# Bibliography management: biblatex package

Barkotel F. Zemenu
Department of Physics, Yale College, New Haven, CT 06511, USA
Weizmann Institute of Science, Department of Particle Physics and
Astrophysics, 76100 Rehovot, Israel111Kupcinet-Getz International Summer
School 2023

# Searching for Variable Stars Using the Large Array Survey Telescope (LAST)

Barkotel F. Zemenu
Department of Physics, Yale College, New Haven, CT 06511, USA
Weizmann Institute of Science, Department of Particle Physics and
Astrophysics, 76100 Rehovot, Israel111Kupcinet-Getz International Summer
School 2023

###### Abstract

This paper introduces a novel variability report generator developed for the
Large Array Survey Telescope (LAST), a cost-effective multi-purpose telescope
array conducting a wide survey of the variable sky in the visible-light
spectrum. Designed to automate variability detection, the report generator
identifies candidate variable stars by employing adjustable thresholds to
detect periodic and non-periodic variables. The program outputs a visual and
tabular photometric report for each candidate variable source from a given
LAST sub-image. Functioning as a whitepaper, this document also provides a
concise overview of LAST, discussing its design, data workflow, and
variability search performance.

###### Contents

  1. I Introduction
  2. II LAST Observatory
  3. III Data Reduction Pipeline
    1. III.1 Preliminary Image Calibration
    2. III.2 Sub-image Processing
  4. IV Variability Search
    1. IV.1 Light Curve
    2. IV.2 Periodogram
    3. IV.3 plotRMS
  5. V Variability Report Generator
  6. VI Conclusion
  7. A Matched Filtering
  8. B Photometry
    1. B.1 Aperture photometry
    2. B.2 PSF-fit photometry

## I. Introduction

Sky surveys are abundant yet imperfect – from probing unexplored regions of
the sky to conducting surveys more efficiently, there exist several areas of
improvement. The science reach of such surveys is manifold, ranging from the
discovery of exoplanets to follow up studies of gravitational waves to
investigation of variable stars. As the underlying astrophysics of the latter
– celestial objects with fluctuating brightness – spans both intrinsic (eg.
pulsation, accretion) and extrinsic (eg. rotation, eclipsing) phenomena,
investigating variable stars carries considerable scientific interest. While
large-scale sky surveys return a wealth of information about variability in
the sky, their infrastructure cost remains an exceedingly prohibitive factor
in the effort to improve upon performances of existing sky surveys. The Large
Array Survey Telescope (LAST) is designed to address this issue. Relying on a
system of multiple small telescopes constructed with readily available
components – instead of a single large custom-designed equivalent – LAST
boasts orders of magnitude improvement in cost-effectiveness compared to
existing sky survey systems. Operating in southern Israel, LAST’s geographic
location not only takes advantage of the Negev desert’s clear night sky, but
also allows for a probe of sky regions not covered by prior wide-field
surveys.

Figure 1: Top view of 32 installed LAST telescopes at Neot-Semadar, Israel

This paper provides an overview of LAST’s variability search performance. In
Section II, the technical aspects of the LAST observatory describing the
imaging strategy and system design are discussed. Section III details the data
reduction pipeline, a crucial component in processing the vast amounts of data
generated by LAST. The pipeline’s calibration and processing steps are further
outlined. In Section IV, some techniques employed in variability searches are
explored, along with their significance in identifying periodic and non-
periodic variable stars. In Section V, the variability report generator is
presented, a novel custom-built code that automates the identification and
reporting of candidate variable sources, streamlining the analysis process.
Section VI concludes with a brief summary, followed by an appendix and a
dedicated glossary provided to ensure accessibility for novice researchers
consulting advanced technical papers on LAST, such as those on its science
goals [1], system overview [2] and data reduction pipeline [3].

## II. LAST Observatory

The LAST observatory consists of 48 telescopes (32 of which have been
installed by summer 2023; see Fig. 1), each equipped with affordable CMOS
detectors that allow for high temporal cadence with an image quality in the
order of an arc-second and an image resolution roughly better than five iPhone
13 Pro cameras.22261 Mpix (LAST camera) vs. 12 Mpix (iPhone 13 Pro) The
telescopes are installed on robot-like structures – better known as equatorial
mounts– which allow for smooth telescope motion. Designed for precise tracking
of celestial objects in the sky, each mount hosts four LAST telescopes that
capture either a contiguous field of view or point to the same position in the
sky. While two computers serve as the brain of each mount, two powerful motors
assume the role of a muscle, one controlling telescope movement along the hour
angle, akin to moving the telescope left and right, while the other controls
the movement of the telescope along the declination, moving the telescope up
and down. The motors are equipped with sensors that constantly monitor the
telescope’s position, allowing the mount to know exactly where the telescope
is pointing at any given time.

Figure 2: LAST raw image of a spiral galaxy

## III. Data Reduction Pipeline

LAST operates using a “20 x 20” imaging strategy whereby a telescope camera
captures 20 successive images (called epochs) – each with 20 seconds exposure
time– with multiple visits across a designated field of view. The
implementation of this strategy not only allows for the detection of phenomena
with high temporal variability in short time frames but also for the study of
dynamic celestial events with detectable changes in brightness. While
scientifically advantageous, however, this imaging strategy generates an
astounding amount of data at a staggering rate of 2.2 Gigabits per second
(GBits/s).333This is akin to transferring the contents of over a 1000 phone
pictures ($\sim 2$MBits) every second! The enormously high data rate presents
unique challenges, necessitating a robust data reduction pipeline that
efficiently handles tasks ranging from raw image processing to processed
imaged calibration to variability detection.

