In this paper we developed a method for the mathematical modeling for the near-zero apparent motion (NZAM) detection of astronomical objects in series of CCD-frames using the methods of statistical and in situ modeling. Such method helps to evaluate the accuracy of computer vision in scope of the NZAM detection of astronomical objects. We have described all especial variables and preconditions for the methods of statistical and in situ modeling. The method with a maximum likelihood criterion and the method with the Fisher distribution were selected as specific algorithms for a NZAM detection of astronomical objects in scope of the research. A method for the mathematical modeling for a NZAM of objects of objects in a series of CCD-frames was developed using the C++ programming language. The modeling results were analyzed using the especial quality indicator, like a conditional probability of true detection, so the selected detection algorithms were evaluated using both statistical and in situ imitation modeling techniques. Statistical modeling, imitation modeling, in situ modeling, computer vision, OLS-evaluation, F-test, maximum likelihood criterion, near-zero motion detection, series of images COLINS-2023: 7th International Conference on Computational Linguistics and Intelligent Systems, April 20-21, 2023, Kharkiv, Ukraine ORCID: 0000-0001-9434-1081 (S. Khlamov); 0000-0001-8840-8278 (V. Savanevych); 0000-0001-6629-4927 (I. Tabakova); 0000-00032689-2679 (T. Trunova); 0000-0001-8086-237X (I. Levykin)
Nowadays the modern algorithms for a series of images processing should save the balance
between speed and quality of processing of such huge amount of information, which is produced by
data streams from the different sources. To proof the high quality of the object’s detection, such
algorithms should be tested not only in the real situation, but also in the virtual simulation using the
predefined dataset. This can be achieved using the statistical imitation [
The modern detection algorithms should detect and recognize objects with the different apparent motion: zero motion (fixed object), near-zero motion, normal motion, high-speed motion, etc. There is no one unified algorithm, which can detect and recognize all objects with motion in all described above cases. So, in our paper we focused on the detection algorithms for the objects that have a nearzero apparent motion (NZAM).
The object, which has a NZAM is the kind of objects, which has a very small shift in pixels
between frames at the moment of capturing. And this shift is commensurate with the measuring error
of its position. Such object has a velocity between frames in series that is less than or equal to 3 root
mean square (RMS) errors (3σ) of measurements of their positions [
2023 Copyright for this paper by its authors.
moving objects with a NZAM in the series of frames. Such types of moving objects are drones [
The main point to detect the object in an image is to recognize it after image filtering [
The purpose of this paper is to develop a method for the mathematical modeling for the NZAM detection of astronomical objects in series of frames using the methods of statistical and in situ modeling. Such method will help to evaluate the accuracy of computer vision in scope of the NZAM detection.
In general, the hypothesis H0 and alternative hypothesis H1 are verified by the maximum likelihood criterion [23] or other criterion from the statistical checking group called Bayesian [24]. In this case, the likelihood ratio will be like a final value of statistic for the appropriate criteria. Such value in the common case assimilates with the predefined critical values (calculated or even from the table) [25].
There are several various situations for the maximum likelihood ratio. Almost all of them depend on the knowledge of the variance σ2 of the object’s position. detection of a NZAM can be used:
So, in general, the following variations of the substitutional methods for the maximum likelihood variance σ2 of the object’s position is known; variance σ2 of the object’s position is unknown and only the estimation of such variance σ2est • • • can be used;
external variance estimation σ2out of the object’s position can be used based on the previous calculations according to the accuracy of previous measurements sets (for example, the already known instrumental error during observation).
In some case it can be known, otherwise the external estimation of the variance σ2 of object’s position is used for this purpose. Such external estimation in the common case is calculated from the estimation accuracy from the previous array of positional measurements.
The method for the object’s NZAM detection with known variance σ2 of the object’s position can
be presented as the following formula [
= ( 0, ) , = ( 0, ) are the vectors of object’s parameters along each coordinate; 0, 0 are the positional coordinates of object at 0 time; , are the velocities of object along coordinates x and y; cr is the threshold of a likelihood ratio. 2.2.
02 − 12 ≥ ,
In case when it is not possible and realistic to use the known variance of the object’s position, authors suggested using the developed detection algorithm based on the Fisher f-criterion [27]. Using the F-test it is possible to check a statistical significance of the object’s velocity along two axes (coordinates x and y). In such case a statistic of the f-distribution has no dependencies on the distribution of errors of the object’s position [28].
Also, the Fisher distribution statistics has already predefined values from the table [26, 29]. The method for the object’s NZAM detection based on the Fisher f-criterion can be presented as the following formula [22]:
12 − where r is the rank of a plan matrix Fx (Fx = r ≤ min(m, Nmea)) [26]; w = 1 is an amount of factors of a linear regression model (only apparent motion of object); m = 4 is the number of estimated parameters of the object’s motion along 2 axes: coordinates (x0, y0) at time τ0 of base frame’s timing and velocities (Vx, Vy) along each coordinate;
Nmea is a count of measurements of the investigated object from each image in series; fcr is a threshold of the Fisher distribution from the table [29].
