# Identifying Black Holes Through Space Telescopes and Deep Learning

Yeqi Fang<EMAIL_ADDRESS>Center for Theoretical Physics, College of
Physics, Sichuan University, Chengdu, 610065, China  Wei Hong
<EMAIL_ADDRESS>Department of Astronomy, Beijing Normal University,
Beijing 100875, People’s Republic of China  Jun Tao<EMAIL_ADDRESS>Center
for Theoretical Physics, College of Physics, Sichuan University, Chengdu,
610065, China

###### Abstract

The EHT has captured a series of images of black holes. These images could
provide valuable information about the gravitational environment near the
event horizon. However, accurate detection and parameter estimation for
candidate black holes are necessary. This paper explores the potential for
identifying black holes in the ultraviolet band using space telescopes. We
establish a data pipeline for generating simulated observations and present an
ensemble neural network model for black hole detection and parameter
estimation. The model achieves mean average precision [0.5] values of 0.9176
even when reaching the imaging FWHM ($\theta_{c}$) and maintains the detection
ability until $0.54\theta_{c}$. The parameter estimation is also accurate.
These results indicate that our methodology enables super-resolution
recognition. Moreover, the model successfully detects the shadow of M87* from
background noise and other celestial bodies, and estimates its inclination and
positional angle. Our work demonstrates the feasibility of detecting black
holes in the ultraviolet band and provides a new method for black hole
detection and further parameter estimation.

††preprint: APS/123-QED

##  I INTRODUCTION

The Event Horizon Telescope (EHT) Collaboration released the first shadowed
image of M87* [1, 2, 3, 4, 5] in April 2019 and of Sagittarius A* [6, 7, 8,
9], the black hole at the center of the Milky Way, in May 2022. These images
provide concrete proof of black holes’ existence, which is a key tenet of
general relativity [10].

The event horizon of a Schwarzschild black hole is defined by
$r_{s}=GM/c^{2}$, where $G$ is the gravitational constant, $c$ is the speed of
light, and $M$ is the mass of the black hole. Any particle (including photons)
that enters this range will inevitably fall into the black hole’s singularity.
However, that does not mean it can not be observed using a telescope. We can
still observe it through its accretion disk, which is the ring of gas and dust
surrounding a black hole. Objects falling into the black hole are subjected to
the strong gravitational force of the black hole and then rotate around it at
high speed while being heated to extremely high temperatures and emitting
electromagnetic waves [11]. The projection of its unstable photon region on an
observer’s sky is called a black hole shadow [12]. Accretion disks emit light
across many wavelengths. For most black holes in the universe ($\sim
10M_{\odot}$), the radiation consists mainly of X-rays, but for larger mass
black holes ($\sim 10^{4}M_{\odot}$), the main electromagnetic waves radiated
are ultraviolet (UV) to X-rays [13]. For supermassive black holes such as M87*
and Sagittarius A*, the main mode of radiation is synchrotron radiation, which
falls inside the radio band wavelength [14].

The EHT has tested the probability of detecting black holes using a radio
interferometer [15]. With the development of interferometers, optical
interferometer arrays such as COAST [16], NPOI [17] and IOTA [18] have
achieved higher resolution in infrared and even visible wavelengths. However,
some smaller black holes might emit higher-frequency waves [19], which are out
of the observable range of radio and optical interferometers [20]. Therefore,
these black holes are better observed using optical telescopes, which can
cover visible and UV light. Among the candidate wavelengths, the short
wavelength of UV light corresponds to higher imaging resolution. Moreover,
compared to X-rays and $\gamma$-rays, UV is easier to be focused by optical
instruments, making it possible for humans to detect black holes in this band.
At present, some UV space telescopes have been successfully launched and
operated, such as the Ultra Violet Imaging Telescope (UVIT) [21], Far
Ultraviolet Spectroscopic Explorer (FUSE) [22], Hubble Space Telescope [23]
and so on.

The black hole shadow provides valuable information about the gravitational
environment on event horizon scales, enabling verification or modification of
general relativity [24, 25, 26, 27]. High accuracy is crucial for both the
detection and parameter estimation of the black hole [28, 29, 30]. According
to Torniamenti et al [31], black holes may exist as close as 80 pc from Earth,
within the observational range of optical telescopes. However, they may be
hidden in a large number of images from current space telescopes.
Distinguishing them from other celestial bodies is challenging due to their
far distance and proximity to other objects. Moreover, the diffraction limit
presents a fundamental constraint on the resolution of optical telescopes,
requiring more accurate detection and recognition methods. This is where
machine learning (ML) can be useful [32]. Sophisticated ML algorithms enable
astronomers to automatically search for celestial objects and enhance the
resolution of astronomical images beyond what is possible with conventional
optics alone [33]. Techniques such as super-resolution imaging and image
reconstruction algorithms trained on simulated data enable astronomers to
effectively enhance the resolution of telescope images, offering a glimpse
into previously unseen details of celestial objects [34]. ML is a powerful
tool for addressing various astronomical physics issues, and neural networks
(NNs) are increasingly being used for this purpose. For instance, they have
been instrumental in improving the resolution of the M87 black hole image [35]
and is used for the identification and classification of celestial objects
such as galaxies, stellar, and supernovae [36]. In addition, machine learning
methods are aiding in the identification of infrequent and hard-to-find
astronomical occurrences by analyzing large datasets to uncover subtle
patterns and signals that may otherwise be overlooked. [37].

In recent years, convolutional neural networks (CNNs) have been considered one
of the most effective tools in the field of image recognition [38], and have
an increasingly wide range of applications in the field of astrophysics, such
as the detection of strong gravitational lenses [39] by deep CNNs, the input
of time-domain spectrograms into CNNs for the detection of gravitational waves
[40], the detection and classification of gravitational waves [41],
gravitational wave noise reduction [42] and so on. CNNs have also been used to
identify black holes in radio telescope observation images and recover black
hole parameters [43] such as accretion rate, inclination, and positional
angle. In Ref. [44], telescope observation images are mapped to the U-V plane
and then recognized by CNNs.

Figure 1: Data simulation pipeline and ensemble NN model.

For black hole simulations, previous studies for radio band observation often
use general relativistic magnetohy drodynamical (GRMHD) to simulate the
accretion disk and then generate images of black hole shadows [45]. In the
imaging of Sgr A*, the EHT collaboration constructs the relationship between
theoretical black hole shadows and the observation of ring-like images using a
library of simulations and then uses the CLEAN algorithm and Bayesian method
to estimate the parameters as well as the confidence level [46, 47].

Unlike the above methods, what we use in this paper is an ensemble model for
both detection and parameter estimation. We first calculate the trajectory of
photons in relativistically curved spacetime and then render the image by ray-
tracing methods [48, 5, 49] to establish the data pipeline for the subsequent
model. Then we present an ensemble NN model with the backend of You Only Look
Once (YOLO) [50] and EfficientNet [51]. For black hole detection, our detector
can accurately distinguish black holes in observation images from tens to
hundreds of other celestial objects and determine their positions in the image
with a confidence level. For parameter estimations, it can infer the parameter
of black holes from the shadow, where four parameters are selected, including
inclination $i$, mass of black hole $M$, position angle $\phi$, and accretion
disk temperature $T$.

This paper is organized as follows: In section II we render black hole
accretion disks using ray tracing and then use the simulated telescope to get
the observation images. In section III we introduce the ensemble NN model for
both detection and parameter estimation of black holes. In section IV we test
the validity of our model using the image of M87*. Finally, in section V, we
summarize the results and discuss the feasibility of real-time detecting black
holes of candidate black holes and further parameter estimation. The flow
chart of the whole work is shown in Fig. 1.

##  II OBSERVATION SIMULATION

###  II.1 Black hole accretion disk simulation

To render the image of black holes, ray-tracing for the accretion disk of the
Schwarzschild black hole is used, whose metric has the form,

$ds^{2}=-\left(1-\frac{r_{\mathrm{s}}}{r}\right)c^{2}dt^{2}+\left(1-\frac{r_{\mathrm{s}}}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$
(1)

where $r_{s}=GM/c^{2}$ and $d\Omega^{2}=\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)$. From this equation, the photon trajectories outside the
black hole can be solved numerically using the fourth-order Lunger-Kutta
algorithm with $\theta=\pi/2$,

$\frac{d^{2}u(\phi)}{d\phi^{2}}=\frac{3}{2}u(\phi)^{2}-u(\phi),$ (2)

where $u(\phi)=1/r(\phi)$. The result is shown in Fig. 2. The innermost stable
circular orbit (ISCO) is the smallest edgewise-stable circular orbit in which
particles can be stabilized to orbit a massive object in the theory of general
relativity. No particle can maintain a stable circular orbit smaller than
$r_{\text{ISCO}}$. In that case, it would fall into the event horizon of the
black hole while rotating around it. For a Schwarzschild black hole,
$r_{\text{ISCO}}=3r_{s}$. Typically, this is where matter can generate an
accretion disk [52, 53, 54], which corresponds approximately to the center of
the accretion disk in this work.

