# Photometry of Saturated Stars with Machine Learning

Dominik Winecki Department of Computer Science and Engineering, The Ohio State
University, Columbus, OH 43210, USA  Christopher S. Kochanek Department of
Astronomy, The Ohio State University, 140 W. 18th Ave., Columbus, OH 43210,
USA  Center for Cosmology and AstroParticle Physics, The Ohio State
University, 191 West Woodruff Ave., Columbus, OH 43210, USA

###### Abstract

We develop a deep neural network (DNN) to obtain photometry of saturated stars
in the All-Sky Automated Survey for Supernovae (ASAS-SN). The DNN can obtain
unbiased photometry for stars from $g\simeq 4$ to $14$ mag with a dispersion
(15%-85% $1\sigma$ range around median) of $0.12$ mag for saturated ($g<11.5$
mag) stars. More importantly, the light curve of a non-variable saturated star
has a median dispersion of only $0.037$ mag. The DNN light curves are, in many
cases, spectacularly better than provided by the standard ASAS-SN pipelines.
While the network was trained on g band data from only one of ASAS-SN’s 20
cameras, initial experiments suggest that it can be used for any camera and
the older ASAS-SN V band data as well. The dominant problems seem to be
associated with correctable issues in the ASAS-SN data reduction pipeline for
saturated stars more than the DNN itself. The method is publicly available as
a light curve option on ASAS-SN Sky Patrol v1.0.

††thanks: Email<EMAIL_ADDRESS>
##  1 Introduction

Projects such as the All-Sky Automated Survey for Supernovae (ASAS-SN, Shappee
et al. 2014,Kochanek et al. 2017, Hart et al. 2023), the Asteroid Terrestrial-
impact Last Alert System (ATLAS, Tonry et al. 2018a) and the Zwicky Transient
Facility (ZTF, Bellm et al. 2019) currently monitor all (ASAS-SN and ATLAS) or
large fractions (ZTF) of the visible sky and provide public access to their
data. For ASAS-SN, this consists of its catalog of variable stars (Jayasinghe
et al. 2019), ASAS-SN Sky Patrol (Kochanek et al. 2017), which will provide an
uncensored light curve for any user requested sky coordinate, and ASAS-SN Sky
Patrol v2.0 (SP2, Hart et al. 2023), which provides continuously updated light
curves of roughly 100 million sources from asteroids to stars and quasars. In
the near future, the Vera Rubin Observatory (e.g., Ivezić et al. 2019) will
provide ZTF-like coverage and cadence for far fainter sources.

Professional astronomy has lost, however, the ability to monitor the bright
sky – the need for both funding and efficient operations drive projects
towards fainter and more numerous sources. Yet these very bright sources are
the ones which will have the best stellar spectroscopic observations from
projects like SDSS/APOGEE (e.g., Abdurro’uf et al. 2022) or Gaia (e.g., Gaia
Collaboration et al. 2023) including either indications of binarity or full
orbital solutions.

The problem is that charge coupled device (CCD) detectors have a finite
dynamic range above which a pixel is saturated. A typical dark time All-Sky
Automated Survey for Supernovae (ASAS-SN) image has a sky background of $\sim
200$ counts and saturates at $\sim 60,000$ counts. The PSF has a FWHM of
roughly 2 pixels (Shappee et al. 2014, Kochanek et al. 2017), leading to a
point source dynamic range of roughly 7 magnitudes from $g\sim 11.5$ to $g\sim
18.5$ mag. This leaves some $\sim 10^{6}$ stars that are saturated in ASAS-SN
observations. The excess charge of the saturated pixels then bleeds into
pixels along the read direction of the detector.

ASAS-SN inherited from the original All-Sky Automated Survey (ASAS, Pojmanski
2002) one method of trying to obtain photometry of saturated stars. The
pipeline tries to identify the flux in the bleed trails of the saturated stars
and then adds a Gaussian with that flux at the star’s location. The bleed
trails are then replaced by linear interpolation of the adjacent, unsaturated
pixels. This approach has also been applied to Swift/UVOT data by Page et al.
(2013). As shown by some of the examples in Kochanek et al. (2017), this
procedure frequently works surprisingly well. An alternative approach is to
model the unsaturated wings of the PSF (e.g., Su et al. 2022, Zhou et al.
2023).