According to the known count of measurements of the investigated object from each image in series Nmea, it is easy to determine the degrees of freedom for the f-distribution [29]. Also, the predefined significance level α helps to select the appropriate threshold of the f-distribution from the table. (2)
In common case, the errors of experimental frequencies in the mathematical modeling are defined by estimates of CPFD γ0 and CPTD γ1. The acceptable values for them were predefined by authors, so γ0accept = α/10 and γ1accept = 10−3. Also, the dependence of the number of experiments and the errors of experimental frequencies in the mathematical modeling were defined by the following formulas: N0exp = 102 / γ0accept, N1exp = 102 / γ1accept = 106.
According to the research purposes only 103 of the smallest values of decisive statistics [30] were selected. Such data set also can be used for the Wavelet coherence analysis [31] as an alternative method of data analyzing. 3.2.
To perform the mathematical modeling the following preconditions were defined: • rectangular coordinate system (CS) with zero point (0;0) was used during the mathematical modeling; • velocity module V is presented in the RMS error of measurement deviations of the position of object (V = kσ);
object’s apparent motion is uniform and linear, so the velocity module is = √ 2 + 2; • modeling of the appropriate velocity module V is based on the angle , so the velocity projections are the following: Vx = Vsin and Vy = Vcos; • preliminary calibration sessions prepare the external variance σout2, which is used as a known variance σ2 of the object’s position for the statistical and in situ modeling for method (1); • modeling of the hypothesis H0 (V = 0) provides a possibility to calculate a threshold cr for the method (1) in accordance with the predefined significance level α; • in accordance with the predefined significance level α and the appropriate degrees of freedom (1, 4), the threshold fcr for the f-distribution in method (2) is also predefined and can be selected from the table [29]. 3.3.
The appropriate test data for in situ modeling were selected in scope of the current research from the following real observatories: ISON-NM and ISON-Kislovodsk with unique observatory codes “H15” and “D00” accordingly. The information about these observatories is provided in the Table 1. The observatory codes are unique and approved by the Minor Planet Center (MPC) [32] of the International Astronomical Union (IAU) [33].
Test data for in situ modeling are consist of the different series of CCD-frames that were collected during the regular observations by the various CCD-cameras. The information about the CCDcameras that are installed on the telescopes from the observatories list above is presented in the Table 1. This table contains the following information about CCD-camera: model and its parameters, like resolution, pixel size and exposure time.
Each series of CCD-frames includes the different investigated objects in each frame of series. The main restriction during the test data preparation was selection only series of frames that contain the appropriate investigated object in each of them. The number of frames Nimg in series was from four to eight. The average time between such frames was about ten minutes. 3.4.
The random values for the method of statistical modeling are normally distributed [34] and generated using the Ziggurat method [35]. This is a method for a sampling of the pseudo-random numbers. It belongs to the methods type for sampling rejection and its underlying source is related to the uniform distributed random numbers. Ziggurat method in general is a pseudo-random number generator, which uses the already predefined tables for randomization of numbers.
Ziggurat method generates the appropriate values that have a probability distribution, which always monotonically decreases. It can also be applied for the normal distribution as a symmetric unimodal distribution by selecting the value from one half of the distribution and then randomly selecting what part of the value will be drawn from. The common value created by the Ziggurat method requires only a generating of the one random float point and one random index of the table. After this the appropriate table will be looked up with the further multiply operation and one comparison.
To generate the random value distributed by the normal law Nx(m, 2) with mathematical expectation m and standard deviation , the randomized variable by the normal law Nx(0, 1) should be multiplied with standard deviation and then added to the mathematical expectation m. 3.5.
To perform the mathematical modeling the following constants were used for modeling: • significance level (error of the 1st kind) α = (10−3; 10−4); • count Nimg of frames of the investigated series Nimg = (4; 6; 8; 10; 15); • velocity coefficient k = (0; 0.5; 1; 1.25; 1.5; 1.75; 2; 3; 4; 5; 10); • mean of the external estimation of RMS error of position m(σout) = 0; • external estimation of RMS error of position σ(σout) = (0.15; 0.25); • angular direction of the object’s apparent motion = 45°; • thresholds of a f-distribution with (1; 4) freedom degrees are: fcr = 74.13 (α = 10−3) and fcr = 241.62 (α = 10−4) [29]. 3.6.
Mathematical modeling for a NZAM detection of objects in the series of CCD-frames is described
in the papers [
Each stage of modeling for verification of both hypotheses has the common sequence of actions: • determining of the experiment parameters; • modeling of the appropriate number of experiments including the OLS-evaluation of the object’s motion parameters; • determining of both final values of the likelihood ratio of the methods (1) and (2); • determining of both thresholds of the likelihood ratio and the f-distribution according to the appropriate significance levels; • comparing of both final values of the likelihood ratio of the methods (1) and (2) with the appropriate thresholds that were calculated during modeling of the hypothesis H0; • determining of the CPTD: Dtrue = Nexc / N1exp, where Nexc is an amount of exceeding of critical values.
