###### Abstract

Quality inspection applications in industry are required to move towards a
zero-defect manufacturing scenario, with non-destructive inspection and
traceability of 100 % of produced parts. Developing robust fault detection and
classification models from the start-up of the lines is challenging due to the
difficulty in getting enough representative samples of the faulty patterns and
the need to manually label them. This work presents a methodology to develop a
robust inspection system, targeting these peculiarities, in the context of
solar cell manufacturing. The methodology is divided into two phases: In the
first phase, an anomaly detection model based on a Generative Adversarial
Network (GAN) is employed. This model enables the detection and localization
of anomalous patterns within the solar cells from the beginning, using only
non-defective samples for training and without any manual labeling involved.
In a second stage, as defective samples arise, the detected anomalies will be
used as automatically generated annotations for the supervised training of a
Fully Convolutional Network that is capable of detecting multiple types of
faults. The experimental results using 1873 EL images of monocrystalline cells
show that (a) the anomaly detection scheme can be used to start detecting
features with very little available data, (b) the anomaly detection may serve
as automatic labeling in order to train a supervised model, and (c)
segmentation and classification results of supervised models trained with
automatic labels are comparable to the ones obtained from the models trained
with manual labels.

###### keywords:

Anomaly detection; Electroluminescence; Solar cells; Neural Networks

Anomaly Detection and Automatic Labeling for Solar Cell Quality Inspection
Based on Generative Adversarial Network Julen Balzategui *, Luka Eciolaza and
Daniel Maestro-Watson Julen Balzategui, Luka Eciolaza and Daniel Maestro-
Watson Balzategui, J.; Eciolaza, L.; Maestro-Watson, D. Correspondence:
<EMAIL_ADDRESS>
##  1 Introduction

Quality inspection applications in industry are becoming very important. It is
a requirement to move towards a zero-defect manufacturing scenario, with
unitary non-destructive inspection and traceability of produced parts. This is
one of the applications where image analysis with deep learning (DL) methods
is showing its full potential. DL has proven to greatly improve the results of
solutions obtained using traditional vision techniques, regarding precision,
robustness, and flexibility. These improvements allow models to be adapted to
incorporate new features of interest, transfer learned models between
different domains, and to speed-up the design and development of models for
new tasks.

However, the quality inspection environment in the industry has peculiarities
that must be taken into account when applying DL-based solutions. Thus, it is
not easy to generate large enough data sets with representative images of the
different characteristics of interest. Manual labeling of each of the examples
must be carried out, which is usually an arduous task that takes large amounts
of time and resources. Furthermore, the scarcity of available examples, and
the fact that the images of industrial products manufactured successively are
very similar to each other, can make the DL algorithm prone to overfitting.
Finally, in new applications or industrial processes, there are no defective
data samples from the beginning, so it would be necessary to wait a long time
to be able to have DL models capable of identifying the faults that may
appear.

Thus, this work presents a methodology to deal with these peculiarities. This
methodology should work as a guide towards robust classification and
segmentation models, giving rise to fault detection models that are able to
detect anomalous feature patterns from the start-up of a new line. The
methodology will make use of an anomaly detection model which allows anomalous
patterns to be detected in the produced parts, and in addition, the detected
anomalies will serve as automatic annotations making the labeling of the
images much faster. The work will be tailored to the solar cell manufacturing
industry; however, it could be extrapolated to different domains.

In the last decade, about 2.6 trillion dollars have been invested in renewable
energies, half of it in solar energy, with the objective of developing
efficient alternatives to traditional energy sources such as oil or gas
(Centre/BNEF, 2019). The development of the technology has reduced solar
electricity generation cost per kilowatt-hour by 81%. This cost reduction has
turned solar energy into an attractive source of energy for electricity
production, increasing the installation of Photovoltaic (PV) cells by 36.8%
between 2010 and 2018 (Centre/BNEF, 2019). This investment trend is expected
to continue in the coming years (IEA, 2019).

During the assembly of the panels, different events, such as excessive
mechanical stress on the panel or a soldering failure, can lead to defects
that can harm the long-term energy generation capacity of the module. A defect
that covers 8% of the total cell area may not have a significant impact on the
performance if the cell is isolated. However, the same area can have a
significant impact when cells are connected and soldered to each other in cell
arrays (Köntges et al., 2011), which is the most common layout. The defective
area may spread with time, breaking the cell and considerably reducing the
energy production capacity of the module. As cell production increases,
quality inspection becomes critical to avoid defective cells being assembled
into the final panel, and thus, to ensure high efficiency and reliable
performance of the produced panels.

Nowadays, different imaging techniques are used during PV module inspection to
obtain images where defects appear highlighted, for example,
Electroluminescence (EL) (Bartler et al., 2018; Chen et al., 2020),
Photoluminescence (PL) (Demant et al., 2016; Nos et al., 2016), or
Thermography (Pierdicca et al., 2018; Vaněk et al., 2016). During the assembly
stage, EL is one of the predominant techniques. In EL, the cells emit light
under electrical current by the phenomenon of Electroluminescence. This light
is then captured in high-resolution images where defective areas, with less
current flow, appear darker than the remaining parts of the cell (Fuyuki and
Kitiyanan, 2009). The most common defects that may arise are cracks, breaks,
and finger interruptions (Tang et al., 2020). Figure 1 shows the appearance of
these kinds of defects in EL images. This technique requires a high degree of
control over environmental conditions since the images have to be taken in
total darkness. This requirement makes the application of EL unfeasible for
outdoor panel inspection, but suitable for inspection in the manufacturing
phase with controlled environmental conditions. EL provides high-resolution
images where defects are highlighted.

Figure 1: Different type of defects highlighted in Electroluminescence images.
The defects are: a) microcrack, b) crack, c) bad soldering, d) break, and e)
finger interruptions.

Despite these enhanced images, the defect detection process has to be done by
checking each of the cells individually. This process is currently done to a
great extent by human operators, who are prone to error, as it is hard for
humans to meet industrial production cycle-times. For example, a panel
composed of 60 PV modules must be examined in under 30 seconds, which means
half a second for each module. Also, human subjectivity is inevitable when
deciding if a cell is defective or not, affecting the quality inspection
effectiveness. In recent years, several proposals have been made towards the
automation of quality inspection. By automating the inspection, all the cells
can be checked faster and always using the same objective criteria, overcoming
the previous limitations.

The proposed approaches for automatic PV module inspection can be grouped into
three categories according to the required level of human intervention: 1)
traditional image processing based approaches, where the procedures used to
highlight and binarize defective areas in the images must be manually defined,
2) shallow learning approaches, where machine learning techniques are used for
defect identification based on meaningful features that must be obtained
through manual feature engineering, and 3) deep learning techniques, where the
features are automatically obtained from the data. Note that higher levels of
human intervention in the image processing algorithm implementation implies
larger development times in order to adapt it to new requirements.

The remainder of the paper is organized as follows: Section 2 presents some
background and related works in the field of photovoltaic cell inspection. In
Section 3, the unsupervised and supervised training are explained. Section 4
details the dataset, metrics, and hardware and software specifications used in
the experiments. Section 5 describes the performed experiments and their
results. Finally, Section 6 provides some conclusions about the work.

##  2 Related works

In this section, some of the proposed approaches for the automatic detection
of defects in images of PV modules are going to be summarized.

The traditional image processing methods are mainly based on manual feature
engineering. In this process, the discriminating characteristics of the
defects are used to process and binarize the images to highlight the defects.
For example, using anisotropic diffusion filters (Ko and Rheem, 2010; Anwar
and Abdullah, 2014) or modified steerable filters (Chen et al., 2019, 2018),
the background in the modules is smoothed such that only defects remain. Or
inversely, applying anisotropic diffusion (Tsai et al., 2010) or filters in
the frequency domain (Tsai et al., 2012) to remove the defects in the cells,
so then the difference between the filtered and the original image is used to
highlight the defects.

In other works, the manual feature extraction is combined with shallow
learning methods: In (Tsai et al., 2013) and (Zhang et al., 2013), they
extract Independent Component Analysis basis (ICA) from defect-free solar
cells samples to construct a demixing matrix. At the inspection stage, the
images are reconstructed using the learned basis images and the reconstruction
error is used for detecting the presence of defects. In Rodriguez et al.
(2021), 20 different LoG-Gabor Filters are used to extract 81 features for
each pixel in the images. Then, Principal Component Analysis (PCA) is used to
refine these features, and finally, a Random Forest model classifies each
pixel as non-defective or defective. In (Tsai et al., 2015), they extract
characteristics of local grains patterns and clusterize them using Fuzzy
C-means. At testing, the distance of the grains from the samples to the
clusters is used to decide if the grain is defective or not. Similarly, in (Su
et al., 2019), they use a modified Center-Symmetric Local Binary Patterns (CS-
LBP) feature descriptor to extract features from the defective areas in the
cells, which are then used to train the K-means algorithm. The cluster
centroids from training samples are employed to generate global feature
vectors to train a classification algorithm, such as a Support Vector Machine
(SVM).

