# ECAFormer: Low-light Image Enhancement using Cross Attention

IEEE Publication Technology  Yudi Ruan, Hao Ma, Weikai Li∗ and Xiao Wang∗
Weikai Li and Xiao Wang are corresponding authors.Yudi Ruan is with the School
of Information Science and Engineering, Chongqing Jiaotong University,
Chongqing, 400074, China (e-mail: yudi.ruan@mails.cqjtu.edu.cn).Hao Ma is with
the School of Artificial Intelligence, Anhui University, Hefei, AnHui, 230039,
China (e-mail: WA22301107@stu.ahu.edu.cn).Weikai Li is with the School of
Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074,
China, MIIT Key Laboratory of Pattern Analysis and Machine Intelligence,
Nanjing, 210016, China (e-mail: leeweikai@outlook.com).Xiao Wang is with
School of Artificial Intelligence, Anhui University , Hefei, AnHui, 230039,
China (e-mail: x.wang@atu.edu.cn).Manuscript received April 19, 2021; revised
August 16, 2021.

###### Abstract

Low-light image enhancement (LLIE) is vital for autonomous driving. Despite
the importance, existing LLIE methods often prioritize robustness in overall
brightness adjustment, which can come at the expense of detail preservation.
To overcome this limitation, we propose the Hierarchical Mutual Enhancement
via Cross-Attention transformer (ECAFormer), a novel network that utilizes
Dual Multi-head Self Attention (DMSA) to enhance both visual and semantic
features across scales, significantly preserving details during the process.
The cross-attention mechanism in ECAFormer not only improves upon traditional
enhancement techniques but also excels in maintaining a balance between global
brightness adjustment and local detail retention. Our extensive experimental
validation on renowned low-illumination datasets, including SID and LOL, and
additional tests on dark road scenarios. or performance over existing methods
in terms of illumination enhancement and noise reduction, while also
optimizing computational complexity and parameter count, further boosting SSIM
and PSNR metrics. Our project is available at
https://github.com/ruanyudi/ECAFormer.

###### Index Terms:

Image Restoration, Cross Attention, Transformer, Feature fusion.

##  I Introduction

Capturing images in low-light conditions frequently leads to a variety of
photographic challenges, such as the loss of fine details, a reduction in
color intensity, a decrease in contrast and dynamic range, and uneven
exposure. These issues can degrade the quality and clarity of the visual data,
which in turn can significantly hinder subsequent vision-based tasks. For
instance, they can impede the performance of autonomous driving systems [1, 2,
3], nighttime surveillance efforts [4], and remote sensing applications [5].
Enhancing the visibility of objects and details in low-light images is,
therefore, a critical issue with broad implications for a range of
applications.

To enhance the low-light images, numerous algorithms have been proposed to
address the challenges encountered in low-light environments. These algorithms
can be broadly categorized into two groups: knowledge-driven methods and data-
driven methods.

Knowledge-driven methods for low-light enhancement include Histogram
Equalization-based methods [6, 7] and Retinex model-based methods [8, 9, 10,
11, 12, 13, 14, 15]. Among these, the Retinex model-based methods have
received relatively more attention. A typical Retinex model-based approach
decomposes a low-light image into a reflection component and an illumination
component using priors or regularizations. However, models based on
theoretical principles often overlook factors present in the real environment
due to the idealization of the theory, and thus may not produce satisfactory
results.

Data-driven methods methods often use convolutional neural networks (CNNs) to
learn complex image features and generate enhanced images based on these
features. Typical examples include AutoEncoder (AE)[16], generative
adversarial networks (GANs)[17]. These deep learning models are also divided
into end-to-end models and theoretical prior-based models, while the latter
use prior experience to constrain the neural network[18, 19]. Due to the
defects in the ideal assumptions, prior-based methods may fall into bias.
Compared to the explosive development of Transformer models in other fields,
the potential of Transformer models in the LLIE field has not yet been fully
tapped. Additionally, low-light image enhancement demands spatially varying
operations[20], although transformers are particularly adept at capturing
long-range information. While SNRNet[20] acknowledged this aspect, it solely
relied on a single layer of global transformer, thereby overlooking the
crucial interaction between information across varying hierarchical levels.

Regarding the current situation in the LLIE field, we proposed a transformer-
based Network with an novel efficient fusion module, fully utilizing
information from different scales to integrate the requirements of long-range
and short-range. Meanwhile, the addition of pre-convolution compensates for
the spatial context induction bias in transformers that rely solely on
positional embeddings. Our main contribution can be summarized as follows:

  * •

Our major idea is to comprehensively combine the local(short-range) detail
feature extraction ability of CNN networks and the global(long-range)
information interaction ability of Transformer networks, and utilize the
multi-scale interaction ability of U-shaped networks to improve the processing
ability of the Enhanced-network.

  * •

We propose a advanced method called ECAFormer to simultaneously and
effectively extract short-range and long-range features. DMSA blocks enable
the network to have powerful feature fusion capabilities. Our work has made
meaningful attempts and innovations in the development of transformer models
in the field of LLIE.

  * •

We validated our proposed method on six datasets, and extensive experimental
results showed that our method has top-notch performance and good robustness
at the same time. Meanwhile, our proposed network has a top-notch improvement
effect at the same computational complexity.

The paper is structured as follows. In Chapter II, we respectively reviewed
the related work in the field of low light enhancement and cross-attention
mechanisms. In Chapter III, we introduced the definition of the problem, the
proposed method, and the details of model optimization. In Chapter IV, we
compared our model with seven publicly available datasets and conducted
ablation experiments to validate its effectiveness. Finally, we summarized in
Chapter V and proposed directions for future work.

##  II Related work

###  II-A LLIE: low light image enhancement

Knowledge-driven methods. CLAHE[21] normalizes the value of each pixel in an
image based on a histogram function, which lacks local adaptability and can
easily lead to overexposure or underexposure in images. Earlier works
addressed this issue by optimizing the parameters in the equalization
function, such as [22], which enhanced the adaptability of the method by
altering one or two parameters using a generalization of histogram
equalization. Subsequent extensive research has begun using priors to
constrain the equalization process. Brightness Preserving Dynamic Histogram
Equalization (BPDHE)[6] extends traditional Histogram Equalization (HE) by
producing an output image whose mean intensity closely matches that of the
input. This method effectively fulfills the requirement to maintain the
original mean brightness of the image. [23] adjust the level of contrast
enhancement by introducing specifically designed penalty terms through noise
robustness, white/black stretching and mean-brightness preservation. The
Retinex model[24] and its multi-scale variant[25] decompose brightness into
illumination and reflectance components, which are then processed
independently. LIME[11] introduced a structural prior to refine the initial
illumination map, ultimately synthesizing the enhanced image in accordance
with Retinex theory. Additionally, there are some works based on gamma
correction, such as an adaptive gamma correction (AGC)[26] that is proposed to
enhance the image contrast appropriately. The parameters of AGC are
dynamically set based on the image information.

However, these manually crafted constraints and priors lack sufficient self-
adaptivity to accurately recover image details and colors, often leading to
the obliteration of details, local under- or over-saturation, uneven exposure,
or the emergence of halo artifacts around objects. Given the demonstrated
advantages of deep learning across various low-level visual tasks, the field
of low-light image enhancement has increasingly embraced learning-based
methodologies.

Data-driven methods. In recent years, the rapid development of low-light image
dataset collection has prompted the introduction of numerous deep learning-
based enhancement methods[27]. These methods significantly improve the
restoration capabilities over traditional approaches, offering better
accuracy, robustness, and speed. The first study[28], validated the
effectiveness of deep learning methods in the field of low-light enhancement.
[16] proposed a variant of the stacked sparse denoising autoencoder to enhance
degraded images, laying a foundational framework for the application of deep
learning in image enhancement. RetinexNet[19] utilized a deep Retinex-based
architecture to enhance low-light images by decomposing them into illumination
and reflectance components, which were processed independently. Despite its
innovative approach, this method faced limitations in image decomposition,
which were subsequently addressed by [29] through the development of three
subnetworks based on Retinex for layer decomposition, reflectance restoration,
and illumination adjustment, thereby enhancing the network’s decomposition
capabilities. Zero-DCE[18], FlexiCurve[30], CuDi[31], and ReLLIE[32] redefined
the task of low-light enhancement as an image-specific curve estimation
problem with a fixed default brightness value, notable for its rapid
processing speed.RUAS[33] employed a sophisticated search unfolding technique
based on a Retinex architecture. EnlightenGAN[17] innovatively utilized a
generative inverse network as the main framework, initially training with
unpaired images. LEDNet, introduced by [34], is a robust network specifically
designed to tackle the dual challenges of low-light enhancement and deblurring
simultaneously.[35] introduces PIE, a sophisticated physics-inspired
contrastive learning paradigm specifically designed for enhancing low-light
images. With the significant advancements of transformer models in image
processing, SNRNet[20] employed a signal-to-noise ratio-aware transformer and
a CNN with spatially varying operations for restoration. But it only utilizes
a single global Transformer at the lowest level of the U-Net.