Here, we experiment with a new approach, namely, training a deep neural
network (DNN) to do photometry of saturated stars. Convolutional neural
networks (CNNs) have been used successfully for a range of astronomical
problems, such as star–galaxy classification (Kim & Brunner 2017), transient
classification (Qu et al. 2022) and photometric redshifts (Pasquet et al.
2019). There have been some applications of CNNs to photometry (Yang et al.
2023; Yuan et al. 2023) but not to the specific problem of saturated star
photometry. Here we use DNNs because they retain the position information,
which seems relevant to photometry of an already identified object. The theory
is that a network can simply learn the sensor-specific behavior for stars of
different levels of saturation and then predict the true brightness. In
Section 2 we describe the construction of the training set, and in Section 3
we describe the model and the training process. We present the results in
Section 4 with comparisons to the results produced with aperture photometry on
subtracted images by ASAS-SN Sky Patrol v2.0 (Hart et al. 2023). We discuss
the results, known problems, potential solutions and possible future
improvements in Section 5.

##  2 Training Data

We selected stars from Gaia DR3 (Gaia Collaboration et al. 2016, Gaia
Collaboration et al. 2023) that were observed by the “bi” camera of the ASAS-
SN Bohdan Paczynski mount at CTIO in Chile. We selected a random 40,000 non-
variable stars (based on Gaia DR3, Eyer et al. 2023) over the sky per one
magnitude wide bin starting at $G=2$ mag. This provides a training set roughly
uniformly distributed in magnitude. Obviously, the very brightest bins have
fewer than 40,000 stars and so we were simply including all available stars.
We kept stars observed by bi, with defined $G$, $B_{P}$ and $R_{P}$
magnitudes.

Each target then has 100s of epochs of ASAS-SN observations, so we randomly
selected $\sim 4$ images per target and extracted a 21 pixel square “postage
stamp” image of the target. We used single exposures interpolated to the
astrometric frame of the reference image. Only images flagged as having been
taken in good conditions were used and the postage stamp edges had to be at
least 50 pixels from the detector edges. We generated $\sim 332$ thousand
postage stamps spanning $G=3$ to $15$ mag.

We predict the g band magnitudes from the Gaia DR3 magnitudes because the Gaia
magnitudes homogeneously span the full range we consider. We first applied the
Gaia EDR3 saturation corrections from Riello et al. 2021 (these affect $G<8$
for G, $G<3.94$ for $B_{P}$ and $R_{P}<3.45$ for $R_{P}$) and estimate the
g-band magnitude from the corrected $G$, $B_{P}$ and $B_{P}$ magnitudes
following Riello et al. (2021). We then convert to the estimated counts in an
image using the zero point $Z$ of the ASAS-SN reference image and the mean
transparency correction $t$ between fluxes on the reference image and the
target image determined by ISIS (Alard & Lupton 1998, Alard 2000) as part of
the standard ASAS-SN image subtractions (the values in sum_kernel). This gives
predicted counts of

$\log N_{s}=-0.4(g-Z)-\log t.$ (1)

The ASAS-SN point spread function has a FWHM of roughly 16 arcsec. At this
resolution, the flux in any ASAS-SN photometric aperture can be a blend of the
fluxes from multiple stars. This is not important for the brightest stars, but
becomes increasingly important for fainter stars, particularly at lower
Galactic latitudes. To mitigate this, we found all $G<20$ mag Gaia DR3 stars
within 1 arcmin of each target star. We computed a G-band flux correction
$f_{c}$ factor to go from the flux of the target star to the total flux of all
stars within $8$ arcsec. Assuming that the network will “learn” about
background subtraction we need to correct this contaminating flux for the
median stellar background flux contribution, $b_{c}$, to the signal region.
Hence, we rescale the prediction for the number of counts as

$N_{s}\rightarrow N_{s}(1+f_{c})-b_{c}.$ (2)

When we added bias subtraction to the ASAS-SN pipelines, we started to damage
the cores of the images of saturated stars, a problem that remains to be
fixed. Such damaged stars are identifiable by the presence of pixels that are
exactly unity because the final consequence of the damage is that the modified
ASAS pipeline declares them to be bad pixels and resets them to unity. Postage
stamps with more than four such pixels are rejected for training and
validation.