Overall, both traditional image processing methods and traditional methods in
combination with shallow machine learning techniques can achieve high defect
detection rates. However, manual feature engineering is usually time-consuming
and requires high domain knowledge. In addition, inspection systems based on
these approaches are commonly very case-specific solutions that lack
adaptability. A change in the data can mean a substantial change in the
inspection system, which would require additional time-consuming manual
feature engineering labor to adapt it.

In more recent works, DL methods have been widely applied in the solar cell
inspection field. These methods can directly extract meaningful features from
the raw data without any feature engineering work, thus making these methods
more flexible to changes. The following works (Bartler et al., 2018; Akram et
al., 2019; Dunderdale et al., 2020; Deitsch et al., 2019; Akram et al., 2020;
Chen et al., 2020) are some examples of how Convolutional Neural Networks
(CNN) have been employed for classifying solar cells as defective or defect-
free during quality inspection. In addition to classification, in some cases,
the location of the defects in the cells is also provided. For example, in our
previous works, we used the sliding window approach with a CNN designed for
classification to process cell images by patches and accumulate the results in
a heatmap-like image, highlighting areas with a high probability of being
defective (Balzategui et al., 2019). Or we explicitly train a Fully
Convolutional Network to perform pixel-wise classification (Balzategui et al.,
2020). Additionally, other researchers have also proposed other types of
defect location, using bounding boxes (Liu et al., 2019; Zhang et al., 2020),
or by visualizing the activation maps from the last network layer (Deitsch et
al., 2019; Mayr et al., 2019).

Nevertheless, to obtain high detection rates, the networks are trained using
supervised learning, which requires a considerable amount of annotated
defective data. The quality of the results (i.e., detection rate) in
supervised learning is directly proportional to the amount of employed
annotated data. However, this represents a challenge in many industrial
applications as sufficient defective samples may be difficult to obtain in an
industrial setting. Thus, the creation of accurate inspection models may be
difficult, as a new manufacturing line will need time to generate a
representative dataset with enough examples. There may also be certain very
rare defect types that might be difficult to gather for the dataset.

To tackle the problem of insufficient defective data, several researchers have
proposed different solutions. One of the approaches is transfer learning
(Akram et al., 2020; Demirci et al., 2019; Qian et al., 2020), where the
neural network is initialized using weights from a previously trained network.
Then, the model is refined using a few case specific samples. Transfer
learning is limited by the similarity between the source and target domains.
Currently, the available pre-trained weights have been mainly trained on
natural images rather than on industrial datasets, which can limit their use
in industrial cases.

Another approach consists in generating synthetic data to compensate
imbalanced datasets employing variants of the Generative Adversarial Network
(GAN) (Goodfellow et al., 2014). These architectures have shown remarkable
capabilities in learning latent representations of real data to generate
realistic synthetic samples. In this way, synthetic defective samples are
generated and employed along with real samples to train a conventional CNN.
This approach alleviates the risk of overfitting and improves the
generalization capability of the network. This strategy has been successfully
employed to generate realistic human faces Choi et al. (2018), synthetic
machinery faulty signals Zhang et al. (2020), and also defective solar cell
samples (Tang et al., 2020; Luo et al., 2019). Nonetheless, both Transfer
Learning and GANs still require defective data.

In other domains, researchers have used an anomaly detection approach to avoid
the need for defective data. The objective of this approach is to train a
network to learn the probabilistic distribution of normal data. Then, the
learned features can be used to discriminate samples that will be far from
what is considered normal, and thus, detect defective samples. In anomaly
detection, just defect-free samples are used during training and there is no
need for annotations. These features make anomaly detection an interesting
approach for industrial applications. Anomaly detection has been applied in
different industrial cases, e.g., (Haselmann et al., 2018; Staar et al., 2019)
and also in the medical domain, e.g., (Schlegl et al., 2019; Chen and
Konukoglu, 2018), where it is also difficult to obtain anomalous data for
training. However, these approaches usually result in less accurate models
than those obtained with supervised training.

In the case of solar cell inspection, anomaly detection approaches have been
proposed in (Qian et al., 2020) and (Qian et al., 2020), where they train a
Stacked Denoising AutoEncoder (SDAE) to extract features from defect-free
samples using the sliding window method. In (Qian et al., 2020), they extend
the network architecture with a pre-trained VGG16 network that works as an
additional feature extractor. This extra branch extracts additional features
which are fused with the already extracted feature maps enhancing the obtained
information. At testing time, the same procedure is applied and the extracted
features are processed using matrix decomposition to localize the defects in
the cells. After that, some morphological processing is applied to improve the
results. However, in these works, only the detection of cracks is targeted.
Furthermore, the images are processed using the sliding window method. This
procedure slows down the inspection process limiting its deployment into a
real production environment.

An inspection system should be able to detect the maximum number of defects,
be fast to meet the established inspection time, and require the minimum human
intervention in order to save resources and time. The main contribution of
this work is a methodology that tries to meet these requirements by combining
the accuracy of supervised models with the benefits of an anomaly detection
approach, i.e., that it only requires defect-free samples for training and
avoids the need for data labeling. The approach has been tailored for the
detection and segmentation of different types of defects such as cracks,
microcracks, or finger interruptions in EL images of solar cells; however, it
should also be applicable in other industrial inspection tasks. The
methodology is illustrated in Figure 2 and consists of two stages:

Figure 2: General schema of the proposed methodology. 1) In the unsupervised
training, the network for anomaly detection is trained. 2) In anomaly
detection, the network from the previous step is applied to detect and locate
defects, and thus, generate an automatically labeled dataset. 3) Finally, the
generated dataset is employed to train a supervised model that improves the
detection rates.

  1. 1.

First, using an anomaly detection approach, defect-free samples can be
employed to obtain an initial inspection model that from the very beginning of
a new production line can detect and segment anomalies in EL images of cells.
For this purpose, f-AnoGAN (Schlegl et al., 2019), a GAN based anomaly
detection network that has been shown to work well with medical images, is
adapted for inspection. The original architecture has been modified such that
instead of using a sliding window method, the images can be processed as a
whole, reducing the processing time drastically. Also, a modified training
scheme is proposed which improves the defect detection rates with respect to
the results with the original training scheme.

  2. 2.

Then, as defective cells arise, the anomaly detection model will separate them
from the defect-free ones and it will generate pixel-level annotations without
any human intervention. The experiments have shown that these segmentation
results can be used as pixel-wise labels for the supervised training of a
U-Net Ronneberger et al. (2015) based model that improves the defect detection
rates of the anomaly detection model.

##  3 Methodology

This section details how the different networks used in the methodology are
trained.

###  3.1 Unsupervised Model for Anomaly Detection

In this stage, the objective is to train an anomaly detection model that can
detect and locate anomalous patterns within solar cell images. This is
achieved by training f-AnoGAN network to encode and reconstruct only defect-
free samples, so then, when processing defective samples, it will output a
defect-free version of them. The differences between the original and the
reconstructed defect-free version will highlight anomalies in the cells.

f-AnoGAN is composed of three different sub-networks (a generator $G$, a
discriminator $D$, and an encoder $E$) that are trained in two phases.

In the first training phase, the generator and discriminator are trained in an
adversarial manner to learn a latent space of normal data variability using
just normal data. In this work, defect-free samples are considered as normal
data and defective samples as anomalous data.

In the second phase, the encoder is trained to map normal data from the image
space to the learned latent space while the Generator and Discriminator are
kept unaltered. Once these two phases have finished, the encoder can map test
images from the image space to the latent space, and the generator can
reconstruct the encoded version of the images from the latent space back to
the image space. As the network is trained on normal data, it only learns to
encode and reconstruct correctly normal features, thus when processing
anomalous samples, deviations from the reconstructed images can be used for
anomaly detection and location.

####  3.1.1 Phase 1 - WGAN Training

The objective of the first training phase consists in learning the variability
of normal data. For this purpose, a Wasserstein GAN (WGAN), composed of a
generator and a discriminator, is optimized to learn the normal data
probability distribution. The optimization is achieved using the gradient
penalty based loss shown in Equation 1 proposed by (Gulrajani et al., 2017),
where the Wasserstein distance between the real normal data probability
distribution $P_{r}$, and generator synthesized data probability distribution
$P_{g}$ is minimized.