Totally, these methods predominantly utilize CNN models as their backbone,
failing to fully leverage the emerging Transformer models that have recently
gained prominence in image processing. This paper proposes a transformer model
equipped with a cross-attention module, which harnesses the global perceptual
capabilities of the transformer to deeply integrate the visual and semantic
features extracted by CNN models.

###  II-B Cross Attention

In recent years, the efficacy of cross-attention modules in advanced
downstream tasks has been extensively validated. In the realm of image
classification, CrossVit[36] has developed a fusion module based on cross-
attention that integrates tokens from both small-patch and large-patch
configurations, enhancing model accuracy with minimal increases in FLOPs. In
the field of multi-modal matching, [37] propose the innovative MultiModality
Cross Attention (MMCA) Network for image and sentence matching. The MMCA model
is adept at leveraging not only the intra-modality relationships within each
modality but also the inter-modality relationships between image regions and
sentence words, thereby augmenting their mutual capabilities for more
effective matching. In image segmentation, [38] introduced the U-Transformer
network, which combines a U-shaped architecture with both self- and cross-
attention mechanisms from the Transformer framework. In the area of few-shot
classification, [39] designed a module based on cross-attention that generates
cross-attention maps for each pair of class feature and query sample feature,
thus highlighting the target object regions and rendering the extracted
features more discriminative. In the field of few-shot object detection,
TIDE[40] introduced a symmetrical cross-attention module to enhance the
interaction between support features and query features, ultimately achieving
notable results.

In summary, the cross-attention module has demonstrated formidable
capabilities in feature fusion. Consequently, this paper embeds the cross-
attention module within the Transformer block, facilitating deep integration
of visual and semantic features extracted by the CNN network. By combining a
multi-scale U-net network, it interacts features across different scales,
ultimately achieving state-of-the-art (SOTA) performance.

Figure 1: The flowchart of ECAformer mainly consists of three parts: (1)
Visual-Semantic convolution modulesIII-B2, which output short-range features
(visual-feature) and long-range features (semantic-feature). (2)The U-shaped
cross-attention Transformer4 engages with multi-scale and multi-range input
features through DMSA, concurrently propagating these features. (3) Mapping
convolution, where the module projects the interacted features back to image
features.${\in\mathbb{R}^{\emph{C}\times\emph{H}\times\emph{W}}}$.

##  III Methodology

In this section, we will provide a detailed description of each network module
we proposed and the loss functions utilized.

Our paper proposes a transformer network with a U-shaped architecture, as
illustrated in figure 1. The low light input image is first subjected to a
convolutional filter to extract detailed features, obtaining semantic
information and visual detail information respectively. Then, the enhanced
image is obtained through the interaction enhancement of a U-shaped network
using semantic- and visual-feature. The network model is optimized through VGG
perceptual loss and Charbonnier L1 loss.

###  III-A Problem Definition

Low light image enhancement aims to address the challenges of low brightness,
low contrast, noise, and artifacts in images captured under insufficient
lighting conditions. These issues stem from the loss of visual information due
to inadequate illumination or complex environments when the images are taken.
The goal of low light image enhancement is to restore and improve low light
images $\mathbb{L}$ through algorithms and technical methods, making them
closer to normal images $\mathbb{N}$ and more comprehensible to human visual
systems.

The problem can be mathematically modeled as follows:

$\hat{I}=f(I,\theta)$ (1)

where ${I\in\mathbb{L}^{C\times H\times W}}$ represents the input images,
${\hat{I}\in\mathbb{R}^{C\times H\times W}}$ represents the enhanced images,
and ${\theta}$ represents the parameters of the network function $f$.

The objective of the optimization process is to find the optimal parameters
${\theta}$ that minimize the differences between the enhanced images and the
reference high-light images. This can be formulated as:

$\hat{\theta}=\mathop{\arg\min}\limits_{\theta}\ell(\hat{I},R)$ (2)

where ${R\in\mathbb{N}^{C\times H\times W}}$ denotes the well-exposed
reference images, and ${\ell}$ denotes the loss function used to guide the
network.

###  III-B Network framework

####  III-B1 Preliminary

Convolutional Neural Networks (CNNs) are adept at extracting detailed local
features, whereas Transformer networks excel in capturing valuable global
feature information from complex environments. In CNNs, as the depth of the
convolutional layers increases, the extracted features progressively embody
richer semantic information. Consequently, features obtained from initial
shallow convolutional operations are predominantly visual in nature, while
those derived from deeper convolutional layers contain more sophisticated
semantic insights. To leverage these characteristics, we have designed a
sophisticated convolutional extractor:

$\displaystyle feat_{visual}=Conv_{1}(I_{input})$ (3) $\displaystyle
feat_{semantic}=Conv_{2}(feat_{visual})$

$feat_{visual}$ retains an abundance of detailed visual features, while
$feat_{semantic}$ preserves advanced semantic information. Both attributes are
pivotal for achieving the enhanced final result.

The attention mechanism allows each element to interact with others, thereby
enhancing global feature extraction capabilities significantly. The
conventional self-attention mechanism is outlined as follows Eq. 4. The
vectors $Q$, $K$, and $V$ represent query, key, and value, respectively, all
derived from a single input. This operation selectively concentrates on
pivotal information, optimizing resource usage and rapidly capturing the most
pertinent data. Leveraging this benefit, we have developed a unique dual-
branch multi head self-attention module, $\bf{DMSA}$, to facilitate the fusion
of two distinct features. This module processes two inputs and enables their
interaction via the attention mechanism while preserving their dimensional
integrity. We will provide a detailed discussion of this module in section 4.

$Attention(Q,K,V)=softmax\left(\frac{QK^{T}}{\sqrt{d_{k}}}\right)V$ (4)

We have integrated a U-shaped network architecture enhanced with residual
connections, specifically designed to amplify the network’s capability for
multi-scale interaction. This architecture facilitates a comprehensive
synthesis of features across different scales. In the downsampling phase, we
strategically employed a 2$\times$ scaling factor, compelling the network to
engage in a rigorous compression process. This forced compression is critical
as it enables the network to extract more refined global information,
essential for understanding broader contextual cues. The methodology and
specifics of the downsampling stage are represented in Eq. 5.

(i) $\displaystyle=$ (5) $\displaystyle
Resample(DMSA\\_Block_{i}([feat_{visual},$ $\displaystyle
feat_{semantic}]^{(i-1)}))$

where $\\{i\in{1,2\\}}$ denotes the step of the downsample stage, DMSA_Block
is composed of different quantities of DMSA modules at different steps. At the
bottom of the U-shaped network, we used a bottleneck consisting of two DMSA
modules, and during the upsampling process, we employed operations
proportional to downsampling. During the upsampling phase, we incorporated
residual connections to effectively preserve and restore the intricate details
that are often lost during the downsampling process. These residual
connections first perform cross attention with their corresponding features,
and then the two types of features undergo cross attention via the
CSDMSA_block. This process plays a pivotal role in maintaining the fidelity of
the feature representations, ensuring that the reconstructed outputs closely
mirror the original inputs.Finally, we obtain the final output through
concatenation and mapping convolution. The final output can be represented by
Eq. 6

$\hat{I}=\mathcal{A}_{agg}\left(feat_{visual}^{(T)},feat_{semantic}^{(T)}\right)$
(6)

Figure 2:  From top to bottom are the shallow outputs and deep outputs of the
network, respectively. It is observable that ${feat_{visual}}$ focuses more on
fine details, while ${feat_{semantic}}$ emphasizes broader connections and
contextual relationships within the image. Figure 3: Visual-Semantic
Convolution Module. Deeper features focus more on semantic information, while
shallower features focus more on visual information.

####  III-B2 Visual-Semantic Convolution Module

Inspired by the potent capability of convolutional layers to enhance local
features, we devised a visual-semantic convolution module specifically
designed to capture local characteristics. In CNNs, as the number of
convolutional layers increases, the receptive field of the model progressively
enlarges, leading to the extraction of increasingly complex semantic features.
However, this expansion often results in the attenuation of fine-grained
detail within the features. To address this, we introduce a dedicated
convolutional module that outputs two distinct types of features:
${feat_{visual}}$ and ${feat_{semantic}}$. Here, $feat_{visual}$ is derived
from shallow convolution layers, capturing detailed visual features, whereas
$feat_{semantic}$ emanates from deeper convolution layers, encapsulating
higher-level semantic information.Additionally, convolutional neural networks
are inherently capable of capturing periodic and local spatial features,
addressing the spatial context induction bias in transformers that rely solely
on positional embeddings. We employ depthwise separable convolution within
this module to enhance the rate of forward propagation without compromising
accuracy. Upon processing through this module, the network then employs
attention mechanisms to facilitate a dynamic interaction between these two
distinct features.

