A centering method for low signal-to-noise ratio severely undersampled star images

Abstract

The application relates to a centering method for low signal-to-noise ratio seriously undersampled star images, belonging to the technical field of astronomical image star image centering, which comprises the following steps: counting star image contour parameters and determining a fixed shape parameter; constructing a sub-pixel point diffusion function integral model, introducing the fixed shape parameter as a constant, and coupling a background parameter to establish a star image theoretical gray value calculation model; combining the actual gray value of a target star image, using a least square method to solve the optimal solution of the to-be-estimated parameter, so that the corresponding star image center position is obtained; the fitting freedom is reduced through the fixed shape parameter, the pixel phase system error is eliminated by combining the sub-pixel integral model, and high-precision centering of low signal-to-noise ratio seriously undersampled star images can be realized in a high background noise environment.

Description

Technical Field

[0001] This invention relates to the field of astronomical image star centering technology, specifically to a centering method for stars with low signal-to-noise ratio and severe undersampling. Background Technology

[0002] In astronomical observation, the precise determination of the center position of stars is one of the core tasks of astrometry. With the continuous development of observational equipment, modern space-based astronomical observation systems often operate under severe undersampling conditions to obtain a larger field of view, meaning the full width at half maximum (FWHM) of the star image is less than 1.5 pixels. Simultaneously, in the observation of faint targets, the acquired star images often have extremely low signal-to-noise ratios due to limitations in target brightness. Under these extreme conditions, traditional star centering methods face severe challenges. Currently, commonly used star centering methods mainly include the centroid method, the corrected moment method (CMM), the Gaussian fitting method (GS), and the effective point spread function method (ePSF).

[0003] The centroid method is simple to calculate, but it is extremely sensitive to background noise and suffers from a significant pixel phase effect, meaning that the centering result will produce periodic systematic errors depending on the position of the star center within the pixel. Although the corrected moment method suppresses the influence of noise to some extent through a weighting function, its systematic error remains significant under severe undersampling conditions, making it difficult to meet the requirements of high-precision measurement.

[0004] Gaussian fitting estimates the center position by fitting the gray-scale distribution of a star. The fitting parameters typically include five parameters (star center position, total luminosity, x-direction variance, and y-direction variance). Under conditions of low signal-to-noise ratio and severe undersampling, the limited information contained in the star and an excessive number of parameters to be fitted can easily lead to divergence in the fitting process, and the fitted shape parameters are often unstable, thus seriously affecting the centering accuracy.

[0005] The effective point spread function (ePSF) method uses an oversampled point spread function model for fitting, resulting in high accuracy. However, ePSF fitting relies on a high-precision effective point spread function model, which needs to be constructed based on a set of observational images taken with good jitter at the sub-pixel scale of the telescope platform. This places high demands on telescope attitude control and is not easy to implement.

[0006] In summary, existing star centering techniques generally suffer from large systematic errors, poor noise resistance, or excessive computational complexity when dealing with extreme conditions of overlapping low signal-to-noise ratios and severe undersampling. These issues fail to meet the high-precision centering requirements of next-generation large-field-of-view astronomical surveys for faint targets. Therefore, a high-precision star centering method is urgently needed that can effectively suppress noise and eliminate pixel phase systematic errors under conditions of low signal-to-noise ratios and severe undersampling. Summary of the Invention

[0007] The purpose of this invention is to provide a centering method for star phenomena with low signal-to-noise ratio and severe undersampling, aiming to solve the problems of large systematic errors, poor noise resistance and insufficient fitting stability in the existing technology when processing star phenomena with low signal-to-noise ratio and severe undersampling.

[0008] To solve the above-mentioned technical problems, the present invention adopts the following technical solution: a centering method for severely undersampled star images with low signal-to-noise ratio, wherein for astronomical images with a full width at half maximum (FWHM) of less than 1.5 pixels, the following operations are performed:

[0009] Statistical analysis of the constellation outline parameters, and determination of a fixed shape parameter;

[0010] A sub-pixel diffusion function integral model is constructed, the fixed shape parameter is introduced as a constant, and a star theory gray value calculation model is established by coupling the background parameter.

[0011] Based on the aforementioned astrological theory grayscale value calculation model, and combined with the actual grayscale value of the target constellation, the optimal solution for the parameters to be estimated is obtained using the least squares method, thereby obtaining the corresponding constellation center position.

