3D Multi-Phase Sub-Pixel PSF Estimation Based on Space Debris Detection System

Abstract

The distribution of diffuse spot energy can be used to sensitively evaluate the aberrations and defects of optical systems. Therefore, the objective and quantitative measurement of diffuse spot parameters is an important means to control the detection quality of space debris detection systems. At present, the existing optical system dispersion measurement method can only judge whether the energy distribution meets the index. However, these methods ca not provide an objective quantitative basis to guide the installation process. To solve this problem, a mathematical simulation model of 3D multi-phase sub-pixel PSF distribution is proposed. According to the relation between the CCD target plane and the theoretical image plane (focal plane, defocus, and deflection), the diffuse spot distribution of the optical system is simulated with different phase combinations. Then, Pearson Correlation Coefficient (PCC) is used to evaluate the matching similarity of the diffuse spot image. The simulation results show that when the PCC is greater than 0.96, the distribution of the two diffuse spots can be identified as matching. This also confirms the accuracy of the proposed PSF model. Then, the focusing deviation of the system being tested can be analyzed according to the phase size of the diffuse spot simulation image. This method can quickly and accurately guide the focal surface installation and testing of the system. Therefore, the purpose of improving the detection accuracy of space debris is achieved. It also provides a quantitative basis for the engineering application of optical detection systems in the future.

1. Introduction

With the increasing frequency of space launch activities, the amount of space debris increases rapidly. It makes orbital resources become very scarce and seriously threatens the safety of spacecraft in orbit [1,2]. Therefore, in order to develop and utilize space resources safely and continuously, it is necessary to continuously improve the tracking and monitoring technology of space debris, enhance the detection and identification ability of space debris, and master the target characteristics and distribution dynamics of space debris [3].

Optical observation is an important means of space debris monitoring, and the detection performance of optical systems is crucial to the quality of space debris research. For some detection optical systems such as telescopes (ground-based and space-based), it is necessary to acquire the model of the PSF. The PSF model is one of the important technical indexes to evaluate the detection optical system. If the PSF model is built accurately, the accurate star point diffuse spot distribution can be simulated accurately [4,5,6].

At present, there are probably two mainstream methods for PSF modeling estimation. One is the parametric method. Physical parameters of the detection system can be used to build a PSF model, such as the relative aperture, focal length, wavelength, and other parameters. Firstly, the estimation precision is set according to the task requirement, and then the model complexity is determined. Generally, the higher the accuracy requirement of the PSF, the higher the complexity of the model will be. The PSF model is used to simulate the star target image. The second method is not to use the parameter model. The star point target at different positions of the field of view is used as the measurement value of the PSF across the field of view, and the PSF of the star point or other positions is estimated based on the data.

In 1969, Moffat proposed the classic Moffat PSF model [7]. The PSF model is represented by a function of N empirical parameters, where N is not fixed. Usually, the Moffat model is accurate with only two parameters. Moffat distribution can accurately describe the point spread function in star point simulation, but neither Gaussian distribution nor Lorentz distribution can accurately describe the characteristics of the point spread function in both wings. Generally, the PSF appears as a sharp correction peak and a wide halo extension, and the Moffat model is still widely used today because of its good approximation to the PSF peak. Simon Zieleniewski proposed a PSF modeling method using fine parameters, which interpolates PSFs of different wavelengths [8,9], as shown in Figure 1. This model method can use harmonics to create accurate and computationally efficient PSF data cubes. The PSF varies significantly with the wavelength and type of AO system, so it is necessary to create a detailed PSF in all wavelength channels for accurate simulation. The AO simulations showed that a single PSF introduced misleading features into the simulated observations of HARMONI, so an analytical model was used to parameterize the PSF as a function of the wavelength for interpolation. Serre et al. show that the PSF of the VLT telescope MUSE can be accurately modeled from the Moffat profile, where two parameters vary smoothly with wavelength. Using the point extension function fitting code eltPSFfit developed by J. Liske, Simon Zieleniewski et al. could fit the radial profile of each PSF using several analytic functions. Domestic Mingying et al. carried out modeling of the PSF using the Cauchy distribution [10].