###  III.1 Preliminary Image Calibration

The initial step in the LAST pipeline involves a series of calibration
procedures to ensure the data’s reliability. We outline key components of the
initial calibration in this section.

#### Bias Image Subtraction

The bias image accounts for the fluctuating noise of photon counts in the raw
images. This image is captured with zero-exposure time and a closed shutter,
effectively measuring the baseline noise inherent in the detector. By
incorporating the bias image, we mitigate negative values that may arise due
to noise during the image readout, ensuring a more stable data foundation.

#### Dark Image Subtraction

The dark image is taken by covering the camera lens with an opaque material
(or with closed shutter) in order to determine the light produced by the
camera itself. As a result, the dark image represents the combination of the
baseline bias level and the dark current generated by the detector in the
absence of light. Subtracting the dark image allows us to correct for both the
bias level and the impact of dark current. The dark image is obtained by the
“20 $\times$ 20” strategy every few weeks. The 20 images are combined into a
“master” dark image via a sigma-clipped mean in order to reject outlier pixels
and suppress the effect of random noise.

#### Flat Image Subtraction

Completing the calibration trio, the flat image compensates for the non-
uniformity of light on the camera’s sensor. This calibration image is captured
during twilight, a time when the light is assumed to be evenly distributed
across the camera’s field of view. Through normalizing the raw image by the
flat image, variations due to non-uniform illumination in individual pixel
sensitivities are accounted for, ensuring that each pixel’s response is
calibrated to represent the true intensity of the incoming light. Just like
the dark image, the flat image is also obtained via the “20 $\crossproduct$
20” strategy. The individual images are normalized by their own image median
before the combined flat image is yet again normalized by its own image
median. The median pixel – now equal to one – would then serve as a reference
point for determining the distribution of pixel values.

Figure 3: Flat image taken near twilight

#### Bit Mask Representation

To flag various pixel properties in the raw image, a 32-bit mask image is
generated. Based on their bit positions, the mask image can identify 32
different pixel states. Each bit position corresponds to a specific pixel
attribute that is populated by propagation throughout the pipeline process.
For instance, the Saturation bit (“Is pixel saturated?”), the DarkHighVal bit
(“Is dark value too high?”) and the FlatLowVal bit (“Is normalized flat value
too low?”) are populated by the raw image, dark image and flat image
respectively.

#### Additional Calibrations

We perform additional calibration steps to further refine the raw image data.
Chief among these is the issue of dark current, which refers to the random
electrical signal present in the detector even when no light is incident on
it. This dark current can introduce unwanted noise into the image, potentially
affecting the quality of our data. To mitigate this, we subtract the global
median value of the darkened pixels, effectively removing the residual dark
current and normalizing the dark signal across the entire image. We also
tackle the non-linear response of the detector’s potential well, the region
where the charges generated by incoming photons are collected and measured.
Due to the non-linearity of this process, the recorded digital values may not
precisely correspond to the actual number of photons detected. To account for
this, we apply a correction that ensures the brightness levels in the
calibrated image accurately represent the true intensity of the incoming
light. Finally, we address the detector gain, which refers to the conversion
factor between the number of incident photons and the recorded digital values.
The gain varies across different areas of the detector and can introduce non-
uniformity in the image’s intensity. By multiplying the pixel values with the
appropriate detector gain factor, we achieve a uniform intensity level in the
calibrated image. This step also compensates for any variations or non-linear
effects introduced during the calibration process, ensuring the final image
accurately reflects the original signal with the correct amplitude.

###  III.2 Sub-image Processing

Following preliminary image calibration, the LAST pipeline follows the
procedure below for image processing:

  1. 1.

Partition the resulting image into 24 sub-images, accounting for pixel overlap
and near-edge effects

     * •

The reduction of the field of view increases calibration accuracy, improves
performance speed and renders the file size more manageable

  2. 2.

Subtract background from each sub-image, where background could emanate from
the atmosphere, light pollution, etc

     * •

Background is characterized by the mode pixel value as it typically refers to
the relatively homogeneous and uniform intensity level representing the
underlying scene

     * •

Applying linear interpolation, map the total background and subtract it off
from each sub-image

  3. 3.

Generate a matched-filter response (see Appendix A) by convolving each sub-
image with five Gaussian filter templates of varying widths to enhance signals
well-correlated with the templates

     * •

Normalize each matched-filter image by its standard deviation (StD) such that
the resulting image is in units of StD

     * •

Generate a maximized-pixel image by combining the highest value for each pixel
across all five matched-filter images

     * •

Search for source candidates by imposing a 5$\sigma$ threshold on local maxima
in the combined matched-filter image

  4. 4.

Create a catalog for each sub-image by identifying candidate sources and
calculating their signal-to-noise ratio (SNR) 444LAST requires both the
detection SNR (used for detecting sources; disregards noise contribution of
source) and the measurement SNR (used for measuring source properties; takes
into account noise arising from source) to provide a measure of the detection
significance

  5. 5.