####  III-B3 DMSA

Figure 4: DMSA: A highly symmetrical module is depicted in the figure, which
illustrates the process of computing the $\alpha$ features guided by $\beta$
($\alpha^{’}_{\beta_{guided}}$). The $\beta^{’}_{\alpha_{guided}}$ can be
obtained through a symmetrical approach.

Self-attention is highly efficient at directing concentrated attention towards
pivotal information, thereby optimizing resource utilization and rapidly
acquiring the most relevant data. However, its standard configuration is
restricted to processing only a single input, which can limit its
applicability in scenarios requiring complex interactions between multiple
data streams. To address this constraint, we have innovated the Dual Multi-
head Self Attention (DMSA) module, a sophisticated mechanism designed to
effectively manage and integrate dual inputs. In this module, inputs $\alpha$
and $\beta$ are processed to generate distinct sets of $Q$, $K$, and $V$
vectors through separate mappings. We strategically cross the key vectors of
both inputs within this module to enhance their interconnectivity, fostering a
richer, more comprehensive interaction. This is achieved by the multiplication
of the query vector with the corresponding crossed key vector, producing an
attention map that encapsulates the intricate dynamics between the two inputs.
Enhanced by the adaptive sampling operations of the U-shaped network, this
setup facilitates a profound interaction across multiple scales, allowing for
a nuanced integration of features that is essential for complex analysis and
interpretation tasks.

$\displaystyle
DMSA(Q_{\alpha},K_{\beta},V_{\alpha})=softmax(Q_{\alpha}K_{\beta}^{T}*\zeta_{rescale})V_{\alpha}$
(7)

The scaling factor ${\zeta_{rescale}}$ is determined through optimization. To
preserve the spatial positional relationships among pixels, we have
incorporated a position encoding module, executed via convolutional
operations. Consequently, the output of the Dual Multi-head Self Attention
module can be formally defined in Eq. 8.

$\displaystyle[\alpha^{\prime},\beta^{\prime}]=$ (8) $\displaystyle[$
$\displaystyle DMSA(Q_{\alpha},K_{\beta},V_{\alpha})+PosEmbed(V_{\alpha}),$
$\displaystyle DMSA(Q_{\beta},K_{\alpha},V_{\beta})+PosEmbed(V_{\beta})]$

###  III-C Loss function

We incorporated two types of loss functions that better align with human
visual perception and facilitate faster model training. TotalLoss can be
represented as Eq. 9, $\lambda\in{[0,1]}$.

$TotalLoss=\lambda*\mathcal{L}_{p}+(1-\lambda)*\mathcal{L}_{c}$ (9)

where $\mathcal{L}_{p}$ is the loss of perceptual and $\mathcal{L}_{c}$ is the
loss of charbonnier.

####  III-C1 Perceptual Loss

Perceptual Loss[41] adopts a efficient approach by quantifying the
discrepancies through the squared error between features extracted from
specific layers or an aggregation of multiple layers after both the ground
truth and the reconstructed image have traversed the same pre-trained neural
network. This technique transcends traditional pixel-based difference
calculations, offering a superior method for handling anomalies and enhancing
the overall robustness of the model. By focusing the loss calculations on the
variances within the deeper layers of feature representations, Perceptual Loss
adeptly captures and emphasizes high-level semantic nuances. Consequently,
this approach produces images that more accurately reflect human visual
perceptions, thereby aligning the output more closely with the intricacies of
human sight.

$PerceptualLoss=\sum_{i=1}^{n}\frac{1}{C_{i}H_{i}W_{i}}\left|F_{i}^{l}(f(\hat{I}))-F_{i}^{l}(R)\right|^{2}$
(10)

Among them, _f_ symbolizes the enhancement network. ${F_{i}^{l}}$ indicates
the i-th feature map in the $l$-th layer. We have utilized a VGG-19 network
pre-trained on ImageNet and employed the output feature maps from its initial
five ReLU layers to compute the loss. This method leverages the deep network’s
architecture to extract rich, complex feature representations that are
critical for assessing the perceptual quality of the enhanced images.

Figure 5: Perceptual Loss: Implementing loss on the features extracted from a
frozen VGG19 network

####  III-C2 Charbonnier $L_{1}$ Loss

Compared to the conventional $L_{1}$ Loss, Charbonnier $L_{1}$ Loss exhibits
superior robustness and stability, particularly when dealing with outliers.
Additionally, the computational efficiency of Charbonnier $L_{1}$ Loss is
enhanced due to its reliance on a single square and root operation, in
contrast to the ordinary $L_{1}$ Loss which necessitates an absolute value
computation. This streamlined calculation not only speeds up the processing
but also contributes to smoother gradients, facilitating more effective
optimization during model training.

$CharbonnierLoss=\sum_{i=1}^{n}\sqrt{\left(\hat{I}-R\right)^{2}+\epsilon^{2}}$
(11)

Benefiting from the addition of epsilon, the phenomenon of gradient vanishing
when $\hat{I}$ and _R_ are very close has been alleviated, making the model
easier to train.

##  IV Experiment

###  IV-A Experimental configurations

Datasets. We conducted comparisons across multiple public datasets:
LOL-v1[19], LOL-v2[42], SID[43], SMID[44], and SDSD[45].

Traffic-297. We have gathered 297 images captured by traffic video cameras
and, following the LOL methodology, have created corresponding image pairs.
These images encompass diverse weather conditions and traffic scenarios. Among
them, 50 images are dedicated for testing purposes, while 247 are allocated
for training.

LOL-v1. It represents the inaugural dataset comprising paired low- and normal-
light images captured in real-world scenarios. The low-light images within
this collection are meticulously gathered by adjusting exposure times and ISO
settings. The dataset encompasses 500 pairs of low- and normal-light images,
each with dimensions of 400×600 and preserved in RGB format, offering a
comprehensive resource for enhancing low-light image processing techniques.
The training and testing sets are split in proportion to 485:15.

LOL-v2. The LOL-v2 dataset consists of two distinct subsets: LOL-v2-real and
LOL-v2-synthetic. The LOL-v2-real subset is captured in real scenes by
altering ISO settings and exposure times. In the LOL-v2-synthetic subset, low-
light images are synthesized from RAW images by analyzing the illumination
distribution in low-light conditions.The training and testing sets are split
in proportion to 689:100 and 900:100.

SID. The Sony $\alpha$7S II subset of the SID dataset is utilized for
evaluation purposes. It consists of 2,697 pairs of short/long-exposure RAW
images. The low-/normal-light RGB images are derived by applying the SID’s in-
camera signal processing techniques to convert RAW images to RGB format. Of
these, 2,099 image pairs are designated for training, while 598 pairs are
reserved for testing.

SMID. The SMID benchmark amasses a collection of 20,809 short-/long-exposure
RAW image pairs. We convert these RAW datasets into low-/normal-light RGB
image pairs, utilizing a consistent transformation process to ensure data
integrity. For model training, 15,763 pairs are utilized, while the remaining
pairs are set aside for rigorous testing.

SDSD. We utilize the static version of the SDSD dataset, which was captured
using a Canon EOS 6D Mark II camera equipped with an ND filter. The dataset
comprises both indoor and outdoor subsets. For the SDSD-indoor subset, we
allocate 62 low-/normal-light video pairs for training and 6 pairs for
testing. Similarly, for the SDSD-outdoor subset, 116 pairs are used for
training while 10 pairs are reserved for testing.

Training. We implemented the ECAFormer model using PyTorch. The model was
trained on an NVIDIA RTX 4090 with 24 GB of VRAM using the Adam optimizer
($\beta$1 = 0.9 and $\beta$2 = 0.999), across a total of 250,000 iterations.
The initial learning rate was set at 2×10-4 and gradually reduced to 1×10-6
through a cosine annealing schedule during the training process. Training
samples were generated by randomly cropping 128×128 patches from pairs of
low-/normal-light images. The batch size was set to 8. The training data was
augmented with random rotations and flips to enhance variability and
robustness. The training objective was to minimize both $PerceptualLoss$III-C1
and $CharbonnierL_{1}Loss$III-C2 between the enhanced images and their
corresponding ground truths, ensuring high fidelity in the restoration of
image details and color accuracy.

Metrics. We employ the peak signal-to-noise ratio (PSNR) Eq. 12 and structural
similarity index (SSIM) Eq. [46]13 as the primary metrics for evaluation.