$L_{WGAN}=\underset{\tilde{\mathbf{x}}\sim\mathbb{P}_{g}}{\mathrm{\mathbb{E}}}[D(\tilde{\mathbf{x}})]-\underset{\mathbf{x}\sim\mathbb{P}_{r}}{\mathrm{\mathbb{E}}}[D(\mathbf{x})]+\lambda\underset{\hat{\mathbf{x}}\sim\mathbb{P}_{\hat{\mathbf{x}}}}{\mathrm{\mathbb{E}}}[{(\lVert{\nabla_{\hat{\mathbf{x}}}D\hat{\mathbf{x}})\rVert}_{2}-1)}^{2}],$
(1)

where $\tilde{\mathbf{x}}=G(\mathbf{z})$,
$\hat{\mathbf{x}}=\alpha\mathbf{x}+(1-\alpha)\tilde{\mathbf{x}}$ with
$\alpha\sim U(0,1)$ and $\lambda$ is the penalty coefficient. During training,
the generator is fed with a noise input vector $\mathbf{z}$, sampled from a
latent space $\mathcal{Z}$, and tries to learn the mapping from that latent
space to the image space $\mathcal{X}$. The synthesized data $G(\mathbf{z})$
should follow as closely as possible the real data distribution $P_{r}$.
Simultaneously, the discriminator is given the generated sample
$\tilde{\mathbf{x}}$ and the real sample $\mathbf{x}$ so it outputs a scalar
measure of how close both distributions are. The training and the components
in this phase are illustrated in Figure 3.

Figure 3: The schema of phase 1 of f-AnoGAN training. The Generator takes a
vector $\mathbf{z}$ and tries to generate an image that follows the same
distribution of the real data. Then, the Discriminator measures the difference
between the generated data distribution and the real data distribution. This
schema was inspired by the images from Schlegl et al. (2019).

After the first phase of training, 1) a latent space that represents the
variability of the normal data, 2) a generator that can map samples from this
latent space to image space, and 3) a discriminator that can detect samples
that do not follow the normal data distribution are obtained.

However, at this phase, there is no network component that can perform the
inverse mapping, i.e., from image space to latent space. The next phase will
focus on learning this mapping.

####  3.1.2 Phase 2 - Encoder Training

In the second training phase, illustrated in Figure 4, the objective is to
make the encoder learn to map a real image to the latent space such that the
generator can map it back to the image space. During this phase, both the
generator’s and the discriminator’s weights remain unaltered. This network
configuration is denoted as $izi$ in (Schlegl et al., 2019). In this case, the
encoder is optimized by minimizing the Mean Square Error (MSE) with respect to
the difference between the original image $\mathbf{x}$ and the reconstructed
one $G(E(\mathbf{x}))$. Additionally, the reconstruction error from the $izi$
architecture loss is extended by including feature residuals from an
intermediate layer in the discriminator, yielding the $izi_{f}$ architecture.
By taking into account these residuals in the feature space, the
reconstruction is improved (Schlegl et al., 2019). The loss function of
$izi_{f}$ is defined by Equation 2:

$L_{izi_{f}}=\frac{1}{n}\lVert\mathbf{x}-G(E(\mathbf{x}))\rVert_{2}+\frac{k}{n_{d}}\lVert
f(\mathbf{x})-f(G(E(\mathbf{x})))\rVert_{2},$ (2)

where $f(\cdot)$ corresponds to the discriminator’s intermediate layer
features, $n_{d}$ is the dimensionality of the intermediate feature
representation, and $k$ is a weighting factor.

Figure 4: The schema of phase 2 of f-AnoGAN training. In the second phase, the
Generator and Discriminator are kept unaltered while an Encoder is added and
trained to learn to encode the images to the latent space so the Generator can
reconstructed them back. This schema was inspired by the images from Schlegl
et al. (2019).

####  3.1.3 Anomaly Detection

Once the training has finished, all the components are fixed and ready to be
used for anomaly detection. At this point, the images are processed as in the
encoder training. First, the encoder maps the images to the latent space, and
then, the generator maps them back to the image space. Finally, the difference
between the reconstructed and the original image defined in Equation 3 is used
for anomaly detection.

$A(\mathbf{x})=A_{R}(\mathbf{x})+k\cdot A_{D}(\mathbf{x}),$ (3)

where
$A_{R}(\mathbf{x})=\frac{1}{n}\lVert\mathbf{x}-G(E(\mathbf{x}))\rVert_{2}$,
$A_{D}(\mathbf{x})=\frac{1}{n_{d}}\lVert
f(\mathbf{x})-f(G(E(\mathbf{x})))\rVert_{2}$ and $k$ is a weighting factor
from Equation 2.

Only defect-free cell samples have been used for training, therefore the
network will have only learned to reconstruct normal samples. In the case of
defect-free samples, the network outputs an image similar to the input image,
thus there is not much deviation when subtracting one image from the other.
Instead, when processing a defective cell, the output is a defect-free version
of the input sample. As a consequence, the deviation between the original and
reconstructed images can be used to detect anomalous parts. This behavior is
shown in Figure 5.

Figure 5: Example anomaly detection with f-AnoGAN. In a) the final structure
of the network used for anomaly detection, and in b) some example results
obtained when the network process a defect-free cell and a defective cell.
This schema was inspired by the images from Schlegl et al. (2019).

The absolute value of the pixel-wise difference between the original and the
reconstructed image, $\lvert\mathbf{x}-G(E(\mathbf{x}))\rvert$, is used for
pixel-wise anomaly detection. By applying a threshold $c$, defined in Equation
4, to the residuals image obtained from
$\lvert\mathbf{x}-G(E(\mathbf{x}))\rvert$, the binary image
$\mathbf{y}\in\\{0,1\\}$ is obtained.

$\mathbf{y}=\begin{cases}1,&\lvert\mathbf{x}-G(E(\mathbf{x}))\rvert\geq c\\\
0,&otherwise.\end{cases}$ (4)

This binary image can be considered as a pixel-wise annotation of defective
samples as described in Section 3.2.

In this work, two modifications have been made to the original f-AnoGAN
network to adapt it for anomaly detection in photovoltaic cell manufacturing.

With f-AnoGAN, the images are processed in patches of size 64x64 pixels, which
requires multiple executions of the network, increasing the time to process an
entire cell. As a consequence, the network does not meet the industrial
production cycle time (under half a second per cell). In order to reduce the
inspection time, the encoder input and the generator output layers’ dimension
was increased. Thus, whole cell images will be processed in a single pass,
reducing processing time drastically with respect to the original sliding
window approach.

In addition, the training scheme was also modified. In f-AnoGAN, the generator
is frozen during the second training phase in Section 3.1.2, thus, only
encoder weights are modified. This can limit the network capability in terms
of reconstructing the input image. In order to maintain a stable training
without restricting the reconstruction capability, the generator is also
trained at a certain number of the encoder training iterations with a lower
learning rate, while keeping the discriminator unaltered. By training the
generator, the reconstruction of defect-free samples will improve. Therefore,
the deviation between the original and the reconstructed images of normal data
will be reduced. Consequently, both the anomaly score and the pixel
differences will be lower for defect-free samples, but higher for defective
ones, thus the model’s detection rate will improve.

###  3.2 Supervised Model for Defect Segmentation

In anomaly detection, the model is taught to find everything that is not
considered normal. In supervised training, the model is instead trained with
labels to search for specific defective patterns in the data, which usually
yields more precise models for defect detection. Using the anomaly detection
approach as an automatic labeling method, one may benefit from the precision
of supervised learning models avoiding the time-consuming, and not always
trivial, pixel-level labeling task, thereby considerably reducing the effort
dedicated to the setup of a new inspection system.

This way, in the first stage of the inspection system development, where lots
of defect-free cell samples and few defective cell samples are available, an
initial inspection model can be obtained using anomaly detection. Then, as
defective cells arise, the trained anomaly model will process the samples and
output pixel-wise annotations avoiding the time-consuming data annotation
task. After some time, when there are enough annotated defective cell samples,
a model will be trained in a supervised manner to search for specific features
in the images as in our previous works (Balzategui et al., 2019, 2020),
obtaining more accurate models.

For the supervised training, as in our previous work (Balzategui et al.,
2020), U-net (Ronneberger et al., 2015), an end-to-end trainable Fully
Convolutional Neural Network (FCN), was used. This network has been shown to
work well on biomedical image segmentation with low amounts of data. The
network follows an encoder-decoder shape where, after successive downsampling
and then upsampling steps, features from the images are extracted to finally
output a segmentation map of the same size of the input. Additionally, skip
connections connect blocks in the encoder and decoder parts helping to recover
fine-grained details lost during the downsampling and improving the final
results.

##  4 Experimental setup

To validate the proposed methodology, several experiments were carried out
regarding unsupervised training and supervised training. In this section, the
characteristics of the employed industrial dataset, the metrics used to
evaluate the performance of the models, and the hardware and software
specifications will be described

###  4.1 Dataset

The employed dataset is composed of Electroluminescence images of 1873
monocrystalline solar cells extracted from 31 panels. The images were provided
by Mondragon Assembly S. Coop. and were taken at the quality inspection stage
during the assembly of the solar panels. The cells have a size of 15x15 cm and
the images have an average resolution of 840x840 pixels.

