the theorem. ∎

According to Theorem 3.1, we can give an alternative proof of Bollobás’ result
for $n\equiv{0}\pmod{3}$.

######  Corollary 3.1.

Let $G$ be a cancellative $3$-graph on $n$ vertices with $n\equiv{0}\pmod{3}$.
Then $|E(G)|\leq t_{3}(n)$ with equality if and only if $G=T_{3}(n)$.

###### Proof.

Denote by $\bm{z}$ the all-ones vector of dimension $n$. In view of (2.1), we
deduce that

$\lambda^{(3)}(G)\geq\frac{P_{G}(\bm{z})}{\|\bm{z}\|_{3}^{3}}=\frac{3|E(G)|}{n}.$

On the other hand, by Theorem 3.1 we have

$\lambda^{(3)}(G)\leq\lambda^{(3)}(T_{3}(n))=(t_{3}(n))^{2/3}.$

As a consequence,

$|E(G)|\leq\frac{n}{3}\cdot(t_{3}(n))^{2/3}=t_{3}(n).$

Equality may occur only if
$\lambda^{(3)}(G)=(t_{3}(n))^{2/3}=\lambda^{(3)}(T_{3}(n))$, and therefore
$G=T_{3}(n)$ by Theorem 3.1. ∎

Next, we will prove Theorem 1.2 for the case $p>3$ as stated in Theorem 3.2.

######  Lemma 3.7 ([14]).

Let $p\geq 1$ and $G$ be an $r$-graph with $m$ edges. Then the function

$f_{G}(p):=\bigg{(}\frac{\lambda^{(p)}(G)}{rm}\bigg{)}^{p}$

is non-increasing in $p$.

######  Theorem 3.2.

Let $p>3$ and $G$ be a cancellative $3$-graph on $n$ vertices. Then
$\lambda^{(p)}(G)\leq\lambda^{(p)}(T_{3}(n))$ with equality if and only if
$G=T_{3}(n)$.

###### Proof.

Assume that $p>3$ and $G$ is a $3$-graph in
$\operatorname{Ex}_{sp}(n,\\{F_{4},F_{5}\\})$ with $m$ edges. It is enough to
show that $G=T_{3}(n)$. By Lemma 3.7, we have

$\bigg{(}\frac{\lambda^{(p)}(G)}{3m}\bigg{)}^{p}\leq\bigg{(}\frac{\lambda^{(3)}(G)}{3m}\bigg{)}^{3},$

which, together with $\lambda^{(3)}(G)\leq(t_{3}(n))^{2/3}$ by Theorem 3.1,
gives

$\lambda^{(p)}(G)\leq(3m)^{1-3/p}\cdot(\lambda^{(3)}(G))^{3/p}\leq(3m)^{1-3/p}\cdot(t_{3}(n))^{2/p}.$

On the other hand, we have

$\lambda^{(p)}(G)\geq\lambda^{(p)}(T_{3}(n))=\frac{\big{(}27\cdot
t_{3}(n)\big{)}^{1-1/p}}{9}.$

We immediately obtain $m\geq t_{3}(n)$. The result follows from Theorem 1.1. ∎

Finally, we shall give a proof of Theorem 1.2 for the remaining case $p=1$. In
what follows, we always assume that $\bm{x}\in\mathbb{S}_{1,+}^{n-1}$ is an
eigenvector such that $\bm{x}$ has the minimum possible number of non-zero
entries among all eigenvectors corresponding to $\lambda^{(1)}(G)$. Before
continuing, we need the following result.

######  Lemma 3.8 ([5]).

Let $G$ be an $r$-graph and $S$ be the support set of $\bm{x}$. Then for each
pair vertices $u$ and $v$ in $S$, there is an edge in $G[S]$ containing both
$u$ and $v$.

######  Theorem 3.3.

Let $G$ be a cancellative $3$-graph. Then $\lambda^{(1)}(G)=1/9$.

###### Proof.

Assume that $G$ is a cancellative $3$-graph with support set $S$. Let
$H:=G[S]$. By Lemma 3.8, for any $u,v\in S$ there is an edge in $H$ containing
both $u$ and $v$. Hence, for any two edges for each pair of edges of $H$ has
at most one common vertex by $H$ being cancellative. So the shadow graph of
$H$ is the complete graph $K_{|S|}$. Since $H$ is cancellative, the link
graphs $L_{H}(u)$ and $L_{H}(v)$ are edge-disjoint graphs for any distinct
vertices $u,v\in S$. It follows from (2.2) that

$|S|\cdot\lambda^{(1)}(G)=\sum_{uv\in
E(\partial(H))}x_{u}x_{v}\leq\frac{1}{2}\Big{(}1-\frac{1}{|S|}\Big{)},$ (3.6)

where the last inequality follows from Motzkin–Straus Theorem [11]. On the
other hand, set

$z_{v}=\begin{cases}1/|S|,&v\in S,\\\ 0,&\text{otherwise}.\end{cases}$

We immediately have

$\lambda^{(1)}(G)\geq 3\sum_{e\in E(H)}\bm{z}(e)=\sum_{v\in
V(H)}\bigg{(}z_{v}\cdot\sum_{f\in
L_{H}(v)}\bm{z}(f)\bigg{)}=\frac{|S|-1}{2|S|^{2}},$

where the last inequality follows from the fact that $d_{H}(v)=(|S|-1)/2$ for
$v\in V(H)$. Combining with (3.6) we get

$\lambda^{(1)}(G)=\frac{|S|-1}{2|S|^{2}}.$

Clearly, $(|S|-1)/|S|^{2}$ attains its maximum at $|S|=3$ when $|S|\geq 3$.
Hence, we see $\lambda^{(1)}(G)\leq 1/9$. Finally, noting that
$\lambda^{(1)}(G)$ is at least the Lagrangian of an edge $K_{3}^{(3)}$, i.e.,

$\lambda^{(1)}(G)\geq\lambda^{(1)}(K_{3}^{3})=\frac{1}{9},$

we obtain $\lambda^{(1)}(G)=1/9$, as desired. ∎

######  Remark 3.1.

For an $r$-graph $G$ on $n$ vertices, it is well-known that
$\lambda^{(1)}(G)/r$ is the Lagrangian of $G$. In [16], Yan and Peng present a
tight upper bound on $\lambda^{(1)}(G)$ for $F_{5}$-free $3$-graphs, see [16]
for details.

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