A Method for Constructing an Imaging Quality Prediction Model for Optical Systems that Takes into Account Both Surface Shape Error and Pose Error

Abstract

This invention relates to the field of optical system imaging, specifically to a method for constructing an optical system imaging quality prediction model that takes into account both surface shape error and pose error. The method employs Zernike polynomials to accurately fit the mirror surface shape error caused by bolt preload, and uses the finite element method to simulate the bolt tightening process of the optical system. Based on this, a joint simulation method for the assembly and imaging of a two-mirror optical system is proposed. Using Zernike polynomial fitting as the basic parameter, various mirror surface shape errors and assembly pose deviations are fully considered when using Zemax for optical path imaging simulation. Energy concentration is used as a quantitative evaluation index for imaging quality. An SVR surrogate model containing local and global hybrid kernel functions is established, and sensitivity analysis is performed on mirror surface shape and assembly pose parameters. An assembly dataset is obtained through joint assembly and imaging simulation, and the model is trained to obtain an imaging quality prediction model, enabling rapid and accurate prediction of imaging quality.

Description

Technical Field

[0001] This invention relates to the field of optical system technology, and in particular to a method for constructing an optical system imaging quality prediction model that takes into account both surface shape error and pose error. Background Technology

[0002] A seeker is a device that carries a detection unit, signal processing equipment, and servo control system. It is an important component of weaponry and equipment. Its main function is to detect, identify, and track targets. Its performance directly affects the accuracy of the weaponry. Seekers generally use a coaxial two-mirror optical system, where the primary mirror and secondary mirror are located on the same axis, in order to effectively reduce optical distortion and improve imaging resolution.

[0003] In practice, most optical systems consist of a primary mirror, secondary mirrors, and a support frame, assembled using bolted connections. Bolts and nuts are used to secure the primary and secondary mirrors to the support frame. However, assembly stress inevitably causes structural deformation of the mirror surface, resulting in surface shape errors in the optical components and affecting the imaging quality of the optical system. Therefore, quantitatively evaluating the imaging quality of an optical system is a crucial step in assessing its service performance. Traditional imaging quality evaluation methods include the star method, knife-edge method, slit method, and Langie method. However, these methods require placing each optical component in a specified order and distance during testing, which is inefficient and consumes significant time and manpower. Therefore, it is necessary to establish a rapid and accurate method for predicting imaging quality. Summary of the Invention

[0004] To address the aforementioned problems, embodiments of the present invention provide a method for constructing an optical system imaging quality prediction model that takes into account both surface shape error and pose error.

[0005] This invention provides a method for constructing an imaging quality prediction model for an optical system that takes into account both surface shape error and pose error, comprising:

[0006] S1. The Zernike polynomial is used to fit the mirror surface shape in the optical system to obtain the Zernike polynomial coefficients used to characterize the mirror surface shape error.

[0007] S2. Establish a finite element model of the optical system, simulate the tightening process of the bolt connection in the finite element model, and obtain the mirror deformation.

[0008] S3. Import the mirror deformation into the optomechanical-thermal coupling analysis module and insert Zernike polynomial coefficients to obtain the mirror surface shape error parameters and mirror assembly pose deviation.

[0009] S4. Import the mirror surface shape error parameters and mirror assembly pose deviation into the optical simulation module to perform optical path imaging simulation and obtain the energy concentration used to characterize the imaging quality.

[0010] S5. Construct a dataset based on mirror surface shape error parameters, mirror assembly pose deviation, and energy concentration, and establish an SVR proxy model that includes local and global hybrid kernel functions;

[0011] S6. Use the dataset to train the SVR surrogate model to obtain the optical system imaging quality prediction model.

[0012] Optionally, S1 uses Zernike polynomials to fit the mirror surface shape in the optical system to obtain Zernike polynomial coefficients used to characterize the mirror surface shape error. These coefficients specifically include:

[0013] Let u be the amount of translation of the mirror surface relative to the ideal position along the X, Y, and Z axes after deformation. x u y and u z The rotations around the X, Y, and Z axes are θx, θy, and θz, respectively, and cosθx=cosθy=cosθz=1, sinθx=θx, sinθy=θy, sinθz=θz; according to the homogeneous coordinate transformation method, the coordinate transformation before and after the mirror deformation is shown in equation (1):

[0014]