Determine the average position (first moment) and moment of inertia (second
moment) of each source

     * •

Moments are calculated based on pixel intensities within each sub-image using
a list of each source candidate’s integer-pixel position

     * •

The first moment provides the two-dimensional Cartesian coordinates that
indicate the center location of the object while the second moment is related
to the object’s size, shape, orientation and elongation

     * •

Moments are calculated iteratively, beginning with a flat weight function
(i.e. each pixel’s contribution considered equally regardless of intensity)
until narrowing down to a pre-defined Gaussian width (i.e. the weight assigned
to each pixel becomes a normal distribution)

  6. 6.

Reject sources that resemble cosmic rays, i.e. with a delta function signature

     * •

These pixels are identified using hypothesis testing to delta function and
flagged in the bit mask image as cosmic rays

  7. 7.

Perform photometry (see Appendix B)

     * •

To work in flux units, normalize the filtered image and determine each
source’s flux value at its whole pixel position (e.g. $(3.2,6.7)\to(3,6)$)

     * •

Perform aperture photometry and Point Spread Function (PSF) fit photometry to
determine the magnitude of each source

  8. 8.

Perform astrometric calibration

     * •

Identify the precise celestial coordinates (viz., right ascension and
declination) for each remaining stellar source (i.e. with a PSF – and not
delta function – signature) by aligning each sub-image with the GAIA-DR3
catalog reference frame

     * •

Determine the astrometric solution by matching star patterns in an image
catalog to a reference catalog, without prior knowledge on the image’s sky
position, rotation, or scale555Suffice to remark that the pattern matching
process entails a two-step fitting procedure with an affine transformation and
a 3rd order polynomial.

  9. 9.

Perform photometric calibration

     * •

Match the sources within each sub-image to corresponding sources in the GAIA-
DR3 catalog

     * •

Set a limit on the photometric error and SNR to ensure that the matched
sources have accurate measurements and are not affected by significant noise

     * •

Fit for photometric parameters (namely the zero point and color coefficient
using GAIA’s Vega-AB magnitude and passband)

     * •

Extend the photometric calibration to all instrumental magnitudes so that the
measured brightness values in the sub-images are consistent with the GAIA
catalog666At this step, the photometry procedure also performs interpolation
over pixels that are either saturated, dominated by dark current or
numerically invalid (negative or NaN) using local convolution to obtain
cleaner images

  10. 10.

Match all sources in the 20 epochs of a visit (in a sub-image) to produce a
visit-matched source catalogue

     * •

Populate the first image with new sources that appear in any one of the 19
other epochs

     * •

Match each source in the first image against all the sources in the other 19
images to generate a matrix – for every measured property777These could be
astrometric measurements (such as the declination or first-moment),
photometric measurements (such as the PSF magnitude or background flux
measurement), or bit-mask flag information – with 20 rows for the epochs and
however-many columns for the sources

     * •

Generate a merged catalog table using information from all 20 epochs, where
each source populates a row and every relevant source information populates a
column

  11. 11.

Populate the merged catalog table with proper motion fits and variability
estimators

     * •

Fit source positions both to a no-proper-motion and linear-proper-motion model
to statistically investigate888The statistical investigation involves, among
others, a reduced $\chi^{2}$ for the null hypothesis, the Student’s
t-statistic, an adaptive threshold and a false alarm probability positional
variation of sources, and identify the moving sources

     * •

Apply a relative photometry zero point using the first image as a reference
frame, i.e. align the magnitude system across all images by anchoring it to
the photometric calibration of the first image

     * •

Search for variable sources using variability estimators across the 20
epochs999This step is described in further detail in Section IV

  12. 12.

Cross-match each source with nearly two dozen external catalogs of sources

     * •

Identify which external catalog with which each source was matched using a bit
integer mask

  13. 13.

Coadd the 20 epochs, refine the photometry and astrometry, and save the
coadded catalog

     * •

Register the epochs in each sub-image using the World Coordinates System (WCS)
to ensure the images are properly aligned with respect to their celestial
coordinates

     * •

Refine the photometry by summing the pixel values of corresponding pixels in
the registered images, coadding using a 5$\sigma$ clipping method, and
estimating the background and variance in the coadd image

     * •

Refine the astrometry by improving the positional information accuracy of the
identified sources in the coadd image

  14. 14.

Analyze the outputs from this pipeline process

     * •

A single visit from each telescope’s camera produces (1) a sky image data, (2)
an associated bit mask image, and (3) a catalog of detected sources; these
three data products are saved for each of the 24 sub-images obtained from each
of the 20 epochs, resulting in a total of 1440 data files

     * •

The sources in each sub-image are then matched across epochs to produce a
total of 24 merged catalog tables (in FITS file format) and 24 merged catalog
matrices (in HDF5 file format) as described in 10 and 11

     * •

Finally, each coadded sub-image of a visit produces (1) a data image, (2) a
bit mask, (3) source catalog, and (4) PSF-fit photometry catalog (obtained via
the refined photometry), resulting in a total of 96 files from a single visit
using one telescope

## IV. Variability Search

The calibration and processing steps described in III are intended to make raw
data collected by LAST telescopes ripe for LAST’s science goals.101010LAST’s
science goals are thoroughly discussed in [1]. Sifting through data collected
over several visits, each with 20 epochs and 24 sub-images, we seek to detect
and study variable sources. We describe below some of the tools we employ in
this variability search.