$\text{PSNR}=10\cdot\log_{10}\left(\frac{\text{MAX}_{I}^{2}}{\text{MSE}}\right)$
(12)
$\text{SSIM}(x,y)=\frac{(2\mu_{x}\mu_{y}+c_{1})(2\sigma_{xy}+c_{2})}{(\mu_{x}^{2}+\mu_{y}^{2}+c_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+c_{2})}$
(13)

Comparison Methods. To validate the effectiveness of the model, we conducted
extensive comparisons with state-of-the-art models in recent years. The
compared models are as follows: SID[43],UFormer(UF)[47], RetinexNet(RN)[19],
EnlightenGAN(EnGAN)[17], RUAS[33], DRBN[48], KinD[29], Restormer[49],
MIRNet[50], SNR-Net[20].

###  IV-B Results analysis

Quantitative Results. We compared our model across seven public datasets using
two metrics, PSNR and SSIM, with the results presented respectively in Table
III and Table IV. The results indicate that our model achieves state-of-the-
art (SOTA) performance while maintaining a relatively low parameter count and
computational complexity.

Figure 6: Comparative Visual Effects on the Test Set Figure 7: Visual Detail Comparison on the Test Set Figure 8: Visual Detail Comparison on the Traffic-297. We tested models trained on the synthesised dataset under real-world nighttime scenarios. While other methods introduce noise and lose details, our model exhibits excellent visual performance. TABLE I: Results on Traffic-297 | RetinexNet | ZeroDCE++ | EnGAN | SNR | ECAFormer
---|---|---|---|---|---
PSNR | 18.56 | 18.35 | 18.19 | 30.06 |  32.17
SSIM | 0.77 | 0.884 | 0.623 | 0.965 |  0.975

Quality Results. In Figures 6 and 8, we present visual comparison charts of
the results. The recent fully supervised learning method, SNR, did not retain
sufficient detail and exhibited blurring. ZeroDCE++, which adjusts the pixel
curves, introduced excessive noise. The results demonstrate that the ECAFormer
proposed in this paper produces high-quality outcomes that closely approximate
the true images.

###  IV-C Ablation study

To validate the effectiveness of the various modules within the model proposed
in this article, we adjusted the model’s structure and reported the PSNR
results on the SDSD-out dataset.

CNN_feat. We validated the effectiveness of the shallow-deep convolution
module. When this module is not utilized, the input image is directly reshaped
to the format required by the MGMSA for input.

VGG_CbLoss. Verified the effectiveness of the loss functions by comparing the
Mean Squared Error (MSE) loss with the two types of losses used in this study.

DMSA. Replaced the DMSA with a classic self-attention mechanism to validate
the performance of this module in feature fusion.

TABLE II: Ablation study results on SDSD-out CNN_feat | VGG_CbLoss | DMSA | PSNR(avg)
---|---|---|---
| ✓ | ✓ | 28.56
✓ |  | ✓ | 29.41
✓ | ✓ |  | 28.38
✓ | ✓ | ✓ | 29.57
TABLE III: Comparison of PSNR Across Eight Public Datasets Using Different
Methods. ECAFormer achieved promising results with the least number of
parameters.

| SID | UF | RN | EnGAN | RUAS | DRBN | KinD | RSTM | MIRNet | SNR | ECA
---|---|---|---|---|---|---|---|---|---|---|---
LOL-v1 | 14.35 | 16.36 | 16.77 | 17.48 | 18.23 | 20.13 | 20.86 | 22.43 | 24.14 | 24.61 | 24.24
LOL-v2-r | 13.24 | 18.82 | 15.47 | 18.23 | 18.37 | 20.29 | 14.74 | 19.94 | 20.02 | 21.48 | 21.99
LOL-v2-s | 15.04 | 19.66 | 17.13 | 16.57 | 16.55 | 23.22 | 13.29 | 21.41 | 21.94 | 24.14 | 25.86
SID | 16.97 | 18.54 | 16.48 | 17.23 | 18.44 | 19.02 | 18.02 | 22.27 | 20.84 | 22.87 | 24.69
SMID | 24.78 | 27.2 | 22.83 | 22.62 | 25.88 | 26.6 | 22.18 | 26.97 | 25.66 | 28.49 | 29.34
SDSD-in | 23.29 | 23.17 | 20.84 | 20.02 | 23.17 | 24.08 | 21.95 | 25.67 | 24.38 | 29.44 | 29.11
SDSD-out | 24.9 | 23.85 | 20.96 | 20.1 | 23.84 | 25.77 | 21.97 | 24.79 | 27.13 | 28.66 | 29.57
FLOPS | 13.73 | 12 | 587.47 | 61.01 | 0.83 | 48.61 | 34.99 | 144.25 | 785 | 26.35 | 27.34
Params | 7.76 | 5.29 | 0.84 | 114.35 | 0.003 | 5.27 | 8.02 | 26.13 | 31.76 | 4.01 | 2.5

TABLE IV: Comparison of SSIM Across Eight Public Datasets Using Different
Methods

| SID | UF | RN | EnGAN | RUAS | DRBN | KinD | RSTM | MIRNet | SNR | ECA
---|---|---|---|---|---|---|---|---|---|---|---
LOL-v1 | 0.436 | 0.771 | 0.56 | 0.65 | 0.72 | 0.83 | 0.79 | 0.823 | 0.83 | 0.842 | 0.85
LOL-v2-r | 0.442 | 0.771 | 0.567 | 0.617 | 0.723 | 0.831 | 0.641 | 0.827 | 0.82 | 0.849 | 0.853
LOL-v2-s | 0.61 | 0.871 | 0.798 | 0.734 | 0.652 | 0.927 | 0.578 | 0.83 | 0.876 | 0.928 | 0.931
SID | 0.591 | 0.577 | 0.578 | 0.543 | 0.581 | 0.577 | 0.583 | 0.649 | 0.605 | 0.625 | 0.66
SMID | 0.718 | 0.792 | 0.684 | 0.674 | 0.744 | 0.781 | 0.634 | 0.758 | 0.762 | 0.805 | 0.81
SDSD-in | 0.703 | 0.859 | 0.617 | 0.604 | 0.696 | 0.868 | 0.672 | 0.827 | 0.864 | 0.894 | 0.874
SDSD-out | 0.693 | 0.748 | 0.629 | 0.616 | 0.743 | 0.841 | 0.654 | 0.802 | 0.837 | 0.866 | 0.862

The results indicate that all three modules positively impact the model’s
performance and are effective. Among them, the DMSA module demonstrates the
most significant performance in terms of feature fusion.

##  V Conclusion

In this paper, we propose a one-stage transformer-based method, specifically
utilizing DMSA blocks for enhancing low-light images. Furthermore, the network
of ECAFormer combines the benefits of the local detailed feature information
extraction ability in Convolutional Neural Networkthe and global information
extraction ability in the Transformer Network. Therefore, ECAFormer can
extract global information and local information from the image. Extensive
experiment results verify the competitive advantages of ECAFormer compared
with other state-of-the-art algorithms.

Future work. In recent years, models trained on extensive datasets have
progressively evolved. The integration of prior knowledge from large models
can mitigate the limitations of current loss functions and extend to
unsupervised learning. The two-dimensional Fourier transform holds substantial
potential in handling image details and could be incorporated into the DMSA
module proposed in this paper, enabling interactions in both the frequency and
time domains.

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# AoA Estimation for OAM Communication Systems With Mode-Frequency Multi-Time
ESPRIT Method

Wen-Xuan Long, Rui Chen,
Marco Moretti, and Jiandong Li Copyright (c) 2015 IEEE. Personal use of this
material is permitted. However, permission to use this material for any other
purposes must be obtained from the IEEE by sending a request to pubs-
<EMAIL_ADDRESS>work was supported in part by Natural Science Basic
Research Program of Shaanxi (Program No. 2021JZ-18), Natural Science
Foundation of Guangdong Province of China under Grant 2021A1515010812, the
open research fund of National Mobile Communications Research Laboratory,
Southeast University under Grant number 2021D04, the Fundamental Research
Funds for the Central Universities, and the Innovation Fund of Xidian
University.Wen-Xuan Long is with the State Key Laboratory of Integrated
Service Networks (ISN), Xidian University, Shaanxi 710071, China, and also
with the University of Pisa, Dipartimento di Ingegneria dell’Informazione,
Italy (e-mail: wxlong@stu.xidian.edu.cn).Rui Chen is with the State Key
Laboratory of Integrated Service Networks (ISN), Xidian University, Shaanxi
710071, China, and also with the National Mobile Communication Research
Laboratory, Southeast University, Nanjing 210018, China (e-mail:
rchen@xidian.edu.cn).Marco Moretti is with the University of Pisa,
Dipartimento di Ingegneria dell’Informazione, Italy (e-mail:
marco.moretti@iet.unipi.it).Jiandong Li is with the State Key Laboratory of
Integrated Service Networks (ISN), Xidian University, Shaanxi 710071, China
(e-mail: jdli@mail.xidian.edu.cn).

