A method for star map blind restoration for elliptic arc trajectory

Abstract

The application discloses a star map blind restoration method when star points are in elliptic arc trajectories under compound motion with three-axis angular velocities all being non-zero, which comprises the following steps: S1, elliptic fitting is performed on the elliptic arc trajectories of the star points; S2, elliptic arc trajectory and circular arc trajectory transformation based on projection transformation is performed; S3, circular arc trajectory and straight line segment trajectory transformation based on polar coordinate transformation is performed; S4, straight line segment trajectory restoration based on motion blurred image blind restoration is performed; S5, coordinate inverse transformation and projection inverse transformation are performed to obtain the final restoration result of the star map with the original trajectory being the elliptic arc trajectory. The application realizes the star map blind restoration of the elliptic arc trajectory by transforming the elliptic arc trajectory into the circular arc trajectory through the projection transformation, transforming the circular arc trajectory into the straight line segment trajectory through the polar coordinate transformation, and then realizing the star map restoration of the elliptic arc trajectory through the straight line segment restoration and the coordinate inverse transformation.

Description

Technical Field

[0001] This invention relates to the field of fuzzy reconstruction technology of moving star maps, and in particular to a method for blind reconstruction of star maps when the star sensor is in a composite motion in which the angular velocities of the three axes are all non-zero, and the trajectory of the star point is an elliptical arc. Background Technology

[0002] A star sensor is an instrument that performs high-precision attitude measurement by observing stars. It first captures images of the night sky and then calculates the spacecraft's attitude based on the imaging information. When a spacecraft moves at high speed, the imaged star points produce a trailing effect, and different motion patterns result in different trailing trajectories. At this time, starlight energy is dispersed across multiple pixels, causing a decrease in the image signal-to-noise ratio, reducing the accuracy of star point extraction, and consequently reducing the accuracy of attitude calculation. Therefore, eliminating motion blur of trailing star points under the high dynamic conditions of a spacecraft is of great significance for the attitude measurement of star sensors.

[0003] To address the motion blur problem mentioned above, one approach is to mitigate the image quality degradation caused by motion blur through hardware optimization before the star sensor captures the image. This includes methods such as time-delay integration, servo tracking platform technology, electron multiplication charge-coupled devices (CCDs), enhanced CCDs, and multi-field star sensors. However, these improvements are relatively difficult and costly; therefore, the second type of method for solving motion blur is often more commonly used.

[0004] The second type of method involves optimizing the image using software after the star sensor captures the image, thereby eliminating star trails. This type of method mainly falls into two categories: non-blind restoration and recovery. Non-blind restoration involves restoring motion-blurred images using other instruments that can measure the spacecraft's angular velocity. Researchers such as Wu Xiaojuan from Beijing University of Aeronautics and Astronautics and Xing Fei from Tsinghua University have used data from deeply integrated SINS (strap-down inertial navigation) or MENS (micro electro-mechanical system) gyroscopes to remove star motion blur. Chen Nan added an extended Kalman filter (EKF) to the above coupling system. Sun Ting from Tsinghua University, building on Chen Nan's work, coupled the star sensor with an INS (inertial navigation system). Ma Liheng from the National University of Defense Technology used multiple sub-star region growing techniques to eliminate star trails by correlating frames based on integrated MEMS gyroscope data. All these methods require angular velocity measurement instruments.

[0005] Another software-based method for star image motion restoration is blind restoration. Blind restoration is performed solely based on images captured by a star sensor, thus requiring no prior knowledge of the motion pattern before image motion restoration. For blind restoration of straight line trajectories, existing methods typically employ Radon transform, Hough transform, and differential operator construction to obtain the motion direction, and curve fitting, spectral analysis, and dual-spectral methods to obtain the trailing length. Then, a blur kernel function is generated based on the motion direction and trailing length. Finally, image restoration algorithms such as LR filtering, Wiener filtering, and inverse filtering are used to achieve blurred star point restoration of the straight line trajectory. In addition, there are a series of optimized and improved algorithms for angle and length detection of straight lines and image restoration filtering algorithms.