##  3 Methods

In our dataset, the target star never covered more than a few pixels in any
direction, even at the highest brightness levels. As a pre-processing step, we
reduced the analyzed region of each postage stamp to a 5 pixel radius circle
around the center. This is to help focus the model training on the relevant
region and to give less context for a model to memorize. A sampling of the
brightest stars found that a 5 pixel radius would capture most visible detail
with a $\sim$1-2 pixel margin, as shown in Fig. 1.

Figure 1: A log scale image of the postage stamp of a 99.9th percentile bright
star with a red contour drawn to show which pixels are used in our model.

As an easy way to expand the training set, we also considered reflections of
the postage stamps, adding three new postage stamps flipped horizontally,
vertically, or both. This grows the training set from $\sim 332$ thousand
stamps to $\sim 1.33$ million stamps. To the extent that the ASAS-SN point
spread function is not circularly symmetric, this is like moving the postage
stamp to the corresponding reflection of its position on the detector. We
expected this to be a safe way of trivially multiplying the training set. We
tracked the results for both the original and permuted orientations and were
unable to find significant differences in the results.

Neural networks operate on the data turned into “tensor data” which is simply
a one dimensional array. We can also append metadata to the end of the
training data, where we experimented with adding the reference image zeropoint
$Z$ and the estimated sky brightness of the individual image $S$. We discuss
other possible metadata that could be used in the discussion.

The overall training set is then divided into training (60%), testing (20%)
and validation sets (20%). We were concerned about the potential for the
network to memorize the expected properties of a particular star based on the
pattern of surrounding background stars. To make sure this was not an issue in
terms of validating the results, all images of a particular star were placed
in one of the three subsamples.

##  4 Results

Figure 2: Final Model Architecture

We carried out our analysis using PyTorch (Paszke et al. 2019), optimizing the
mean squared error (MSE) loss function between the input estimate of $\log
N_{s}$ and the output estimate from the network. We began by experimenting
with different DNN architectures. We started with 2-4 smaller 64-wide layers,
but found that increasing the width led to faster convergence. We ultimately
settled on 256-wide layers as increasing past this led to minimal
improvements. The final architecture is shown in Fig. 2.

After selecting the architecture, we evaluated PyTorch’s Adam, AdamW,
Adadelta, Adagrad, Adamax, RMSProp, and SDG optimizers. We trained each model
to 150 epochs with each optimizer multiple times. The Adam optimizer had the
lowest losses, with AdamW coming second. The other optimizers were less
reliable for this task, with some never converging. We used the Adam optimizer
of Kingma & Ba (2017) for our final models, with a learning rate of 0.0001 and
a batch size of 128. We trained $24$ models using this architecture including
the hyper-parameters in the training and selected the one with the best MSE.

The performance of the standard model for the verification data set is shown
in Fig. 3. The results show negligible bias over 10 magnitudes (a factor of
$10^{4}$ in flux) from $g\simeq 4$ mag to $g=14.5$ mag. The median difference
is $-0.004$ mag, with a clear bias appearing only for the $g>14$ mag stars.
The magnitude differences encompassing 68% and 95% of the stars are $0.12$ and
$0.34$ mag and the RMS dispersion is $0.168$ mag.

We also considered several possible elaborations. First, we considered whether
ensemble results would perform better than the best model. For example, if we
defined the ensemble model as the average of the results from the four
individual models with the lowest MSE values, the RMS residual did not
improve. Second, we tried averaging the results for the input image and its
three permutations, but again found no significant improvement. As noted
earlier, many saturated stars have pixels reset to unity due to some
unanticipated consequences of adding overscan corrections to the pipeline. We
built an independent network trained only on objects with at least one pixel
set to unity to see if this would do better than simply mixing the two
populations. This model had an RMS residual of $0.360$ mag, far worse than the
general analysis. We believe the small size of the resulting training set
likely drives this. From this point on, we focus on our standard model.