###### Abstract

Radio orbital angular momentum (OAM) communications require accurate alignment
between the transmit and receive beam directions. Accordingly, a key feature
of OAM receivers is the ability to reliably estimate the angle of arrival
(AoA) of multi-mode OAM beams. Considering the limitations of existing AoA
estimation techniques, in this paper, we propose an easier-to-implement AoA
estimation method based on applying multiple times the estimating signal
parameters via rotational invariance techniques (ESPRIT) algorithm to the
received training signals in OAM mode and frequency domains, which is denoted
as the mode-frequency (M-F) multi-time (MT)-ESPRIT algorithm. With this
method, the misalignment error of real OAM channels can be greatly reduced and
the performance approaches that of ideally aligned OAM channels.

###### Index Terms:

Orbital angular momentum (OAM), uniform circular array (UCA), angle of arrival
(AoA) estimation, estimating signal parameter via rotational invariance
techniques (ESPRIT).

##  I Introduction

Since the discovery in 1992 that vortex light beams can carry orbital angular
momentum (OAM) [1], a significant research effort has been focused on vortex
electromagnetic (EM) waves [2]. The phase front of a wave carrying OAM rotates
with azimuth exhibiting a helical structure $e^{j\ell\phi}$ in space, where
$\phi$ is the transverse azimuth and $\ell$ is an unbounded integer defined as
OAM _topological charge_ or OAM _mode number_ [1]. Due to inherent
orthogonality among different OAM modes, OAM wireless communications represent
a novel approach for multiplexing a set of orthogonal signals on the same
frequency channel to achieve the high spectral efficiency [3, 4, 5, 6, 7].
However, there are still some technical challenges for the practical
application of OAM wireless communications.

Figure 1: The OAM communication system model and the geometrical relationship
between the transmit and receive UCAs in the non-parallel misalignment case.

One special challenge for OAM wireless communications is that they require
accurate alignment between the transmit and receive antennas or at least
accurate alignment between the transmit and receive beam directions [8]. If
this condition is not precisely met, the system performance quickly
deteriorates, so that the accurate estimation of the angle of arrival (AoA) is
essential for OAM wireless communication systems. Although AoA estimation of
planar waves has been well studied [9, 10, 11], AoA estimation for OAM beams
is still an open field of investigation. To the best of our knowledge, [7] is
the only work addressing the problem of AoA estimation for OAM wireless
communication systems. One major shortcoming of [7] is that it needs to
process the amplitude of the received OAM training signals exploiting the
knowledge of the parameters of transmit antennas, which may be difficult to
obtain at the receiver.

$\displaystyle x_{m}(\ell_{u},k_{p})$
$\displaystyle=-j\frac{\mu_{0}\omega_{p}}{4\pi}\sum_{n=1}^{N}e^{i\ell_{u}\varphi_{n}}\int|\mathbf{d}_{mn}|^{-1}e^{ik_{p}|\mathbf{d}_{mn}|}d\widetilde{V}_{n}s(\ell_{u},k_{p})+z_{m}(\ell_{u},k_{p})$
$\displaystyle=-j\frac{\mu_{0}\omega_{p}}{4\pi}\sum_{n=1}^{N}e^{i\ell_{u}\varphi_{n}}\int|\mathbf{r}-\mathbf{\widetilde{r}}_{n}+\mathbf{\widetilde{r}}_{m}|^{-1}e^{ik_{p}|\mathbf{r}-\mathbf{\widetilde{r}}_{n}+\mathbf{\widetilde{r}}_{m}|}d\widetilde{V}_{n}s(\ell_{u},k_{p})+z_{m}(\ell_{u},k_{p})$
$\displaystyle\approx
e^{i\mathbf{k}_{p}\cdot\mathbf{\tilde{r}}_{m}}\bigg{[}-j\frac{\mu_{0}\omega_{p}d}{4\pi}\frac{e^{ik_{p}r}}{r}\sum_{n=1}^{N}e^{-i(\mathbf{k}_{p}\cdot\mathbf{\tilde{r}}_{n}-\ell_{u}\varphi_{n})}s(\ell_{u},k_{p})\bigg{]}+z_{m}(\ell_{u},k_{p})$
$\displaystyle\approx
e^{ik_{p}R\sin\theta\cos(\varphi-\varphi_{m})}\bigg{[}-j\frac{\mu_{0}\omega_{p}dNi^{-\ell_{u}}}{4\pi}\frac{e^{ik_{p}r}}{r}e^{i\ell_{u}\gamma}{J_{\ell_{u}}}(k_{p}R\sin\theta)s(\ell_{u},k_{p})\bigg{]}+z_{m}(\ell_{u},k_{p}),$
(3)

The major contribution of this paper is to propose an easier-to-implement AoA
estimation method for OAM communication systems, which applies multiple times
the estimating signal parameters via rotational invariance techniques (ESPRIT)
algorithm to the received training signals in OAM mode and frequency domains,
thus being denoted as the mode-frequency (M-F) multi-time (MT)-ESPRIT
algorithm. Simulation results validate that the proposed method is indeed able
to accurately and reliably estimate the AoA of multi-mode OAM beams.

##  II OAM Communication System Model

Employing uniform circular arrays (UCA) is a popular way to generate and
receive radio OAM beams due to simple structure and multi-mode multiplexing
ability [4]. Therefore, we consider a UCA-based OAM communication system, as
shown in Fig.1, where the arrays at the transmitter and receiver are equipped
with $N$ elements, and the transmitted signals are multiplexed on $U$
($1<U\leq N$) OAM modes at $P$ subcarriers. To generate the $\ell$-th mode OAM
component, the $N$ elements of the transmit UCA are fed with the same input
signal but with successive phase shifts
${\phi_{n}}=\ell\varphi_{n}=2{\pi}(n-1){\ell}/N,n=1,2,\ldots,N$, so that after
a full turn the phase has the increment of $2\pi\ell$. For an arbitrary point
$\bar{P}(\bar{r},\bar{\varphi},\bar{\theta})$ in the far field, the electric
field intensity $E(\mathbf{\bar{r}},k,\ell)$ generated by the transmitter can
be written as [12, 6]

$\displaystyle E(\mathbf{\bar{r}},k,\ell)$
$\displaystyle=-j\frac{\mu_{0}\omega}{4\pi}\sum_{n=1}^{N}e^{i\ell\varphi_{n}}\int|\mathbf{\bar{r}}-\mathbf{\widetilde{r}}_{n}|^{-1}e^{ik|\mathbf{\bar{r}}-\mathbf{\widetilde{r}}_{n}|}d\widetilde{V}_{n}$
$\displaystyle\overset{(a)}{\approx}-j\frac{\mu_{0}\omega
d}{4\pi}\frac{e^{ik\bar{r}}}{\bar{r}}\sum_{n=1}^{N}e^{-i(\mathbf{k}\cdot\mathbf{\tilde{r}}_{n}-\ell\varphi_{n})}$
$\displaystyle\overset{(b)}{\approx}-j\frac{\mu_{0}\omega
dNe^{ik\bar{r}}e^{i\ell\bar{\varphi}}}{4\pi\bar{r}}i^{-\ell}{J_{\ell}}(kRsin\bar{\theta}),$
(1)

where $j$ is the amplitude of the constant current density of the dipole, $i$
is the imaginary unit, $\mu_{0}$ is the magnetic conductivity in the vacuum,
$d$ is the electric dipole length, $\omega=2\pi f$ is the circular frequency,
$\mathbf{k}$ is the wave vector and $|\mathbf{k}|=k=\frac{2\pi c}{f}$, $c$ is
the speed of light in vacuum and $f$ is the frequency, $J_{\ell}(\cdot)$ is
the $\ell$th-order Bessel function of the first kind, $R$ is the radius of
UCA. In (II), (a) applies the approximation
$|\mathbf{\bar{r}}-\mathbf{\tilde{r}}_{n}|\approx\bar{r}$ for amplitudes and
$|\mathbf{\bar{r}}-\mathbf{\tilde{r}}_{n}|\approx\bar{r}-\mathbf{\hat{r}}\cdot\mathbf{\tilde{r}}_{n}$
for phases in the far field, where $\mathbf{\hat{r}}$ is the unit vector of
$\mathbf{\bar{r}}$,
$\mathbf{\tilde{r}}_{n}=R\left(\mathbf{x^{\prime}}\cos\varphi_{n}+\mathbf{y^{\prime}}\sin\varphi_{n}\right)$,
$\mathbf{x^{\prime}}$ and $\mathbf{y^{\prime}}$ are the unit vectors of x-axis
and y-axis of the coordinate system at the transmitter, respectively, and (b)
holds when $N$ is large enough.