###  IV.1 Light Curve

The light curve– a plot of source brightness as a function of time – visually
represents the changes in a star’s brightness over time, enabling the survey
to identify recurring patterns or irregularities of interest. For instance, a
pulsating star will show periodic increases and decreases in brightness,
creating a characteristic light curve. To enhance interpretability of the
light curve, we employ two key techniques: binning and folding.

#### Binning

When we observe a star over time, we collect numerous data points representing
its brightness at different moments. These data points can sometimes exhibit
random fluctuations, making it challenging to discern any periodic variation.
Binning involves grouping these individual data points into evenly spaced bins
and then calculating an average value for each bin. By doing so, we reduce the
noise in the data, resulting in a smoother representation of the star’s
brightness variations over time. This smoothed version of the light curve
makes it easier to identify trends, patterns, and periodic behaviors that may
be present in the data.

Figure 4: Binned Light curve

#### Folding

When a star exhibits periodic behavior, its brightness variations repeat at
regular intervals, forming a repetitive pattern. Folding aligns multiple
cycles of the light curve on top of each other based on the star’s period as
determined from the dominant frequency in the periodogram. The process of
folding involves taking each data point in the light curve and placing it at
its corresponding phase within one period. The phase represents where each
data point occurs in the repeating cycle of the star’s brightness variation,
normalized to a range of 0 to 1. By aligning the data points in this way, the
folded light curve highlights the repeating pattern more clearly.

Figure 5: Folded Light curve

###  IV.2 Periodogram

The periodogram (“period” \+ “diagram”) identifies dominant
frequencies/periods associated with the periodic variations of a source. By
converting time-domain data – the light curve – to the frequency domain, the
periodogram represents the dataset as a sum of sinusoidal functions with
varying frequencies. The resulting periodogram plot displays the distribution
of power across different frequencies. While frequencies run along the x-axis
of the plot, the y-axis of the periodogram represents the significance level
of the peaks, measured in sigma, which indicates the statistical confidence of
the identified frequencies. Peaks observed in the periodogram correspond to
dominant frequencies associated with the periodic variations in the star’s
brightness. Analyzing these peaks enables us to accurately determine the
period of variation for the star, which often reveals information about the
star’s physical properties, including its size, mass, and evolutionary stage.

Figure 6: Periodogram (Amplitude vs. Frequency)

###  IV.3 plotRMS

The root mean square (RMS) plot (abbreviated plotRMS) provides a measure of
the scatter of brightness measurements around the mean value for each star by
calculating the RMS of the differences between the observed brightness and the
average brightness over all the epochs in all visits. The plotRMS is
particularly useful for identifying variable stars that exhibit irregular
variability – in contrast to stars with periodic variations – as the
periodogram will not identify them. Variable stars with larger RMS values
indicate greater fluctuations in brightness, indicating a higher level of
variability over time. Furthermore, The plotRMS assists in distinguishing true
variable sources from potential noise or systematic errors. It allows us to
set appropriate thresholds for variability detection and select significant
variable candidates for further analysis.

Figure 7: Root Mean Square (RMS) vs Magnitude

## V. Variability Report Generator

In this section, we describe a novel variability report generator developed by
the author using the AstroPack/MAATv2 package in MATLAB. Designed to
streamline the process of identifying variable sources in LAST’s dataset, our
custom-built code efficiently analyzes each source within a given sub-image,
utilizing specific thresholds to identify candidate variable stars.
Specifically, the report generator employs a periodogram cutoff, evaluating
the power spectrum of each source to determine whether it surpasses a
designated sigma value. Additionally, an RMS threshold is applied to detect
sources with photometric variations further away from the median, thereby
flagging them as potential (albeit non-periodic) variable stars. By
incorporating these criteria, the generator automates the identification of
variability in a LAST image, saving valuable time and effort in the analysis
process. Upon identifying the candidate variable sources, the program
automatically generates visual photometric reports for each source. These
include a binned light curve, a folded light curve, a plotRMS and a
periodogram depicting the dominant periods associated with each candidate
variable source. In addition to the visual reports, the program compiles a
photometric tabular report containing essential information for each candidate
variable source, providing details such as period, amplitude, mean, RMS and
other relevant properties. Such a report eases the process of identifying the
few proverbial needles exhibiting variability in the haystack of candidate
variables. In this section, we offer a walkthrough of the MATLAB code used for
generating a variability report from a LAST sub-image. The script encompasses
various steps, starting from calibration to detection to report generation.

### Step 1: Creating Matched Sources Objects

The script begins by creating two matched sources (MS) objects, namely MS1 and
MSall. MS1 contains matched sources from a single visit, while MSall contains
matched sources from all visits (24 subimages). The data are read from the
specified directory path, and the user has the option to provide a custom path
or use the default dataset.

### Step 2: Matching Sources by Sub-Image

Next, the script matches sources based on the sub-image number (n). It
utilizes the mergeByCoo function to match sources from the MSall object to the
corresponding MS1 object for the specified sub-image.

### Step 3: Zero-Point Calibration

The script proceeds with zero-point calibration for the matched sources based
on aperture or PSF-fit photometry. It calculates the calibration parameters
(R) and applies the zero-point correction to create a new ZP-calibrated
matched sources object named MSr. The MAG_APER_3 field with a 6-pixel radius
aperture is used for error calibration in both aperture and PSF-fit
photometry.

### Step 4: Binning the Data

To group successive epochs together, the script bins the data using the
binning function. The user can input a custom binning size or use the default
value (10). The binned data are stored in the MSr_binned object.

