Technological progress in the production of cameras as charge-coupled devices (CCD) [1] and telescopes demonstrates continuous acceleration, reflecting modern trends in scientific and engineering developments. New materials and more efficient components, such as photodetectors and optical systems, contribute to the creation of cameras and telescopes with increased resolution and sensitivity.
Today's digital cameras possess impressive characteristics, including resolutions exceeding 100 megapixels, significantly surpassing those of previous models. Telescopes are also keeping pace: it is expected that the Large Synoptic Survey Telescope (LSST) (Figure 1) will have a resolution of around 3.2 gigapixels [2] The speed of data acquisition and processing has also increased noticeably. Modern cameras are capable of recording and processing data orders of magnitude faster than their predecessors, achieving shooting speeds of up to 20 frames per second while maintaining full resolution. The development of parallel computational algorithms contributes to more efficient processing of such a big data [3] obtained from cameras and telescopes, opening up new possibilities for scientific research and practical applications in various fields including astronomy [4]. __________________________ As the resolution of cameras and telescopes has increased over time, allowing for the registration and analysis of more detailed images of the night sky, this process directly impacts the ability to detect and study various objects in space [5]. Additionally, wide-angle cameras allow for an expanded field of view, significantly increasing the number of objects captured in a single frame. This is particularly useful when studying areas with high stellar density or when searching for distant galaxies and cosmic objects [6]. However, even with the increase in resolution and field of view, the quality of astronomical imaging of objects, especially those located at great distances and with low brightness, remains a challenge for modern cameras and telescopes. One of the main factors affecting image quality is the level of noise. When imaging faint objects, even a small amount of noise on the camera sensor can significantly distort the image and hinder its interpretation. Despite significant improvements in noise reduction in modern cameras, this aspect remains a problem when dealing with very faint and distant objects [7].
Another important factor is atmospheric conditions. Atmospheric turbulence and atmospheric distortions can significantly affect the quality of images and its typical shape or form [8], especially when working with high magnifications. This limitation can be overcome by using space telescopes, which are located beyond the Earth's atmosphere.
It is also worth considering the technical limitations of cameras and telescopes, such as limited sensitivity to certain wavelengths or limitations on dynamic range. Research and development efforts continue with the aim of overcoming these limitations; however, this remains a relevant area of research in astronomy and optics.
With these improvements in cameras and telescopes comes an increase in the volume of data in astronomical catalogs [9]. This is due to the increasing number of observations conducted, the use of more advanced instruments, and the expansion of the coverage area of the celestial sphere. So, it is necessary to develop the mathematical methods for the automated data mining [10] of the reference stars [11] from astronomical CCD-frames received from a such big volume of astronomical data.
In recent decades, astronomy has undergone a data processing and analysis revolution due to the implementation of large astronomical projects and the use of advanced observational tools. These modern instruments collect and process vast amounts of data, extracting valuable information about celestial objects and phenomena [12].
One of the main sources of data is ground-based telescopes and observatories. These facilities, located around the world, gather data on stars, galaxies, and other celestial objects using various observation methods, including optical, infrared, and ultraviolet spectra. Thanks to ground-based telescopes and Virtual Observatories [13], astronomers have access to a wide range of objects and can study their properties and evolution [14].
New telescopes are capable of generating huge volumes of high-resolution data. This includes not only images but also spectroscopic data and information on temporal changes in the brightness of objects. Modern big data processing techniques and machine learning allow for efficient extraction of information from data arrays, identification of interesting objects, and automatic compilation of astronomical catalogs [15]. Additionally, international projects and collaborations are a contributing factor. Astronomers from around the world join forces to conduct joint observations and create extensive catalogs, leading to more comprehensive coverage of the celestial sphere and enrichment of data [16]. This growth of data in astronomy provides unique opportunities for scientific research.
Pattern recognition of both celestial objects and patterns themselves is an important tool for analyzing data obtained from astronomical surveys and observations [17]. Pattern recognition involves the process of analyzing and identifying astronomical objects in images obtained from telescopes and space observatories. This process may include the detection and classification of stars, galaxies, cosmic objects, and other astronomical phenomena based on their characteristics such as position [18], brightness [19], shape, spectral features, and others. Initially, data collection and preparation occur -this may include observational information from telescopes, including images, spectra, and time series. Then, data processing, including noise filtering, background alignment [20], image quality enhancement, and identification of objects of interest.