[0012] Preferably, a number of high signal-to-noise ratio star samples are selected from astronomical images, and the star contour parameters are calculated and statistically analyzed.

[0013] Preferably, the contour parameters of each star sample are calculated, outliers are removed, the average value of the star contour parameters is calculated based on the signal-to-noise ratio, and the average value of the star contour parameters is used as the fixed shape parameter.

[0014] Preferably, the star profile parameters include x-direction profile parameters. and y-direction profile parameters :

[0015] ;

[0016] ;

[0017] in, The first one within the range of the star sample The center coordinates of each pixel; The estimated center location of the star sample; It is the net brightness of the i-th pixel within the range of the star sample.

[0018] Preferably, the sub-pixel diffusion function integral model is:

[0019] ;

[0020] in, To target pixels Divided into After the nth uniform sub-pixel, the th The center coordinates of each sub-pixel; The total luminosity of the celestial body to be determined; This represents the total number of sub-pixels within a single pixel. For fixed shape parameters; The coordinates of the star center to be solved; This indicates the division of the interior of the target pixel. The values ​​of each sub-pixel are accumulated to simulate the total throughput received by that pixel.

[0021] Preferably, the average value of the star background is coupled with the sub-pixel diffusion function integral model:

[0022] ;

[0023] in, It is the average value of the star background. For in pixels The theoretical grayscale value of the star at a given location is determined by the interplay of star phenomena and background noise at the pixel level. These are the components together.

[0024] Preferably, the target constellation is a low signal-to-noise ratio constellation.

[0025] Preferably, the signal-to-noise ratio of the low signal-to-noise ratio constellation is less than or equal to 10.

[0026] Preferably, the astrological theory grayscale value calculation model is converted into a linear model;

[0027] ;

[0028] in, These are the model values ​​under the initial parameters; This is the correction amount for the parameter;

[0029] An observation equation is constructed using the least squares method combined with the grayscale values ​​of the target star sample. Then, the least squares method is used iteratively to solve the equation: a residual vector is constructed, the elements of which are the differences between the observed grayscale values ​​of each pixel and the theoretically calculated grayscale values ​​under the current parameters. In each iteration, the parameter correction amount that minimizes the sum of squared residuals is solved, and the parameter vector is updated. This process is repeated until the parameter correction amount converges to less than a preset threshold, at which point the iteration stops. The coordinates of the star center obtained at this point are the corresponding star center positions.

[0030] Beneficial Effects: This invention significantly reduces the number of free parameters in the fitting process (from 5 to 3) by introducing fixed shape parameters determined by high signal-to-noise ratio (SNR) star images, effectively avoiding the fitting divergence problem caused by too many parameters under severe undersampling conditions. Simultaneously, the constructed sub-pixel spread function integral model eliminates pixel phase systematic errors at the physical level. Experiments show that for star images with a full width at half maximum (FWHM) of 0.6 pixels and an SNR of 5, the systematic bias of this method is only -0.002 pixels, improving accuracy by two orders of magnitude compared to the traditional corrected moment method (CMM). It eliminates the need to construct a complex oversampled effective point spread function (ePSF), making it particularly suitable for real-time processing of massive undersampled data generated by large field-of-view survey telescopes. Attached Figure Description

[0031] Figure 1 This is a flowchart illustrating the centering method for star phenomena with severe undersampling and low signal-to-noise ratio according to the present invention.

[0032] Figure 2 This is a centering result image of a severely undersampled star system obtained using the method of this invention. Detailed Implementation

[0033] To make the objectives and advantages of this invention clearer, the invention will be specifically described below with reference to embodiments. It should be understood that the following text is merely used to describe one or more specific embodiments of the invention and does not strictly limit the scope of protection specifically claimed by the invention.

[0034] Under severe undersampling conditions (e.g., star full width at half maximum (FWHM) < 1.5 pixels), the amount of information contained in a single frame image is extremely limited. Directly fitting all parameters (including center position, shape parameters, and total luminosity) to each star is prone to divergence due to excessive freedom. This paper proposes a centering method for severely undersampled stars with low signal-to-noise ratios, utilizing the similarity in point spread function (PSF) shapes of stars within the same image or observation sequence, referencing... Figure 1 As shown, the following centering operation is performed using a "statistics first, then fixation" strategy:

[0035] Step 1: Calculate the constellation outline parameters and determine a fixed shape parameter;

[0036] In one embodiment, several bright stars with high signal-to-noise ratios (e.g., SNR > 50), unsaturated and isolated (without surrounding clutter) are selected as samples from the target image. The second moment of each sample star is calculated to estimate the contour parameters in the x-direction. and y-direction profile parameters :

[0037] , ;

[0038] in, The first within the range of the star The center coordinates of each pixel; The estimated center position of the constellation (can be calculated using centering methods such as the corrected moment method and Gaussian profile fitting); It is the first in the range of the stars The net brightness at each pixel (i.e., the difference between the pixel's grayscale value and the average background value of the star); "It is a summation operator that sums over the entire celestial sphere."