### Step 5: Generating Periodograms

The script generates periodograms for all sources in the MSr object. It sets
specified frequency limits and peak power thresholds to identify potential
periodic variables. The user can customize the thresholds, or default values
are used – $(100,1000)$ per day for the frequencies and $13\sigma$ for the
periodogram – if not provided.

### Step 6: Polynomial Subtraction

Systematic effects are removed and variability is extracted by performing
polynomial subtraction on both the unbinned and binned light curves. The
script fits a third-degree polynomial to the data and subtracts it to obtain
the subtracted magnitudes (M_os for unbinned and M_bs for binned).

### Step 7: Plotting Periodograms and RMS

Periodograms (power spectra) are plotted for all sources and for selected
sources that surpass the peak power threshold. The power spectra are
represented as ampltiude ($\sigma$) vs. frequency. Additionally, the script
plots RMS (Root Mean Square) vs. magnitude for all sources, where RMS serves
as a measure of variability.

### Step 8: Selecting Sources

The user is prompted to enter a source index or (RA, Dec) coordinates to
select a specific source for detailed analysis. If left empty, the script
automatically chooses the default source with the highest peak in the
periodogram.

### Step 9: Plotting Light Curves

The script plots both the unbinned and binned light curves for the selected
source. The unbinned light curve shows data points using M_os while the binned
light curve displays the data points after binning (using M_bs).

### Step 10: Folded Light Curve

A folded light curve is generated for the selected source using the frequency
of the peak in the periodogram. The script folds the data points according to
the dominant period of the source’s variability and plots the folded light
curve, revealing the repeating pattern of its brightness fluctuations.

### Step 11: Tabular Photometric Report

Finally, the script generates a tabular photometric report for selected
sources with information such as period, amplitude, mean magnitude, median
magnitude, robust standard deviation, standard deviation of magnitude, and
RMS. The report is displayed in a table format for easy analysis and
comparison.

### Improvement Areas

The code author has identified certain areas for improvement as of August
2023. These include implementing polynomial subtraction on the power spectrum,
setting an RMS threshold for identifying non-periodic variable stars, flagging
problematic sources based on bit masks, including $\Delta\chi^{2}$ based on
polynomial fit in the table report, addressing inconsistencies in binning
methods, and debugging the window function for sampling periodicity.

### Usage and Acknowledgment

The code was authored by Barkotel Zemenu in August 2023 and is originally
designed for analyzing data from LAST. Users can execute the script as
demonstrated in the example below:

    >> generatevariabilityreport
    Enter sub-image number [1, 24]: 10
    Enter the source index or (RA, Dec) coordinate:
    (3.022022977394895e+02, 65.349260622926622)

### Output

The program has five input options (sub-frame number, bin value,
minimum/maximum periodogram frequency, and peak power threshold), only one of
which is not optional (sub-frame number). With just a single input, the
program generates a visual report for a selected source and a tabular report
for all the candidate variable sources (see Fig. 8).

Figure 8: Visual Photometric Report for Candidate Variable Source

Figure 9: Tabular Photometric Report for Automatically Identified Candidate
Variables

## VI. Conclusion

This paper provided a comprehensive look at the Large Array Survey Telescope
(LAST)’s variability search performance. The technical aspects of the LAST
observatory were reviewed, focusing on its imaging strategy and system design.
The data reduction pipeline was described in detail, highlighting the
calibration procedures that ensure accurate data for analysis. Variability
search tools were explored, with emphasis on analyzing light curves,
periodograms, and RMS plots. Finally, the variability report generator, a
custom-built program aiding variability searches, was introduced. The
program’s efficiency and automation capabilities streamline the process of
identifying and characterizing candidate variable sources, making it an
indispensable tool for efficiently extracting variability insights from the
vast amount of observational data produced by LAST.

## Acknowledgments

The author expresses gratitude to the LAST research team in general and Prof.
Eran Ofek, Dr. David Polishook and Yarin Meir Shani in particular. This
research was conducted as part of the Kupcinet-Getz International Summer
School at Weizmann Institute of Science in summer 2023. For official LAST
publications, please consult [1], [2], [3] and the Weizmann Astrophysical
Observatory official website.

##  Appendix A Matched Filtering

Matched-filtering is a signal processing technique we employ to improve the
detection of particular signals present in LAST images. It involves convolving
an image image with a template (or filter) that is designed to match the
expected shape of the signal of interest. In LAST, the template used is a
Gaussian filter. Below are some key concepts related to matched-filtering:

Gaussian Filter Template: A Gaussian filter is characterized by its center
position, width (standard deviation), and amplitude. The Gaussian filter is
chosen as our template for its desirable properties, such as smoothing and
preserving the spatial information of the image.

Convolution: The next step is to convolve the image with the Gaussian filter
template. Convolution is a mathematical operation that combines the pixel
values of the image with the template in a way that emphasizes regions where
the template closely matches the image. The discretized convolution operation
can be represented by the equation:

$M^{\prime}(\mathbf{x})=\sum_{\mathbf{k}}T(\mathbf{k})\cdot
I(\mathbf{x}-\mathbf{k}),$ (1)

where $M^{\prime}(\mathbf{x})$ is the matched-filter response image,
$T(\mathbf{k})$ is the Gaussian filter template, and
$I(\mathbf{x}-\mathbf{k})$ is the image being convolved with the template.