As we know, the modeling methods of the PSF both at home and abroad are mainly based on parameter modeling. The substantial advantage of the parametric PSF is to compress all the important information about the physical PSF into a small number of parameters. The values of these parameters can be used for comparison, correlation, or any statistical analysis. In the aspect of data-driven non-parametric PSF modeling, a large amount of test data is needed as a reference for estimation. However, there are often errors in the test data, and the resulting estimates are not very accurate.

The multi-phase sub-pixel PSF simulation analysis method proposed in this paper not only uses the parameters of the optical system, but also comprehensively considers the phase distribution characteristics of the point source array. Based on different phase point sources, sub-pixel sampling is performed to realize sub-pixel PSF reconstruction. The simulated diffuse spot distribution is more accurate using this proposed method. Through comparison and analysis with the actual star distribution map, it can provide more accurate and reliable interpretation data for focusing work.

Therefore, according to the optical dispersion theory, this paper theoretically analyzes the possible relationship between the focal plane and the theoretical image plane of the CCD and proposes a 3D multi-phase sub-pixel PSF estimation method. This proposed method is not only using the parameters of the optical system, but the phase distribution characteristics of the point source array are also considered comprehensively. Based on the results of the comparison between two traditional algorithms and the proposed algorithm, the advantages of the 3D multi-phase sub-pixel PSF estimation method are obvious. In addition, this paper also proposes a comparison of the energy distribution of the dispersion spot simulation image with the actual image. Then, the energy distribution deviation of the star spot image is given quantitatively. Compared with the traditional focusing method, the subjective factors are eliminated, the inspection standard is unified, and the quantitative basis is provided for the focal plane adjustment and testing of the optical system.

2. Multi-Phase Sub-Pixel PSF Estimation

2.1. Principle of Star Detection

According to the Fraunhofer diffraction theory, the light intensity distribution of the diffraction image is no longer an ideal image point but a diffuse spot after passing through the optical system. That is called the Point Spread Function (PSF). In the ideal star point imaging, the light intensity before and after the image plane is symmetrically distributed and changes with the different field of view. However, in the actual optical system imaging, aberration can easily destroy this symmetry. The dispersion energy distribution can reflect the optical aberrations and defects very sensitively, so the imaging quality of the optical system can be controlled via quantitative measurement of the dispersion parameters [11,12].

The intensity distribution of the point source depends on the shape of the pupil. For an ideal optical system, the intensity distribution of the star image only depends on the pupil shape [13]. In the case of a circular pupil, the light intensity distribution of the star image in the focal plane is the airy spot’s light intensity distribution. The energy formula of the diffuse spot is defined as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{I}{I_0}} = {\left[{\displaystyle \frac{2J_1\left(\psi \right)}{\psi }}\right]}^2\\ \psi = {\displaystyle \frac{2\pi }{\lambda }}·h·\theta = {\displaystyle \frac{\pi }{\lambda }}·{\displaystyle \frac{D}{f^{\prime }}}·r = {\displaystyle \frac{\pi }{\lambda }}·{\displaystyle \frac{r}{F}}\end{array}\right.$$

(1)

where I/I₀ is the relative intensity (the center of the star diffraction image is defined as 1.0), $J_1\left(\psi \right)$is the first-order Bessel function, $D/f^{\prime }$ is the relative aperture, and $r$ is the radial distance away from the center of the star diffraction image. The physical significance of the above parameters is shown in Figure 2 [14,15]. The airy spot consists of a central bright spot and a few concentric outer rings with rapidly diminishing brightness [16].

The expression for the proportion of energy falling in the circle is expressed as follows [17]:

$$L\left(\psi \right)=1-{J_0}^2\left(\psi \right)-{J_1}^2\left(\psi \right)$$

(2)

where $J_0\left(\psi \right)$is the 0 order Bessel function, and $J_1\left(\psi \right)$is the 1 order Bessel function.

2.2. Focal Plane Distribution of Optical System

According to the light intensity distribution of the image plane and the characteristics of the optical path, the relationship between the CCD target plane and the theoretical image plane can be divided into three types, namely the focal plane, defocus, and deflection, as shown in Figure 3.