We looked for correlations of the residuals with distance from the detector
center and Galactic latitude. We examined the distance from the center because
a source of a given flux produces $\sim 50\%$ fewer counts in the extreme
corners of the detectors compared to the center due to vignetting. We examined
the Galactic latitude because the stellar density increases rapidly for lower
absolute latitudes, so the results depend more on correctly accounting for
crowding by other sources under the ASAS-SN PSF. We found no significant
correlation of the residuals with either the detector location or Galactic
latitude.

Figure 3: The differences between the input ($g_{in}$) and output ($g_{out}$)
magnitudes for the verification stars. The red curves show the median, 68% and
95% ranges of the differences in bins of $0.25$ mag. Figure 4: The light curve
dispersions of approximately $10^{3}$ non-variable sources as a function of
$g$ magnitude analyzed using the current SP2 pipeline (black squares) or by
the neural network (red triangles). The dispersion is defined as one-half of
the 15-85% ($1\sigma$) range of the residuals about the median. Figure 5:
Light curves of the non-variable verification sample stars closest to $g=8$
mag to $11.5$ mag in steps of $0.5$ mag. They were also required to have 68%
distribution widths less than $0.053$ mag – 85% of $g<11.5$ mag stars have
smaller dispersions. The horizontal lines are the Gaia-estimated g magnitudes.
The number gives the dispersion estimated from the 15-85% ($1\sigma$) range of
the points about their median.  Figure 6: Light curves of Mira variables from
the DNN analysis (left) and Sky Patrol V2.0 (right, SP2). While the saturated
star corrections of the ASAS-SN pipeline produce surprisingly good results in
many cases, the DNN pipeline results are smoother and have fewer outliers like
the ones prominently seen for the SP2 light curve of X Oct. X Oct is also the
example which is most saturated at peak brightness. The vertical scales are
the same for both the DNN and SP2 light curves.  Figure 7: Light curves of
very saturated Cepheid variables from the DNN analysis (left) and Sky Patrol
V2.0 (right, SP2). The SP2 light curves are shifted to brighter magnitudes by
the amount next to the variable name. The red points lying in a line along the
bottom are outliers fainter than the minimum magnitudes of the panels. The
horizontal red line is the g magnitude predicted from the mean Gaia
magnitudes, which should be relatively close to the true mean magnitude. Both
pipelines produce many faint outliers for W Sgr and R Cru, but the DNN light
curves are arguably cleaner except for $\ell$ Car at peak. The vertical scales
are the same for both the DNN and SP2 light curves.  Figure 8: Light curves of
eclipsing binaries from the DNN analysis (left) and Sky Patrol V2.0 (right,
SP2). The SP2 light curves are shifted to brighter magnitudes by the amount
next to the variable name. The bright magnitude limit is the same for the DNN
and SP2 panels, but the magnitude range is generally much larger for the SP2
panels. The red points lying in a line along the bottom are outliers fainter
than the minimum magnitudes of the panels. The horizontal red line is the g
magnitude predicted from the mean Gaia magnitudes.  Figure 9: Light curves of
eclipsing binaries from the DNN analysis (left) and Sky Patrol V2.0 (right,
SP2). The SP2 light curves are shifted to brighter magnitudes by the amount
next to the variable name. The bright magnitude limit is the same for the DNN
and SP2 panels, but the magnitude range is generally much larger for the SP2
panels. The red points lying in a line along the bottom are outliers fainter
than the minimum magnitudes of the panels. The horizontal red line is the g
magnitude predicted from the mean Gaia magnitudes. VY Ret has a highly
eccentric orbit leading to the asymmetric eclipse phases.  Figure 10: DNN
light curves of the Miras and Cepheids from Figs. 6 and 7 (black squares) as
compared to their DNN light curves using both V and g band data from all the
ASAS-SN cameras observing each star. Outliers are marked at the bottom in red.
Figure 11: DNN light curves of the eclipsing binaries from Figs. 8 and 9
(black squares) as compared to their light curves using both V and g band data
from all the ASAS-SN cameras observing each star. Outliers are marked at the
bottom in red.