Template Matching: During convolution, the Gaussian filter template is slid
across the image, and at each position, the template is multiplied element-
wise with the corresponding image pixels. The resulting values are summed to
generate the matched-filter response image $M^{\prime}(\mathbf{x})$.

Signal Detection: The matched-filter response image highlights regions in the
original image that closely resemble the shape and characteristics of the
Gaussian filter template. It effectively enhances the presence of signals or
features that match the template. Higher values in the matched-filter response
image indicate a stronger correlation between the template and the
corresponding regions in the original image.

Thresholding and Analysis: Once the matched-filter response image is obtained,
further analysis can be performed. This may include applying a threshold to
identify significant detections or extracting quantitative measurements of the
detected signals, such as their positions, sizes, or intensities.

The matched-filtering technique with a Gaussian filter template is often used
for various purposes, such as detecting and characterizing point sources
(e.g., stars), identifying extended structures (e.g., galaxies), or enhancing
specific features in the data (e.g., faint signals or low contrast
structures). Applying the matched-filtering technique with a Gaussian filter
template effectively enhances the detectability of specific signals or
features of interest, enabling more accurate analysis in LAST images.

##  Appendix B Photometry

By photometry, we are referring to the measurement of the brightness of
sources in LAST images. The process involves quantifying the amount of light
coming from celestial objects, such as stars, galaxies, or other astronomical
sources, captured from all the visits. While photometry can be performed in
several ways, two key methods are utilized in LAST: aperture photometry and
PSF-fit photometry.

###  B.1 Aperture photometry

Aperture photometry involves measuring the total amount of light within a
predefined circular – and sometimes elliptical – aperture centered on the
object of interest. This method provides a simple and intuitive way to
estimate the object’s brightness, but it can be affected by background noise
and contamination from nearby sources. To perform aperture photometry, we
first select a circular or elliptical region around the target object in the
astronomical image. The size of the aperture is typically chosen based on the
expected size of the object and the level of background noise in the image.
The aperture is centered on the object’s position to ensure that we measure
the majority of the light coming from the source. Next, we sum the pixel
values within the aperture to obtain the total flux or brightness of the
object. This sum includes both the light from the object itself and any
background noise or contamination from nearby sources that falls within the
aperture. To account for the background noise, we often subtract the average
background value per pixel from the total sum. The resulting value is the
aperture photometry measurement, representing the estimated brightness of the
object. However, it is essential to consider factors like the aperture size
and background estimation carefully, as they can significantly influence the
accuracy and precision of the photometric measurement. Aperture photometry is
particularly useful for bright and well-isolated sources in the image, where
contamination from neighboring objects is minimal. It provides a quick and
straightforward way to estimate the brightness of celestial objects and is
commonly used for studies involving point sources, such as stars. However,
despite it being relatively straightforward, aperture photometry has
limitations. For extended or faint sources, the choice of aperture size can
become challenging, as larger apertures may introduce more background noise,
while smaller apertures may miss some of the light from the source. In such
cases, PSF-fit photometry is often preferred as it offers a more sophisticated
approach to handle varying source sizes and background conditions.

###  B.2 PSF-fit photometry

In real-life situations, when we observe sources (like stars or galaxies),
there is something called a point spread function (PSF) that affects how they
appear in our images. This means that different pixels in the image containing
the source will have different amounts of light, or photon counts. Because of
this, we can assign different weights to different pixels to account for these
variations. A more naïve – but sometimes justified111111If the source noise
dominates over the background, readout and/or dark current noise, for
instance. – approach would be to equally weigh all pixels assuming that the
source uniformly. One important aspect we need to consider, however, is the
signal-to-noise ratio (SNR), which tells us how strong the signal (the source
we want to detect) is compared to the background noise. Our ultimate goal, of
course, is to choose a weight that maximizes the SNR. In the case of
background-dominated Gaussian noise, the best weights to use (read: those
reducing source detection uncertainty) are the ones that match the shape of
the PSF. This means that we give more importance to pixels that are closer to
the center of the source. Of course, this is assuming we know a source’s PSF
in the first place, which is not often true.121212For a Gaussian PSF – which
approximates the PSF (Airy Function) quite well – the maximal SNR can actually
be determined in closed form. On the other hand, PSF photometry takes into
account the shape of the point spread function, which describes how light from
a point source spreads out in the image due to the effects of optics and
atmosphere. The PSF is typically modeled as a Gaussian or a more complex
function estimated based on bright sources. PSF photometry involves fitting
the PSF model to the observed data around the object and extracting the flux,
position and even background from the best-fit model. This technique provides
a more accurate estimation of the object’s brightness, especially in crowded
fields where there are faint sources.

Convolution-based PSF photometry is a particular type of PSF photometry
employed in LAST. Think of convolution as a way to combine the information in
the image with the information in the PSF. We start with the image we
captured, which consists of pixels that represent different amounts of light.
Then, we take the PSF, which describes how the light spreads, and we apply it
to every pixel in the image. The PSF is like a “template” that we slide over
each pixel in the image. As it slides, it combines the light from that pixel
with the light from its neighboring pixels, just like how the actual spreading
of light happens. After going through every pixel, we end up with a new image
where the light has been “convolved” or blended according to the PSF. This new
image gives us a better understanding of how the light from the object is
distributed. Now, to measure the amount of light coming from the object, we
can examine the convolved image. We look at a specific spot in the image that
corresponds to the object’s position. By reading the value of the pixel at
that spot, we can estimate the amount of light coming from the object. In sum,
convolution helps us account for the spreading of light in the image by
combining the information from the image with the PSF to create a convolved
image. By analyzing the convolved image, we can estimate the amount of light
coming from sources in a LAST image.