The temperature of the accretion disk determines the wavelength of black body
radiation, which in turn determines whether a black hole can be observed
through a telescope within a certain wavelength range. The temperature of the
accretion disk is [13, 19]:

$T=\left[6.27\times
10^{7}\mathrm{~{}K}\right]\alpha^{\frac{1}{4}}\left(\frac{M}{3M_{\odot}}\right)^{-\frac{3}{4}}\left(\frac{\dot{M}}{10^{17}\mathrm{g}/\mathrm{s}}\right)^{\frac{1}{2}},$
(3)

where $M_{\odot}$ is the mass of solar, $\dot{M}$ is the accretion rate and
$\alpha$ is the standard alpha viscosity, and $0<\alpha<1$ is a dimensionless
coefficient, assumed by Shakura and Sunyaev to be a constant [55]. To reduce
dimensions of the parameter space, we set $\alpha=0.1$ and $\dot{M}=7.26\times
10^{16}g$ $\text{sec}^{-1}$. We can assume that the accretion disk is
radiatively efficient, i.e. the rate of accretion is small enough so that any
heat generated by viscosity can be immediately converted into light energy and
radiated outward. It is also supposed that the accretion disk is very thin,
resulting in all accreted material being on the equatorial plane. [19].

Figure 2: Photon trajectories, event horizont ($r_{s}$), photon ring
($1.5r_{s}$) and ISCO, where the initial direction makes 26.5°angle with
horizontal axis. Note: the colors are only for demonstration and don’t
indicate the wavelengths.

To render a more realistic image of the black hole, gravitational lensing [56]
and the Doppler effect should also be considered [57]. The Doppler color shift
is given by

$(1+z)_{\text{Doppler}}=\frac{1-\beta\cos(\gamma)}{\sqrt{1-\beta^{2}}},$ (4)

where $\gamma$ is the angle between the ray direction and the disk’s local
velocity [49]. The redshift from the relative motion of the black hole to the
observer can be ignored since in our simulations the Earth is typically about
several hundred light-years away from the black hole.

For simplicity, one can consider blackbody radiation and disregard other
radiation, such as synchrotron radiation. According to Planck’s formula for
blackbody radiation [58], it can be calculated that the intensity of radiation
at a certain wavelength is
$f(\lambda)=\frac{1}{\lambda^{5}}\frac{1}{\exp(hc/\lambda k_{B}T-1)}$. Since
we assume that the telescope operates at a single wavelength, the brightness
observed by the telescope is also proportional to $f(\lambda)$, and to
simplify the calculations, we ended up simplifying the telescope photo to a
black-and-white photo and normalizing the radiant intensity over [0, 255]. The
result is shown in the first column of Fig. 3.

Figure 3: Simulated image, wherein the first column is the simulated generated
image, the second and third column is the recolored image, and the fourth and
fifth columns are the blurred images in the simulated telescope.

Note that the radiation used to simulate the black body of a black hole is UV
light with a narrow spread of wavelengths. Therefore it can be seen as
monochromatic light, so the image is shown in grayscale. To demonstrate
intuitively, it is mapped to be a colored image in the second and third column
of Fig. 3. We can see that the light that black holes emit is not symmetrical.
That is because the gravitational force of the black hole bends the light,
which makes the accretion disk twist into the shape of a “mushroom”.

We can simulate the stellar through the stellar mass-luminosity relation [59]:

$\frac{L}{L_{\odot}}=\left(\frac{M}{M_{\odot}}\right)^{a},\quad
0.43M_{\odot}<M<2M_{\odot},$ (5)

where $M_{\odot}$ and $L_{\odot}$ are the mass and luminosity of the Sun and
$1<a<6$. We take $a=4$ in this simulation, which is the most probable range
for stellar in the universe.

###  II.2 Telescope simulation

The diffraction limit is the fundamental constraint on telescope resolution.
According to the Rayleigh criterion, two objects are considered just
resolvable if the first minimum (dark fringe) of the diffraction pattern
created by one object coincides with the central peak of the pattern created
by the other. The imaging FWHM of a telescope is
$\theta_{c}=\frac{1.22\lambda}{\mathrm{D}}$, where $\lambda$ is the wavelength
and D is the diameter of the telescope. Throughout the observing range of
optical telescopes, UV light has the highest resolution. Electromagnetic waves
with smaller wavelengths, such as X-rays and $\gamma$-rays, are no longer
possible for an optics telescope to observe because of the difficulty in
focusing. To prevent atmosphere absorption of UV light, a telescope has to be
placed on satellites. In our work, the configuration of the simulated
telescope follows the Hubble Telescope [60], with a resolution of 10 $\mu$as
(1000$\mu$as = $1\text{\,}\mathrm{\SIUnitSymbolArcsecond}$), as shown in Table
1.

Table 1: Telescope configuration Symbol | Value | Explanation
---|---|---
D | 2.4m | Diameter
F | 57.6m | Focal length
$L_{\text{CCD}}$ | $2\mu m$ | Size of the pixels on the detector
$N_{\text{CCD}}$ | 3072 | Number of pixels of the CCD
SNR | 10 | Signal-to-Noise Ratio
$\theta_{c}$ | $0.01\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ | Angular resolution in arcsecond

After generating the simulated image of the black hole and stellar, PSFs of
the telescope for different angular sizes of images are calculated. Then, the
PSF of the telescope is convolved with the simulated image to obtain the
observed results. The shadows with different angular sizes are shown in the
fourth and fifth columns of Fig. 3. We define the angular size of the whole
image as $\theta$, the angular size of the outer edge of the accretion disk as
$\theta_{\text{AD}}=\arcsin\left(2r_{\text{AD}}/r_{\text{obs}}\right)$, and
the angular size of ISCO as
$\theta_{\text{ISCO}}=\arcsin{(2r_{\text{ISCO}}/r_{\text{obs}})}$, where
$r_{\text{obs}}$ is the distance between the black hole and the observer. The
doughnut-like shadow and size relations are shown in Fig. 5. There is almost
no light distribution inside the event horizon ($r_{s}$), and the ISCO
($3r_{s}$) is approximately the center of the accretion disk. $r_{\text{AD}}$
is approximately 2 times $r_{\text{ISCO}}$.

(a) No diffraction

(b) $\theta_{\text{AD}}=10\theta_{c}$

(c) $\theta_{\text{AD}}=5.4\theta_{c}$

(d) $\theta_{\text{AD}}=3.1\theta_{c}$

(e) $\theta_{\text{AD}}=1.2\theta_{c}$

Figure 4: The shadows from the simulated telescope for different angular sizes
of the black hole.

When $\theta_{\text{ISCO}}>\theta_{c}$, the shadow is a doughnut-shaped bright
spot with unequal brightness on both sides, which is easy to distinguish, as
shown in Fig. 4 (b, c, d). When $\theta_{\text{ISCO}}<\theta_{c}$, the shadow
is connected to form a circular facula, which is difficult to recognize by the
naked eye, as shown in Fig. 4 (e) The overview of the process for telescope
simulation is shown in Fig. 6.

Figure 5: Schematic of the size relationship between the black hole event
horizon, accretion disk and ISCO, where $\theta_{\text{AD}}=4.8\theta_{c}$,
$i=32.4\degree$ and $\phi=0\degree$

To match the real observation as closely as possible, noise should also be
considered, which is determined by the SNR of the telescope with ${\rm
SNR}=N/\Delta N$, where N is the number of photons released by the source, and
$\Delta N$ is the noise. In optics and telescopes, the Charge Coupled Device
(CCD) serves as a sensitive detector capturing light from celestial objects
and converting it into digital signals for analysis. Suppose the number of
photo-electrons detected from the object, sky background and dark current is
$S_{o},S_{b}$ and $S_{d}$ respectively, with the time-independent readout
noise $R$, the CCD SNR equation is written as [61]

$\mathrm{SNR}=\frac{S_{o}Qt}{\sqrt{S_{o}Qt+S_{b}Qtn_{p}+S_{d}tn_{p}+R^{2}n_{p}}},$
(6)

where $n_{p}$ is the number of pixels that the object is spread over, $t$ is
the exposure time in seconds and $Q$ is the quantum efficiency of the CCD,
expressed as a number between 0 and 1. Referring to the parameters of the
Hubble Telescope as well as its historical observations [60], we use Gaussian
noise and make all simulated observations satisfy $\text{SNR}<10$.

(a) Image before convolution (b) PSF of the telescope (c) Image after
convolution

Figure 6: Schematic of the telescope simulation, where (a) is the image before
convolution, (b) is the PSF of the telescope and (c) is the simulated
observation image generated by convolution of (a) and (b). The image is zoomed
in to show the details. In (a) and (c), the left zoomed image is a black hole
and the right zoomed image is a stellar.

##  III ENSEMBLE MODEL FOR DETECTION AND RECOGNITION

To ensure clarity and coherence, it is essential to introduce some concepts
relevant to our discussion. Detection refers to the model’s ability to
identify black holes in observation images. This includes distinguishing black
holes from other celestial objects and locating their positions. Recognition
involves estimating parameters for both continuous and discrete variables.
Regression focuses on predicting continuous variables, while classification
focuses on discrete variables.

###  III.1 Datasets

In this paper, two NN models for black hole detection and parameter estimation
share the same data generation pipeline but with different configurations. The
former corresponds to datasets where black holes and stellar are generated in
one image with the size of $1024\times 1024$, while the latter has datasets
that fix black holes in the center, with different sizes of accretion disk,
inclinations, position angles and temperatures, with an image size of
$240\times 240$.

For the detection task, multiple data groups are generated with different
$\theta$, each containing 1000 observation images. Each image has a
corresponding text file with metadata on the bounding circles that define the
objects in the image. The metadata for each object includes its class, x-y
coordinates, and radius of the bounding circle. There is either zero or one
black hole and 3 to 100 stellar in one image.