Fig. 3 is not really a test of our primary goal, which is to produce improved
light curves rather than improved absolute photometry over the current ASAS-SN
pipelines as implemented in Sky Patrol v2.0 (SP2, Hart et al. 2023). ASAS-SN
epochs are generally comprised of three 90 second exposures. SP2 does aperture
photometry on the co-added, subtracted images obtained for each epoch and then
adds the flux of the source in the reference image. The DNN analyzes
individual images since we are limited by systematic problems not photon
statistics for the saturated stars. For the final DNN light curves we used the
median of the results for the individual epoch images (the average if only two
images, and the value if there is just one image). This procedure noticeably
reduces the light curve dispersions.

To test the performance on light curves, we randomly selected 1000 (Gaia) non-
varying sources uniformly in magnitude over $8<g<14.5$ mag and extracted their
light curves using both the best DNN model and SP2. We again limited the
analysis to good images and kept only images analyzed by both pipelines. Fig.
4 compares the dispersions of the DNN and SP2 light curves with at least 50
points as a function of magnitude. The dispersions are defined by one-half of
the 15-85% ($1\sigma$) range about the light curve median. The standard
pipeline has obvious problems for saturated ($g<11.5$) images, with a large
jump in the scatter from a median of $0.016$ mag for $g>11.5$ to a median of
$0.22$ mag for brighter sources. The band of sources with scatters above one
magnitude are the ones most badly affected by the effects of the bias
subtraction problem. The performance of the DNN is fairly uniform with a
median scatter of $0.037$ mag for the brighter sources and $0.048$ for the
fainter sources. A few of the high scatter outliers are high proper motion
stars, but most are cases where the bleed trail correction algorithm has put
the flux from the bleed trail in the wrong location. An obvious issue for
future work is to explore why the DNN has three times the scatter of the
standard pipeline for the unsaturated stars. Fig. 5 shows eight examples of
these DNN light curves. They were selected to have 15-85% ($1\sigma$) ranges
about their medians of $<0.053$ mag, which is true for 85% of the saturated
stars (see Fig. 4), but were otherwise just chosen to be the closest star to
$8.0$, $8.5$ $\cdots$ $11.5$ mag.

Figs. 6, 7, 8 and 9 compare the DNN light curves of Mira, Cepheid and
eclipsing binaries to those from SP2. These variables were not selected to
make the DNN results look good, but simply to have interesting magnitudes and
amplitudes for illustrating results. The Miras illustrate the ability of the
DNN analysis to smoothly track the brightness of variable stars from well
below the saturation limit near $g\simeq 11.25$ mag to significantly above it.
While the SP2 light curves are frequently good, they start to produce
significant outliers at the brightest magnitudes, particularly for X Oct. The
examples of classical Cepheids include the extremely bright $\ell$ Car, the
still naked eye visible W Sgr and the slightly fainter R Cru. The DNN $\ell$
Car light curves are producing significant numbers of outliers near peak,
while the SP2 light curves struggle near minimum. The W Sgr and R Cru light
curves have significant numbers of faint outliers, but the DNN produces
generally smoother light curves if we ignore the outliers. The DNN magnitudes
are also closer to the g magnitudes predicted from the Gaia mean magnitudes.
Note that since they are mean magnitudes, they should be relatively immune to
any variability. The SP2 light curves are significantly fainter than the
predictions and are shifted brighter by $\sim 1$ mag in order to use the same
magnitude ranges for both the DNN and SP2 light curves.

The driver of these problems for the bright Cepheids are two failure modes in
the pipeline saturated star correction. One failure mode is to place the flux
from the bleed trails at the wrong location, usually a relatively bright (but
not nearly as bright) star lying nearby on the bleed trail. The second, which
we do not understand, but appears to be the dominant failure for W Sgr, is to
have not placed the missing flux anywhere. The saturated target appears to
have been treated as part of the bleed trail and filled in by interpolation
across the estimated trail.

The eclipsing binary comparisons provide the most dramatic illustration of the
improvements from the DNN approach - the eclipses are clearly visible in the
DNN light curves and essentially invisible in the SP2 light curves. Even CY
Hyi, with its fairly high amplitude and continuous variability with phase, has
a barely discernable variability pattern in the SP2 light curve. The shallow
eclipses of del Cir and KV Lup are clear for the DNN light curves, as are the
very narrow eclipses of TZ Men and VY Ret. The mean DNN magnitudes are close
to the predicted means, while the SP2 means are again significantly fainter
than predicted. There are outliers in the DNN light curves, but most could
probably be eliminated with some variant of sigma clipping.