Space telescopes and missions are also important sources of astronomical data. Space telescopes such as Hubble [21], Kepler, and the upcoming James Webb Space Telescope provide high-quality data on distant galaxies, exoplanets, and other objects in the Universe. These missions allow astronomers to explore cosmic objects without the influence of Earth's atmosphere and expand our knowledge of the Universe [22].
Radioastronomical observations also play a crucial role in astronomical research. Using radio telescopes such as the Very Large Array (VLA) and ALMA, astronomers study radio sources and processes occurring in the radio spectrum. These observations provide information on various phenomena, including active galactic nuclei, radio pulsars, and cosmic microwave background radiation [23].
Recently, the detection of gravitational waves has become a new source of astronomical data. Using interferometers such as LIGO and Virgo, astronomers detect events such as black hole mergers and neutron star mergers. These observations provide new data on cosmic phenomena that were previously inaccessible to observation.
All astronomical data sources even small play a key role in scientific research, providing astronomers with valuable information about cosmic objects and processes. Modern observational tools and data processing technologies allow researchers to expand our knowledge of the Universe and address important scientific questions.
Data mining, in the context of astronomical research, is a method of data analysis that relies on computational algorithms to discover patterns, trends, and structures in space data [24]. With the increasing volume of astronomical data generated by advancements in observational technologies and expansion of spatial coverage, data mining becomes a crucial tool for extracting valuable information from these data arrays.
Astronomical data is often characterized by high dimensionality, complex structure, and a significant amount of noise. Data mining enables efficient processing and analysis of such data, revealing hidden patterns that may not be apparent at first glance. Data mining methods encompass various approaches such as clustering, classification, regression, associative rules, and others.
Applied to astronomical data, data mining can serve various purposes, including [25]:
• discovery of new object classes: using clustering and classification methods, data mining can unveil new classes of astronomical objects hidden within the dataset.
• identification of correlations and dependencies: by analyzing multiple parameters and characteristics of astronomical objects, data mining can help identify correlations and dependencies among them, leading to new scientific discoveries and understanding of physical processes.
• prediction of temporal changes: using regression and time series methods, data mining can be employed to predict temporal changes in luminosity or other characteristics of astronomical objects. Thus, data mining represents a powerful tool for the analysis of astronomical data [26], playing a significant role in unraveling the mysteries of the Universe and deepening scientific understanding of space.
Pattern recognition of celestial objects [27] and patterns is closely related to concepts such as data mining and knowledge discovery in databases (KDD) in the context of astronomical research.
Data mining is the process of automated analysis of large volumes of data to identify interesting and non-obvious patterns, templates, and trends. In astronomy, where data on astronomical objects are extremely extensive and multiparametric, data mining plays a key role in processing and analyzing this data. Pattern recognition of objects and stars is one of the stages of this process, where machine learning algorithms [28] and statistical methods [29] are applied for classification and clustering of objects in images or observational data.
On the other hand, KDD encompasses a wide range of methods and techniques for identifying and interpreting patterns and new knowledge from databases. In astronomy, where data often have a complex structure and may contain noise, KDD helps researchers identify hidden relationships between various parameters of astronomical objects, leading to the discovery of new physical laws and understanding of the Universe [30]. Thus, for pattern and star recognition, the use of techniques such as data mining as well as KDD in astronomy is an important aspect as they help researchers gain valuable knowledge from cosmic astronomical data. Knowledge discovery in databases in astronomy is a methodology for analyzing and interpreting extensive datasets collected from various observational sources in space (Figure 2). This approach involves the application of various algorithms and models to identify important patterns, trends, and structures within astronomical data, enabling astronomers to extract new knowledge and draw scientific conclusions. Knowledge discovery in databases in astronomy leads to a number of scientific outcomes, including:
• classification of stellar spectra: utilizing machine learning methods and data analysis, astronomers can classify stars based on their spectral characteristics. For example, clustering spectral data enables the identification of various types of stars and determining their evolutionary stages.
• discovery of new classes of galaxies: analyzing astronomical catalogs using knowledge discovery algorithms can lead to the discovery of new types of galaxies, such as galaxies with unusual shapes or structures, which require further investigation and explanation.
• prediction of gamma-ray bursts: time series methods and statistical analysis can be employed to forecast temporal changes in the activity of gamma-ray bursts, enabling astronomers to prepare for observations and studies of such phenomena.