[0006] Current blind star image restoration algorithms are only applicable to cases where the star's trajectory is a straight line. However, when the star sensor's motion is complex, especially when the angular velocities of all three axes are not zero, its trajectory becomes an ellipse, and the above methods are not applicable. Summary of the Invention

[0007] The present invention is a method for blind star map reconstruction of elliptical arc trajectories in order to overcome the defects of the prior art.

[0008] To achieve the above objectives, the present invention adopts the following technical solution: A method for blind star map reconstruction of elliptical arc trajectories includes the following steps: S1: Perform ellipse fitting on the elliptical arc trajectory of the star point; S2: Perform elliptical arc trajectory and circular arc trajectory transformation based on projective transformation; S3: Perform the transformation between circular arc trajectory and straight line segment trajectory based on polar coordinate transformation; S4: Perform trajectory restoration of straight line segments based on blind restoration of motion-blurred images; S5: Perform inverse coordinate transformation and inverse projective transformation to obtain the final restoration result of the star map with the original trajectory being an elliptical arc trajectory.

[0009] Furthermore, step S1 specifically includes: S101: Expressing the elliptical arc trajectory in a general form, defining the objective function as the algebraic distance from the coordinate point to the elliptical arc trajectory; S102: Collecting the pixel coordinates on the trajectory; S103: Substituting the coordinate values, simplifying the ellipse fitting problem into a problem that can be solved using the least squares method, and solving for the general expression of the elliptical arc trajectory.

[0010] Furthermore, the specific content of S2 is as follows: S201: Project the elliptical arc trajectory on the image plane onto a plane with its composite angular velocity as the normal vector, and its projected trajectory becomes a circular arc trajectory with an eccentricity of 0; at the same time, the trajectories of stars at other different positions on this projection plane will also become circular arc trajectories with an eccentricity of 0; S202: Collect the coordinates of four sets of corresponding points on any elliptical arc trajectory on the image plane and the corresponding circular arc trajectory on the projection plane; S203: Substitute the coordinate values, and obtain the projective transformation matrix from the image plane to the projection plane through least squares estimation and singular value decomposition, so as to realize the transformation from the elliptical arc trajectory to the circular arc trajectory.

[0011] Furthermore, the specific content of S3 is as follows: S301: Select the optimal polar coordinate transformation rotation angle quantization unit; S302: Transform the circular arc trajectory in the rectangular coordinate system into the straight line segment trajectory in the polar coordinate system through polar coordinate transformation, effectively solving the problem of light spot deformation in the polar coordinate inverse transformation process and ensuring restoration accuracy.

[0012] Furthermore, the specific content of S4 is as follows: S401: For the converted straight line segment star point trajectory, its corresponding motion blur angle is 0, so only the motion length needs to be detected; the classic Steger algorithm is used to extract the light stripe, and then the motion blur length is obtained; S402: Based on the angle and length information of the motion blur, a blur kernel function is generated; S403: The Lucy-Richardson algorithm is used to restore the straight line segment trajectory in the polar coordinate system.

[0013] Furthermore, the specific content of S5 is as follows: S501: Through inverse polar coordinate transformation, the image pixel information is transformed from polar coordinates to the rectangular coordinates corresponding to the plane with the rotation axis as the normal vector; S502: Another inverse projective transformation is performed to transform the rectangular coordinates corresponding to the projection plane to the rectangular coordinates corresponding to the image plane. The corresponding transformation matrix is ​​the inverse matrix of the transformation matrix from the image plane to the projection plane, thereby obtaining the final restoration result of the star map whose original trajectory is an elliptical arc trajectory.

[0014] The present invention has the following advantages: Compared with existing technologies, this method has the following advantages: 1. This invention transforms the elliptical arc trajectory into a circular arc trajectory through projective transformation, transforms the circular arc trajectory into a straight line segment trajectory through polar coordinate transformation, and then restores the star map of the elliptical arc trajectory through straight line segment restoration and inverse coordinate transformation.

[0015] 2. This invention solves the problem of light spot deformation after restoration by selecting the optimal polar coordinate transformation quantization unit, thus ensuring restoration accuracy.

[0016] 3. The proposed method was verified and tested through simulation and experiments. The experimental results show that the method is effective and can perform blind reconstruction of star charts of elliptical trajectories, with accuracy comparable to existing star chart reconstruction methods for straight line segment trajectories. Attached Figure Description

[0017] Figure 1 This is a flowchart of the star map blind restoration algorithm for elliptical arc trajectories.