Figure 7: Determining the range of $T$ and $M$ for NN model

For the parameter estimation task, we also generated several groups of data
according to the different $\theta$. Each group has 27018 images, which is
three times the number of its parent dataset. The temperature is determined by
Wien’s displacement law from the observable range of the telescope, and the
mass of a black hole is inferred from its size of accretion disk by Eq. (3).
This process is shown in Fig. 7. The parameters to be estimated and their
range are shown in Table 2.

Table 2: Parameter range of the black hole Parameter | Range | Explanation
---|---|---
$i$ | $[-90\degree,90\degree]$ | Inclination
$\phi$ | $[0\degree,360\degree]$ | Position angle
$M$ | $[1\times 10^{4}M_{\odot},5\times 10^{4}M_{\odot}]$ | Mass
$T$ | [1.91 2.06 2.69 3.47] ($\times 10^{5}\mathrm{K}$) | Temperature

The generated samples were randomly split into training and validation sets
with ratios $N_{\text{train}}:N_{\text{validation}}:N_{\text{test}}=6:2:2$. To
ensure the accuracy of training, the data in the training set is rounded up to
an integer multiple of the batch size, and the excess is divided into the
validation set.

###  III.2 Model introduction

In computer vision (CV), object detection is typically defined as the process
of locating and determining whether specific instances of a real-world object
class are present in an image or video. In recent years, a large range of
sophisticated and varied CNN models and approaches have been developed. As a
result, object detection has already been widely used in a variety of
automatic detection tasks, such as the auto-count of the traffic flow or the
parking lot [62, 63, 64], making it the best choice for us to detect black
holes from the images of telescopes. Among all object detection models, YOLO
is considered one of the most outstanding due to its highly accurate
detection, classification, and super-fast computation [50]. The YOLO family
comprises a series of convolution-based object detection models that have
demonstrated strong detection performance while being incredibly light [65,
66]. This enables real-time detection tasks on devices with limited
computational resources. In particular, we make use of the Ultralytics package
for the YOLO model [67], which implements these models using the Python
environment and PyTorch framework. Aside from offering a variety of model
architectures with differing pre-trained parameters and sizes of the model,
Ultralytics can also provide a wealth of functionality for training, testing,
and profiling these models. Various tools are also available for transforming
the trained models into different architectures. This facilitates the redesign
of our model for the detection of black holes with the YOLO backend.

After obtaining the location of the black hole by the above BH detection
model, it is also important to determine the parameters of the black hole and
its accretion disk (e.g., $i,\phi,M,T$, etc.), which is also performed by the
deep CNN model in this work. There are lots of famous deep CNNs for image
recognition, such as VGG [68], ResNet [69], DenseNet [70] and EfficientNet
[71]. After trial and error for almost all the commonly used CNN models,
EfficientNet-b1 turns out to have the highest accuracy and low computational
resource consumption. Similar to YOLO, EfficientNet is a family of models
consisting of 8 models ranging from b0 to b7. Each successive model number has
more parameters associated with it. In addition to higher accuracy, this model
also has a significant advantage in terms of scalability. It is based on the
concept of compound scaling, which balances the depth, width, and resolution
of the network. This results in a more accurate and efficient model compared
to its predecessors. To attain the best outcomes, the model can be scaled by
modifying the parameters of EfficientNet to suit the input image’s size. This
is dissimilar to traditional models that necessitate a uniform input size and
may lose information when compressing larger images. However, in astronomical
observations, every piece of information is exceedingly valuable and scarce.
Therefore, the advent of EfficientNet is a noteworthy advancement. The ideal
size of input image varies from 224 to 600 pixels, from b0 to b7.

###  III.3 Method

####  III.3.1 Black hole detection model

Figure 8: The structure of EfficientNet used in this paper, where the gray
tensor denotes the changes of the input image, the blue tensor denotes the
convolution kernel, the red arrows denote the convolution process, and the
black arrows denote the fully connected layer, with the depth of each tensor
and the size of the convolution kernel not plotted to actual scale.

The YOLO v5, v7 and v8 [72, 73, 74] were trained and tested on the simulated
datasets, and YOLOv5 has the best performance in terms of mAP, F1 score and
speed. While the YOLO model is a popular tool for object detection, its
application in astrophysics is limited. YOLO uses bounding boxes to locate
objects, which is incompatible with circle-based celestial bodies. To address
this gap, we have made enhancements to the YOLO backend and got a model that
is specialized in detecting circle-shaped celestial bodies for astronomical
applications. The computational resources are conserved and accuracy is
enhanced by reducing the parameter space to three dimensions (x, y, and
radius) compared to the traditional bounding boxes’ four dimensions (x, y,
width and height). Furthermore, the inherent rotational symmetry of circles
ensures consistent results regardless of orientation changes, which is a
critical aspect of astronomical observations, such as luminosity calculations.
Additionally, the channels of the convolutional kernel have been reduced to
handle monochrome imagery, alleviating computational stress. The loss function
is changed to circular Intersection over Union (IoU) calculation instead of
rectangular to align with the model’s focus on circle detection, as explained
in the Appendix,

The metrics used in this paper are as follows: Precision is calculated as the
ratio of true positives (TP, instances correctly identified as positive) to
the sum of TP and false positives (FP, instances incorrectly identified as
positive). Recall is calculated as the ratio of TP to the sum of TP and false
negatives (FN, instances incorrectly identified as negative),

$\text{Precision}=\frac{\text{TP}}{\text{TP +
FP}},\quad\text{Recall}=\frac{\text{TP}}{\text{TP+FN}}.$ (7)

Accuracy is calculated by dividing the total number of instances by the ratio
of properly predicted instances. It is the most commonly used metric in
classification. However, it may not be suitable for our situations, because
there is an imbalanced class distribution. Stellars are far more than black
holes, making accuracy a misleading metric. In contrast, F1 score is suitable
to deal with this situation. It is the harmonic mean of precision and recall
[cf. Eq. (8)]. It provides a more impartial assessment of the model’s efficacy
by taking into account both FP and FN,

$\text{F}_{1}=\frac{2\times\mathrm{Precision}\times\text{Recall}}{\text{Precision}+\text{Recall}}.$
(8)

Intersection over Union (IoU) is a measure of the overlap between the
predicted bounding circle and the ground truth bounding circle. When the IoU
is 0.5 or greater, the prediction is considered a true positive. For the
detailed formula see Eq. (12) and (13) in the Appendix.

Figure 9: Evolution of four training metrics as a function of the epochs,
including precision, recall, mAP[0.5] and mAP[0.5:0.95], where the pink lines
show the original value and the blue dots show the smoothed value.

Mean Average Precision (mAP): There are two versions of mAP, The first one,
mAP[0.5], is calculated by considering predictions with an IoU threshold of
0.5 or higher as correct detections. The mAP[0.5] evaluates how well the
algorithm performs when the bounding circles have at least a 50% overlap with
the ground truth. Another version: mAP[0.5:0.95], considers a range of IoU
thresholds, specifically from 0.5 to 0.95 with some interval (here we use 0.05
intervals). It provides a more detailed evaluation by taking into account
detections at various levels of overlap with the ground truth. So it gives a
more comprehensive view of the algorithm’s performance across different levels
of precision and recall. Considering mAP[0.5:0.95] is more accurate and
comprehensive [75], the model is evaluated by 90% of mAP[0.5:0.95] and 10% of
mAP[0.5].

Since the effective variable affecting the resolution is the angular size of
the accretion disk $\theta_{\text{AD}}$, we fix the observation distance and
vary the size of the black hole accretion disk in practice, with the
assumption that the accretion disk size is proportional to the black hole
mass. Four metrics are selected to measure the accuracy of the model, which
are mAP[0.5] and mAP[0.5:0.95] for positioning capacity, and precision and
recall for classification capacity. We have fixed the training period to 100
and the total images to 1000. For detailed configurations and hyperparameters
of the model, see Table 7 in the Appendix. The validation metrics with the
change of training epoch are shown in Fig. 9, where the angular size of the
black hole is $1.78\theta_{c}$. It indicates that our model has a stable
training process and a converged result.

####  III.3.2 Parameter estimation model

To reduce computing time and power consumption, we utilized transfer learning
for the convolutional layer in our model. Specifically, we used pre-trained
weights from EfficientNet trained on ImageNet dataset for the convolutional
layer in our regression and classification model. This approach resulted in
improved accuracy values compared to using raw models with randomly
initialized parameters. We chose the B1 model with 7.8 million parameters,
which is practical for our experimental setup compared to the B5, B6, and B7
models with 30M, 43M, and 66M parameters, respectively.

Figure 10: mAP[0.5], mAP[0.5:0.95], precision and recall for different
$\theta$, where each color range denotes a region of iconic angular size,
which is explained in section IV.2. Note that, the x-axis is reversed to show
the super-resolution region on the right.

The four fully connected layers are designed by ourselves, and the final
output is the predicted parameter (e.g. $i$,$M$,$\phi$). Considering there are
many ways to implement the model, the specific network architecture is shown
in Fig. 8. The parameters of input and output are shown in Table 8 in the
Appendix, where N is the batch size. Every fully connected layer follows a
ReLU activation function and a dropout layer with a dropout rate of 0.5.