The dataset is composed of 1498 images of cells considered defect-free by the
company and 375 defective cells containing cracks, microcracks, and finger
interruptions. The distribution of the dataset is shown in Table 4.1. Each
defective sample has its manually labeled pixel-wise binary annotation
$\\{0,1\\}$ as shown in Figure 6.

[H] Dataset sample distribution.

|  | Total
---|---|---
Defect-free | 1498
Defective | 375
Crack | 18
Microcrack | 240
Finger interruptions | 117

Figure 6: Defective monocrystalline samples with their pixel-level
annotations.

###  4.2 Metrics

The results from the experiments were quantitatively and qualitatively
measured. For the defect detection performance assessment in the unsupervised
part experiments, the anomaly score from Equation 3 was used to construct the
Receiver Operating Characteristic (ROC) curves and to calculate the Area Under
Curve (AUC). The results were analyzed taking into account all the defects as
if they were from the same class, and also considering each defect type
separately.

Note that there is a great unbalance between samples from each defective class
and defect-free samples as shown in Table 4.1. So, to avoid misleading
conclusions when interpreting the ROC curves, the data was balanced by first
taking all defective samples of the class that was being analyzed, and then,
randomly selecting the same amount of defect-free samples.

Then, the results were binarized to calculate Precision, Recall, Specificity,
and F1-score metrics, defined in Equations 5, 6, 7, and 8 respectively.

$Precision=\frac{TP}{TP+FP}$ (5) $Recall=\frac{TP}{TP+FN}$ (6)
$Specificity=\frac{TN}{TN+FP}$ (7) $F1\\_score=2\cdot\frac{Precision\cdot
Recall}{Precision+Recall}$ (8)

where TP stands for True Positive, TN for True Negative, FN for False
Negative, and FP for False Positive. In this work, defective samples belong to
the Positive class and defect-free samples to the Negative class.

As there were few defective samples left for testing the supervised part, the
performance of the models was mainly evaluated at a qualitative level,
although quantitative analysis has also been reported at image-level. The
Recall, Precision, and Specificity metrics have been applied considering the
cells as defective if they contained more than 20 defective pixels.

###  4.3 Hardware and software

Two Nvidia GeForce RTX 2080 GPUs were used in the experiments. The models’
training in the unsupervised part required both GPUs, while the anomaly
detection and supervised training only required one GPU. For all f-AnoGAN
based models, the publicly available code
111https://github.com/tSchlegl/f-AnoGAN was used, which is written in
Python2.7 and employs Tensorflow 1.2 and CUDA 8. Instead, in the Convolutional
Deep Autoencoder and supervised U-net training, Python 3.6, Tensorflow 1.14,
and CUDA 10 were used.

##  5 Experiments

In this section, the experiments and the results from each part (unsupervised
and supervised) are going to be described.

###  5.1 Unsupervised Model for Anomaly Detection

In this part, the impact of the proposed network modifications with regards to
the processing time and defect detection rates is evaluated. First, the
technical details about the experiments are going to be explained, and then,
the results obtained from the experiments will be analyzed and compared.

####  5.1.1 Experimental Design

First, the network in (Schlegl et al., 2019) was applied in our dataset to
ensure its applicability in this specific industrial context. Then, the
modifications regarding the input size and training scheme were incorporated.
These two models are referred to as f-AnoGAN-64 for the original network
configuration, and f-AnoGAN-256 for the model with the modifications.

The hyperparameters in both models were all kept the same as in (Schlegl et
al., 2019): The $\mathbf{z}$ vector was sampled from a Normal distribution and
had a size of 128, the value of $\lambda$ parameter for the gradient penalty
was 10, and the value for the weighting factor $k$ in Equation 2 was set to 1.
The optimization algorithm for the first training phase was Adam (Kignma and
Ba, 2014) and for the second RMSprop (Hinton et al., 2012). For both models,
all the images were rescaled to a range [-1,1] as in the original work
(Schlegl et al., 2019). In this way, the pixels in the images will match the
range of the Generator output layer activation function (i.e., tanh), and it
will help the network have a stable training (Chintala et al., 2016). The only
hyperparameter that was modified was the batch size for f-AnoGAN256 training
in order to fit the model in memory. This was due to the increase in trainable
parameters resulting from the modification of the architecture. The training
took a different number of iterations depending on the phase. Both models
required 40k iterations in the first phase and 70k iterations in the second
phase to converge.

In addition to the mentioned models, two Convolutional Deep Autoencoders were
also trained. These models were used to establish a base with which the
results from the previous two models could be compared. In addition, the
results from these two base models served to check if a more simple network
architecture could be enough to obtain high defect detection rates in this
specific context. Following the two approaches from previous models, one
Autoencoder was trained to process the images in patches, and the other
Autoencoder was trained to process the images in an image-wise setup. These
models will be referred to as AE-64 and AE-256.

Regarding the architectures, both networks are composed of an encoder and a
decoder with several convolutional layers. In the case of AE-64, the encoder
has two convolutional layers with 64-32 filter distribution, followed by 4
Fully Connected layers of 128 units each, and finally a decoder with the
inverted shape of the encoder part. In the case of AE-256, the architecture is
two convolutional layers deeper than the AE-64 such that the output dimension
before the Fully Connected layers is the same. The filter distribution is
8-16-32-64. After each Fully Connected layer, a dropout layer with a drop rate
of 0.25 was set. Both networks were optimized with the MSE loss function and
Adam as the optimization algorithm. The AE-64 model training took about 30k
iterations with a batch size of 32, and the AE-256 model training took about
6k iterations with a batch size of 8.

In the experiments in this part, only defect-free cell images were required
for training. The defect-free samples in Table 4.1 were separated into the
train, validation, and test sets. In addition, the test set also contained 375
defective samples. The dataset distribution used in this part is illustrated
in Table 5.1.1.

[H] Dataset sample distribution for unsupervised part experiments. Train Val
Test Total Defect-free 750 373 375 1498 Defective - - 375 375 Crack - - 18 -
Microcrack - - 240 - Finger inter. - - 117 -

f-AnoGAN-64 and AE-64 were designed to process the images patches-wise. For
these cases, each image was split into 256 patches using a sliding window. The
final train, validation, and test sets were composed of 192000, 95488, and
192000 images, respectively. For the other networks, the images were resized
to the network input size (i.e., 256x256 pixel resolution).

To compare the performance between the models, the results from the patch-
based models were post-processed. While in the image-wise models it was enough
to apply a single threshold so as to classify a sample as defective or non-
defective, in patch-based models the errors of all patches belonging to the
cell must be taken into account. So, in the latter, the same threshold was
applied to every patch and if a single patch was evaluated as defective, the
entire cell was also evaluated as defective.

####  5.1.2 Results

Regarding the image-level results, the network modifications had a positive
impact on the results. f-AnoGAN-256 was able to detect more defects (higher
Recall values) than the original f-AnoGAN-64 without incurring in more False
Positive cases (higher Precision and Specificity values). This is particularly
visible in Table 5.1.2 for the case of the finger interruption and microcrack
defect classes where all the metrics improved over 10 points. This improvement
can also be appreciated in the ROC curves and the AUC values in Figure 7 where
the curve reflecting the performance of f-AnoGAN-256 appears closer to the top
left corner that represents the perfect classifier, and the AUC value that
changed accordingly.

Figure 7: ROC curves from the different models in the unsupervised part
experiments. a) Results considering all defects as they belong to the same
class, b) results with defective samples with cracks, c) results with
defective samples with microcracks, d) results with defective samples with
finger interruptions.

[!ht] The results of anomaly detection at the image-level. Precision tells how
accurate the classifier is when classifying a sample as defective. Recall
tells how many samples have been correctly classified as defective from all
defective samples. Specificity describes how many defect-free samples have
been correctly classified as defect-free samples. The F1-score is the harmonic
mean of the Precision and Recall. In all metrics, the higher the value, the
better the classifier is.

| Model | AUC | Precision | Recall | Specificity | f1-score
---|---|---|---|---|---|---
All test samples |  |  |  |  |  |
| f-AnoGAN-64 | 66 | 61.3 | 62.8 | 61 | 62
| f-AnoGAN-256 | 81.5 | 75 | 78 | 75 | 77
| AE-64 | 72 | 65.6 | 64 | 68 | 65
| AE-256 | 73 | 68.4 | 58 | 72 | 63
Cracks |  |  |  |  |  |
| f-AnoGAN-64 | 99 | 66.7 | 100 | 50 | 80
| f-AnoGAN-256 | 100 | 95 | 100 | 94 | 97
| AE-64 | 98 | 78 | 100 | 100 | 87.7
| AE-256 | 100 | 95 | 100 | 94 | 97
Micro |  |  |  |  |  |
| f-AnoGAN-64 | 63 | 58.7 | 59 | 59 | 58.9
| f-AnoGAN-256 | 78 | 73 | 73 | 74 | 73
| AE-64 | 71 | 66.5 | 63.7 | 67.9 | 65
| AE-256 | 70 | 66 | 53 | 72 | 59
Finger int. |  |  |  |  |  |
| f-AnoGAN-64 | 70 | 66 | 64.9 | 66.7 | 65.5
| f-AnoGAN-256 | 86 | 78 | 85 | 75 | 81
| AE-64 | 69.7 | 61.9 | 59.8 | 63 | 60.8
| AE-256 | 75 | 69 | 63 | 71 | 66

If the results of f-AnoGAN models are compared with the ones from the
Autoencoders, it is further underlined that the incorporated of network
modifications brought an improvement in defect detection rates. Setting aside
the case of finger interruptions, f-AnoGAN-64 obtained worse detection rates
than its Autoencoder counterpart (i.e., AE-64) and also AE-256. But, when the
proposed changes were incorporated, the obtained results surpassed the ones
from the Autoencoders for all the classes and all the metrics, which means
higher True Positives cases and lower False Positive cases for all defect
classes.