[0039] In one embodiment, outliers are removed, and the average value of the star contour parameters is calculated based on a signal-to-noise ratio weighted average. This average value of the star contour parameters is then used as a fixed shape parameter. In subsequent centering of faint targets, these two parameters will be used as constant inputs and will no longer be involved in the fitting; the constrained star shape enables the algorithm to converge robustly even in low signal-to-noise ratio (SNR≤10) environments, overcoming the defect of traditional Gaussian fitting methods being prone to failure under noise interference.

[0040] Step 2: Traditional fitting methods often directly approximate the grayscale value of the entire pixel using the function value at the pixel center. This introduces a significant "pixel phase error" when there is severe undersampling. To eliminate this error, a sub-pixel diffusion function integral model is constructed, introducing a fixed shape parameter as a constant. This model is noise-resistant and can eliminate the systematic error of the centered pixel phase in high background noise environments. Its expression is:

[0041] ;

[0042] in, To target pixels Divided into After the nth uniform sub-pixel, the th The center coordinates of each sub-pixel; The total luminosity of the celestial body to be determined; This represents the total number of sub-pixels within a single pixel. The average value of the star contour parameters obtained in step S1; The coordinates of the star center to be solved; This indicates the division of the interior of the target pixel. The values ​​of each sub-pixel are accumulated to simulate the total flux received by that pixel; in this example, The value is 5.

[0043] A celestial theory grayscale value calculation model is established based on the constructed sub-pixel diffusion function integral model coupled with background parameters:

[0044] ;

[0045] in, It is the average value of the star background. For in pixels The theoretical grayscale value of the star at a given location is determined by the interplay of star phenomena and background noise at the pixel level. The model is composed of multiple parts; it accurately describes the energy distribution at the sub-pixel level and eliminates the systematic errors caused by pixelation from a physical mechanism perspective.

[0046] Step 3: Perform Taylor series expansion on the parameters to be estimated in the astrological theory gray value calculation model (including the coordinates of the star center and the total luminosity) to transform the nonlinear model into a linear model;

[0047] ;

[0048] in, These are the model values ​​under the initial parameters; This is the correction amount for the parameter.

[0049] An observation equation is constructed using the least squares method combined with the grayscale values ​​of target star samples (0.6 pixels at half maximum width and height, with a signal-to-noise ratio covering 5 to 50). The least squares method is then used iteratively to solve the equation. First, a residual vector is constructed, whose elements are the differences between the observed grayscale value of each pixel and the theoretically calculated grayscale value under the current parameters. In each iteration, the parameter correction amount that minimizes the sum of squared residuals is calculated, and the parameter vector is updated. This process is repeated until the parameter correction amount converges to a preset small threshold (e.g., 1 × 10⁻⁻⁶). 6 When the pixel value reaches 1, the iteration stops; the coordinates of the star center obtained at this point are the high-precision centering result that eliminates the systematic error of pixel phase effect.

[0050] In this embodiment, the target constellation is a low signal-to-noise ratio constellation, with a signal-to-noise ratio (SNR) of less than or equal to 10.

[0051] This application also provides a specific embodiment that comprehensively analyzes the effectiveness and superiority of the method provided above; the following settings are made:

[0052] The image shows the centering residual results for severely undersampled constellations with a full width at half maximum (FWHM) of 0.6 pixels at different signal-to-noise ratios. The rhombus, square, circular, and equilateral triangle scatter points represent the statistical average values ​​of the centering residual results for the x-axis measurement coordinates of all 1000 constellations at the corresponding SWHM using the Corrected Moment Method (CMM), Gaussian Profile Fitting Method (GS), Effective Point Spread Function Fitting Method (ePSF), and the method of this invention (OurMethod). Since the phase effect of the centering pixel of the undersampled constellation is more significant at (0.25, 0) pixels, the true phase value of the center pixel of all constellations is (0.25, 0) pixels.