The loss function for $i,M$ is mean square error (MSE),

$L=\frac{1}{N}\sum_{n=1}^{N}(x_{n}-y_{n})^{2},$ (9)

where $N$ is number of objects and $x,y$ is prediction and ground truth
respectively. For $\phi$ ranging from $[0,2\pi]$, the loss function is
periodic MSE,

$L=\frac{1}{N}\sum_{n=1}^{N}\min\\{\left(x_{n}-y_{n}\right)^{2},\left(360\degree-
x_{n}+y_{n}\right)^{2}\\},$ (10)

and the metric for the regression task is mean absolute error (MAE)111We have
also tested training with MAE as the loss function, but both the training
speed and validation accuracy are not as good as MSE.: $\quad
l_{n}=\left|x_{n}-y_{n}\right|$ and $\text{MAE}=\operatorname{mean}(l_{n})$.
For the classification task, the loss function is cross-entropy loss,

$L=-\sum_{j=1}^{4}y_{j}\log p_{j},$ (11)

where $p_{j}$ is the predicted probability, $y_{j}$ is a bool value indicating
whether the class label $j$ is the proper classification. In our work, there
are four distinct temperatures of the accretion disk. And the metric for
classification is accuracy.

The model is trained with the No. epochs of 100 and the No. images of 27018.
We have used Bayesian optimization to select the optimal hyperparameters such
as learning rate, L2 regularization coefficients, and dropout rate during the
training of the model, and all subsequent results are from the models with
optimal hyperparameters. The training system utilized a Gen Intel (R)
i9-13900K with 24 vCPU cores and 128 GB of RAM, in addition to a single NVIDIA
GeForce RTX 4070 with a 12 GB graphical processing unit. The environment
consisted of Windows 11, Python 3.9.12, Torch 2.2.1, and other relevant
software.

##  IV TESTS

###  IV.1 Unbalanced datasets

In real observations, one of the challenges is that the datasets are
unbalanced, where most of the objects are stellar and few are black holes. In
these unbalanced datasets, conventional accuracy may be a misleading
indicator, making our model evaluation a major challenge. Our solution is to
make the black hole a positive class and set the proper confidence level to
have a larger F1 score. The F1 scores of black holes, stellar and overall with
the change of confidence are shown in Fig. 11. The F1 score reaches the
maximum of 0.97 when the confidence level is 0.625, which is close to the
desired neutral 0.5. The F1 score between 0.2 and 0.8 is flat, which indicates
our model is insensitive to the change of confidence. These prove the good
performance of our model in unbalanced datasets. So we simply choose the
confidence level as 0.5 in the subsequent discussion.

Figure 11: F1 scores with the change of confidence, where
$\theta_{\text{AD}}=2.09\theta_{c}$ and the BH/stellar ratio is 1/10.

To test the ability of our model to handle unbalanced datasets, we generate
three groups of datasets, with the BH/stellar ratio of 1/3, 1/10 and 1/100
respectively, and $\theta_{\text{AD}}=1.6279\theta_{c}$. All other
configurations are identical to the training process in section III.3.1. The
results are shown in Table 3. When the ratio decreases, mAP also decreases
because the unbalanced datasets cause unbalanced training. Precision is
increasing because more stellar means fewer false positives, and recall is
decreasing because fewer black holes mean more false negatives. Even if the
dataset is unbalanced, the result remains satisfactory, indicating that our
model is robust to unbalanced datasets.

Figure 12: Performance of recognition model for different angular sizes of images, where each color range denotes a region of iconic angular size, which is explained in section IV.2. The first row displays the model’s normalized MAE for inclination and position angle. The second row displays the normalized MAE and accuracy for mass and temperature, respectively. Note that, the x-axis is reversed to show the super-resolution region on the right. Table 3: Four metrics with the change of BH/stellar ratios, where $\theta_{\text{AD}}=1.6279\theta_{c}$. BH/stellar | mAP[0.5] | mAP[0.5:0.95] | Precision | Recall
---|---|---|---|---
1/3 | 0.97036 | 0.74807 | 0.91688 | 0.92440
1/10 | 0.95035 | 0.69731 | 0.95712 | 0.88908
1/100 | 0.90464 | 0.70239 | 0.95275 | 0.85548

###  IV.2 Angular size metrics

It is important to analyze the influence of the resolution on the performance
of the model. As a result, the model is trained under different $\theta$. We
define the following regions: ISCO range denotes
$\min{\theta_{\text{ISCO}}}<\theta_{c}<\max{\theta_{\text{ISCO}}}$, and AD
range denotes $\min{\theta_{\text{AD}}}<\theta_{c}<\max{\theta_{\text{AD}}}$.
They are all ranges rather than points because the masses of black holes in
images are different. Transition range refers to the region in between. Normal
resolution (Super resolution) denotes that the black hole is larger (smaller)
than $\theta_{c}$.

Considering the model has different metrics for different output parameters,
we should have a unified metric defined in the range [0, 1]. For the detection
model, the performance is defined as the mAP[0.5]. For the regression model,
the performance is calculated in a normalized way:
$1-\text{MAE}/\text{MAE}_{max}$, where $\text{MAE}_{max}$ is MAE of a random
guess. For the classification of temperature, the performance is defined as
the normalized accuracy: $\left(Acc-
Acc_{min}\right)/\left(Acc_{max}-Acc_{min}\right)$. If the model’s performance
is lower than the midpoint (mean of the max and the min), it is deemed to have
lost its screening capability. To describe the requirement for the
resolutions, we also define the midpoint angle as $\theta_{half}$ where the
model has half of the performance. For example, $\theta_{half}$ for mAP is
where $\text{mAP}=(\text{mAP}_{\text{max}}+\text{mAP}_{\text{min}})/2$. The
results are shown in Table 4. The first row is the model, the second column is
the corresponding $\theta_{half}$, the third column is the value at
$\theta_{half}$, and the fourth column is the value at $\theta_{c}$.

Table 4: Models’ half performances $\left(\text{in the range }[0,1]\right)$ and corresponding angle $\theta_{half}$. Model | Model’s half performance | Corresponding $\theta_{half}$
---|---|---
Detection222Calculated by mAP[0.5] | 0.596 | $0.54\theta_{c}$
Regression333Calculated by the normalized MAE of inclination | 0.445 | $1.48\theta_{c}$
Classification444Calculated by the model’s accuracy of temperature | 0.515 | $0.69\theta_{c}$

Figure 13: The ability of our model for parameter estimations. The first row
shows the scatter plot of $i$, where the x-axis is the prediction and the
y-axis is the ground truth. Green, blue-green and blue indicate errors belong
to [0, $0.2$MAE], [$0.2$MAE, $0.5$MAE], $>0.5$MAE, respectively. The second
row is a group of violin plots. The x-axis is the ground truth of size (px),
which are integers ranging from 64 to 75. Each “violin” shows the distribution
of prediction for each ground truth, and the yellow dots are the baselines.
The third row shows the confusion matrix of the classification for $T$. Figure
14: Skymap of absolute error of inclination and positional angle, denoted by
latitude and longitudes respectively. The color indicates the absolute error.

Each metric with the change of $\theta$ for detection and recognition is shown
in Fig. 10 and Fig. 12, respectively. For the detection model and
classification model, the performance would retain a lot even when
$\theta_{\text{AD}}=\theta_{c}$. For the BH detection, the model doesn’t lose
its ability until $\theta_{\text{AD}}=0.54\theta_{c}$ in terms of mAP[0.5].
For the classification of temperature, the model still has the accuracy of 89%
when $\theta_{\text{AD}}=\theta_{c}$ and retains its functionality until
$\theta_{\text{AD}}=0.69\theta_{c}$. The result shows that even if the shadow
is indistinguishable in the context of the classical Rayleigh criterion, it
can still be identified by our NN model, suggesting the properties of super-
resolution detection of NNs [76], which also indicates that our model has an
exciting ability to extract every little information from the super-blurred
image. However, for the regression model, the model has not reached the edge
of super resolution. The model has half of its functionality when
$\theta_{\text{AD}}\approx 2\theta_{c}$ or
$\theta_{\text{ISCO}}\approx\theta_{c}$. And it almost loses all of its
ability when $\theta_{\text{AD}}$ reaches $\theta_{c}$.

The probable reasons are as follows: The detection and classification of black
holes are mainly based on the outline shape and color scale of the image, so
even if $\theta_{\text{AD}}<\theta_{c}$, some part of the information will
still be retained. For the regression of $i$ and $\phi$, the ability of our
model starts to decline after the diffraction limit of the black hole ISCO is
reached ($\theta_{\text{ISCO}}<\theta_{c}$). When
$\theta_{\text{ISCO}}<\theta_{c}$, the shadow will be connected to a facula
and thus difficult to distinguish. As for the estimation of black hole mass
(infer from the size of the shadow), when $\theta_{\text{AD}}>\theta_{c}$, the
size of PSF is much larger than that of the shadow, so the size of the shadow
in the image no longer depends on the size of the shadow itself but on the
size of the PSF, which makes it difficult to estimate.

We have visualized the degree of conformity between prediction and ground
truth for $i,M$ and $T$, see Fig. 13. The first row is the scatter plot for
$i$. The “X” shaped plots indicate that the MAE of $i$ goes up as $|i|$
increases. The second row is the violin plot for $M$ (inferred by the size of
its shadow), which shows the distribution of prediction on the y-axis for each
ground truth on the x-axis. The predictions gradually go diffuse and
inaccurate as $\theta$ increases. The third row shows the confusion matrices
for the classification of $T$. The data is distributed on the diagonal and
spread out when $\theta$ increases. The error is shown as skymaps in Fig. 14,
where latitude and longitudes denote $i$ and $\phi$ respectively. These plots
show that errors are mainly distributed in the part with a larger inclination
angle. The data of the skymap is obtained by piecewise linear interpolator for
interpolation and nearest neighbor interpolator for extrapolation in scipy.
The former is a method of triangulation of the input data using Qhull’s method
[77], followed by the linear center of gravity interpolation on each triangle.