[0015] Among them, X i ,Y i Z i (i = 1, 2…n) are the coordinates of the nodes before the mirror deformation, (X i ',Y i ',Z i ')(i=1,2…n) are the node coordinates after mirror deformation, and T is the rigid body displacement of the mirror deformation, which is obtained based on the least squares method;

[0016] Multiply the left side of the mirror-distorted node coordinates by the transpose matrix T. -1 The coordinates of the surface shape error nodes after removing the displacement of the mirror cylinder are obtained, as shown in equation (2):

[0017]

[0018] Using Zernike polynomials as basis functions to fit the surface shape error of optical elements, solving for the Zernike polynomial coefficients is equivalent to solving for the surface shape error function W. N The Zernike polynomial linear combination coefficients of (x,y) are shown in equation (3):

[0019]

[0020] Among them, Z i (x,y) is the i-th Zernike polynomial in Cartesian coordinates, C i Let n be the coefficient of the i-th Zernike polynomial, and n be the number of Zernike polynomial terms.

[0021] Substitute the actual measured discrete point displacements into equation (3), solve the Zernike polynomial coefficient matrix using the least squares method, and evaluate the fitting accuracy of the coefficient matrix through the root mean square error. The coefficient matrix obtained after fitting accuracy is the polynomial coefficient that characterizes the surface shape error of the mirror.

[0022] Optionally, the process of establishing the finite element model of the optical system in S2 includes:

[0023] A finite element model of an optical system including a threaded connection structure is established, and a free tetrahedral mesh is created for the primary mirror, secondary mirror, and support.

[0024] The mathematical expressions for the cross-sectional profiles of internal and external threads are established as follows:

[0025] The thread profile is divided into three different parts according to the thread parameters: the thread root part, the thread side part, and the thread crest part; the three parts are extended to a plane to obtain the thread profile along the transverse section. The thread root is selected as the starting point, and θ represents the rotation angle of a point on the thread cross-sectional profile relative to the starting point. The expression of the thread cross-sectional profile is established as shown in Equation (4):

[0026]

[0027] in, d represents the nominal diameter, d1 represents the minor diameter of the thread, r represents the distance from the thread profile to the bolt center axis, and ρ represents the radius of the arc at the root of the thread.

[0028] Based on the mathematical expressions of the cross-sectional profiles of internal and external threads and the rotational invariance of the thread cross-sectional profiles, a thread mesh generation method is used to establish mesh models of internal and external threads.

[0029] Optionally, the process of importing the mirror surface shape error parameters and mirror assembly pose deviation into the optical simulation module for optical path imaging simulation in step S4 includes:

[0030] Establish the corresponding reflecting surfaces of the primary mirror and secondary mirror, where the primary mirror is a parabolic surface of revolution and the secondary mirror is an even-order hyperboloid. At the same time, set the opening size at the center of the primary mirror, the distance between the primary and secondary mirrors, the thickness tolerance of the primary and secondary mirrors, the surface shape error, and the concentricity.

[0031] The mirror assembly posture deviation is superimposed with the reflective surface to generate a composite surface, thereby fitting the mirror deformation to the mirror surface;

[0032] The mirror surface shape error is set by inserting mirror coordinate discontinuities to achieve the eccentricity and tilt of the primary and secondary mirrors.

[0033] Optionally, the process of constructing the dataset in S5 based on mirror surface shape error parameters, mirror assembly pose deviation, and energy concentration includes:

[0034] The mirror surface shape error parameters were selected from the Zernike polynomial coefficients, including defocus, first-order astigmatism-Y-axis, and first-order cloverleaf aberration-Y-axis. The mirror assembly pose deviation was selected from the eccentricity error of the primary and secondary mirrors along the X and Y directions and the tilt error around the X and Y directions. A total of 11 sensitive parameters were selected as characteristic variables, and energy concentration was selected as the target variable.

[0035] The feature variables and target variables are normalized, and the normalized feature variables and target variables are divided into training set and test set according to a set ratio.

[0036] Optionally, the process of building an SVR proxy model in S5 that includes a mixture of local and global kernel functions includes:

[0037] The Gaussian kernel function is selected as the local kernel function, and the polynomial kernel function is selected as the global kernel function. The hybrid kernel function is constructed as shown in equation (5):

[0038] K mix =λK local +(1-λ)K global (5)

[0039] Among them, K local Refers to the local kernel function, K global The global kernel function is used to adjust the ratio between the local and global kernel functions. λ is a scaling factor used to adjust the weight between the local and global kernel functions.