[0018] Figure 2 It is a schematic diagram of the projection of the star point trajectory onto a plane whose rotation axis is the normal vector.

[0019] Figure 3 This is a schematic diagram of the transformation from a rectangular coordinate system to a polar coordinate system.

[0020] Figure 4 This is a schematic diagram of the light spot after polar coordinate transformation based on conventional quantization units.

[0021] Figure 5 This is a schematic diagram of the light spot after polar coordinate transformation based on the optimal quantization unit.

[0022] Figure 6 This is a schematic diagram of the simulation and experimental results after image processing in each step of the present invention. Detailed Implementation

[0023] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0024] A method for blind star map reconstruction of elliptical arc trajectories is shown in the attached figure. Figure 1 As shown, it includes the following steps: S1: Perform ellipse fitting on the elliptical arc trajectory of the star point; S2: Perform elliptical arc trajectory and circular arc trajectory transformation based on projective transformation; S3: Perform the transformation between circular arc trajectory and straight line segment trajectory based on polar coordinate transformation; S4: Perform linear trajectory restoration based on blind restoration of motion-blurred images; S5: Perform inverse coordinate transformation to obtain the final restored result of the star map with the original trajectory being an elliptical arc trajectory.

[0025] Furthermore, step S1 specifically includes: S101: Expressing the elliptical arc trajectory using a general formula, as shown in equation (1): In the process of ellipse fitting, the objective function can be defined as the algebraic distance from the input coordinate point to the elliptical arc trajectory, as shown in equation (2): Define vector Then the objective function can be written as equation (3): in, Represents trajectory points The vector formed; express The vector formed by the coefficients; Represents trajectory points Substitute the coefficients as The objective function formed by the elliptic equation of the vector is the first... One equation; S102: Pixel coordinates on the acquisition trajectory; S103: Substituting the coordinate values ​​simplifies the ellipse fitting problem into a problem that can be solved using the least squares method as shown in equation (4). Solve for the general expression of the elliptical arc trajectory.

[0026] Furthermore, the specific content of S2 is as follows: S201: as attached Figure 2 As shown, when the elliptical arc trajectory on the image plane is projected onto a plane with its composite angular velocity as the normal vector, its projected trajectory becomes a circular arc trajectory with an eccentricity of 0; at the same time, the trajectories of stars at other different positions on this projection plane will also become circular arc trajectories with an eccentricity of 0. S202: Projection from the Cartesian coordinate system containing the image plane to the axis of rotation as the normal vector. The transformation matrix of a plane can be expressed as a projective transformation matrix with nine parameters. The relationship between points on the image plane and points on the projection plane can be expressed by the following equation (5): in, because The last line and The result of multiplication is 1, which yields an additional constraint equation. Therefore, the degrees of freedom of the projective transformation parameter matrix are 8. Thus, it is only necessary to perform the transformation on any elliptical arc trajectory in the image plane and its corresponding... The matrix can be obtained by substituting the coordinates of four corresponding points collected on the circular arc trajectory in the plane into the solution. To improve accuracy, four sets of corresponding points are preferentially selected on the trajectory in the image plane that are far from the image center, as shown in the attached figure. Figure 2 In Meanwhile, to ensure calculation accuracy, four specific points on the elliptical trajectory in the image plane are preferentially selected for substitution into the solution, such as... Figure 2 middle The trajectory lies at four points A, B, C, and D on the ellipse. Let M be the intersection of the rotation axis and the image plane. A and C form the trajectory. The intersection of the ellipse and its major axis; B and D are the line segments passing through M and perpendicular to the major axis of the ellipse. The intersection points of the ellipses. Then, based on the projective geometry of the star vectors, the points corresponding to A, B, C, and D on the image plane are determined. Points on a plane .

[0027] Substituting the four sets of coordinate values, the following equation (6) is given. Equations in the form of: in, S203: Substitute the coordinate values ​​and obtain the column vector solution through least squares estimation and singular value decomposition. Thus, the projective transformation matrix that transforms the elliptical arc trajectory into a circular arc trajectory is obtained. This allows for the transformation from an elliptical arc trajectory to a circular arc trajectory.