• identification of gravitational lenses: Analyzing large astronomical databases using knowledge discovery algorithms can assist in identifying and classifying gravitational lenses, which is crucial for studying dark matter and furthering our understanding of cosmology. These examples illustrate the significance of knowledge discovery in databases in astronomy, as it plays a crucial role in scientific research and helps expand our understanding of the Universe.
The uniformity of the standard form of the image of objects [8] is an important factor influencing the subsequent process of identification with the astronomical catalogue [31]. Therefore, it is necessary to conduct an in-depth analysis of literature data to compare methods for preparing images for the identification process itself. Such methods are expected to reduce the shift in the positional coordinates of the frame center between the frames themselves in the series.
For example, classical methods of computer vision [32] and object image recognition [16] are not able to provide the required level of processing speed. These methods require the analysis of all pixels of potential objects to determine their typical shape. However, when the standard form is heterogeneous, objects are confused, which increases the processing and identification time. Methods for estimating image parameters [33] are based on the analysis of only those pixels that potentially belong to the object under study. Their disadvantage is the inability to determine specific pixels and reject those whose intensity exceeds a specified limit value initially accurately.
In the study [34], the authors use automatic selection of a reference point to select calibration frames. However, this is not a requirement for the identification process itself. Because if there are artifacts in the image, these control points may be false. Thus, the accuracy of identification with real objects from the astronomical catalog decreases. The works [35] propose segmentation method. However, it only work with single images of objects. That is, in the case of a variety of standard shapes (stroke, extended, circular), this method will not provide the necessary accuracy due to the ambiguity in the number of brightness peaks.
This variety of typical shapes also influences various methods of Wavelet transform [36] and time series analysis [37]. The disadvantage of these methods is that they can only work with "pure" measurements, so image heterogeneity will greatly spoil the overall indicator. Another implementation is presented in the study [38] in the form of an additional calibration procedure to avoid the internal coma of the telescope's secondary mirror. But, to equalize brightness and remove "highlights", there is a brightness method that is more improved in accuracy and quality using an inverse median filter [39]. However, the disadvantage of these implementations is the poor accuracy of positional coordinate estimates during the process of identification between frames of the same series.
The matched filtering procedure is also known [40], but it uses only an analytical image model. The disadvantage of this procedure is the inaccuracy of identification when the typical image of an object is different in different frames of the series. The classical method of adding frames [41,42] to improve the "super" frame is also ineffective in the case when the SSO image does not have clear boundaries on all digital frames of the series. Therefore, it is necessary to develop the mathematical computational methods for automatic selection of reference stars in astronomical images, which will take into account the peculiarities of digital frame formation.
The preliminary identification procedure [43] allows us to obtain linear plate constants (𝑎 𝑝𝑙1 , 𝑏 𝑝𝑙1 , 𝑐 𝑝𝑙1 ) and (𝑎 𝑝𝑙2 , 𝑏 𝑝𝑙2 , 𝑐 𝑝𝑙2 ), which will determine the relationship between the coordinate system (CS) of the CCD-frame and the tangential (ideal) coordinate system of CCDframe:
where ξ and η -ideal (tangential) coordinates of reference stars; x, y -measured coordinates of reference stars in the coordinate system of CCD-frame. The calculated linear constants of the plate allow us to obtain estimates of the equatorial coordinates of objects in the frame using the following expression:
00 00 00 00 00 arctg (2) where 𝛼 00 , 𝛿 00 -equatorial coordinates of the optical center of the CCD-matrix. In the final conversion from CCD-frame coordinates to equatorial coordinates, a cubic model of plate constants is used, which ensures reliable identification and measurement of positions throughout the frame.
Practice shows that the concentration of bright measurements in a certain area of the CCD-frame (for example, in the center) can increase the identification accuracy in this area by reducing it in other areas of the same CCD-frame (Figure 3, left).
To ensure almost equal accuracy of object coordinate measurements throughout the entire CCDframe, it is advisable to distribute candidates for reference stars evenly throughout the frame.
Thus, a uniform distribution of identified pairs throughout the entire CCD-frame will ensure the necessary uniformity in the accuracy of determining the equatorial coordinates of objects throughout the entire CCD-frame (Figure 3, right).