The results in the ROC curves in Figure 7 and the metrics in Table 5.1.2 show
that all models could detect all samples with cracks, but they could not
detect all samples with microcracks and finger interruptions. This is caused
by the fact that cracks are defects that cover a larger area of the cells than
finger interruptions or microcracks, and therefore have more defective pixels
that result in a higher anomaly score. The same happens in the case of finger
interruptions and microcracks. The first appear in groups of three or more,
whereas the latter appears isolated. Because of this, the sum of defective
pixels in samples with finger interruptions contributes to higher anomaly
scores resulting in higher detection rates.

With respect to the defect location results, Figure 8 shows that all the
models were able to properly locate the different defect classes.
Nevertheless, the segmentation results were more refined in f-AnoGAN-256 and
AE-256 models. Although the patch-based models were able to point out the
presence of defective areas, the borders and shape of the predictions were not
as accurate as the ones from the image-wise models.

Also, the patch-based models have more False Positive cases. An example of
this behavior is the sample from the second row, where the buses in the cell
were mistakenly detected as defects. The same happened on the defect-free
samples, where patch-based models classified defect-free areas as defective
(e.g. samples seven and eight), whereas the image-wise models obtained clean
predictions. Although not illustrated in Figure 8, this behavior was shared
across several other samples in the test set.

Figure 8: Defect localization results from each model.

In addition to the classes in quantitative analysis, the models were also
executed on samples that contained two other defects that were put aside as
they were few available samples for a proper analysis. These defects classes
were breaks and bad soldering present in Figure 8 in the bottom-right of the
first row sample and on the right in the third row sample, respectively. In
the case of the break, image-wise models were able to output a relatively
precise segmentation. The AE-64 model results indicated the defect location,
however, they did not have much precision. Instead, in the case of
f-AnoGAN-64, it can be noticed that at the defect location there is a certain
anomalous pattern but very vaguely segmented. Regarding the bad soldering,
f-AnoGAN-256 was the only model that presented a reasonable segmentation
result.

Regarding the processing time, Table 5.1.2 shows how the architecture
modification made f-AnoGAN able to reduce the time required to process each
cell. While patch-based models required more than half a second to process the
cells (maximum stipulated time per cell), f-AnoGAN-256 and AE-256 required
only 0.05 and 0.02 seconds respectively to process each cell.

[H] Time required to process a cell for each model. Model Time per patch Time
per image f-AnoGAN-64 0.02s 5.12s f-AnoGAN-256 - 0.05s AE-64 0.012s 3.07s
AE-256 - 0.02s

Overall, it can be concluded that the proposed modifications made the network
obtain higher defect detection rates and also reduced the processing time
meeting the established time for industrial inspection.

The results in this part have shown that the anomaly model can yield
relatively high detection rates and defect locations. Among the trained
models, f-AnoGAN-256 has shown the highest detection rates, short enough
processing time for industrial inspection, and precise pixel-level results.
Taking into account that the defect location results in Figure 8 were close
enough to what human experts annotated, this model was selected as the
automatic annotator model to obtain automatic labels for the next stage in the
methodology (i.e., supervised stage).

###  5.2 Supervised Model for Defect Segmentation

The goal of this experiment was to prove the feasibility of using the anomaly
detection approach as an automatic labeling method. To this end, the
segmentation results from a model trained on automatic labels obtained from
the previous stage and a model trained with labels created by experts were
compared. Some samples of the automatic labeling used for training are shown
side by side with their corresponding manually labeled counterpart in Figure
9.

Figure 9: Manual and automatic labeling for different samples. The automatic
labeling kept the segmentation of the labeled defects, but at the same time
introduced some additional areas. This is especially noticeable in the samples
from the first row, where the manual labeling only considered the defect
itself, but the automatic labeling also considered the darker areas created by
the effect of the defect.

In the same way as in the previous section, first the technical details of the
experiments are going to be described, and then, the results are going to be
analyzed.

####  5.2.1 Experimental Design

For both manual and automatic labeling models, U-net was used. The network
configuration was kept as in (Ronneberger et al., 2015), which is composed of
9 convolutional layers blocks: 4 in the encoder, 4 in the decoder, and a
middle block that acts as a bottleneck. The blocks of the encoder are composed
of two convolutional layers followed by a Max-Pooling and a Batch
Normalization layer. At these blocks, features are extracted from the data,
and the output is downsampled to half of the input size. Instead, the blocks
of the decoder are configured in an inverse way, substituting the pooling
operators by upsampling operators which consecutively upscale the extracted
low-resolution features into a final segmentation map of the input size. The
networks were trained to minimize the dice loss in Equation 9 that measures
the difference between the output and the ground truth. The training took 1000
iterations using Adam as the optimization algorithm with a learning rate of
1-e4.

$L_{dice}=-\frac{2\cdot\sum_{j}p_{j}g_{j}}{\sum_{j}p_{j}+g_{j}}$ (9)

where $p\in[0,1]$ is the network output, and $g\in\\{0,1\\}$ is the ground
truth.

Unlike the experiments in the previous section, in the experiments in the
supervised part, just defective samples were used for training. So the
defective samples were split into train, validation and test sets following
the next distribution: 300 for training and validation, and the remaining 75
for testing (4 crack images, 48 microcrack images, and 23 finger interruptions
images). In addition, the defect-free samples used in the evaluation in the
unsupervised part experiments were also employed to evaluate the models in
this section. The final dataset for this part is shown in Table 5.2.1.

[H] Dataset sample distribution for the supervised part experiment. Train Val
Test Total Defect-free - - 375 375 Defective 232 68 75 375 Crack 14 4 4 18
Microcrack 152 50 48 240 Finger inter. 70 24 23 117

####  5.2.2 Results

After training U-net separately with the two versions of the dataset (manual
and automatic), the 75 defective and 375 defect-free samples were employed to
compute the metrics and evaluate the performance of the models. The results
are shown in Table 5.2.2. In addition to the U-net based models, the model
from the previous section (i.e., f-AnoGAN-256) was also executed on the same
test to validate that the supervised training with automatic labels improved
the detection rate compared with the anomaly model.

[H] Image-level results from U-net trained on manually created labels, U-net
trained on automatically created labels, and also, the results from the
anomaly model used for annotation. Model Recall Precision Specificity U-net w/
manual labels 80 95 99 U-net w/ auto. labels 93 81 95 f-AnoGAN-256 79 73 73

As shown in Table 5.2.2, both supervised models yielded higher detection rates
than the anomaly detection models without incurring more False Positive cases.
If supervised models are compared with each other, U-net trained with
automatic labels was able to detect more defective samples (Recall of 93%)
than U-net trained on manual labels (Recall of 80%).

However, automatic labels made U-net have more False Positive cases, making
the Precision and Specificity decrease from 95 to 81 and 99 to 95,
respectively. However, using automatic labels resulted in more False Positive
cases, making the Precision and Specificity values decrease from 95 to 81 and
99 to 95, respectively. Note that the increase of False Positives has a larger
impact on the Precision because of the imbalance of defective and defect-free
samples in the test set.

As for the segmentation results illustrated in Figure 10, it can be seen that
the defects were more thoroughly marked with U-net trained with automatic
labels than with manual labels. The second and third samples in Figure 10 are
an example of this. However, impurities in the cells that were not taken into
account during manual labeling were also detected as defects (e.g black spots
under the crack in the second sample or around the finger in the third
sample). This caused certain defect-free samples with such impurities to be
classified as defective cells, which increased the number of False Positives
resulting in an impact on the image-level metrics. Nevertheless, few defect-
free samples present these False Positive cases.

Figure 10: Results from supervised training models and from the anomaly model
used for annotation for comparison. Samples 1, 2, 3, and 4 are defective
samples with defects contemplated at training and metrics evaluation, and
samples 5 and 6 are defect-free samples. Samples 7 and 8 contain defects that
were not considered during training and testing, but illustrate the effect of
the automatic labels in the segmentation results. Label refers to the
annotation made by experts, manual refers to the segmentation results obtained
from the supervised segmentation model trained with manually labeled samples,
and automatic refers to the segmentation results obtained from the supervised
segmentation model trained with automatically labeled samples as ground truth.