The DNN was trained using data from only a single camera (“bi” on the
Paczynski mount), while ASAS-SN is presently comprised of 20 cameras. ASAS-SN
also initially used V band, while the training data was all g band photometry.
As an experiment, we simply analyzed the data from all cameras and both
filters for the stars in Figs. 6, 7, 8 and 9. We “inter-calibrated” the data
from all the sources using a Damped Random Walk (DRW) Gaussian process for
interpolation and solving for the best light curve means for the individual
cameras and filters (these are a form of the “linear parameters” discussed in
Kozłowski et al. 2010). We found we could reduce the effects of faint outliers
by dropping the faintest 10% of the light curves. Each light curve was then
offset by the difference between the median light curve mean and the mean for
the individual camera and filter. This does assume that the variability
amplitudes and light curves shapes are identical for the two filters, but it
also the most extreme test for seeing how crucial it will be to individually
train by camera and filter.

Figs. 10 and 11 show the results. They are generally very good, suggesting
that the current network can simply be applied generally and that we could
train one DNN using data from all cameras and filters. There are outliers such
as those seen for X Oct, and all the data is struggling with the bright
Cepheids from Fig. 10. The biggest failure is for R Cru, where the huge number
of outliers has confused the inter-calibration procedure. This is seen to a
lesser extend for W Sgr where too few faint outliers were rejected for the
good parts of the light curves to be perfectly lined up. But for R Cae, S Tuc
and the eclipsing binaries in Fig. 11, the results are extremely good.

##  5 Conclusions and Future Work

We developed a deep neural network (DNN) for the accurate photometry of
saturated stars. The network was trained using predictions for the counts
produced by the target star in a sample of $\sim 332$ thousand postage stamp
images of stars roughly uniformly spread in magnitude from $g\sim 3$ to $15$
mag. For the verification sample there is little bias in the magnitude
estimates except for the fainter stars, and 68% (95%) of the magnitude
differences are less than $0.12$ ($0.34)$ mag (see Fig. 3). More importantly,
the typical scatter in the light curve of a saturated star is only $0.037$ mag
(half the 15 to 85 percentile range about the mean) compared to $0.22$ mag for
the SP2 light curves (see Fig. 4). The DNN light curves of many bright
variable stars are dramatically better than their SP2 light curves (see Figs.
6 to 9) although the performance can some times be poor, as illustrated by the
naked eye Cepheids in Fig. 7 (see below). Although the network was trained
using data from only one camera, it appears to work equally well when tested
using data from all cameras and both V and g band data when combined using a
Gaussian process to intercalibrate the light curve means (see Figs. 10, 11).
For some systems, this calibration procedure and/or outlier rejection would
have to be done more carefully, but the results are otherwise very
encouraging.

We analyzed the reduced ASAS-SN images interpolated to the frame of the
reference image. Using the interpolated images meant that the pixel location
of the target stars was fixed and could be accurately determined from the
well-verified astrometry of the reference images (2 arcsec errors relative to
Tycho stars at worst, compared to a 16 arcsec FWHM). We do not see any
evidence that the interpolation causes problems.

Using the as-reduced images creates the problems seen for the bright Cepheids
because of two issues. The first is that an interplay between adding bias
corrections and the inheritance from the ASAS pipeline of reading only integer
fits files can lead to damaging saturated images because of integer overflows.
This manifests as pixels in the star being flagged as bad and given a pixel
value of unity. This obviously causes severe problems for the standard
aperture photometry pipeline. The DNN pipeline largely manages to correctly
interpret the flagged pixels and recover a good estimate of the true flux.
While we understand the origin of the problem, a fix is not trivial and has
yet to be implemented. We believe the performance of a DNN retrained after the
problem is fixed should be improved.

The second problem comes from mistakes made by the saturated star corrections
inherited from the ASAS pipeline. As noted earlier, the pipeline tries to
collect the flux from the bleed trail and add it as a Gaussian at the location
of the saturated star with the FWHM of the data. This generally works very
well, but the flux is sometimes assigned to the wrong location and sometimes
seems not to be assigned to any location. We see this problem here for the
very brightest examples we show (Figs. 7, 10, 11). Determining the origin of
these problems in the pipeline is beyond our present scope.