The expression of the three-dimensional distribution function of the two-dimensional diffuse spot is shown as follows [18]:

$$I(u,v) = I_0{\left[{\displaystyle \frac{2J_1\left(\sqrt{u^2+v^2}\right)}{\sqrt{u^2+v^2}}}\right]}^2$$

(3)

When defocus occurs, there is a ratio of amplification $k$ ($k$ > 1), and the equations are $u = u^{\prime }/k$ and $v=v^{\prime }/k$. Then, the diffuse spot distribution function can be obtained as $I(u^{\prime }/k,v^{\prime }/k)$. When deflection occurs, the equation is $u^{\prime } = kz$. We use linear approximation at the deflection, $u$ is $u^{\prime }/(1 + kz/u^{\prime })$, and $v$ is $v^{\prime }$. The distribution function of the diffuse spot is expressed as${I(u}^{\prime }/(1 + kz/u^{\prime },v^{\prime })$. The distribution of the three types is clearly shown in Figure 4.

2.3. Simulation Analysis of Standard Diffuse Spot Distribution

For conventional detection systems, the energy concentration of the optical system is generally required to account for 80% in 2 × 2 pixels. Combined with the design technical index, the radius of an 80% energy spot is equal to the pixel size of the detector. For ease of calculation and description, the size of 2 × 2 pixels is the largest external square of the 80% energy spot. Although the response area of 2 × 2 is 27.3% and is larger than the 80% energy patch, this does not add much energy as seen from the energy percentage curve [19]. Therefore, it is reasonable to use the pixel whose size is equal to 80% of the radius of the energy spot.

In the simulation process, the phase of the star target source in the image represents its normalized position in one pixel, and the phase value interval is from 0 to 1. When the phase is 0, it means that the point source falls into one pixel exactly [20]. According to the theoretical definition of the PSF, this situation satisfies the direct sampling condition. As shown in Figure 5, the phase of the point source in the pixel ($i$,$j$) is ${\phi }_x$ and ${\phi }_y$ in the $x$ and $y$directions, respectively. The gray level of the star target source is set as ${DN}_{target}$, and $i$ and $j$ represent the position of pixels in the image. Then, the gray level ${DN}_{i,j}$, ${DN}_{i,j + 1}$, ${DN}_{i + 1,j}$, and ${DN}_{i + 1,j + 1}$ of the four pixels under phase action are, respectively, described as follows:

$$\left\{\begin{array}{c}{DN}_{i,j} = {DN}_{target}\left(1-{\phi }_x\right)\left(1-{\phi }_y\right)\\ \begin{array}{c}{DN}_{i,j+1} = {DN}_{target}{\phi }_x\left(1-{\phi }_y\right)\\ \begin{array}{c}{DN}_{i+1,j} = {DN}_{target}{\phi }_y\left(1-{\phi }_x\right)\\ {DN}_{i+1,j+1} = {DN}_{target}{\phi }_x{\phi }_y\end{array}\end{array}\end{array}\right.$$

(4)

In the simulation conditions of this paper, three types of relations between the CCD target plane and the theoretical image plane are simulated. And the three types are the focal plane, defocus, and deflection [21]. In addition, for each relation, it is also necessary to consider the sub-pixel PSF estimation of different phase point sources.

For a point target source in the surface source array, a window with a size of 5 × 5 pixels is extracted. And the normalized pixel position of the point source is defined as (${\phi }_x$, ${\phi }_y$). The local coordinate system is $x^{\prime }o{y}^{\prime }$. The center of the window is the origin. $x^{\prime }$ and $y^{\prime }$ are parallel to the x and y directions, respectively, and the coordinates of $x^{\prime }$ and $y^{\prime }$ take the values of +2, +1, 0, −1, −2, respectively. Each pixel in the window is represented by two horizontal coordinate values ($x_{PSF}$, $y_{PSF}$) in the PSF reconstruction coordinate system as follows [22]:

$$\left\{\begin{array}{c}{x}_{PSF}={\phi }_x-x^{\prime }\\ y_{PSF}={\phi }_y-y^{\prime }\end{array}\right.$$

(5)

According to the target design geometry calculation, each phase (${\phi }_x$, ${\phi }_y$) of the window is determined. Here, take 1/8 pixel accuracy as an example in Figure 6. Each source point target moves to the adjacent point source in a phase of 1/8, such as an interval sequence of 0, 0.125, 0.25, 0.375, and 0.5. And the amplitude corresponding to the arrow represents the sampling value of each pixel. Similarly, 2D PSF reconstruction means that the phase moves in both x and y directions at the same time [23,24,25,26,27].