In practice, accurate alignment between the transmit and receive UCAs may be
difficult to realize. For easier analysis, we consider the non-parallel
misalignment case as shown in Fig.1 [8, 7], where the center of transmit OAM
beams overlaps with the receive UCA center, but the receive UCA plane has an
angular shift relative to the transmit UCA plane. In the geometrical model,
the transmitter coordinate system
$\textrm{Z}^{\prime}-\textrm{X}^{\prime}\textrm{O}\textrm{Y}^{\prime}$ is
established using the transmit UCA plane as the
$\textrm{X}^{\prime}\textrm{O}\textrm{Y}^{\prime}$ plane and the axis through
the point O and perpendicular to the UCA plane as the
$\textrm{Z}^{\prime}$-axis, and the receiver coordinate system
$\bar{\textrm{Z}}-\bar{\textrm{X}}\bar{\textrm{O}}\bar{\textrm{Y}}$ is
established on the plane of the receive UCA by the similar approach. In the
non-parallel misalignment case, the coordinates of the receive UCA center can
be denoted as $\bar{\textrm{O}}(r,0,0)$ in
$\textrm{Z}^{\prime}-\textrm{X}^{\prime}\textrm{O}\textrm{Y}^{\prime}$, and
the coordinates of the transmit UCA center is denoted as
$\textrm{O}(r,\varphi,\theta)$ in
$\bar{\textrm{Z}}-\bar{\textrm{X}}\bar{\textrm{O}}\bar{\textrm{Y}}$, where $r$
is the distance between the transmit and the receive UCA centers, $\varphi$ is
the azimuth angle, $\theta$ is the elevation angle, and _$\varphi$ and
$\theta$ are defined as the AoA of OAM beams_.

Figure 2: The AoA estimation of the OAM wireless communication system with M-F
MT-ESPRIT method

.

To obtain the estimates of $\varphi$ and $\theta$, we have to first build the
coordinate system $\textrm{Z}-\textrm{XOY}$ that is at the transmit UCA and
parallel to
$\bar{\textrm{Z}}-\bar{\textrm{X}}\bar{\textrm{O}}\bar{\textrm{Y}}$, and then
get the expression of (II) at $\bar{\textrm{O}}$ point in
$\textrm{Z}-\textrm{XOY}$ coordinate system. According to the geometrical
model, the angle between $\textrm{Z}^{\prime}$-axis and
$\bar{\textrm{Z}}$-axis is $\pi-\theta$, so as Z-axis and
$\textrm{Z}^{\prime}$-axis. Let B be a point on the
$\textrm{X}^{\prime}$-axis, C is the projection of B on the XOY plane, and D
is the projection of C on the X-axis, then $\angle\textrm{BOC}=\theta$. As the
lines OC and $\bar{\textrm{O}}\textrm{A}$ are parallel,
$\angle\textrm{COD}=\varphi$. Denote the angle between
$\textrm{X}^{\prime}$-axis and X-axis as $\gamma$, then $\gamma$ can be
obtained as

$\displaystyle\gamma$
$\displaystyle=\angle\textrm{BOD}=\arccos\left(\cos(\angle\textrm{BOC})\cos(\angle\textrm{COD})\right)$
$\displaystyle=\arccos\left(\cos\theta\cos\varphi\right).$ (2)

Based on (II) and the geometrical model, the electric field intensity at the
$m$th $(1\leq m\leq N)$ element of receive UCA in the
$\textrm{Z}-\textrm{X}\textrm{O}\textrm{Y}$ coordinate system can be expressed
as (I), where $\mathbf{d}_{mn}$ is the position vector from the $n$th transmit
antenna element to the $m$th receive antenna element,
$\mathbf{\tilde{r}}_{m}=R(\mathbf{\bar{x}}\cos\varphi_{m}+\mathbf{\bar{y}}\sin\varphi_{m})$,
$\mathbf{\bar{x}}$ and $\mathbf{\bar{y}}$ are the unit vectors of
$\bar{\textrm{X}}$-axis and $\bar{\textrm{Y}}$-axis,
$\varphi_{m}=2{\pi}(m-1)/N$, $s(\ell_{u},k_{p})$ is the $\ell_{u}$-th
$(-U_{L}\leq\ell_{u}\leq U_{R},U_{L}+U_{R}=U)$ OAM mode information symbol on
the $u$th timeslot at frequency $f_{p}=k_{p}c/(2\pi)$ $(1\leq p\leq P)$,
$z_{m}(\ell_{u},k_{p})$ is the white Gaussian noise affecting the
$\ell_{u}$-th OAM mode at frequency $f_{p}$. Then, all the signals received by
the $N$ elements of receive UCA in the
$\textrm{Z}-\textrm{X}\textrm{O}\textrm{Y}$ coordinate system are combined,
which is expressed as

$\displaystyle x(\ell_{u},k_{p})$
$\displaystyle=\sum_{m=1}^{N}x_{m}(\ell_{u},k_{p})$
$\displaystyle\approx\beta_{u,p}\frac{e^{ik_{p}r}}{r}e^{i\ell_{u}\gamma}{J_{\ell_{u}}}(k_{p}R\sin\theta){J_{0}}(k_{p}R\sin\theta)$
$\displaystyle\quad+z(\ell_{u},k_{p}),$ (4)

where
$\beta_{u,p}=-j\frac{\mu_{0}\omega_{p}dN^{2}}{4\pi}i^{-\ell_{u}}s(\ell_{u},k_{p})$,
$z(\ell_{u},k_{p})$ is the corresponding additional noise.

As it happens for multiple-input multiple-output (MIMO) communications, also
OAM transmitted frames may carry a combination of training and data symbols.
In the case of MIMO, pilot data is used to perform channel estimation, and it
is well known that the number of training symbols per subcarrier or subchannel
has to be larger than the number of transmit antennas. On the other hand, in
the case of OAM, pilot data is exploited for AoA estimation and only
$U~{}(1\leq U\leq N)$ training symbols per subcarrier or subchannel are
required. Accordingly, we assume that
$\\{s(\ell_{u},k_{p})|U-L\leq\ell_{u}\leq U_{R},1\leq p\leq P\\}$ are the
pilot signals in the training phase known to the OAM receiver.

##  III The AoA Estimation Based on M-F MT-ESPRIT

###  III-A Problem Formulation

The aim of the AoA estimation is to obtain the estimates of the azimuthal
angle $\varphi$ and the elevation angle $\theta$. From (I) and (II), we can
observe that the OAM mode number $\ell_{u}$ and the angle $\gamma$ satisfy the
dual relationship, so do $k_{p}$ and $r$, where
$\gamma=\arccos\left(\cos\theta\cos\varphi\right)$ is an intermediate
parameter related to $\varphi$ and $\theta$. Meanwhile, the elevation angle
$\theta$ is associated with both $\ell_{u}$ and $k_{p}$. Therefore, we propose
to first estimate $r$ and $\gamma$ with the two dual relationships in (II),
and then obtain the estimates of $\varphi$ and $\theta$ according to (I). The
working process of AoA estimation for multi-mode OAM beams is shown in Fig.2.

###  III-B Estimation of $r$ and $\gamma$

To estimate $r$ and $\gamma$, we exploit the fact that we can obtain
independent estimates of these parameters on each mode $\ell_{u}$ and
subcarrier $k_{p}$. Accordingly, at the receiver, we need to extract the
exponentials containing $r$ and $\gamma$ from the signal $x(\ell_{u},k_{p})$
by performing the following operation

$\displaystyle\tilde{x}(\ell_{u},k_{p})$
$\displaystyle=-\frac{x(\ell_{u},k_{p})}{|x(\ell_{u},k_{p})|}\frac{s^{*}(\ell_{u},k_{p})}{|s(\ell_{u},k_{p})|}i^{\ell_{u}}{\rm
sign}(x(\ell_{u},k_{p}))$
$\displaystyle=e^{ik_{p}r}e^{i\ell_{u}\gamma}+\tilde{z}(\ell_{u},k_{p}),$ (5)

where $\tilde{z}(\ell_{u},k_{p})$ is the corresponding noise.

All the signals are received on the $U$ OAM modes at the $P$ frequencies and
can be collected in the matrix

$\mathbf{\tilde{X}}_{U,P}=\begin{bmatrix}\tilde{x}(\ell_{1},k_{1})&\tilde{x}(\ell_{1},k_{2})&\cdots&\tilde{x}(\ell_{1},k_{P})\\\
\tilde{x}(\ell_{2},k_{1})&\tilde{x}(\ell_{2},k_{2})&\cdots&\tilde{x}(\ell_{2},k_{P})\\\
\vdots&\vdots&\ddots&\vdots\\\
\tilde{x}(\ell_{U},k_{1})&\tilde{x}(\ell_{U},k_{2})&\cdots&\tilde{x}(\ell_{U},k_{P})\\\
\end{bmatrix}.$ (6)

For easier analysis, we assume the adopted frequencies and OAM modes satisfy
$k_{p+1}-k_{p}=1$ and $\ell_{{u}+1}-\ell_{{u}}=1$.