To sum up, our model achieves the high performance of black hole detection and
parameter estimation by the maturity of a pre-trained YOLO and EfficientNet
model and our proper modification. According to the results above, minimum
resolvable angular size and maximum observation distances obtained by
different discriminants or models are shown in Table 5, and observed distances
correspond to a fixed black hole mass of $4\times 10^{4}M_{\odot}$.

Table 5: Min of $\theta_{\text{AD}}$ and max observable distance Criterion | Resolution | Max distance
---|---|---
Rayleigh criterion | 10.48$\mu$as | 83.08ly
Black hole detection | 5.659$\mu$as | 153.9ly
Inclination estimation | 15.51$\mu$as | 56.14ly
Mass estimation | 15.93$\mu$as | 54.66ly
Positional angle estimation | 7.126$\mu$as | 122.2ly
Temperature classification | 7.231$\mu$as | 120.4ly

Black holes that were ejected from the Hyades in the last 150 Myr display a
median distance $\sim 80$ pc (260.8 ly) from the Sun [31]. Therefore, the
methodology presented in this work might be capable of detecting black holes,
according to Table 5.

###  IV.3 Model tests with M87*

Although the model performs well in simulated training, validation, and test
sets, its real-world performance in detecting black hole shadows is what truly
matters. To test the model’s ability to detect real black holes, we scaled
down an image of M87* observed by the EHT and added it to the generation
pipeline along with other objects and background noise. The results are
presented in Fig. 15.

Figure 15: Model validation results, where the number after the label denotes
the confidence of the prediction (between $[0,1]$), namely the probability
that the object is a BH (stellar).

In this task, we first convert the M87* [1] black hole captured by the EHT
into a grayscale image and compress it to $40\times 40$, which is then fed
into the data pipeline of the telescope simulation (here it is equivalent to
the image being diffracted twice with loss of information) and makes the black
hole’s angular size to be 20$\mu as$ and rotates it clockwise by 88$\degree$
(the angle is randomly generated), accompanied by 10 stellar and random noise
to ensure the SNR $<10$. The final image is input into the model which has
been trained in the corresponding resolution in section III, to get the output
of the classification, location and confidence level. And the result is shown
in Fig. 15, indicating that the model can successfully classify correctly all
of a black hole and ten stellar, and accurately locate their positions. The
confidence level of the black holes is 0.639, and for all the stellar is above
0.80, according to the output of the BH detection model.

Figure 16: Parameter estimation result of M87*. The estimated value of $\phi$
is misleading because of the periodicity. The output of our model for $\phi$
is $31.9\degree$.

The parameter estimation model is also tested. According to the EHT
collaboration [5], the positional angle of M87* is $288\degree$ and the
inclination angle is $17\degree$. In our coordinate system, take the transform
$i\rightarrow(90\degree-i)$ and $\phi\rightarrow 90\degree-(360\degree-\phi)$
555The spin axis of the accretion disk in this work is vertical while in Ref.
[5] is horizontal. The positive rotation direction for $\phi$ in this work is
counterclockwise while in Ref. [5] is clockwise., and they should be
$i_{\text{true}}=73\degree,\phi_{\text{true}}=18\degree$. The model outputs
$i_{\text{pred}}=55.9\degree,\phi_{\text{pred}}=31.9\degree$. The posterior
distribution of estimated parameters is shown in Fig. 16. The posterior
distribution is obtained by the distribution of ground truth from the test
dataset that satisfies $i_{\text{pred}}\approxeq 55.9\degree$ and
$\phi_{\text{pred}}\approxeq 31.9\degree$, where $\approxeq$ denotes the
difference is less than $10\degree$. Our model performs better in terms of
positional angle but has a larger error for the estimation of the inclination.

##  V DISCUSSION AND CONCLUSION

Our model is based on medium-sized, non-rotating black holes in UV band while
images of M87* taken by EHT [5] are based on supermassive, rotating black hole
in radio band. However, the difference in terms of spin and observation
wavelength might not perform a significant role in the detection and parameter
estimation task. Our NN model recognizes a black hole by its doughnut-like
shape, which is nearly identical in the ratio band (see Fig. 1 in [78]) and UV
band (see Fig. 3). Additionally, according to the GRMHD simulation, the spin
of the black hole mainly affects the size of its shadow rather than its shape
at high temperatures (cf. the first row of Fig. 2 in citeEHT5). That is the
reason why our model can still get a decent result despite the huge difference
between the model’s training data and the real black hole. This indicates that
the model has a certain degree of robustness and generalization ability.

Our model is underestimated by the calculations in section III. Compressing a
3072-pixel image to 1024 pixels during image processing results in some loss
of information in image quality for our model’s input data. There is also a
loss of color information when only considering the luminosity. Additionally,
it is important to note that the actual black hole is a Kerr black hole, and
the accretion disk of a Kerr black hole may be larger than that of a
Schwarzschild black hole depending on the direction of rotation and other
factors [13]. The size and temperature of a black hole’s accretion disk are
determined by various parameters, such as the accretion rate, which can vary
depending on the environment surrounding the black hole [19]. This variability
allows for the existence of larger black holes with larger accretion disks,
which are easier to observe. The advancement of telescope manufacturing has
led to the launch of larger and more advanced telescopes into space, such as
the James Webb Space Telescope (JWST) [79] with a 6.5m aperture. This
development demonstrates that humans can launch larger optical telescopes with
smaller imaging FWMH into space, expanding the observation range of the model.
Additionally, the ensemble NN model is highly versatile. It can be used to
detect black holes, as demonstrated in this paper, and can also be applied to
other tasks, such as identifying other celestial objects or galaxies. One way
to achieve this is by replacing the training data with simulation images of
the objects. The model is applicable to other telescopes, including radio and
optical interferometers operating in the ratio, infrared and visible
wavelength bands. However, the telescope simulations presented in this paper
should be replaced with simulation programs for the corresponding telescopes.

To sum up, this work presents an ensemble NN model with YOLO and EfficientNet
as the backend. The model can detect and recognize black holes in both the
simulated images and the real-world task, which has demonstrated that it can
accurately work in real-world situations for detecting black holes and
estimating parameters for potential candidates.

First, we have constructed a data pipeline consisting of accretion disk ray-
tracing and telescope simulation. Realistically shaped black holes are
obtained through reverse ray tracing. Telescopic simulations were then
conducted, revealing that black holes are indistinguishable when their angular
size is smaller than FWHM. These simulated observations were ultimately used
to train the ensemble NN model.

Using the dataset above, the model structure and loss function are altered,
followed by training until convergence. For black hole detection, the model
has a high detection accuracy, which achieves mAP[0.5] values of 0.9176 even
when $\theta_{\text{AD}}$ reaching the imaging FWHM ($\theta_{c}$), and
doesn’t lose its detection ability until $0.54\theta_{c}$, indicating that our
model can go somewhat beyond the limits of the traditional Rayleigh
diffraction limit. In other words, super-resolution recognition beyond the
traditional optical diffraction criterion is realized. On the other hand,
recognition for $i,\phi$ and $M$ requires a significantly higher resolution
than detector, with a minimum resolution of $\theta_{\text{AD}}\gtrsim
1.5\theta_{c}$ for $i,\phi$ and $\theta_{\text{AD}}\gtrsim 2.5\theta_{c}$ for
$M$, which is natural, since estimating the parameters of black holes is more
sophisticated than simply detection them and thus requires a higher
resolution.

Additionally, the image of M87 *, taken by EHT, is input into the model. The
model successfully distinguishes it and other celestial bodies and accurately
locates their positions. The confidence level of the black hole is 0.639 and
for all the stellar above 0.8. The model can still produce a satisfactory
result even when the image is captured in the radio band rather than the UV
band, further supporting the model’s validity and applicability.

We have demonstrated the ability to detect black holes in the UV band using
space telescopes. The parameter estimate component of the model is challenging
to utilize in real-world simulations because it requires high-resolution
images.

In this paper, we do not consider other luminous objects such as galaxies and
quasars, but they may interfere with the identification of black holes. For
example, some galaxies might also show the shape of black holes, and larger
galaxies affect the imaging quality of the observation picture. To solve this
issue, we can increase the complexity of the celestial body in the training
data. Additionally, interstellar dust may block high-energy ultraviolet rays,
which can affect the accuracy of our observations.

In future work, it may be possible to obtain more realistic and accurate
images of black holes by rendering Kerr black holes. The training data should
include other celestial bodies such as galaxies and quasars to better simulate
the real-world observations. The effect of stardust should also be considered.
Additionally, Bayesian statistics can be used to compute the posterior
distribution of the parameters in parameter estimation, instead of only
computing the parameter values.

##  VI ACKNOWLEDGE

The authors gratefully acknowledge Shangyu Wen for insightful talks and input
throughout the study. This work is supported by the National Natural Science
Foundation of China (NSFC) with Grant Nos. 12175212, 12275183 and 12275184.
The simulation of black holes references GitHub repo with code available at
https://github.com/ebruneton/black_hole_shader.git. The conclusions and
analyses presented in this publication were produced using the following
software: Python (Guido van Rossum 1986) [80], Opencv (Intel, Willow Garage,
Itseez) [81], Scipy (Jones et al. 2001) [82], PyTorch (Meta AI September 2016)
[83], Matplotlib (Hunter 2007) [84], Seaborn [85] and Corner (Daniel et al.
2016) [86]. This work is finished on the server from Kun-Lun in Center for
Theoretical Physics, School of Physics, Sichuan University.