The described above algorithm for the mathematical modeling is presented in the Figure 1 in view of the UML-diagram. According to the described in section 3.1. number of experiments for mathematical modeling the following common sequence of actions is performed for each k-th experiment (Figure 2): • formation the set of positional measurements of objects with a NZAM; • adding the appropriate deviations using random generator for each positional measurement of objects for modeling the hypothesis H0; • adding the appropriate velocity for each positional measurement of objects for modeling the hypothesis H1; • performing the OLS-evaluation of the object’s motion parameters; • performing the interpolation of the objects coordinate’s estimation;
According to the calculated CPTD of the object’s NZAM, the detection curves for the methods (1) and (2) were created as a proof of efficiency of the selected detection algorithms. Such detection curves were plotted using the basic calculations according to the modeling method and the received calculated information in view of CPTD. Such information is provided below in the following sections of this paper. 4.1.
The processing results in terms of the CPTD after statistical modeling stage during mathematical modeling are presented in the Table 3. This table contains the following information: count of frames Nimg, significance level α, list of the different values of generated apparent velocity in the RMS error of measurement deviations of the position of object (V = kσ), where the coefficient k was taken according to the definition above. Count of frames Nimg = 15, significance level α = 10−4 Count of frames Nimg = 4, significance level α = 10−3 Count of frames Nimg = 8, significance level α = 10−3 Count of frames Nimg = 15, significance level α = 10−3
According to the data received after processing and applying the statistical modeling stage during mathematical modeling from the Table 3, the detection curves for the methods (1) and (2) were created.
The Figure 3 shows the detection curves for objects with a NZAM for the method (1) with variance σout = 0.25 (dotted line) and the method (2) (solid line) after the statistical modeling stage during mathematical modeling.
The x-axis is a velocity V of the apparent motion of objects with a NZAM and the y-axis is a CPTD Dtrue. e) f)
Figure 3: Detection curves after the statistical modeling stage during mathematical modeling for the objects with a NZAM for: a) Nimg = 4 and α = 10−4; b) Nimg = 8 and α = 10−4; c) Nimg = 15 and α = 10−4; d) Nimg = 4 and α = 10−3; e) Nimg = 8 and α = 10−3; f) Nimg = 15 and α = 10−3. 4.2.
During the in situ modeling stage of the mathematical modeling as a precondition step the RMS error of position σout of objects in series of CCD-frames was determined from the previous calculations as an instrumental error of the telescopes, which were used under research. The RMS error of position σout is presented in the Table 4 for each used telescope.
Also, the total number of all investigated objects Nobj with nullable motion is presented for the appropriate telescopes in the same table below.
The processing results in terms of the CPTD after in situ modeling stage during mathematical modeling are presented in the Table 5. This table contains the following information: count of frames Nimg, significance level α, list of the different values of generated apparent velocity in the RMS error of measurement deviations of the position of object (V = kσ), where the coefficient k was taken according to the definition above.
The main point of the in situ modeling is that the objects with no apparent motion (fixed objects) were taken from the prepared internal catalogue (IC) with fixed objects in all CCD-frames of the investigated series.
According to the data received after processing and applying the in situ modeling stage during mathematical modeling from the Table 5, the detection curves for the methods (1) and (2) were created.
The Figure 4 shows the detection curves for objects with a NZAM for the method (1) with variance σout = 0.25 (dotted line) and the method (2) (solid line) after the in situ modeling stage during mathematical modeling.
The x-axis is a velocity V of the apparent motion of objects with a NZAM and the y-axis is a CPTD Dtrue.
The architecture of an information system consists of the following main components: telescope -> software for saving the raw data -> server for the data collecting -> developed method for mathematical modeling (Figure 1) -> main detection algorithms -> accuracy indicators analysis.
A method for the mathematical modeling for a NZAM of objects in a series of CCD-frames was developed using the C++ programming language. A few general C++ methods are presented below: • generating the appropriate deviations using random generator for each positional measurement of objects for modeling the hypothesis H0 (Figure 5); • performing the OLS-evaluation of the object’s motion parameters (Figure 6); • modeling of the hypothesis H0 (Figure 7); • modeling of the hypothesis H1 (Figure 8).
We developed a method for the mathematical modeling for a NZAM of objects of objects in a series
of CCD-frames. The methods for statistical [
The especial variables and preconditions for the mathematical modeling were defined as well as their clarification. The method with a maximum likelihood criterion (1) [36, 37] and the method with Fisher distribution (2) [22, 38] we selected as the detection algorithms for a NZAM of objects for our research.
A method for the mathematical modeling for a NZAM of objects in a series of CCD-frames was developed using the C++ programming language (Figures 5-8). The modeling results were analyzed using the especial quality indicator (CPTD).
The obtained results from the Tables 3, 5 and Figures 3, 4 showed that the method (2) for the object’s NZAM detection based on the Fisher f-criterion is more delicate to changes in the object’s apparent motion and is very effective when the object’s velocity is less than V = 3σ. The CPTD is up to 95 percent and only depends on the variance values. Such results were confirmed by both modeling types statistical and in situ.
The received results after processing by the developed method including the generated experiments with statistical and in situ data will be also used for the Wavelet coherence analysis [39, 40].
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