[0028] Furthermore, the specific content of S3 is as follows: S301: The horizontal coordinate of the polar coordinate system is the rotation angle. The vertical axis represents the radius of rotation. If the coordinate transformation is performed using conventional quantization units (i.e., rotation angle quantized in 1° units, and rotation radius quantized in units corresponding to the length of one pixel), the restored spot will exhibit significant tangential stretching deformation when transformed to a Cartesian coordinate system, as shown in the attached figure. Figure 3 As shown. The reason for this phenomenon is that the restored light spot after the trajectory of a straight line segment in the polar coordinate system has a certain size, and the lateral width of the light spot represents the size of the rotation angle. The larger the quantization unit of the horizontal coordinate in the polar coordinate system, the larger the lateral width of the restored light spot. The larger the lateral width of the restored light spot, the longer the arc formed when inversely transformed to the rectangular coordinate system. When the ratio of the arc stretching length to the radial quantization stretching length is greater, the light spot deformation is more obvious. Therefore, this invention calculates the corresponding optimal rotation angle quantization unit based on the rotation radius quantization unit, thereby effectively solving the light spot deformation problem in the polar coordinate inverse transformation process and ensuring restoration accuracy.

[0029] In this invention, the quantization unit for the rotation radius is 1 pixel, and the quantization unit for the rotation angle is... The width of the restored spot in polar coordinates is... , length is The length There are N rotation angle quantization units. In the Cartesian coordinate system, the optimal rotation angle quantization value will vary depending on the radius r of the light spot. Therefore, the optimal rotation angle quantization unit corresponding to half the polar coordinate rotation radius (i.e., 1 / 4 of the image sensor length) is selected as the optimal rotation quantization unit for the entire transformation. Thus, the optimal rotation angle quantization unit value is obtained according to formula (7). .

[0030] In this embodiment of the invention, the restored spot width It has 3 pixels, and the spot length is 4 quantization units, therefore The value is 4. In this embodiment of the invention, the star sensor pixel array is 2048×2048 pixels, therefore... The value is 512 pixels. Accordingly, the optimal rotation angle quantization unit determined in this embodiment of the invention is approximately 0.1°.

[0031] S302: According to the above optimal quantization unit, the pixel values ​​corresponding to the rectangular coordinates in the image are mapped to their corresponding polar coordinate pixel values. The transformation formula is shown in the following formula (8), thereby transforming the arc trajectory in the rectangular coordinate system into the straight line segment trajectory in the polar coordinate system: Appendix Figure 5 This is a schematic diagram of the light spot after polar coordinate inverse transformation according to this optimal quantization unit, compared with the attached diagram. Figure 4 It can be seen that the problem of light spot distortion has been significantly improved.

[0032] Furthermore, the specific content of S4 is as follows: S401: For the converted straight line segment trajectory, the corresponding motion blur angle is 0, so only the motion length needs to be detected; the classic Steger algorithm is used to extract the light stripe, and then the motion blur length is obtained; S402: For the fuzziness caused by uniform linear motion, the fuzzy kernel function is shown in equation (9) below: In the formula It is the motion ambiguity angle, representing the angle between the direction of motion and the positive direction of the horizontal axis; It is the motion blur scale, representing the distance a pixel moves in the direction of motion; therefore, a blur kernel function is generated based on the angle and length information of the motion blur. S403: Using the Lucy-Richardson algorithm, the fuzzy kernel function is substituted into formula (10): In the formula It is the first Image estimation after the next iteration; It is the observed degraded image; thus, the trajectory of a straight line segment in polar coordinates can be restored.

[0033] Furthermore, the specific content of S5 is as follows: S501: Through inverse polar coordinate transformation, the image pixel information is transformed from polar coordinates to rectangular coordinates corresponding to the plane with the rotation axis as the normal vector; in polar coordinates, each pixel contains three pieces of information: the horizontal coordinate rotation angle value. , radius of rotation of the vertical axis and corresponding pixel grayscale value Therefore, this information needs to be transformed back to the rectangular coordinate system. The inverse transformation formula is as follows (11): ; S502: After the polar coordinate inverse transformation is completed, another projective inverse transformation is performed to transform the rectangular coordinates corresponding to the projection plane to the rectangular coordinates corresponding to the image plane. The corresponding transformation matrix is ​​the inverse matrix of the transformation matrix from the image plane to the projection plane. After completing all the above transformations, the final restoration result of the star map with the original trajectory being an elliptical arc trajectory can be obtained.