##  Appendix A APPENDIX

Each output image in section II is of size $3072\times 3072$, exactly the
pixel number of CCD. It is subsequently compressed to size $1024\times 1024$.
The reason for not using the 1024-size image directly is that due to the short
UV wavelength, the continuum spectrum of the PSF will show a very sharp peak,
and if the input image is small, the sampling interval will be too large
during sampling, resulting in sampling distortion. After testing, the size of
$3072\times 3072$ is just enough to meet the requirements, see Fig. 6 (b).

Table 6: Example of a labeled file Class |  $x$ coord. |  $y$ coord. | radius
---|---|---|---
1 | 0.431429 | 0.8350 | 0.015714
0 | 0.240357 | 0.5335 | 0.016429
0 | 0.761071 | 0.6615 | 0.016429
0 | 0.037500 | 0.5605 | 0.010714
0 | 0.325000 | 0.5580 | 0.010000
0 | 0.594643 | 0.0225 | 0.009286

The example is shown in Table 6 and Fig. 17, where the first line of the table
indicates that there is a bounding circle for the black hole in the coordinate
(0.43, 0.83) and radius 0.016, each value relative to the whole image (“1”
accounts for black holes and “2” accounts for stellar).

Figure 17: Simulated observation images containing one black hole. Both BH and
sellers are labeled with bounding circles. For the sake of clarity, we only
label ten stellar and one black hole.

Since we have changed the original bounding boxes of the YOLO model to
bounding circles, recalculation of IoU is needed. Find the distance between
the centers of two circles $d$.

Figure 18: Evolution of training and validating loss as the function of epochs
during model, where the pink lines show the original value and the blue dots
show the smoothed value.

Check for three conditions: If $d>r_{1}+r_{2}$, the circles do not intersect.
If $d\leq|r_{1}-r_{2}|$, one circle is completely inside the other. Otherwise,
the circles intersect, and you need to calculate the area of intersection. The
area of intersection $\left(A_{\text{intersection }}\right)$ :

$\displaystyle A_{\text{intersection }}=$ $\displaystyle
r_{1}^{2}\arccos\left(\frac{d^{2}+r_{1}^{2}-r_{2}^{2}}{2dr_{1}}\right)+$ (12)
$\displaystyle
r_{2}^{2}\arccos\left(\frac{d^{2}+r_{2}^{2}-r_{1}^{2}}{2dr_{2}}\right)-$
$\displaystyle\frac{1}{2}$ $\displaystyle((-d+r_{1}+r_{2})(d+r_{1}-r_{2})$
$\displaystyle(d-r_{1}+r_{2})(d+r_{1}+r_{2}))^{1/2},$

The area of the union $\left(A_{\text{union }}\right)$ is $A_{\text{union
}}=\pi r_{1}^{2}+\pi r_{2}^{2}-A_{\text{intersection }}$. Finally, calculate
the loU:

$\mathrm{IoU}=\frac{A_{\text{intersection }}}{A_{\text{union }}}$ (13)

Table 7: Parameterization of the BH detector model Hyper-Para | Value | Hyper-Para | Value
---|---|---|---
Learning Rate | 0.01 | epochs | 100
Momentum | 0.937 | image size | 1024
Weight Decay | 0.0005 | Augmentation | True
Batch Size | 16 | Pre-Trained | True

The BH detector model is obtained after tuning hyperparameters. The training
is started with the pre-trained weights using the ImageNet Train dataset [87]
provided by the Ultralytics YOLOv5 project. The optimizer is set to the
stochastic gradient descent (SGD) and the optimal parameters are shown in
Table 7:

Table 8: Model structure of recognition model Name | Type | Output size
---|---|---
Initial | Input image | $(N,240,240,3)$
Eff. Top | Pre-trained model | $(N,8,8,1280)$
Avg pool | Global Avg. Pool | $(N,1280)$
Dropout | Dropout layer | $(N,1280)$
FC1 | Linear+ReLU | $(N,256)$
Dropout | Dropout layer | $(N,256)$
FC2 | Linear+ReLU | $(N,32)$
Dropout | Dropout layer | $(N,32)$
FC3 | Linear+ReLU | $(N,1)$

The model is trained for 100 epochs and the change of loss functions is shown
in Fig. 18.