According to the above principles, 3D ($x$-dimension, $y$-dimension, and gray-dimension) multi-phase sub-pixel PSF distribution simulation is carried out by focusing on the three states of focal plane, defocusing, and deflection, respectively [28,29]. Fifteen different sizes of phase shifts are given here, including 0 × 0, 0.125 × 0, 0.25 × 0, 0.375 × 0, 0.5 × 0, 0.125 × 0.125, 0.25 × 0.125, 0.375 × 0.125, 0.5 × 0.125, 0.375 × 0.25, 0.375 × 0.375, 0.5 × 0.5, 0.25 × 0.25, 0.5 × 0.25, and 0.5 × 0.375. The simulated energy distribution in the focal plane state is shown in Figure 7. The simulated energy distribution in the defocusing state is shown in Figure 8. The simulated energy distribution in the deflection state is shown in Figure 9.

3. Collecting Images of the Diffuse Spot Distribution of the Actual Detection System

The actual diffuse spot measuring device of the optical system in this paper is mainly composed of a monochromator light source, optical bench collimator, high-precision turntable, detection system, and image acquisition system, as depicted in Figure 10. The star point is placed on the focal plane of the parallel optical tube and illuminated with a monochromator to form a target at infinite distance. The detection system is fixed on the turntable so that it is coaxial with the parallel optical tube. The turntable is turned during measurement and the image acquisition system is used to collect star point images [30,31]. In this way, the diffuse spot distribution data in different fields of view are obtained. Experimental data can be used as a database for subsequent comparative analysis with simulation images, as seen in Figure 11.

From the distribution of various star diffuse spots as shown in Figure 10, the actual energy distribution of the space debris detection system is relatively complex and diverse. Only through subjective interpretation of the image can we reach the conclusion of defocusing. But it is impossible to clarify the technical direction of focusing. Therefore, we need to compare the dispersion spot distribution of the actual system with the above simulated diffuse spot distribution data template library one by one and objectively give the quantitative basis for focusing. Finally, this optical system quickly meets the target requirements.

4. Comparative Analysis of Standard Diffuse Spot Simulation Images and Actual Detection System Diffuse Spot Images

In order to evaluate the correlation and similarity between simulated dispersion images and the actual dispersion images, the PCC is used here [32,33]. The PCC is an evaluation index that measures the strength of the linear relationship between two continuous variables. The value of the PCC is between −1 and 1. It is expressed by the quotient of their respective covariance and variance.

Assuming that the simulated image and the measured image of the diffuse spot are $X$ and $Y$, respectively, the expression of covariance is described as follows:

$$\mathit{cov}(X,Y)=E(X,E\left(X\right)\left)\right(Y,E\left(Y\right)={\displaystyle \frac{1}{(M\times N{)}^2}}((M\times N){\displaystyle \sum _{i=1}^M}\sum _{j=1}^N{x}_{i,j}{y}_{i,j}-(\sum _{i=1}^M\sum _{j=1}^N{x}_{i,j})(\sum _{i=1}^M\sum _{j=1}^N{y}_{i,j}))$$

(6)

The expression for the variance is defined as follows:

$$D\left(X\right)=E\left(\right(X-E\left(X\right){)}^2)={\displaystyle \frac{1}{(M\times N{)}^2}}((M\times N)\sum _{i=1}^M\sum _{j=1}^N{x}_{i,j}^2-(\sum _{i=1}^M\sum _{j=1}^N{x}_{i,j}{)}^2)$$

(7)

PCC is calculated and derived as follows:

$$\begin{array}{l}\rho (X,Y)={\displaystyle \frac{\mathit{cov}(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}}}\\ ={\displaystyle \frac{((M\times N){\sum }_{i=1}^M{\sum }_{j=1}^N{x}_{i,j}{y}_{i,j}-({\sum }_{i=1}^M{\sum }_{j=1}^N{x}_{i,j})({\sum }_{i=1}^M{\sum }_{j=1}^N{y}_{i,j}))}{\sqrt{(M\times N){\sum }_{i=1}^M{\sum }_{j=1}^N{x}_{i,j}^2-({\sum }_{i=1}^M{\sum }_{j=1}^N{x}_{i,j}{)}^2}\sqrt{(M\times N){\sum }_{i=1}^M{\sum }_{j=1}^N{y}_{i,j}^2-({\sum }_{i=1}^M{\sum }_{j=1}^N{y}_{i,j}{)}^2}}}\end{array}$$