In addition, even not considered during training and when the metrics were
calculated, the automatic labels enable U-net to segment other kinds of
defects. For example, in the seventh sample in Figure 10, both models were
able to detect the microcrack, but the break at the bottom right was only
detected by the models trained with automatic labels. The same happened in the
last row sample where the bad soldering was not segmented when using manual
labels.

Concerning annotations, it seems that annotating dark areas around the defects
has a positive effect on the models’ pixel-level results. For example, in the
first sample in Figure 9 the manual label does not cover the areas around the
defect, whereas with automatic labeling these areas are annotated as
defective. The experts did not consider these areas during the labeling as
they are not part of the defect, but a consequence of the defect itself.
However, these dark areas will not appear in defect-free cells. Because of
that, the anomaly detection network annotated them as defective areas. When
considering these dark areas as part of the labels, the network trained on
automatic labels recognized dark areas around defects as defective.

Consequently, as shown in the eighth sample in Figure 10, even if the class
was not included in the training, the dark area in the right that belongs to a
bad soldering defect was segmented when using automatic labels and not when
using manual annotations. The same happened with the break in the first
sample. Moreover, the segmentation of other defects, for example, the finger
interruption in the third sample and the microcrack in the fourth, have been
more accurately segmented. Nonetheless, by annotating dark areas as defective,
certain impurities that were not considered as defects were also segmented.
So, including dark areas as part of the labels was beneficial for pixel-level
results and to detect more defective samples, even if it made some new False
Positive cases arise.

##  6 Conclusions

In this work, an anomaly detection-based methodology has been proposed for the
development of a quality inspection system of monocrystalline solar cells.
With anomaly detection, only defect-free samples are required to obtain a
model for inspection which can detect and locate defects in the cells. This
feature is key for the development of a PV module inspection system as it
permits companies to have an inspection model from the very beginning stage of
a new production line setup, without waiting for defective data to appear.
Furthermore, it also avoids expending time in the annotation of the samples
which saves a lot of effort concerning data preparation when constructing an
inspection system.

In order to apply anomaly detection for industrial inspection, a GAN proposed
to detect and locate anomalies in the medical domain has been adapted. The
adaptations have been two-fold: First, the architecture has been modified such
that the images can be processed in a single step instead of processing them
by patches. In this way, less time is required to process a cell; therefore
the established inspection time mark of less than half a second per cell has
been met. And second, the training scheme has also been modified. This
modification has resulted in an improvement in the defect detection
capabilities of the model.

In addition, it has been experimentally demonstrated that the results from the
anomaly detection are potential pixel-wise labels that can be used for
supervised training. In the experiment, the defect localization results
obtained from a model trained with labels generated by experts and a model
trained with automatically generated labels have been compared. The comparison
has shown that using automatic labels is comparable to using manual
annotations; thus, it is feasible to use anomaly detection as an automatic
annotator which saves time and resources.

The proposed methodology is rooted in the use of GANs, which are known for
their difficult training process. In industry, most of the quality inspection
cases are related to homogeneous parts that can alleviate to some extent the
training instability that the network can face. However, less homogeneous
parts might need some modifications in the training or in the network, in
order to learn the data distribution and obtain high quality image
reconstruction for the anomaly detection.

Lastly, although the experimental results already demonstrate the feasibility
of the proposed method for inspection of solar cells, we plan to explore
different architectures and parameters for optimizing the methodology in
future works. It would also be interesting to test it in other industrial
contexts.

Conceptualization, J.B., L.E. and D.M. ; Methodology, J.B. and L.E.;
Validation, J.B.; Formal Analysis, J.B.; Investigation, J.B.; Data Curation,
J.B.; Software, J.B.; Writing – Original Draft Preparation, J.B.; Writing –
Review & Editing, J.B., L.E. and D.M.; Visualization, J.B.; Supervision, L.E.;
Funding Acquisition, L.E.

###### Acknowledgements.

This research was developed as part of the Elkartek project ENSOL 2
(KK-2020/00077) supported by the Basque Government. Also, we will like to
thank Jean Philippe Agerre and Jon Altube from Mondragon Assembly S. Coop. for
providing us with an industrial dataset. The authors declare that they have no
known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper. References yes

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# AdS Black Hole with Cylindrical Symmetry

Mehdi Sadeghi , Ramin Anvari Asl , Mohammad Shamseh
Department of Physics, School of Sciences,
Ayatollah Boroujerdi University, Boroujerd, Iran
Corresponding author: Email:
mehdi.sadeghi@abru.ac.irr.anvari11<EMAIL_ADDRESS>
###### Abstract

In this paper, we consider Einstein-Hilbert gravity in the presence of
cosmological constant with cylindrical symmetry to introduce the black hole
solution of this model. Here, we solve the Einstein’s vacuum field equation,
and then we calculate the appropriate metric for this problem.

Keywords: Black hole; Einstein’s field equations; Cylindrical symmetry

##  1 Introduction

General relativity is a theory of gravity formulated by Albert Einstein in
1915. In this theory, gravity is massless and spin-2 [1]-[3]. General
relativity is a gauge theory and it is invariant under diffeomorphism. The
presence of matter induces gravity by curving the space-time in general
relativity. Solving Einstein’s field equations in the different situations is
an interesting subject. For example, massive gravity[4], higher derivative
gravity[5]-[6] and Rastall gravity[7], as modifications of gravity, were
introduced for explanation of dark matter and dark energy.
Levi-Civita [8] and Weyl [9] studied the cylindrical symmetry in general
relativity for the first time and its black hole solution in an anti-de Sitter
(AdS) background has been studied in [10]-[11]. Cylindrical symmetry is the
most simple capable symmetry to produce gravitational waves.
In this paper, we have tried to solve Einstein’s field equations for a mass
line in AdS space-time with cylindrical symmetry.
The resulted state in a gravitational collapse of astrophysical massive
objects is called the black hole. In other words, a black hole is a solution
of Einstein’s equations with an event horizon. Event horizon is a null hyper-
surface and one-dimensional. There is not a causal relation between the two
sides of the event horizon.
Cosmic strings [12], the role of conical singularities, space-time topological
defects, gravitational waves and the observable acceleration of high energy
jets in galaxies [13] are the concepts that could be explained better by the
symmetric aspects of cylindrical space-time. Therefore, it motivates us to
study the model with axial symmetry.

##  2 Einstein AdS Black Hole Solution with Cylindrical Symmetry

The action for this model is written as follows,

$S=\frac{1}{2\kappa}\int{d^{4}x\sqrt{-g}(R-2\Lambda)},$ (1)

where, the first term, $R$, is the Ricci scalar and it represents the kinetic
term of Einstein gravity. The second term ,$\Lambda$, is the cosmological
constant expressed as,

$\Lambda=-\frac{3}{l^{2}},$ (2)

where $l$ is the radius of AdS space-time. Negative sign in $\Lambda$
guarantees that we are in AdS space-time.
We choose the following metric as an ansatz for the line element,

$ds^{2}=e^{\alpha(r)}dt^{2}-e^{\beta(r)}dr^{2}-r^{2}e^{\gamma(r)}d\varphi^{2}-e^{\gamma(r)}dz^{2}.$
(3)

The equations of motion can be derived by variation of the action (1) with
respect to the metric,

${G_{\mu\nu}}+\Lambda{g_{\mu\nu}}=0,$ (4)

where $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ is the Einstein tensor.
We can write Eq.(4) as the following,

${R_{\mu\nu}}=\Lambda{g_{\mu\nu}}.$ (5)

The (t,t) component of Eq.(5) is written as the following by using the metric
ansatz (3),

$\frac{1}{4r}\Bigg{(}2r\alpha^{\prime}(r)\gamma^{\prime}(r)-r\alpha^{\prime}(r)\beta^{\prime}(r)+r\alpha^{\prime}(r)^{2}+2r\alpha^{\prime\prime}(r)+2\alpha^{\prime}(r)\Bigg{)}=-\Lambda\,e^{\beta(r)}.$
(6)

The (r,r) component of Eq.(5) is given by,

$\displaystyle\frac{1}{4r}\Bigg{(}-r\alpha^{\prime}(r)\beta^{\prime}(r)+r\alpha^{\prime}(r)^{2}+2r\alpha^{\prime\prime}(r)$
$\displaystyle-2r\beta^{\prime}(r)\gamma^{\prime}(r)-2\beta^{\prime}(r)+2r\gamma^{\prime}(r)^{2}+4\gamma^{\prime}(r)+4r\gamma^{\prime\prime}(r)\Bigg{)}=-\Lambda\,e^{\beta(r)}.$
(7)