## References

  * Collaboration  _et al._ [2019a] T. E. H. T. Collaboration, K. Akiyama, and A. A. et al, First m87 event horizon telescope results. i. the shadow of the supermassive black hole, The Astrophysical Journal Letters 875, L1 (2019a).
  * Collaboration  _et al._ [2019b] T. E. H. T. Collaboration, K. Akiyama, A. Alberdi, and W. A. et al, First m87 event horizon telescope results. ii. array and instrumentation, The Astrophysical Journal Letters 875, L2 (2019b).
  * Collaboration  _et al._ [2019c] T. E. H. T. Collaboration, K. Akiyama, A. Alberdi, and W. A. et al, First m87 event horizon telescope results. iii. data processing and calibration, The Astrophysical Journal Letters 875, L3 (2019c).
  * Collaboration  _et al._ [2019d] T. E. H. T. Collaboration, K. Akiyama, A. Alberdi, and W. A. et al, First m87 event horizon telescope results. iv. imaging the central supermassive black hole, The Astrophysical Journal Letters 875, L4 (2019d).
  * Collaboration  _et al._ [2019e] T. E. H. T. Collaboration, K. Akiyama, A. Alberdi, and W. A. et al, First m87 event horizon telescope results. v. physical origin of the asymmetric ring, The Astrophysical Journal Letters 875, L5 (2019e).
  * Akiyama  _et al._ [2022] K. Akiyama, A. Alberdi, W. Alef, J. C. Algaba, R. Anantua, K. Asada, R. Azulay, U. Bach, A.-K. Baczko, D. Ball, _et al._ , First sagittarius a* event horizon telescope results. i. the shadow of the supermassive black hole in the center of the milky way, The Astrophysical Journal Letters 930, L12 (2022).
  * Collaboration  _et al._ [2022a] E. H. T. Collaboration, K. Akiyama, A. Alberdi, W. Alef, and J. C. Algaba, First sagittarius a* event horizon telescope results. ii. eht and multiwavelength observations, data processing, and calibration, The Astrophysical Journal Letters 930, L13 (2022a).
  * Collaboration  _et al._ [2022b] E. H. T. Collaboration, K. Akiyama, A. Alberdi, and W. Alef, First sagittarius a* event horizon telescope results. iii. imaging of the galactic center supermassive black hole, The Astrophysical Journal Letters 930, L14 (2022b).
  * Collaboration [2022] T. E. H. T. Collaboration, First sagittarius a* event horizon telescope results. vi. testing the black hole metric, The Astrophysical Journal Letters 930 (2022).
  * Schwarzschild [1916] K. Schwarzschild, On the gravitational field of a mass point according to Einstein’s theory, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1916, 189 (1916), arXiv:physics/9905030 .
  * Nättilä and Beloborodov [2022] J. Nättilä and A. M. Beloborodov, Heating of magnetically dominated plasma by alfvén-wave turbulence, Phys. Rev. Lett. 128, 075101 (2022).
  * Perlick and Tsupko [2022] V. Perlick and O. Y. Tsupko, Calculating black hole shadows: Review of analytical studies, Physics Reports 947, 1 (2022).
  * Penna  _et al._ [2012] R. F. Penna, A. Sadowski, and J. C. McKinney, Thin-disc theory with a non-zero-torque boundary condition and comparisons with simulations, mnras 420, 684 (2012), arXiv:1110.6556 [astro-ph.HE] .
  * Yang  _et al._ [2020] X. Yang, S. Yao, J. Yang, L. C. Ho, T. An, R. Wang, W. A. Baan, M. Gu, X. Liu, X. Yang, _et al._ , Radio activity of supermassive black holes with extremely high accretion rates, The Astrophysical Journal 904, 200 (2020).
  * Psaltis [2019] D. Psaltis, Testing general relativity with the event horizon telescope, General Relativity and Gravitation 51, 10.1007/s10714-019-2611-5 (2019).
  * Baldwin  _et al._ [1996] J. Baldwin, M. Beckett, R. Boysen, D. Burns, D. Buscher, G. Cox, C. Haniff, C. Mackay, N. Nightingale, J. Rogers, _et al._ , The first images from an optical aperture synthesis array: mapping of capella with coast at two epochs., Astronomy and Astrophysics, v. 306, p. L13 306, L13 (1996).
  * Armstrong  _et al._ [2013] J. Armstrong, D. Hutter, E. Baines, J. Benson, R. Bevilacqua, T. Buschmann, J. CLARK III, A. Ghasempour, J. Hall, R. Hindsley, _et al._ , The navy precision optical interferometer (npoi): an update, Journal of Astronomical Instrumentation 2, 1340002 (2013).
  * Carleton  _et al._ [1994] N. P. Carleton, W. A. Traub, M. G. Lacasse, P. Nisenson, M. R. Pearlman, R. D. Reasenberg, X. Xu, C. M. Coldwell, A. Panasyuk, J. A. Benson, _et al._ , Current status of the iota interferometer, in  _Amplitude and Intensity Spatial Interferometry II_ , Vol. 2200 (SPIE, 1994) pp. 152–165.
  * Abramowicz and Fragile [2013] M. A. Abramowicz and P. C. Fragile, Foundations of black hole accretion disk theory, Living Reviews in Relativity 16, 10.12942/lrr-2013-1 (2013).
  * Quirrenbach [2001] A. Quirrenbach, Optical interferometry, Annual Review of Astronomy and Astrophysics 39, 353 (2001).
  * Tandon  _et al._ [2017] S. N. Tandon, J. B. Hutchings, S. K. Ghosh, A. Subramaniam, G. Koshy, V. Girish, P. U. Kamath, S. Kathiravan, A. Kumar, J. P. Lancelot, P. K. Mahesh, R. Mohan, J. Murthy, S. Nagabhushana, A. K. Pati, J. Postma, N. K. Rao, K. Sankarasubramanian, P. Sreekumar, S. Sriram, C. S. Stalin, F. Sutaria, Y. H. Sreedhar, I. V. Barve, C. Mondal, and S. Sahu, In-orbit performance of uvit and first results, JOURNAL OF ASTROPHYSICS AND ASTRONOMY 38, 10.1007/s12036-017-9445-x (2017).
  * Moos  _et al._ [2000] H. Moos, W. Cash, L. e. . a. Cowie, A. Davidsen, A. Dupree, P. Feldman, S. Friedman, J. Green, R. Green, C. Gry, _et al._ , Overview of the far ultraviolet spectroscopic explorer mission, The Astrophysical Journal 538, L1 (2000).
  * Scoville  _et al._ [2007] N. Scoville, R. Abraham, H. Aussel, J. Barnes, A. Benson, A. Blain, D. Calzetti, A. Comastri, P. Capak, C. Carilli, _et al._ , Cosmos: Hubble space telescope observations, The Astrophysical Journal Supplement Series 172, 38 (2007).
  * He  _et al._ [2022] A. He, J. Tao, P. Wang, Y. Xue, and L. Zhang, Effects of born–infeld electrodynamics on black hole shadows, The European Physical Journal C 82, 10.1140/epjc/s10052-022-10637-x (2022).
  * Wen  _et al._ [2023] S. Wen, W. Hong, and J. Tao, Observational appearances of magnetically charged black holes in born–infeld electrodynamics, The European Physical Journal C 83, 10.1140/epjc/s10052-023-11431-z (2023).
  * Hong  _et al._ [2021] W. Hong, J. Tao, and T. Zhang, Method of distinguishing between black holes and wormholes, Phys. Rev. D 104, 124063 (2021).
  * Meng and Wang [2003] X. Meng and P. Wang, Modified Friedmann equations in r^ -1 -modified gravity, Class. Quant. Grav. 20, 4949 (2003), arXiv:astro-ph/0307354 .
  * Doeleman  _et al._ [2008] S. S. Doeleman, J. Weintroub, A. E. E. Rogers, R. Plambeck, R. Freund, R. P. J. Tilanus, P. Friberg, L. M. Ziurys, J. M. Moran, B. Corey, K. H. Young, D. L. Smythe, M. Titus, D. P. Marrone, R. J. Cappallo, D. C.-J. Bock, G. C. Bower, R. Chamberlin, G. R. Davis, T. P. Krichbaum, J. Lamb, H. Maness, A. E. Niell, A. Roy, P. Strittmatter, D. Werthimer, A. R. Whitney, and D. Woody, Event-horizon-scale structure in the supermassive black hole candidate at the galactic centre, Nature 455, 78–80 (2008).
  * Broderick  _et al._ [2014] A. E. Broderick, T. Johannsen, A. Loeb, and D. Psaltis, Testing the No-hair Theorem with Event Horizon Telescope Observations of Sagittarius A*, Astrophys. J. 784, 7 (2014), arXiv:1311.5564 [astro-ph.HE] .
  * Johannsen [2016] T. Johannsen, Testing the no-hair theorem with observations of black holes in the electromagnetic spectrum, Classical and Quantum Gravity 33, 124001 (2016).
  * Torniamenti  _et al._ [2023] S. Torniamenti, M. Gieles, Z. Penoyre, T. Jerabkova, L. Wang, and F. Anders, Stellar-mass black holes in the Hyades star cluster?, Monthly Notices of the Royal Astronomical Society 524, 1965 (2023), https://academic.oup.com/mnras/article-pdf/524/2/1965/50883407/stad1925.pdf .
  * Baron [2019] D. Baron, Machine learning in astronomy: A practical overview, arXiv preprint arXiv:1904.07248 (2019).
  * Chen  _et al._ [2023] C. Chen, Y. Wang, N. Zhang, Y. Zhang, and Z. Zhao, A review of hyperspectral image super-resolution based on deep learning, Remote Sensing 15, 10.3390/rs15112853 (2023).
  * Soo  _et al._ [2023] J. Y. H. Soo, I. Y. K. A. Shuaili, and I. M. Pathi, Machine learning applications in astrophysics: Photometric redshift estimation, in  _AIP Conference Proceedings_ (AIP Publishing, 2023).
  * Medeiros  _et al._ [2023] L. Medeiros, D. Psaltis, T. R. Lauer, and F. Özel, The image of the m87 black hole reconstructed with primo, The Astrophysical Journal Letters 947, L7 (2023).
  * Wang  _et al._ [2018] K. Wang, P. Guo, F. Yu, L. Duan, Y. Wang, and H. Du, Computational intelligence in astronomy: A survey, International Journal of Computational Intelligence Systems 11, 575 (2018).
  * Carrasco Kind and Brunner [2013] M. Carrasco Kind and R. J. Brunner, Tpz: photometric redshift pdfs and ancillary information by using prediction trees and random forests, Monthly Notices of the Royal Astronomical Society 432, 1483–1501 (2013).
  * Gu  _et al._ [2018] J. Gu, Z. Wang, J. Kuen, L. Ma, A. Shahroudy, B. Shuai, T. Liu, X. Wang, G. Wang, J. Cai, _et al._ , Recent advances in convolutional neural networks, Pattern recognition 77, 354 (2018).
  * Schaefer  _et al._ [2018] C. Schaefer, M. Geiger, T. Kuntzer, and J.-P. Kneib, Deep convolutional neural networks as strong gravitational lens detectors, Astronomy and Astrophysics 611, A2 (2018).
  * Chatterjee  _et al._ [2021] C. Chatterjee, L. Wen, F. Diakogiannis, and K. Vinsen, Extraction of binary black hole gravitational wave signals from detector data using deep learning, Physical Review D 104, 064046 (2021).
  * Qiu  _et al._ [2023] R. Qiu, P. G. Krastev, K. Gill, and E. Berger, Deep learning detection and classification of gravitational waves from neutron star-black hole mergers, Physics Letters B 840, 137850 (2023).
  * Murali and Lumley [2023] C. Murali and D. Lumley, Detecting and denoising gravitational wave signals from binary black holes using deep learning, Physical Review D 108, 043024 (2023).
  * van der Gucht  _et al._ [2020] J. van der Gucht, J. Davelaar, L. Hendriks, O. Porth, H. Olivares, Y. Mizuno, C. M. Fromm, and H. Falcke, Deep horizon: A machine learning network that recovers accreting black hole parameters, Astronomy and Astrophysics 636, A94 (2020).