We suspect that our approach would have worked well and avoided this problem
using data without these attempts to correct saturated stars. Testing this is,
however, an involved process. The original raw images are available, but they
would have to be reprocessed without the saturated star corrections (or at
least the postage stamp images needed for each star would have to be
reprocessed). The pixel positions of stars would also now vary from image to
image, so we would need to be fully confident of the astrometry of the
individual images. Checking this astrometry has not been a priority because
all the current ASAS-SN results depend only on the astrometry of the reference
image for a field and not on the astrometry of the individual images.

One likely area where our results could be improved is in the estimation of
the predicted counts for the training set. As a reminder, we predicted SDSS g
magnitudes from Gaia DR3 magnitudes, converted them to counts using the zero
point of the ASAS-SN reference image and the image subtraction estimate of the
transparency difference between the images defining the zero point and the
current image, and attempted to correct for the flux from other stars within
the large ASAS-SN PSF. The Riello et al. (2021) transformation from Gaia
magnitudes to g band is not great, with a reported scatter of $0.075$ mag.
This is, however, not large enough to be a major contributer to the scatter
seen for the verification data. Nonetheless, it might be better to have used
ATLAS REFCAT (Tonry et al. 2018b) since it was built in part to systematically
estimate $g$ band magnitudes.

The crowding corrections were computed based only on the Gaia G band
magnitudes, and they should probably have been based on the estimated g band
magnitudes. The thought was that these corrections should mostly be small, so
modest fractional errors in them would not be important. Retrospectively, this
may be need to be treated more carefully for the non-saturated stars. For the
saturated stars which are the primary target of the project, this is less of
an issue simply because the stars are so bright and the standard pipeline
performs so poorly. Moreover, the scatter seen for the unsaturated stars in
the verification data does not seem to depend on Galactic latitude, which
suggests that problems with confusion cannot be a dominant driver of the
scatter.

If a primary driver of the scatter was simply the accuracy of the conversion
from estimated counts to magnitudes driven by zero point and transparency
errors, then we would expect the fluctuations in the light curves of constant
sources lying in the same field to be correlated. If so, this would then lead
to a natural approach to improving the image calibrations by using the
correlated variability to estimate an image calibration correction. We
inspected this question for the trial light curves of non-varying sources
lying in the same field and found, unfortunately, no obvious correlations.

Probably the most important issues to be addressed are the problems created by
the ASAS-SN pipeline, fixing the damage done to saturated stars and either
understanding the failures of the saturated star corrections or switching to
using images without the corrections. We need to more extensively test how
well the DNN works on multiple cameras to determine if we can simply train a
single DNN for all cameras, perhaps with the camera identification as an
additional item of metadata.

This method is now available as an option on ASAS-SN Sky Patrol v1.0
(https://asas-sn.osu.edu/, Kochanek et al. 2017) as the “Saturated Stars
(Machine Learning)” photometry option. As should be clear from the examples,
the results become less predictable for stars approaching naked eye visibility
and brighter. The present method supplies no magnitude uncertainties, so these
are simply reported as the typical scatter seen in Fig. 4. Better error
estimates of the relative errors can be derived from the light curves, but
this requires some information on the nature of any variability (e.g. periodic
or non-periodic), making it difficult to estimate automatically. We anticipate
that the machine learning results will steadily improve, particularly once the
reduction pipeline problems for saturated stars are corrected. It is also
considerably faster than the most equivalent mode of Sky Patrol v1.0
(“Aperture Photometry”), which takes roughly 5 times longer to produce the
light curves shown in Figs. 6-9.

We thank Dustin Perzanowski of Ohio State’s ASCTech group for adding the
machine learning option to Sky Patrol v1.0. CSK is supported by NSF grants
AST-2307385 and AST-1908570. ASAS-SN is funded in part by the Gordon and Betty
Moore Foundation through grants GBMF5490 and GBMF10501 to the Ohio State
University, and also funded in part by the Alfred P. Sloan Foundation grant
G-2021-14192.

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