In the estimation of $r$, we first denote the $u$th row of
$\mathbf{\tilde{X}}_{U,P}$ as a column vector $\mathbf{\tilde{x}}_{a}^{u}$,
i.e.,

$\displaystyle\mathbf{\tilde{x}}_{a}^{u}=\mathbf{\tilde{X}}_{U,P}(u,:)=[\tilde{x}(\ell_{u},k_{1}),\tilde{x}(\ell_{u},k_{2}),\cdots,\tilde{x}(\ell_{u},k_{P})]^{\mathrm{T}}.$

In the ESPRIT-based distance estimation method [7], we can obtain an estimate
of $r$ based on the $u$th row of the matrix $\mathbf{\tilde{X}}_{U,P}$, which
we denote as $\hat{r}_{u}$. Hence, there are $U$ estimates of $r$ in total,
which can be expressed as

$\displaystyle\hat{r}_{u}=$ $\displaystyle
r+\varepsilon_{u},{u}=1,2,\cdots,U,$

where $\\{\varepsilon_{u}|u$$=$$1,2,\cdots,U\\}$ represent the estimation
errors at $U$ modes. Suppose that $\\{\varepsilon_{u}|u=1,2,\cdots,U\\}$ have
the same average variance $\textrm{Var}(\varepsilon_{r})$, thus,

$\displaystyle\textrm{Var}\left(\frac{1}{U}\sum_{u=1}^{U}\hat{r}_{u}\right)=\frac{\textrm{Var}\left(\varepsilon_{r}\right)}{U}.$
(7)

Therefore, $\hat{r}=\frac{1}{U}\sum_{u=1}^{U}\hat{r}_{u}$ is adopted as the
estimate of $r$.

In the estimation of $\gamma$, we similarly denote the $p$th column of
$\mathbf{\tilde{X}}_{U,P}$ as

$\displaystyle\mathbf{\tilde{x}}_{b}^{p}=\mathbf{\tilde{X}}_{U,P}(:,p)=[\tilde{x}(\ell_{1},k_{p}),\tilde{x}(\ell_{2},k_{p}),\cdots,\tilde{x}(\ell_{U},k_{p})]^{\mathrm{T}}.$

After that, by following the same method employed for $\hat{r}$ we can obtain
$\hat{\gamma}$, the estimate of $\gamma$.

Algorithm 1 AoA Estimation by M-F MT-ESPRIT Algorithm

Input: $\mathbf{\tilde{X}}_{U,P}$ and $\mathbf{\tilde{x}}_{c}$
Output: $\varphi$ and $\theta$

1:procedure

2:
$\mathbf{\tilde{x}}_{a}^{u}\leftarrow\mathbf{\tilde{X}}_{U,P}(u,:),u=1,2,\cdots,U$

3: $\hat{r}_{u}\leftarrow$ ESPRIT algorithm for
$\mathbf{\tilde{x}}_{a}^{u},u=1,2,\cdots,U$

4: $\hat{r}\leftarrow$ $\frac{1}{U}\sum_{u=1}^{U}\hat{r}_{u}$

5:
$\mathbf{\tilde{x}}_{b}^{p}\leftarrow\mathbf{\tilde{X}}_{U,P}(:,p),p=1,2,\cdots,P$

6: $\hat{\gamma}_{p}\leftarrow$ ESPRIT algorithm for
$\mathbf{\tilde{x}}_{b}^{p},p=1,2,\cdots,P$

7: $\hat{\gamma}\leftarrow$ $\frac{1}{P}\sum_{p=1}^{P}\hat{\gamma}_{p}$

8:
$\mathbf{R}_{\mathbf{\tilde{x}}_{c}}\leftarrow\mathbb{E}\left\\{\mathbf{\tilde{x}}_{c}{\mathbf{\tilde{x}}_{c}^{\mathrm{H}}}\right\\}$

9: Q, $\bm{\Lambda}_{\leftarrow}$ decompose
$\mathbf{R}_{\mathbf{\tilde{x}}_{c}}$ such that
$\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{\mathrm{H}}$$=$$\mathbf{R}_{\mathbf{\tilde{x}}_{c}}$

10: $\lambda_{max}\leftarrow\max\\{\lambda_{p}|p=1,2,\cdots,P\\}$

11: $\mathbf{q}\leftarrow$the column of Q corresponding to $\lambda_{max}$

12: $\mathbf{q}_{1}$, $\mathbf{q}_{2}\leftarrow$ the first and the last $P-1$
elements of $\mathbf{q}$

13: $e^{i\hat{\xi}}\leftarrow{\mathbf{q}_{1}}^{\dagger}\mathbf{q}_{2}$

14:
$\hat{\varphi}\leftarrow\arccos\sqrt{\big{(}\frac{\hat{\xi}-\hat{r}}{R}\big{)}^{2}+\cos{\hat{\gamma}}^{2}}$

15:
$\hat{\theta}\leftarrow\arctan\frac{\hat{\xi}-\hat{r}}{R\cos{\hat{\gamma}}}$

16: return $\varphi$ and $\theta$

17:end procedure

###  III-C Estimation of $\varphi$ and $\theta$

Having estimated $r$ and $\gamma$, we can obtain the azimuth angle $\varphi$
and the elevation angle $\theta$. According to (I), the signal received by the
reference element $(\varphi_{1}=0)$ of the receive UCA on the zero OAM mode
and $p$th frequency can be simplified as

$\displaystyle\tilde{x}^{\prime}(k_{p})$
$\displaystyle=-\frac{x_{1}(0,k_{p})}{|x_{1}(0,k_{p})|}\frac{s^{*}(0,k_{p})}{|s(0,k_{p})|}{\rm
sign}(x_{1}(0,k_{p}))$
$\displaystyle=e^{ik_{p}\xi}+\tilde{z}^{\prime}(k_{p}),$ (8)

where $\xi=r+R\sin\theta\cos\varphi$ is an intermediate parameter related to
$\varphi$ and $\theta$, $\tilde{z}^{\prime}(k_{p})$ is the corresponding
noise.

Then, the signals received on the $P$ frequencies can be collected in the
vector

$\mathbf{\tilde{x}}_{c}=[\tilde{x}^{\prime}(k_{1}),\tilde{x}^{\prime}(k_{2}),\cdots,\tilde{x}^{\prime}(k_{P})]^{\mathrm{T}},$
(9)

which can be expressed in a compact form as

$\mathbf{\tilde{x}}_{c}=\mathbf{c}+\mathbf{z}_{c},$ (10)

where $\mathbf{c}$ $=$
$[e^{ik_{1}\xi},e^{ik_{2}\xi},\cdots,e^{ik_{P}\xi}]^{T}$ and $\mathbf{z}_{c}$
is the corresponding noise vector.

The covariance matrix of $\mathbf{\tilde{x}}_{c}$ can be written as

$\mathbf{R}_{\mathbf{\tilde{x}}_{c}}=\mathbb{E}\left\\{\mathbf{\tilde{x}}_{c}{\mathbf{\tilde{x}}_{c}}^{H}\right\\}=\mathbf{c}\mathbf{c}^{H}+\mathbf{R}_{\mathbf{z}_{c}},$
(11)

where
$\mathbf{R}_{\mathbf{z}_{c}}=\mathbb{E}\left\\{\mathbf{z}_{c}{\mathbf{z}_{c}}^{\mathrm{H}}\right\\}$.
The eigenvalue decomposition (EVD) of $\mathbf{R}_{\mathbf{\tilde{x}}_{c}}$ is

$\mathbf{R}_{\mathbf{\tilde{x}}_{c}}=\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{H},$
(12)

where $\mathbf{Q}$ is an $P\times P$ unitary matrix and
$\mathbf{\Lambda}=\textrm{diag}\\{\lambda_{1},$
$\lambda_{2},\cdots,\lambda_{P}\\}$. Denote as
$\lambda_{max}=\max\\{\lambda_{p}|p=1,2,\cdots,P\\}$, and as $\mathbf{q}$ the
eigenvector corresponding to $\lambda_{max}$, so that it is

$\displaystyle\mathbf{R}_{\mathbf{\tilde{x}}_{c}}\mathbf{q}=\lambda_{max}\mathbf{q}.$
(13)

Now, the subspace spanned by $\mathbf{q}$ is the signal subspace spanned by
$\mathbf{c}$ so that the following relationship holds true

$\mathbf{c}=\textrm{T}\mathbf{q},$ (14)

where T is a non-zero parameter. If we consider the two vectors
$\mathbf{c}_{1}$ and $\mathbf{c}_{2}$, obtained by taking the first and the
last $P-1$ elements of $\mathbf{c}$, respectively, it is
$\mathbf{c}_{2}={\Phi}\mathbf{c}_{1}$, where $\Phi=e^{i\xi}$. To obtain the
estimate of $\xi$, we construct the two vectors $\mathbf{q}_{1}$ and
$\mathbf{q}_{2}$, composed by the first and by the last $P-1$ elements of
$\mathbf{q}$, respectively. Then, exploiting the fact that
$\mathbf{c}_{1}=\textrm{T}\mathbf{q}_{1}$ and
$\mathbf{c}_{2}=\textrm{T}\mathbf{q}_{2}$, one obtains

$\mathbf{q}_{2}={\Phi}\mathbf{q}_{1},$ (15)

which leads to

$e^{i\hat{\xi}}={\mathbf{q}_{1}}^{\dagger}\mathbf{q}_{2},$ (16)

where $\hat{\xi}$ is the estimate of $\xi$ based on the signal vector
$\mathbf{\tilde{x}}_{c}$.