Therefore, it is necessary to fragment the frame into 𝑀 𝑟𝑒𝑔 × 𝑀 𝑟𝑒𝑔 areas (sections) of equal area for uniform distribution of identified pairs on the CCD-frame when selecting candidates for reference stars. In each frame fragment, the same number of objects with a bright image (stars) is selected.
The number of measurements of the frame N mea and stars from the forms of the astronomical catalog N st obtained during observations and intra-frame processing is divided by the number of frame sections.
Next, in each such area 𝑁 𝑚𝑒𝑎 ⁄ 𝑀 𝑟𝑒𝑔 2 of the brightest measurements of the frame and 𝑁 𝑠𝑡 ⁄ 𝑀 𝑟𝑒𝑔 2 of the brightest stars in the catalog are selected.
At each stage of selecting guide stars, measurements of nearby objects are excluded from the set of candidates. This means that the distance between them does not exceed the previously specified value 𝑟 (𝑚𝑒𝑎_𝑔𝑟𝑜𝑢𝑝) . That is, the i-th and m-th measurements of the CCD-frame are excluded from candidates for reference stars if the following condition is met:
where x meamnfr and y meamnfr are the positional coordinates of the measurement of a nearby object in the SC of CCD-frame.
Like (3), measurements of nearby stars from clusters/compact groups of stars in the astronomical catalog are excluded from consideration. This is the case when a nearby object has a comparable or greater brilliance. The criterion for such membership is the presence of a nearby star at a distance less than a predetermined value 𝑟 (𝑚𝑒𝑎_𝑔𝑟𝑜𝑢𝑝) :
where α cat and δ cat -positional coordinates on the celestial sphere in the astronomical catalog form. Another important criterion for rejecting candidates is the absence of a brightness peak in the image of the object on the CCD-frame. The criterion for such absence of a peak can be considered the approximate equality of the brightness of the potential peak and the brightness of the pixels 𝐴 𝑖𝑘 from the region 𝛺 𝑝𝑒𝑎𝑘 of size 𝐶 𝑝𝑒𝑎𝑘 × 𝐶 𝑝𝑒𝑎𝑘 of pixels centered in the potential peak. Approximate equality is the difference between the brightness of the pixels of a potential peak and the brightness of the region by no more than 𝑁 𝐴𝑝𝑒𝑎𝑘 brightness units.
In this way, sets of measurements are formed from the side of the CCD-frame and the astronomical catalog, which in the subsequent stage will take part in vaporization and testing of hypotheses about the correspondence of the "measurement-formula" identification pair.
The architecture of data mining process of the automated reference stars selection includes the following sequence of operations (Figure 4). of the brightest measurements of the CCD-frame and catalog forms, respectively, are additionally selected. 4.2. Rejection of selected candidates for guide stars is performed.
4.2.1. Rejection of measurements of objects close to each other (3). 4.2.2. Rejection of measurements of stars in the astronomical catalog if they belong to clusters or compact groups of stars (4). 4.2.3. Rejection of measurements of objects with images without brightness peaks (5). 4.3. Identification of selected frame measurements and catalog forms with the formation of identified pairs. 4.4. At each iteration, calculation/refinement of linear constants of the plate with a higher degree model.
Rejection of identified pairs based on the total deviation 𝛼𝛿𝑖𝑗𝑘 between estimates of equatorial coordinates in the identified «measurement-form» pair:
where
-the average modulus of deviation of an identified pair in equatorial coordinates on the set of selected identified pairs;
K rej -coefficient of the condition for rejecting «measurement-form» pairs from the set; k -number of the identified «measurement-form» pair; N cou -number of identified «measurement-formula» pairs used to calculate plate constants. 5. Final calculation of linear constants of the plate.
The object of study are the images of the Solar System objects (SSO) (like stars, asteroids, comets) and any other space objects (like space robots [44], drones [45], satellites [46]) detected in a series of CCD-frames. The initial series for the study were obtained from a variety of telescopes installed at observatories in Ukraine and around the world. Namely, the ISON-NM observatory, the SANTEL-400AN telescope (New Mexico, USA); Vihorlat Observatory, VNT telescope (Humenne, Slovakia); Odesa-Mayaky Observatory, OMT-800 telescope (Mayaki, Ukraine); Cerro Tololo observatory, PROMPT-8 telescope (La Serena, Chile) [47]. All mentioned above observatories were approved and confirmed by the Minor Planet Center (MPC) as an official organization for the observing and reporting on minor planets or SSOs under the auspices of the International Astronomical Union (IAU) [48].