[0034] Appendix Figure 6 The experimental results of each step of the present invention show that the present invention can effectively restore the star map of elliptical arc trajectory with a restoration accuracy of about 0.2 pixels, which is comparable to the star map restoration accuracy of existing straight line segment trajectories.

[0035] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered by the protection of the present invention.

Claims

1. A method for blind star map reconstruction of elliptical arc trajectories, characterized in that, Includes the following steps: S1: Perform ellipse fitting on the elliptical arc trajectory of the star point; S2: Perform elliptical arc trajectory and circular arc trajectory transformation based on projective transformation; S3: Perform the transformation between circular arc trajectory and straight line segment trajectory based on polar coordinate transformation; S4: Perform trajectory restoration of straight line segments based on blind restoration of motion-blurred images; S5: Perform inverse coordinate transformation and inverse projective transformation to obtain the final restoration result of the star map with the original trajectory being an elliptical arc trajectory.

2. The method for blind star map reconstruction of elliptical arc trajectories according to claim 1, characterized in that, The specific steps of S1 are as follows: S101: Express the elliptical arc trajectory in general form, and define the objective function as the algebraic distance from the coordinate point to the elliptical arc trajectory; S102: Pixel coordinates on the acquisition trajectory; S103: Substitute the coordinate values ​​to simplify the ellipse fitting problem into a problem that can be solved using the least squares method, thereby obtaining the general expression for the elliptical arc trajectory.

3. The method for blind star map reconstruction of elliptical arc trajectories according to claim 1, characterized in that, The specific content of S2 is as follows: S201: Projecting the elliptical arc trajectory on the image plane onto a plane with its composite angular velocity as the normal vector, the projected trajectory becomes a circular arc trajectory with an eccentricity of 0; at the same time, the trajectories of stars at other different positions on this projection plane will also become circular arc trajectories with an eccentricity of 0. S202: Collect the coordinates of four corresponding points on any elliptical arc trajectory in the image plane and on the circular arc trajectory in the corresponding projection plane; S203: Substitute the coordinate values ​​and obtain the projective transformation matrix from the image plane to the projection plane through least squares estimation and singular value decomposition, thereby realizing the transformation from elliptical arc trajectory to circular arc trajectory.

4. The method for blind star map reconstruction of elliptical arc trajectories according to claim 1, characterized in that, The specific content of S3 is as follows: S301: Determine the optimal polar coordinate transformation quantization unit to effectively solve the problem of light spot deformation during the polar coordinate inverse transformation and ensure restoration accuracy; S302: Transform the circular arc trajectory in the rectangular coordinate system into the straight line segment trajectory in the polar coordinate system through polar coordinate transformation.

5. The method for blind star map reconstruction of elliptical arc trajectories according to claim 1, characterized in that, The specific content of S4 is as follows: S401: For the converted straight line segment star point trajectory, its corresponding motion blur angle is 0, so only the motion length needs to be detected; the classic Steger algorithm is used to extract the light stripe, and then the motion blur length is obtained; S402: Generate a fuzzy kernel function based on the angle and length information of motion fuzzing; S403: The Lucy-Richardson algorithm is used to restore the trajectory of a straight line segment in polar coordinates.

6. The method for blind star map reconstruction of elliptical arc trajectories according to claim 1, characterized in that, The specific content of S5 is as follows: S501: Through inverse polar coordinate transformation, the image pixel information is transformed from polar coordinates to rectangular coordinates corresponding to the projection plane with the rotation axis as the normal vector; S502: Perform another inverse projective transformation to transform the rectangular coordinates corresponding to the projection plane to the rectangular coordinates corresponding to the image plane. The corresponding transformation matrix is ​​the inverse matrix of the transformation matrix from the image plane to the projection plane, thus obtaining the final restoration result of the star map whose original trajectory is an elliptical arc trajectory.