  * Popov  _et al._ [2021] A. Popov, V. Strokov, and A. Surdyaev, A proof-of-concept neural network for inferring parameters of a black hole from partial interferometric images of its shadow, Astronomy and Computing 36, 100467 (2021).
  * Gammie  _et al._ [2003] C. F. Gammie, J. C. McKinney, and G. Tóth, Harm: a numerical scheme for general relativistic magnetohydrodynamics, The Astrophysical Journal 589, 444 (2003).
  * Mościbrodzka  _et al._ [2016] M. Mościbrodzka, H. Falcke, and H. Shiokawa, General relativistic magnetohydrodynamical simulations of the jet in m 87, Astronomy & Astrophysics 586, A38 (2016).
  * Davelaar  _et al._ [2018] J. Davelaar, M. Mościbrodzka, T. Bronzwaer, and H. Falcke, General relativistic magnetohydrodynamical $\kappa$-jet models for sagittarius a, Astronomy & Astrophysics 612, A34 (2018).
  * Luminet [1979] J.-P. Luminet, Image of a spherical black hole with thin accretion disk, Astronomy and Astrophysics 75, 228–235 (1979).
  * Bruneton [2020a] E. Bruneton, Real-time high-quality rendering of non-rotating black holes (2020a), arXiv:2010.08735 [cs.GR] .
  * Redmon  _et al._ [2016] J. Redmon, S. Divvala, R. Girshick, and A. Farhadi, You only look once: Unified, real-time object detection (2016), arXiv:1506.02640 [cs.CV] .
  * Tan and Le [2020a] M. Tan and Q. V. Le, Efficientnet: Rethinking model scaling for convolutional neural networks (2020a), arXiv:1905.11946 [cs.LG] .
  * Kawashima  _et al._ [2019] T. Kawashima, M. Kino, and K. Akiyama, Black hole spin signature in the black hole shadow of m87 in the flaring state, The Astrophysical Journal 878, 27 (2019).
  * Dokuchaev and Nazarova [2019] V. I. Dokuchaev and N. O. Nazarova, The brightest point in accretion disk and black hole spin: Implication to the image of black hole m87*, Universe 5, 183 (2019).
  * Krolik and Hawley [2002] J. H. Krolik and J. F. Hawley, Where is the inner edge of an accretion disk around a black hole?, The Astrophysical Journal 573, 754 (2002).
  * Shakura and Sunyaev [1973] N. I. Shakura and R. A. Sunyaev, Black holes in binary systems. Observational appearance., aap 24, 337 (1973).
  * Cunha and Herdeiro [2018] P. V. Cunha and C. A. Herdeiro, Shadows and strong gravitational lensing: a brief review, General Relativity and Gravitation 50, 1 (2018).
  * James  _et al._ [2015] O. James, E. von Tunzelmann, P. Franklin, and K. S. Thorne, Gravitational lensing by spinning black holes in astrophysics, and in the movie interstellar, Classical and Quantum Gravity 32, 065001 (2015).
  * Bramson [1968] M. A. Bramson, Blackbody radiation laws, in  _Infrared Radiation: A Handbook for Applications_ (Springer US, Boston, MA, 1968) pp. 41–72.
  * Salaris and Cassisi [2006] M. Salaris and S. Cassisi, Evolution of stars and stellar populations, Evolution of Stars and Stellar Populations ISBN 978-0-470-09219-4 (2006).
  * Hubble Telescope [2024] Hubble Telescope, Official website of the hubble telescope (2024), [Online; accessed 24-February-2024].
  * Fellers and Davidson [2010] T. J. Fellers and M. W. Davidson, National high magnetic field laboratory, the florida state university (2010), [Online; accessed 24-February-2024].
  * Sultana  _et al._ [2020] F. Sultana, A. Sufian, and P. Dutta, A review of object detection models based on convolutional neural network, Intelligent computing: image processing based applications , 1 (2020).
  * Zaidi  _et al._ [2022] S. S. A. Zaidi, M. S. Ansari, A. Aslam, N. Kanwal, M. Asghar, and B. Lee, A survey of modern deep learning based object detection models, Digital Signal Processing 126, 103514 (2022).
  * Dhillon and Verma [2020] A. Dhillon and G. K. Verma, Convolutional neural network: a review of models, methodologies and applications to object detection, Progress in Artificial Intelligence 9, 85 (2020).
  * Amit  _et al._ [2021] Y. Amit, P. Felzenszwalb, and R. Girshick, Object detection, in  _Computer Vision: A Reference Guide_ (Springer, 2021) pp. 875–883.
  * Jiang  _et al._ [2022] P. Jiang, D. Ergu, F. Liu, Y. Cai, and B. Ma, A review of yolo algorithm developments, Procedia Computer Science 199, 1066 (2022).
  * Jocher  _et al._ [2023] G. Jocher, A. Chaurasia, and J. Qiu, Ultralytics YOLO (2023).
  * Simonyan and Zisserman [2015] K. Simonyan and A. Zisserman, Very deep convolutional networks for large-scale image recognition (2015), arXiv:1409.1556 [cs.CV] .
  * He  _et al._ [2015] K. He, X. Zhang, S. Ren, and J. Sun, Deep residual learning for image recognition (2015), arXiv:1512.03385 [cs.CV] .
  * Huang  _et al._ [2018] G. Huang, Z. Liu, L. van der Maaten, and K. Q. Weinberger, Densely connected convolutional networks (2018), arXiv:1608.06993 [cs.CV] .
  * Tan and Le [2020b] M. Tan and Q. V. Le, Efficientnet: Rethinking model scaling for convolutional neural networks (2020b), arXiv:1905.11946 [cs.LG] .
  * Tang  _et al._ [2023] S. Tang, S. Zhang, and Y. Fang, Hic-yolov5: Improved yolov5 for small object detection (2023), arXiv:2309.16393 [cs.CV] .
  * Wang  _et al._ [2022] C.-Y. Wang, A. Bochkovskiy, and H.-Y. M. Liao, Yolov7: Trainable bag-of-freebies sets new state-of-the-art for real-time object detectors (2022), arXiv:2207.02696 [cs.CV] .
  * Reis  _et al._ [2023] D. Reis, J. Kupec, J. Hong, and A. Daoudi, Real-time flying object detection with yolov8 (2023), arXiv:2305.09972 [cs.CV] .
  * Padilla  _et al._ [2020] R. Padilla, S. L. Netto, and E. A. B. da Silva, A survey on performance metrics for object-detection algorithms, 2020 International Conference on Systems, Signals and Image Processing (IWSSIP) , 237 (2020).
  * Honma  _et al._ [2014] M. Honma, K. Akiyama, M. Uemura, and S. Ikeda, Super-resolution imaging with radio interferometry using sparse modeling, Publications of the Astronomical Society of Japan 66, 95 (2014).
  * Barber  _et al._ [1996] C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, The quickhull algorithm for convex hulls, ACM Transactions on Mathematical Software (TOMS) 22, 469 (1996).
  * Nalewajko, Krzysztof  _et al._ [2020] Nalewajko, Krzysztof, Sikora, Marek, and Różańska, Agata, Orientation of the crescent image of m 87*, Astronomy and Astrophysics 634, A38 (2020).
  * Gardner  _et al._ [2006] J. P. Gardner, J. C. Mather, M. Clampin, R. Doyon, M. A. Greenhouse, H. B. Hammel, J. B. Hutchings, P. Jakobsen, S. J. Lilly, K. S. Long, J. I. Lunine, M. J. Mccaughrean, M. Mountain, J. Nella, G. H. Rieke, M. J. Rieke, H.-W. Rix, E. P. Smith, G. Sonneborn, M. Stiavelli, H. S. Stockman, R. A. Windhorst, and G. S. Wright, The james webb space telescope, Space Science Reviews 123, 485–606 (2006).
  * Van Rossum and Drake [2009] G. Van Rossum and F. L. Drake,  _Python 3 Reference Manual_ (CreateSpace, Scotts Valley, CA, 2009).
  * Itseez [2015] Itseez, Open source computer vision library, https://github.com/itseez/opencv (2015).
  * Virtanen  _et al._ [2020] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1.0 Contributors, SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nature Methods 17, 261 (2020).
  * Paszke  _et al._ [2019] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, Pytorch: An imperative style, high-performance deep learning library, in  _Advances in Neural Information Processing Systems 32_ (Curran Associates, Inc., 2019) pp. 8024–8035.
  * Hunter [2007] J. D. Hunter, Matplotlib: A 2d graphics environment, Computing in Science & Engineering 9, 90 (2007).
  * Waskom [2021] M. L. Waskom, seaborn: statistical data visualization, Journal of Open Source Software 6, 3021 (2021).
  * Foreman-Mackey [2016] D. Foreman-Mackey, corner.py: Scatterplot matrices in python, The Journal of Open Source Software 1, 24 (2016).
  * Deng  _et al._ [2009] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, Imagenet: A large-scale hierarchical image database, in  _2009 IEEE Conference on Computer Vision and Pattern Recognition_ (2009) pp. 248–255.
  * Redmon and Farhadi [2018] J. Redmon and A. Farhadi, Yolov3: An incremental improvement (2018), arXiv:1804.02767 [cs.CV] .
  * Collaboration  _et al._ [2019f] T. E. H. T. Collaboration, K. Akiyama, A. Alberdi, and W. A. et al, First m87 event horizon telescope results. vi. the shadow and mass of the central black hole, The Astrophysical Journal Letters 875, L6 (2019f).
  * El-Badry  _et al._ [2022] K. El-Badry, H.-W. Rix, E. Quataert, A. W. Howard, H. Isaacson, J. Fuller, K. Hawkins, K. Breivik, K. W. K. Wong, A. C. Rodriguez, C. Conroy, S. Shahaf, T. Mazeh, F. Arenou, K. B. Burdge, D. Bashi, S. Faigler, D. R. Weisz, R. Seeburger, S. Almada Monter, and J. Wojno, A sun-like star orbiting a black hole, Monthly Notices of the Royal Astronomical Society 518, 1057–1085 (2022).
  * El-Badry  _et al._ [2023] K. El-Badry, H.-W. Rix, Y. Cendes, A. C. Rodriguez, C. Conroy, E. Quataert, K. Hawkins, E. Zari, M. Hobson, K. Breivik, A. Rau, E. Berger, S. Shahaf, R. Seeburger, K. B. Burdge, D. W. Latham, L. A. Buchhave, A. Bieryla, D. Bashi, T. Mazeh, and S. Faigler, A red giant orbiting a black hole, Monthly Notices of the Royal Astronomical Society 521, 4323–4348 (2023).
  * Müller and Frauendiener [2012] T. Müller and J. Frauendiener, Interactive visualization of a thin disc around a schwarzschild black hole, European Journal of Physics 33, 955–963 (2012).
  * Duong  _et al._ [2020] L. T. Duong, P. T. Nguyen, C. Di Sipio, and D. Di Ruscio, Automated fruit recognition using efficientnet and mixnet, Computers and Electronics in Agriculture 171, 105326 (2020).
  * Hynes  _et al._ [2003] R. I. Hynes, C. Haswell, W. Cui, C. Shrader, K. O’Brien, S. Chaty, D. Skillman, J. Patterson, and K. Horne, The remarkable rapid x-ray, ultraviolet, optical and infrared variability in the black hole xte j1118+ 480, Monthly Notices of the Royal Astronomical Society 345, 292 (2003).
  * Bruneton [2020b] E. Bruneton, Black hole shader, https://github.com/ebruneton/black_hole_shader.git (2020b).
  * Li  _et al._ [2009] L.-X. Li, R. Narayan, and J. E. McClintock, Inferring the inclination of a black hole accretion disk from observations of its polarized continuum radiation, The Astrophysical Journal 691, 847 (2009).