[0040] Configure the parameters of the SVR proxy model, including the regularization parameter C, the fault tolerance parameter ε, the polynomial kernel order, the scaling factor, and the Gaussian kernel width.

[0041] Optionally, the process in S6 of training the SVR surrogate model using the dataset to obtain the optical system imaging quality prediction model includes:

[0042] S61. Train the SVR agent model using the training set data, test the trained SVR agent model using the test set data, and calculate the coefficient of determination R. 2 The root mean square error (RMSE) and the mean relative error (MRE) are shown in equations (6), (7), and (8), respectively:

[0043]

[0044]

[0045]

[0046] Where n is the number of samples in the test set, y i Let y be the true value of energy concentration, and f(x) be the average of the true values ​​of energy concentration. i ) represents the predicted energy concentration, and m represents the number of model training cycles;

[0047] S62. Based on the calculated coefficient of determination, root mean square error, and average relative error, determine whether the prediction accuracy and generalization ability of the SVR surrogate model meet expectations. If they meet expectations, end the training. If they do not meet expectations, adjust the model parameters and execute S63.

[0048] S63. Re-divide the data into training and test sets according to the set ratio and return to execute S61.

[0049] Compared with existing technologies, the advantages of this invention are as follows: It employs Zernike polynomials to accurately fit the mirror surface shape error caused by bolt preload, and uses finite element modeling to accurately simulate the mirror deformation during bolt tightening. This leads to a joint simulation method for the assembly and imaging of a two-mirror optical system. Using Zernike polynomial fitting as the basic parameter, Zemax software is used to fully consider various mirror surface shape errors and assembly pose deviations during optical path imaging simulation, with energy concentration used as a quantitative evaluation index for imaging quality. Next, an SVR surrogate model containing a hybrid local and global kernel function is established to perform sensitivity analysis on mirror surface shape and assembly pose parameters. An assembly dataset is obtained through joint assembly and imaging simulation, and the model is trained. Finally, the established surrogate model is compared with SVR models using a single kernel function / no kernel function, as well as traditional regression models. The results show that the established hybrid kernel function SVR model has the smallest imaging quality prediction error and exhibits excellent imaging quality prediction capability under different assembly error conditions, laying the foundation for real-time assembly and adjustment of optical systems. Attached Figure Description

[0050] The accompanying drawings, which are provided to further illustrate the invention and constitute a part of this invention, are not intended to limit the invention. In the drawings:

[0051] Figure 1 This is a schematic diagram of the principle framework of the present invention;

[0052] Figure 2 This is a Zernike standard polynomial pyramid diagram;

[0053] Figure 3 The two-dimensional wavefront shape represented by the first 16 standard Zernike polynomials;

[0054] Figure 4 The three-dimensional wavefront shape represented by the first 16 standard Zernike polynomials;

[0055] Figure 5 The surface shape error cloud diagram of the primary mirror;

[0056] Figure 6 The fitting residual between the obtained sagittal surface and the mirror deformation;

[0057] Figure 7 This is a structural diagram of a two-mirror optical system;

[0058] Figure 8 The external thread profile with one pitch along the axial section;

[0059] Figure 9 The external thread profile is shown along the transverse section.

[0060] Figure 10 The diagram illustrates the process of modeling internal and external thread meshes, where figure a shows the external thread mesh and figure b shows the internal thread mesh.

[0061] Figure 11 A finite element mesh model of a bolted connection structure;

[0062] Figure 12 A finite element mesh model of a two-reflector optical system;

[0063] Figure 13 Curves showing the torsion relationship under different grid sizes and shapes;

[0064] Figure 14 The distortion cloud images are for the primary mirror and the secondary mirror, where image a is the primary mirror and image b is the secondary mirror.

[0065] Figure 15 For an ideal optical path physical model;

[0066] Figure 16 The MTF curve for an ideal optical path;

[0067] Figure 17 The energy concentration curve for an ideal optical path;

[0068] Figure 18 The images show the imaging results of the optical system under different rotation angles along the Z-axis, where figure a represents a rotation of 0°, figure b represents a rotation of 90°, and figure c represents a rotation of 180°.