7. The method for blind star map reconstruction of elliptical arc trajectories according to claim 2, characterized in that, The elliptical arc trajectory can be expressed in general form, as shown in equation (1): ； In the process of ellipse fitting, the objective function is defined as the algebraic distance from the input coordinate point to the elliptical arc trajectory, as shown in equation (2): ； Define vector Then the objective function can be written as equation (3): Represents coordinate points The vector formed; express The vector formed by the coefficients; Represents coordinate points Substitute the coefficients as The objective function formed by the elliptic equation of the vector is the first... One equation; S102: Pixel coordinates on the acquisition trajectory; S103: Substituting the coordinate values ​​simplifies the ellipse fitting problem into a problem that can be solved using the least squares method as shown in equation (4). ； Solve for the general expression of the elliptical arc trajectory.

8. The method for blind star map reconstruction of elliptical arc trajectories according to claim 3, characterized in that: S202: Projection from the Cartesian coordinate system containing the image plane to the axis of rotation as the normal vector. The transformation matrix of the plane is a projective transformation matrix with 9 parameters. The relationship between a point on the image plane and a point on the projection plane is expressed by the following equation (5): ； in, ; because The last line and The result of multiplication is 1, which yields an additional constraint equation, so the projective transformation parameter matrix... The degree of freedom is 8, therefore it is only necessary to find the corresponding elliptical arc trajectory on any elliptical arc in the image plane. The matrix can be obtained by substituting the coordinates of four corresponding points on the circular arc trajectory of the plane into the solution. Substituting the four sets of coordinate values, the following equation (6) is given. Equations in the form of: ； in, ; S203: Substitute the coordinate values ​​and obtain the column vector solution through least squares estimation and singular value decomposition. Thus, the projective transformation matrix that transforms the elliptical arc trajectory into a circular arc trajectory is obtained. This enables the transformation from an elliptical arc trajectory to a circular arc trajectory.

9. The method for blind star map reconstruction of elliptical arc trajectories according to claim 4, characterized in that: S301: The x-coordinate of the polar coordinate system is the rotation angle. The vertical axis represents the radius of rotation. The quantization unit for the rotation radius is selected as 1 pixel, and the optimal quantization unit for the rotation angle is calculated according to formula (7). ; ； In the formula, and These represent the width and length of the restored light spot in polar coordinates. It is 3-5 pixels long. correspond Each rotation angle quantization unit, when based on the Lucy-Richardson restoration algorithm. It is 4; Take 1 / 4 of the image sensor length; S302: According to the above optimal quantization unit, the pixel values ​​corresponding to the rectangular coordinates in the image are mapped to their corresponding polar coordinate pixel values. The transformation formula is shown in the following formula (8), thereby transforming the arc trajectory in the rectangular coordinate system into the straight line segment trajectory in the polar coordinate system: 。 10. A method for blind star map reconstruction of elliptical arc trajectories according to claim 5, characterized in that: S402: For the fuzziness caused by uniform linear motion, the fuzzy kernel function is shown in equation (9) below: ； In the formula It is the motion fuzziness angle, representing the angle between the direction of motion and the positive direction of the horizontal axis; It is the motion blur scale, representing the distance a pixel moves in the direction of motion; therefore, a blur kernel function is generated based on the angle and length information of the motion blur. S403: Using the Lucy-Richardson algorithm, the fuzzy kernel function is substituted into formula (10): ； This allows for the restoration of the trajectory of a straight line segment in polar coordinates; where It is the first Image estimation after the next iteration; These are the observed degraded images.

11. A method for blind star map reconstruction of elliptical arc trajectories according to claim 6, characterized in that: S501: Through inverse polar coordinate transformation, the image pixel information is transformed from polar coordinates to Cartesian coordinates corresponding to the plane with the rotation axis as the normal vector; in polar coordinates, each pixel contains three pieces of information: the horizontal coordinate rotation angle value. , radius of rotation of the vertical axis and corresponding pixel grayscale value Therefore, this information needs to be transformed back to the rectangular coordinate system. The inverse transformation formula is as follows (11): ； S502: After the polar coordinate inverse transformation is completed, another projective inverse transformation is performed to transform the rectangular coordinates corresponding to the projection plane to the rectangular coordinates corresponding to the image plane. The corresponding transformation matrix is ​​the inverse matrix of the transformation matrix from the image plane to the projection plane. After completing all the above transformations, the final restoration result of the star map with the original trajectory being an elliptical arc trajectory can be obtained.

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