(8)

Based on the above equations, the PCCs between the simulated images and the actual detection system diffuse spot images can be calculated. The simulated image with higher similarity to the actual diffuse spot image can be selected by comparison, so as to obtain the quantitative basis of focal plane adjustment. Figure 12 shows the distribution of the five groups of diffuse spot images using three PSF modeling methods. Column 1 contains the diffuse spot images obtained with the actual detection system. Column 2 contains the simulation images using the proposed 3D multi-phase sub-pixel PSF model. Column 3 contains the simulation images using Simon Zieleniewski’s method. Column 4 contains the simulation images using the classic Moffat PSF model.

Table 1. Calculation of PCCs.

Actual Detection System’s Diffuse Spot Images	Standard Diffuse Spot Simulation Images	PCC-P	PCC-S	PCC-M
20240619-10.raw	0.250 × 0.125.dat	0.9768	0.9630	0.9530
20240619-16.raw	0.250 × 0.250.dat	0.9866	0.9735	0.9572
20240619-21.raw	0.250 × 0.125.dat	0.9875	0.9815	0.9617
20240619-30.raw	0 × 0.dat	0.9937	0.9836	0.9705
20240619-34.raw	0.375 × 0.125.dat	0.9953	0.9897	0.9792

Bold data in PCC-P column represents the maximum value.

shows the PCCs among the five groups of matched images using these three methods. PCC-P represents the PCCs between the actual detection system’s diffuse spot images and images obtained with the proposed method.. PCC-S represents the PCCs between the actual detection system’s diffuse spot images and images obtained with Simon Zieleniewski’s method. PCC-M represents the PCCs between the actual detection system’s diffuse spot images and images obtained with the Moffat PSF model. In Table 1, it can be seen that the star distribution map obtained with the algorithm proposed in this paper is the closest to the actual image. When the PCC of the diffuse spot distribution of the two images is greater than 0.96, it can be concluded that these two diffuse spot distributions are similar.

On one hand, the accuracy of the 3D multi-phase sub-pixel PSF estimation model used in this paper is verified. On the other hand, the simulation parameters of the simulated images provide reference information for the focal plane adjustment of the actual system. Through this method, the energy distribution of the detection system can be judged quickly and effectively, and the image quality of the optical system can be analyzed comprehensively, so as to meet the testing requirements of the optical system image quality in engineering projects.

5. Discussion

The 3D multi-phase sub-pixel PSF simulation analysis method proposed in this paper not only uses the parameters of the optical system, but also comprehensively considers the phase distribution characteristics of the point source array. This method performs sub-pixel sampling based on different phase point sources to realize sub-pixel PSF reconstruction. By using this method, the dispersion spot distribution can be simulated accurately, and the real situation of different phase movement can be simulated, too. Moreover, the data template library of the diffuse spot distribution is enriched. By comparison and analysis with the actual star distribution map, more accurate and reliable interpretation data can be provided for focusing work. Compared with the traditional focusing method, the method eliminates the subjective factors, unifies the inspection standard, and provides a quantitative basis for the focal plane adjustment and testing of optical system.

6. Conclusions

Aiming at the drawbacks of measuring the diffuse spot parameters of a space debris detection system, this paper proposes a mathematical model based on 3D multi-phase sub-pixel PSF estimation. The dispersion energy distribution of a space debris detection system is simulated using a 1/8 mobile phase. Then, based on the PCCs, the energy distribution of the standard diffuse spot simulation images is compared with those of the actual detection system. The deviation of the energy distribution of the star spot image is given quantitatively, the subjective factors are eliminated, and the testing criteria are unified. Therefore, the diffuse spot energy distribution of the optical system can be judged and analyzed quickly and accurately. It provides a quantitative basis for the installation and testing of the optical system. The method presented in this paper has important reference value in engineering testing.