To verify the developed mathematical computational methods for automatic selection of reference stars in astronomical images, testing was carried out on a series of frames containing 27,352 measurements. Such a total number of measurements was successfully identified with the astronomical catalog.
The USNO B1.0 catalog was used as a photometric catalogue. The catalog contains angular positional coordinates and magnitudes of more than one billion SSOs, formed over 3.6 billion measurements.
When conducting research, the following values of the parameters of the developed methods were assumed:
• the number of the brightest measurements of the CCD-frame was 𝑁 𝑚𝑒𝑎 = 400; • the number of the brightest measurements of the catalog forms for selecting candidates for reference stars was 𝑁 𝑠𝑡 = 600; Histogram of the distribution of deviations of the equatorial coordinate (right ascension (RA)) of reference stars from the brightness and coordinates of objects in the rectangular coordinate system of the CCD-frame is presented in Figure 5. The received dependence of deviations of equatorial coordinates from the brightness assessment of objects in frame is presented in Figure 8. The research result based on series of frames with 27,352 measurements is presented in Table 1. 1 indicators show the successful application of the developed methods. The standard deviation of frame identification errors in this case is 5-7 times less than without using the developed methods.
Existing methods for basic image processing [41] and computer vision [32] were analyzed. However, the speed and accuracy of identification by such methods directly depends on the characteristics of the formation of a series of digital frames. There is also a dependence on the completeness of the astronomical catalog with data and on the constancy of the typical image [8] of the object in all frames of the series. Therefore, to develop the methods for automated data mining of the reference stars from astronomical CCD-frames and certain rules and criteria for rejecting candidates at each iteration were proposed.
The obtained research results, as well as the developed mathematical computational methods for automatic selection of reference stars in astronomical images, were implemented in the C++ programming language. This code was implemented at the stage of intra-frame processing of the Lemur software package (Ukraine) [49] for the automated detection of new and maintenance of known objects within the CoLiTec project [50]. The developed mathematical computational methods, implemented in Lemur software (Ukraine), was used during the successful identification of CCD-frames, which contained a total of more than 800,000 SSOs. Their measurements were also successfully identified with known astronomical catalogs.
Obtained in Table 1, the results are determined by the uniform distribution of candidates for reference stars, as well as correctly selected conditions and rejection criteria. It clearly indicates that the assigned tasks have been successfully completed. The research showed that the usage of the developed methods reduces identification errors with cataloged (reference) objects by 5-7 times. This significantly affects the quality and accuracy of a few tasks for detecting the trajectories of objects.
In recent decades, astronomy has undergone a data processing and analysis revolution due to the implementation of large astronomical projects and the use of advanced observational tools. These modern instruments collect and process vast amounts of data, extracting valuable information about celestial objects and phenomena. In astronomical images obtained using telescopes and cameras, there are from 1 to 100 thousand or more stars depending on the resolution and exposure time. These objects are fixed against the background of the frame and have constant positions in the celestial sphere. To determine which part of the sky corresponds more accurately to a given frame, it is necessary to associate the frame with known astronomical astrometric and photometric catalogs.
These catalogs contain millions of position values of various stars as static objects. Having such information in the form of big data, as well as a huge amount of classified and clustered data in the form of databases, computational methods for fast extraction of the necessary data from them need to be developed. For this purpose, classical methods of "knowledge discovery in databases" (KDD) and Data Mining exist. However, for their proper application, it is necessary to classify the input data set for subsequent analysis and rejection. The implementation of these methods is closely related to the developed mathematical computational methods for automatic selection of reference stars in astronomical images.
We presented the developed Lemur software of the CoLiTec (Collection Light Technology) project, which is implemented as a client-server application for the processing of astronomical data using the data mining and KDD methods. As described in this article the KDD with the data mining step is very useful for the data optimization to receive only the helpful data with reference stars.
The research was performed with help of all observatories, useful tools and observers who provided astronomical data for testing the developed Lemur software with implementation of the data mining methods. The research was supported by the Ukrainian project of fundamental scientific research "Development of computational methods for detecting objects with near-zero and locally constant motion by optical-electronic devices" #347 in 2024-2026 years.