[0053] Results Reference Figure 2 As shown, the method of the present invention can effectively eliminate centering system errors compared with other methods. When the star signal-to-noise ratio is 5, the system deviation of the method of this patent is only -0.002 pixels, which is only 1 / 125 of the modified moment method (CMM) and 1 / 61 of the effective point spread function method (ePSF).

[0054] The embodiments of the present invention have been described in detail above with reference to the examples. However, the present invention is not limited to the above embodiments. For those skilled in the art, after learning the contents described in the present invention, several equivalent changes and substitutions can be made without departing from the principle of the present invention. These equivalent changes and substitutions should also be considered to fall within the protection scope of the present invention.

Claims

1. A centering method for star phenomena with severe undersampling and low signal-to-noise ratio, characterized in that: For astronomical images where the full width at half maximum (FWHM) of a star is less than 1.5 pixels, perform the following operations: Statistical analysis of the constellation outline parameters, and determination of a fixed shape parameter; A sub-pixel diffusion function integral model is constructed, the fixed shape parameter is introduced as a constant, and a star theory gray value calculation model is established by coupling the background parameter. Based on the aforementioned astrological theory grayscale value calculation model, and combined with the actual grayscale value of the target constellation, the optimal solution for the parameters to be estimated is obtained using the least squares method, thereby obtaining the corresponding constellation center position.

2. The centering method for severely undersampled star constellations with low signal-to-noise ratio according to claim 1, characterized in that: Several high signal-to-noise ratio star samples were selected from astronomical images, and the star contour parameters were calculated and statistically analyzed.

3. The centering method for severely undersampled star constellations with low signal-to-noise ratio according to claim 2, characterized in that: The contour parameters of each celestial sample are calculated, outliers are removed, and the average value of the celestial contour parameters is calculated based on the signal-to-noise ratio. The average value of the celestial contour parameters is then used as the fixed shape parameter.

4. The centering method for severely undersampled star constellations with low signal-to-noise ratio according to claim 3, characterized in that: The star profile parameters include x-direction profile parameters. and y-direction profile parameters : ； ； in, The first one within the range of the star sample The center coordinates of each pixel; The estimated center location of the star sample; It is the net brightness of the i-th pixel within the range of the star sample.

5. A centering method for severely undersampled star constellations with low signal-to-noise ratio according to claim 1, characterized in that: The sub-pixel diffusion function integral model is as follows: ； in, To target pixels Divided into After the nth uniform sub-pixel, the th The center coordinates of each sub-pixel; The total luminosity of the celestial body to be determined; This represents the total number of sub-pixels within a single pixel. For fixed shape parameters; The coordinates of the star center to be solved; This indicates the division of the interior of the target pixel. The values ​​of each sub-pixel are accumulated to simulate the total throughput received by that pixel.

6. A centering method for severely undersampled star constellations with low signal-to-noise ratio according to claim 5, characterized in that: Couple the average value of the star background with the sub-pixel diffusion function integral model: ； in, This represents the average value of the celestial background. For in pixels The theoretical grayscale value of the star at a given location is determined by the interplay of star phenomena and background noise at the pixel level. These are the components together.

7. A centering method for severely undersampled star constellations with low signal-to-noise ratio according to claim 1, characterized in that: The target constellation is a low signal-to-noise ratio constellation.

8. A centering method for severely undersampled star constellations with low signal-to-noise ratio according to claim 7, characterized in that: The signal-to-noise ratio of the low signal-to-noise ratio constellation is less than or equal to 10.

9. A centering method for severely undersampled star constellations with low signal-to-noise ratio according to claim 7 or 8, characterized in that: The astrological theory grayscale value calculation model is transformed into a linear model; ； in, These are the model values ​​under the initial parameters; This is the correction amount for the parameter; An observation equation is constructed using the least squares method combined with the grayscale values ​​of the target star sample. Then, the least squares method is used iteratively to solve the equation: a residual vector is constructed, the elements of which are the differences between the observed grayscale values ​​of each pixel and the theoretically calculated grayscale values ​​under the current parameters. In each iteration, the parameter correction amount that minimizes the sum of squared residuals is solved, and the parameter vector is updated. This process is repeated until the parameter correction amount converges to less than a preset threshold, at which point the iteration stops. The coordinates of the star center obtained at this point are the corresponding star center positions.

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