## References

  * [1] Ben-Ami, S. al  “The Large Array Survey Telescope – Science Goals”  In _preprint_ , 2023  arXiv:2304.02719 [astro-ph.IM]
  * [2] Ofek, E. O. al.  “The Large Array Survey Telescope—System Overview and Performances”  In _Publications of the Astronomical Society of the Pacific_ 135.1048 IOP Publishing, 2023, pp. 065001  DOI: 10.1088/1538-3873/acd8f0
  * [3] Ofek, E.O. al.  “The Large Array Survey Telescope – Pipeline. I. Basic image reduction and sub visits coaddition”  In _prep_ , 2023

## Glossary

accretion

    process of matter accumulating onto a celestial object, such as a star, planet, or black hole, due to gravitational attraction
adaptive threshold

    a value that adjusts based on certain conditions to handle outliers and ensure that the probability is independent from number of trials; used for finding asteroids in the LAST proper motion search algorithm
aperture photometry

    photometric method that involves measuring the total amount of light from an astronomical source within a predefined circular or elliptical aperture
arc-second

    unit of angular measurement equal to 1/3600th of a degree, commonly used in astronomy to describe small angles on the celestial sphere
astrometric calibration

    process of calibrating astronomical images to establish a precise correspondence between the pixel coordinates in the image and the corresponding celestial coordinates
astrometric solution

    set of transformation parameters that relate pixel coordinates in an astronomical image to their corresponding celestial coordinates (right ascension and declination) obtained through astrometric calibration
bias image

    image obtained by exposing a camera sensor to zero light, ensuring that only the electronic noise inherent in the detector is recorded
calibration

    process of adjusting raw data obtained from astronomical instruments to obtain accurate and scientifically meaningful measurements by applying various corrections.
CMOS

    Complementary Metal Oxide Semiconductor (CMOS), type of semiconductor technology used for photodetection, typically composed of multiple layers, with the light-sensitive part located at the top
coadd

    combine multiple successive sub-images of the same position into a single image after transforming each sub-image to the same reference frame, aligning them in a consistent and standardized manner
color coefficient

    calibration parameter that describes the relationship between the difference in colors (which are, in turn, differences in brightness between two different passbands) of a source and its magnitude, quantifying how changes in the color of a source affect its measured magnitude
cost-effectiveness

    a measure of the efficiency of telescope resource allocation, defined as volume of space a telescope can probe per unit of time per unit cost
dark current

    electrical signal generated within a camera sensor in the absence of incoming light introducing unwanted signal variations in the obtained images
dark image

    image taken with the camera sensor entirely covered and no light reaching it in order to capture the thermal noise or dark current of the detector arising due to the random generation of electrons within the sensor even without incoming light
declination

    celestial coordinate system analogous to latitude on Earth and measures the angle between the object and the celestial equator
detector gain

    measure of the amplification efficiency of a detector by determining the number of electrons generated in the potential well per unit of incident photons
eclipsing

    phenomenon where one celestial object passes in front of another, leading to a temporary decrease in brightness as seen from an observer’s perspective
epochs

    specific time periods when observations are made to capture variations over time
equatorial mounts

    type of telescope mount used to align telescopes with the celestial coordinate system using e two axes, one aligned with the Earth’s rotation axis and the other perpendicular to it
exposure time

    duration for which a telescope camera is exposed to light during an astronomical observation
false alarm probability

    likelihood of concluding that there is a significant effect in the data (e.g. a proper motion) when, in reality, there is no such effect
field of view

    angular extent (measured in degrees sqaure) of the sky that can be captured by a telescope camera (array), determining the area of the celestial sphere that can be observed simultaneously
FITS

    Flexible Image Transport System, a standard file format consisting of a header containing metadata and data arrays organized in a binary format
flat image

    image taken to correct for non-uniform pixel sensitivity and other optical imperfections in the telescope cameras, obtained by imaging a uniformly illuminated, featureless source, such as a dome or the twilight sky
GAIA-DR3 catalog

    third data release of the GAIA mission, a space-based astrometry mission by the European Space Agency (ESA), providing a comprehensive and precise astrometric and photometric database of over a billion stars, galaxies, and other celestial objects in the Milky Way and beyond
Gaussian filter templates

    mathematical functions used in matched-filtering techniques to convolve with astronomical images for signal enhancement; chosen for their ability to smooth the data while preserving spatial information, making them effective in detecting specific signals in images
HDF5