The $(\varphi,\varphi)$ component of Eq.(5) is as,

$\displaystyle\frac{1}{4r}\Bigg{(}r\alpha^{\prime}(r)\gamma^{\prime}(r)+2\alpha^{\prime}(r)-r\beta^{\prime}(r)\gamma^{\prime}(r)$
$\displaystyle-2\beta^{\prime}(r)+2r\gamma^{\prime}(r)^{2}+6\gamma^{\prime}(r)+2r\gamma^{\prime\prime}(r)\Bigg{)}=-\Lambda\,e^{\beta(r)},$
(8)

and the $(z,z)$ component of Eq.(5) becomes,

$\frac{1}{4r}\Bigg{(}r\alpha^{\prime}(r)\gamma^{\prime}(r)-r\beta^{\prime}(r)\gamma^{\prime}(r)+2r\gamma^{\prime}(r)^{2}+2\gamma^{\prime}(r)+2r\gamma^{\prime\prime}(r)\Bigg{)}=-\Lambda\,e^{\beta(r)}.$
(9)

By subtracting Eq.(8) from Eq.(9), we will have,

$2\gamma^{\prime}-\beta^{\prime}+\alpha^{\prime}=0.$ (10)

By plugging Eq.(10) in Eq.(6), Eq.(2) and Eq.(9) we obtain,

$\displaystyle(\alpha^{\prime\prime}r+\alpha^{\prime})=-2\Lambda\,r\,e^{\beta(r)},$
(11)
$\displaystyle(\gamma^{\prime\prime}r+\gamma^{\prime})=-2\Lambda\,r\,e^{\beta(r)},$
(12)
$\displaystyle\gamma^{\prime\prime}r+3\gamma^{\prime}-\beta^{\prime}+\alpha^{\prime}=-2\Lambda\,r\,e^{\beta(r)},$
(13)

Eq.(13) is the same as Eq.(12) by using Eq.(10).
By subtracting Eq.(11) from Eq.(12) we have,

$(\alpha^{\prime\prime}-\gamma^{\prime})r+(\alpha^{\prime}-\gamma^{\prime})=0.$
(14)

Eq.(14) can be written as,

$\alpha^{\prime}-\gamma^{\prime}=\frac{3c}{r},$ (15)

by considering Eq.(10) and Eq.(15) we get,

$\displaystyle\gamma^{\prime}=\frac{1}{3}\,\beta^{\prime}-\frac{c}{r},$ (16)
$\displaystyle\alpha^{\prime}=\frac{1}{3}\,\beta^{\prime}+\frac{2c}{r}.$ (17)

By substituting Eq.(10) in Eq.(7) we will have,

$\beta^{\prime\prime}\,r+\beta^{\prime}=-6\Lambda\,r\,e^{\beta}$ (18)

where, $\beta(r)$ is found as follows,

$\beta(r)=\ln\Bigg{(}-\frac{c_{1}^{2}}{12\Lambda
r^{2}\,{\cos}^{2}(\frac{1}{2}\,c_{1}(\ln
r-c_{2}))}\Bigg{)}=\ln\Bigg{(}-\frac{c_{2}}{12\Lambda
r^{2}\,\cos^{2}\theta}\Bigg{)}.$ (19)

Here, $c_{1}$ and $c_{2}$ are constant and $\theta$ is given by,

$\theta=\frac{1}{2}\,c_{1}\,(\ln r-c_{2}),$ (20)

$\beta(r)$ should be satisfied in Eq.(2),

$r\beta^{\prime\prime}-\frac{1}{3}r\,{\beta^{\prime}}^{2}-\frac{1}{3}\,\beta^{\prime}+\frac{3c^{2}-2c}{r}=-2\Lambda\,r\,e^{\beta}.$
(21)

Eq.(21) can be simplified as,

$\displaystyle\Bigg{(}\frac{-c_{1}}{r}\,\tan\theta+\frac{2}{r}+\frac{c_{1}^{2}}{2r}\,(1+{\tan}^{2}\theta)\Bigg{)}-\frac{1}{3}r(\frac{c_{1}^{2}}{r^{2}}\,\tan^{2}\theta+\frac{4}{r^{2}}-\frac{4c_{1}}{r^{2}}\,\tan\theta)$
$\displaystyle-\frac{1}{3}(\frac{c_{1}}{r}\,\tan\theta-\frac{2}{r})+\frac{3c^{2}-2c}{r}=-2\Lambda
r(\frac{-c_{1}^{2}}{12\Lambda r^{2}}(1+\tan^{2}\theta)).$ (22)

We conclude the following relation,

$c_{1}^{2}+(3c-1)^{2}+3=0.$ (23)

By integrating of Eq.(16) and Eq.(17) with respects to $r$, we will have,

$\gamma=\ln\Bigg{(}-\frac{\sqrt[3]{c_{1}^{2}}r^{-c}}{\sqrt[3]{12\Lambda
r^{2}\cos^{2}\theta}}\Bigg{)},$ (24)
$\alpha=\ln\Bigg{(}-\frac{r^{2c}\sqrt[3]{c_{1}^{2}}}{\sqrt[3]{12\Lambda
r^{2}\cos^{2}\theta}}\Bigg{)}.$ (25)

Therefore, the metric of a mass line in the AdS space is as follows,

$ds^{2}=\Bigg{(}\frac{-r^{2c_{2}}\sqrt[3]{c_{1}^{2}}}{\sqrt[3]{12\Lambda
r^{2}\,{\cos}^{2}\theta}}\Bigg{)}dt^{2}-\Bigg{(}\frac{-c_{1}^{2}}{12\Lambda
r^{2}\,{\cos}^{2}\theta}\Bigg{)}dr^{2}-\Bigg{(}\frac{-r^{-c_{2}}\sqrt[3]{c_{1}^{2}}}{\sqrt[3]{12\Lambda
r^{2}\,{\cos}^{2}}\theta}\Bigg{)}(r^{2}d\varphi^{2}+dz^{2}),$ (26)

where $c_{1}$ and $c_{2}$ are integral constants and they are related by
Eq.(23).
Now, we want to find these constants. By substituting Eq.(23) into Eq.(20) we
have,

$\displaystyle\theta=\frac{i}{2}\,\sqrt{3+(3c-1)^{2}}\,(\ln r-c_{2})=is,$ (27)
$\displaystyle\cos\theta=\cos(is)=\cosh(s)=\frac{e^{s}+e^{-s}}{2}.$ (28)

$e^{2s}$ is found by Eq.(27) as the following,

$e^{2s}=e^{-\sqrt{3+(3c-1)^{2}}\,c_{2}}\,r^{\sqrt{3+(3c-1)^{2}}}.$ (29)

So the line element is as follows,

$\displaystyle ds^{2}=$ $\displaystyle
r^{2c}\Bigg{(}\frac{3+(3c-1)^{2}}{12\Lambda
r^{2}}(\sqrt[3]{\frac{4}{e^{2s}+e^{-2s}+2}})\Bigg{)}dt^{2}$
$\displaystyle-\Bigg{(}\frac{3+(3c-1)^{2}}{12\Lambda
r^{2}}\frac{4}{e^{2s}+e^{-2s}+2}\Bigg{)}dr^{2}$
$\displaystyle-r^{-c}\Bigg{(}\frac{3+(3c-1)^{2}}{12\Lambda
r^{2}}\frac{4}{e^{2s}+e^{-2s}+2}\Bigg{)}^{\frac{1}{3}}(r^{2}d\varphi^{2}+dz^{2}).$
(30)

If $c_{2}$ is real, the solution does not have an event horizon. Since, we
want to find the black hole solution, we consider $c_{2}$ as a complex number
and this condition guaranties $g^{rr}=0$.

$c_{2}=c_{2}^{\prime}+ic_{2}^{\prime\prime}.$ (31)

By inserting Eq.(31) in Eq.(27), we will have,

$\theta=(c_{2}^{\prime}+a_{1}\ln r)i+a_{2},$ (32)

where,

$\displaystyle a_{1}=\frac{1}{2}\,\sqrt{3+(3c-1)^{2}}\,\ln r,$ (33)
$\displaystyle a_{2}=-\frac{c_{2}^{\prime\prime}}{2}\,\sqrt{3+(3c-1)^{2}}.$
(34)

Therefore, we have the following relations,

$\displaystyle\cos\theta=\cos\Bigg{(}a_{2}+(c_{2}^{\prime}+a_{1}\ln
r)i\Bigg{)}$ $\displaystyle=\cos a_{2}\,\cosh(c_{2}^{\prime}+a_{1}\ln r)-i\sin
a_{2}\,\sinh(c_{2}^{\prime}+a_{1}\ln r),$ (35)
$\displaystyle{\cos}^{2}\theta={\cos}^{2}a_{2}\,{\cosh}^{2}(c_{2}^{\prime}+a_{1}\ln
r)-{\sin}^{2}a_{2}\,{\sinh}^{2}(c_{2}^{\prime}+a_{1}\ln r)$
$\displaystyle-2i\,\cos a_{2}\,\sin a_{2}\,\cosh(c_{2}^{\prime}+a_{1}\ln
r)\,\sinh(c_{1}^{\prime}+a_{1}\ln r).$ (36)