After that, according to

$\displaystyle\hat{\varphi}=\arccos\sqrt{\big{(}\frac{\hat{\xi}-\hat{r}}{R}\big{)}^{2}+\cos{\hat{\gamma}}^{2}},$
$\displaystyle\hat{\theta}=\arctan\frac{\hat{\xi}-\hat{r}}{R\cos{\hat{\gamma}}},$
(17)

the estimates of the azimuthal angle $\varphi$ and the elevation angle
$\theta$ can be obtained. Thus, the AoA estimation by M-F MT-ESPRIT algorithm
is completed, and the detailed procedure is summarized in Algorithm 1.
Compared with the existing AoA estimation method [7], the AoA estimation
method based on M-F MT-ESPRIT algorithm can estimate the azimuth and elevation
angles of muiti-mode OAM beams only by utilizing the exponentials of the
received training signals, which is easier to implement in practical
applications.

TABLE I: The complexity of the AoA estimation. Procedure |  Complexity
---|---
Estimating $\hat{r}$ |  $\mathcal{O}\left(UP^{\max\\{2,g\\}}\right)$
Estimating $\hat{\gamma}$ | $\qquad\quad\mathcal{O}\left(PU^{\max\\{2,g\\}}\right)$
Eq.$(11):\mathbf{R}_{\mathbf{\tilde{x}}_{c}}=\mathbb{E}\left\\{\mathbf{\tilde{x}}_{c}{\mathbf{\tilde{x}}_{c}}^{H}\right\\}$ | $\qquad\qquad\quad\mathcal{O}\left(P^{2}\right)$
Eq.$(12):\mathbf{R}_{\mathbf{\tilde{x}}_{c}}=\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{H}$ | $\qquad\qquad\quad\mathcal{O}\left(P^{g}\right)$
Eq.$(16):e^{i\hat{r}_{u}}={\mathbf{q}^{1}_{u}}^{\dagger}\mathbf{q}^{2}_{u}$ | $\qquad\qquad\quad\mathcal{O}\left(P\right)$
The total complexity:
$\mathcal{O}\left(UP^{\max\\{2,g\\}}\right)+\mathcal{O}\left(PU^{\max\\{2,g\\}}\right)$

The computational complexities of the proposed AoA estimation method based on
M-F MT-ESPRIT algorithm is summarized in Table LABEL:Table1, where $g$ is
generally equal to 3, but can be reduced to 2.376 when the Coppersmith and
Winograd algorithm [13] is applied to the eigenvalue decomposition in (12).
Due to the use of M-F MT-ESPRIT algorithm, the dimension of the training
signal vector processed by ESPRIT algorithm each time is greatly reduced. The
total complexity comparison between the existing AoA estimation method [7] and
the proposed AoA estimation method based on M-F MT-ESPRIT algorithm is shown
in Fig.3.

Figure 3: The complexities of the AoA estimation method based on M-F MT-ESPRIT
algorithm and the AoA estimation method in [7] vs. $P$ and $U$ with $g=3$.
Figure 4: The AoA estimation results at 15dB and 20dB.

##  IV Simulation Results and Performance Analysis

We first verify the proposed AoA estimation method by numerical simulations,
and then compare the accuracy of the AoA estimation method proposed in this
paper with that proposed in [7]. Thereafter, we compare the capacity
performance of the OAM channel without beam steering and that with beam
steering [7] based on the estimated AoA. In the simulation, we choose $N=9$,
$P=8$ subcarriers from 2.244GHz to 2.578GHz corresponding to the wave number
$k_{p}=47,48,\dots,54$, number of OAM modes $U=8$ with
$\ell_{u}=-4,-3,\cdots,+3$, $R=10\lambda_{1}$, $\lambda_{1}=2\pi/k_{1}$,
$(r,\varphi,\theta)=(40\textrm{m},7^{\circ},7^{\circ})$. Besides, we assume
the angle range of receive UCA’s main lobe
$[\alpha_{a},\alpha_{b}]=[2^{\circ},8^{\circ}]$, the initial number of
intervals $\mathcal{D}=2$ [7]. Unless otherwise stated, the SNRs in all the
figures are defined as the ratio of the received signal power versus the noise
power.

The AoA estimation results of the proposed M-F MT-ESPRIT method and the method
proposed in [7] are shown in Fig. 4. It can be seen from the figure that with
the increase of SNR the estimated AoAs approach to the actual value. When
SNR$=20$dB, the AoA estimated by M-F MT-ESPRIT method is
$(\hat{\varphi},\hat{\theta})=(7.005^{\circ},6.994^{\circ})$, and the AoA
estimated by the method in [7] is
$(\hat{\varphi},\hat{\theta})=(7.006^{\circ},6.993^{\circ})$, which are very
close to the actual AoA. Besides, the CPU time of M-F MT-ESPRIT method is much
lower than that of the method in [7].

Figure 5: The NMSEs of the estimated $\hat{\varphi}$ and $\hat{\theta}$ vs.
SNR. Figure 6: The capacity comparison of the OAM channel with perfect
alignment, without BS, with BS by the estimated AoA in this paper, and with BS
by the estimated AoA in [7]. BS: beam steering.

Fig.5 illustrates the normalized mean-squared errors (NMSEs) of
$\hat{\varphi}$ and $\hat{\theta}$ obtained by M-F MT-ESPRIT method and the
method in [7]. The NMSE is defined as $\mathbb{E}\\{(\hat{x}-x)^{2}/x^{2}\\}$,
where $\hat{x}$ denotes the estimate of $x$. For the proposed M-F MT-ESPRIT
method, as SNR increases the NMSEs of $\hat{\varphi}$ and $\hat{\theta}$
decline. It follows that the proposed M-F MT-ESPRIT method is able to
accurately estimate the AoA of multi-mode OAM beams. Moreover, due to the
reduction of the dimension of the training signal vector processed by ESPRIT
algorithm each time, the NMSEs of $\hat{\varphi}$ and $\hat{\theta}$ obtained
by M-F MT-ESPRIT method is higher than those obtained by the method in [7].

In Fig.6, we compare the capacity of the OAM channel with perfect alignment,
without beam steering, with beam steering by the estimated AoA in this paper,
and with beam steering by the estimated AoA in [7]. The capacity of the
misaligned OAM channel is greatly improved and approaches the capacity of
accurately aligned OAM channel after applying the beam steering with the AoA
estimation method proposed in this paper and in [7]. Therefore, the accurate
AoA estimation is essential for the OAM receiver to improve the capacity of
the misaligned OAM channel. Moreover, due to the larger estimation error of
$\hat{\varphi}$ and $\hat{\theta}$, the capacity of the OAM channel with beam
steering by the estimated AoA in this paper is lower than that by the
estimated AoA in [7] under the low SNRs.

##  V Conclusions

In this paper, an AoA estimation method based on M-F MT-ESPRIT algorithm for
OAM wireless communication systems is proposed, which can accurately estimate
the azimuth and elevation angles of multi-mode OAM beams only based on the
received training signals. With the accurately estimated AoA, the capacity of
the misaligned OAM channel can be greatly improved by beam steering and
approaches the capacity of accurately aligned OAM channel. Moreover, compared
with the existing AoA estimation method [7], the total complexity of the
proposed AoA estimation method based on M-F MT-ESPRIT algorithm is
significantly decreased, which is more practical.

## Acknowledgment

The authors would like to thank the editor and the anonymous reviewers for
their careful reading and valuable suggestions that helped to improve the
quality of this manuscript.

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