    Hierarchical Data Format 5, a flexible and scalable file format providing a hierarchical structure that can store complex and multidimensional data, making it suitable for large and diverse datasets, such as astronomical images, simulations, and time-series data
hour angle

    angular measurement used to express the westward rotation of a celestial object relative to the observer’s meridian, typically measured in hours, minutes, and seconds to indicate the time elapsed since the object crossed the observer’s meridian
interpolation

    mathematical technique used to estimate values between data points in a dataset to fill in missing data or to obtain continuous functions from discrete observations
Large Array Survey Telescope (LAST)

    cost-effective multi-purpose astronomical observatory equipped with an array of telescopes designed to conduct wide-field imaging surveys of the sky in the visible-light spectrum
light curve

    plot of the flux from an astronomical source on the vertical axis against the corresponding time on the horizontal axis used to graphically represent how the source brightness changes over time
mask image

    binary image used to identify regions of an astronomical image that should be flagged during analysis, typically containing pixels with a value of 1 in regions of interest (e.g., sources or regions affected by near-edge effects) and 0 elsewhere
matched-filter

    signal processing technique used to enhance the detection and measurement of specific signals by convolving the image with a template or filter that designed to match the expected characteristics of the signal of interest
moment

    mathematical measure used to describe the shape, spread, and center of a distribution; the first moment is the mean, while the second moment yields the variance
near-edge effects

    distortions occurring at the edges of astronomical images due to the incomplete exposure of pixels near the image boundary leading to variations in photometric measurements near the image edges
passband

    a “window” that selectively lets through certain colors of light while blocking or attenuating others
periodogram

    mathematical technique used to analyze time series data and identify periodic signals within the data by highlighting prominent frequencies
photometric calibration

    process of converting instrumental magnitudes obtained from astronomical observations into calibrated magnitudes on a standard photometric system by comparing the instrumental magnitudes of reference stars with known magnitudes from photometric standards
photometric error

    uncertainty or variation in the measured brightness of an astronomical source due to various factors, such as instrumental noise, atmospheric turbulence, and variations in the source’s intrinsic brightness
photometry

    process of quantifying the brightness or intensity of light emitted or reflected by astronomical sources by measuring the amount of light captured by a telescope or camera sensor and converting it into a numerical value
pipeline

    series of automated and standardized data reduction and analysis steps applied to raw observational data, systematically calibrating, correcting, and extracting information from large volumes of data
pixel overlap

    region where adjacent pixels in an image (or sub-image) share some common area due to the finite size of the pixels
Point Spread Function (PSF) fit photometry

    photometric method that models the Point Spread Function (PSF) of an astronomical image to perform more accurate source measurements by convolving a theoretical PSF with the observed data and fits it to each detected source
potential well

    region within each pixel where incoming photons generate electrons via the photoelectric effect
proper motion

    apparent motion of an object on the celestial sphere over time due to its actual motion through space. It is typically measured in units of angular displacement per unit time
pulsation

    periodic expansion and contraction of a star, causing its brightness to vary over time due to changes in surface area and, consequently, temperature and light emission
raw image

    unprocessed and uncalibrated digital representation of the data captured by a telescope camera containing the original pixel values recorded by the detector without any corrections for bias, dark current, flat-field, or other instrumental effects
reduced $\chi^{2}$

    a measure of how well a given model fits the data, taking into account the number of data points (degrees of freedom) used in the analysis
register

    align multiple images so that they are properly synchronized and spatially coherent by adjusting the images’ position, rotation, and scale
relative photometry

    technique used to measure the magnitude of sources relative to each other by comparing their brightness in different epochs rather than obtaining absolute measurements of their brightness
root mean square (RMS)

    statistical measure used to quantify the average magnitude of a set of values or a signal calculated by taking the square root of the mean of the squared values
rotation

    spinning motion of a celestial object around its axis, revealing and concealing dark star-spots that cause fluctuations in brightness
sigma-clipped mean

    statistical method used in data analysis to calculate the mean of a set of values while excluding outliers by iteratively removing data points that deviate significantly from the mean (beyond a specified number of standard deviations, known as the sigma threshold) and calculating the mean again
signal-to-noise ratio (SNR)

    quantitative measure used to assess the quality of observational data by representing the ratio of the signal (e.g., the brightness of an astronomical source) to the background noise level
sky surveys

    systematic observations of the celestial sphere to map the distribution of celestial objects and probe astrophysical phenomena
Student’s t-statistic

    a measure that helps assess whether the difference between two groups is statistically significant (or just due to random variation) by taking into account the average difference between the groups and the variability within each group
temporal cadence

    the frequency with which a telescope collects data over time; telescopes with high temporal cadence are designed to minimize the time interval between consecutive observations
variability estimators

    statistical methods and algorithms used to quantify the degree and nature of variation in the brightness (or position) of astronomical sources over time
variability

    property of astronomical objects to exhibit changes in their brightness, luminosity, period, or other observable characteristics; can occur on different timescales, ranging from fractions of a second to billions of years
variable stars

    celestial objects whose brightness fluctuates over time due to intrinsic or extrinsic astrophysical phenomena
Vega-AB magnitude

    a standardized magnitude system used in astronomy to measure the brightness of celestial objects based on two key components: the Vega star (a bright star with a stable apparent magnitude often set to zero in all passbands) and the AB magnitude system (a standardized system that defines magnitudes based on the flux density of light)
visits

    process of observing a specific region of the sky for a predefined duration; multiple visits may be conducted for the same region to enhance data quality and ensure proper coverage
weight

    numerical values associated with each pixel in an image used to represent the degree of contribution of each pixel when calculating the signal or detecting the source; used to account for the variations in light or counts caused by the PSF
World Coordinates System (WCS)

    way to assign precise celestial coordinates (such as right ascension and declination) to each pixel in an image system used in astronomy to assign precise celestial coordinates to each pixel in an image
zero point

    calibration parameter that relates the instrumental magnitudes to a standardized magnitude system; it is essentially an offset or correction factor that brings the instrumental magnitudes onto a common scale