Since the components of metric should be positive, the imaginary part of
${\cos}^{2}\theta$ should be zero. By applying this condition we will have,

$\displaystyle\cos a_{2}=0\qquad\textrm{or}\qquad\sin a_{2}=0.$ (37)

Now, we have two choices:
The first is as follows,

$\cos^{2}\theta=\cosh^{2}(c_{2}^{\prime}+a_{1}\ln r),$ (38)

In this case, we don’t have event horizons since
$\cosh^{2}(c_{2}^{\prime}+a_{1}\ln r)\neq 0$.
The second is as the following,

$\displaystyle\cos^{2}\theta=-\sinh^{2}(c_{2}^{\prime}+a_{1}\ln r),$ (39)

In the above case, we have an event horizon because
$\sinh^{2}(c_{2}^{\prime}+a_{1}\ln r)=0$. So we have,

$r_{+}=e^{-\frac{c_{2}^{\prime}}{a_{1}}}.$ (40)

Finally, the metric of our model is as following,

$\displaystyle ds^{2}=$ $\displaystyle
r^{2c}\Bigg{(}\frac{3+(3c-1)^{2}}{12\Lambda
r^{2}(-{\sinh}^{2}(c_{2}^{\prime}+a_{1}\ln
r))}\Bigg{)}dt^{2}-\Bigg{(}\frac{3+(3c-1)^{2}}{12\Lambda
r^{2}(-{\sinh}^{2}(c_{2}^{\prime}+a_{1}\ln r))}\Bigg{)}dr^{2}$
$\displaystyle-r^{-c}{\Bigg{(}\frac{3+(3c-1)^{2}}{12\Lambda
r^{2}(-{\sinh}^{2}(c_{2}^{\prime}+a_{1}\ln
r))}\Bigg{)}}^{\frac{1}{3}}(r^{2}d\varphi^{2}+dz^{2}).$ (41)

Black hole can be considered as a thermodynamic system since temperature and
entropy are assigned to it. The effect of quantum field on the classical
geometry can be revealed by calculation of thermodynamic quantities [14]-[16].
The temperature [17] is as following,

$\displaystyle
T=\frac{1}{2\pi}\Bigg{[}\frac{1}{\sqrt{-g_{rr}}}\frac{d}{dr}\sqrt{g_{tt}}\Bigg{]}_{r=r_{+}}=\frac{4cc^{\prime}_{2}+2a_{1}(-1+2c)\ln
r_{+}-a_{1}-2c^{\prime}_{2}}{4\pi r_{+}^{1-c}(c_{2}^{\prime}+a_{1}\ln
r_{+})}.$ (42)

To have a finite temperature, we choose $a_{1}=0$. So, by using Eq.(40) event
horizon is located on the infinity.

$T=\frac{r_{+}^{c-1}}{4\pi}(2+4c).$ (43)

The temperature is very large for $c\geq 1$ and the temperature tends to zero
for $c<1$. For $c>\frac{-1}{2}$, we have positive temperature.
The temperature of the black hole is plotted for $c=\frac{4}{3}$ in Fig 1.

Figure 1: $T-r_{+}$ diagram for $c=\frac{4}{3}$.

The Bekenstein–Hawking entropy [18] is as follows,

$\displaystyle S$ $\displaystyle=\frac{A}{4G}=\frac{1}{4G}\int
d^{2}x\sqrt{g}=\frac{V_{2}}{4r_{+}^{c-1}G}{\Bigg{(}\frac{3+(3c-1)^{2}}{-12\Lambda
r_{+}^{2}{\sinh}^{2}(c_{2}^{\prime}+a_{1}\ln r_{+})}\Bigg{)}}^{\frac{1}{3}}.$
(44)

where,

$V_{2}=2\pi lr_{+}^{1-c}{\Bigg{(}\frac{3+(3c-1)^{2}}{-12\Lambda
r_{+}^{2}{\sinh}^{2}(c_{2}^{\prime}+a_{1}\ln r_{+})}\Bigg{)}}^{\frac{1}{3}},$
(45)

by substituting $V_{2}$ in Eq.(44), we have,

$\displaystyle S$ $\displaystyle=\frac{2\pi
l}{4r_{+}^{2c-2}G}{\Bigg{(}\frac{3+(3c-1)^{2}}{-12\Lambda
r_{+}^{2}{\sinh}^{2}(c_{2}^{\prime}+a_{1}\ln r_{+})}\Bigg{)}}^{\frac{2}{3}}.$
(46)

The mass of the black hole is found by Smarr relation[19],

$M=2(TS-PV),$ (47)

the cosmological constant plays as a pressure, $P=-\frac{\Lambda}{8\pi G}$. So
the mass is as the following,

$\displaystyle M$ $\displaystyle=2\pi
l(1+2c)r_{+}^{-c}{\Bigg{(}\frac{3+(3c-1)^{2}}{-12\Lambda
r_{+}^{2}{\sinh}^{2}(c_{2}^{\prime}+a_{1}\ln r_{+})}\Bigg{)}}^{\frac{2}{3}}$
$\displaystyle-\frac{3r_{+}^{1-c}}{2Gl}{\Bigg{(}\frac{3+(3c-1)^{2}}{-12\Lambda
r_{+}^{2}{\sinh}^{2}(c_{2}^{\prime}+a_{1}\ln r_{+})}\Bigg{)}}^{\frac{1}{3}}.$
(48)

To have finite mass, the following relation should be satisfied,

$3+(3c-1)^{2}={\sinh}^{2}(c_{2}^{\prime}+a_{1}\ln r).$ (49)

Therefore, the mass is given by,

$M=\frac{(4c+2)l^{\frac{7}{3}}}{8\sqrt[3]{6}Gr^{c+\frac{1}{3}}}-\frac{\sqrt[3]{3}}{2\
2^{2/3}G\sqrt[3]{l}r^{c-\frac{1}{3}}}.$ (50)

The energy of the black hole is found by using the first law of thermodynamics
[15],[20] as following,

$dE=TdS+PdV.$ (51)

It can be written as,

$dE=T(\frac{\partial S}{\partial r_{+}})dr_{+}+P(\frac{\partial V}{\partial
r_{+}})dr_{+}.$ (52)

Energy of this black hole is found by integrating with respct to $r_{+}$,

$E=\frac{(4c+2)l^{\frac{7}{3}}}{8Gr_{+}^{c+\frac{1}{3}}}+\frac{\sqrt[3]{3}}{4\
2^{\frac{7}{3}}Gl^{\frac{1}{3}}r_{+}^{c-\frac{1}{3}}},$ (53)

The first law of thermodynamics is satisfied. Our solution Eq.(2) is
asymptotically AdS by applying Eq.(49) and considering $c=\frac{4}{3}$.
We will study the Hawking-Page phase transition or the first order phase
transition [21]-[22] in our future work.

##  3 Results and discussion

We solved the Einstein’s equations for a mass line in the 4-dimensional space-
time with negative cosmological constant. The solution of Einstein’s equations
with event horizon is called black hole. In this paper, we introduced the
black hole solution with cylindrical symmetry in AdS space-time. The field
theory dual of this model will be studied by using the fluid-gravity duality
[23]-[26] in our future work.

##  4 Conclusion

In this paper, we studied the static cylindrically symmetric vacuum solutions
of the Einstein equations in AdS space-time. Also, we introduced the
cylindrical black hole or the black string of this model. We obtained the
first law of thermodynamics for this black string by introducing temperature,
Bekenstein-Hawking entropy and pressure. Finally, we mention some future
directions of this model.
One of the key challenges in the study of static cylindrically symmetric
solutions is determining their stability under perturbations. Future works can
focus on analyzing the stability of these solutions and investigating the
conditions under which they are stable.
The study of holographic aspects of this solution is an interesting subject.
The AdS/CFT correspondence is a powerful tool to study the relationship
between gravitational theories and quantum field theories. Using holographic
techniques to study static cylindrical solutions in AdS space-time and
exploring their connection to other areas of physics is an excellent subject.
Since most of the existing literature has focused on cylindrical solutions in
AdS space-time, the extension of cylindrical solutions to other space-times is
an untouched subject.
Another promising direction for future research is the study of cylindrical
solutions in higher-dimensional space-times. This could provide insights into
the behavior of matter and energy in higher-dimensional space-times and
potentially lead to new discoveries in the field of high energy physics.
Therefore, the study of static cylindrically symmetric vacuum solutions of the
Einstein equations in AdS space-time is an active area of research with many
open questions and potential ways for future exploration.

Acknowledgment Authors gratefully thank to Ahmad Moradpouri, Komeil Babaei,
Faramarz Rahmani, Ruben Campos Delgado and the referee of Indian Journal of
Physics for the useful comments and suggestions which definitely help to
improve the manuscript.

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