[0069] Figure 19The Zernike polynomial coefficients for fitting the principal mirror shape under different bolt preloads;

[0070] Figure 20 The diagrams show the optical system under different conditions. In the diagram, a corresponds to the imaging result of the ideal optical path, b and c correspond to the imaging results under the individual influence of mirror surface shape and assembly pose error, respectively, and d is the imaging result under the combined effect of the two.

[0071] Figure 21 The ε-pipeline model is for linear regression;

[0072] Figure 22 A scatter plot of predicted and actual values ​​for the SVR proxy model;

[0073] Figure 23 The mean relative error of the test set under repeated trials;

[0074] Figure 24 A comparison chart of prediction errors for different kernel functions;

[0075] Figure 25 This is a plot of the average relative error for different regression models. Detailed Implementation

[0076] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the embodiments and accompanying drawings. Here, the illustrative embodiments and descriptions of this invention are used to explain the invention, but are not intended to limit the invention.

[0077] This invention provides a method for constructing an imaging quality prediction model for an optical system that takes into account both surface shape error and pose error. The principle framework is as follows: Figure 1 As shown, the specific process includes:

[0078] S1. The Zernike polynomial is used to fit the mirror surface shape in the optical system to obtain the Zernike polynomial coefficients used to characterize the mirror surface shape error.

[0079] In practice, Zernike polynomials are a set of equations expressed in polar coordinates, typically described as:

[0080]

[0081] In the formula, n = 0, 1, 2... is the order of the polynomial, l is an angle-related parameter, the radial coordinate ρ is the normalized radial distance, and θ is the polar coordinate angle; n and l are both odd or even and n ≥ |l|, therefore (nl) is always even; let l = n - 2m, m = 1, 2, 3...n, we can obtain the following expression:

[0082]

[0083]

[0084] Any k-th order wavefront F(ρ,θ) can be fitted using Zernike polynomials, as shown in the following equation:

[0085]

[0086]

[0087] In the formula, a k The coefficients are k-th order Zernike polynomials. The essence of fitting the mirror surface of an optical system with Zernike polynomials is to solve for the coefficients of each order Zernike polynomial.

[0088] Standard Zernike polynomials have two different normalization methods and permutations. This invention uses the magnitude normalization method, setting the maximum magnitude of each unit term to 1. The permutation strategy is as follows: Figure 2 As shown. Standard Zernike polynomials have an infinite number of terms. Considering that too many terms can lead to overfitting, ill-conditioned equations, and low computational efficiency, while too few terms may result in low accuracy in fitting the mirror surface shape, a standard Zernike polynomial with 45 terms is used to fit the mirror surface shape. Table 1 shows the specific meaning of the first 16 standard Zernike polynomials. Figure 3 and Figure 4 The two-dimensional and three-dimensional wavefront shapes represented by Zernike polynomials are shown respectively. It can be seen that each term of the Zernike polynomial has a specific physical meaning, and its key coefficients have a one-to-one correspondence with the Seidel aberration. Moreover, each term is linearly independent and pairwise orthogonal within the unit circle, which can accurately reflect the wavefront aberration of the system or the surface shape changes of optical elements.

[0089] Table 1. The first 16 terms of the standard Zernike polynomial and their meanings.

[0090]

[0091]

[0092] The optical mirror deformation obtained by finite element simulation or actual measurement includes rigid body displacement and surface shape error (surface distortion). Rigid body displacement includes translation, off-axis, and tilt. Rigid body displacement can be eliminated by adjusting the relative position between optical elements, but surface shape error cannot be eliminated. Therefore, when using Zernike polynomial fitting for surface shape error, it is necessary to remove the rigid body displacement of the mirror first.

[0093] Based on the above analysis, the translation amounts of the mirror surface relative to the ideal position along the X, Y, and Z axes after deformation can be set as u. x u y and u z The rotations around the X, Y, and Z axes are θx, θy, and θz, respectively, and cosθx=cosθy=cosθz=1, sinθx=θx, sinθy=θy, sinθz=θz; according to the homogeneous coordinate transformation method, the coordinate transformation before and after the mirror deformation is shown in equation (1):

[0094]

[0095] Among them, X i ,Y i Z i (i = 1, 2…n) are the coordinates of the nodes before the mirror deformation, (X i ',Y i ',Z i ')(i=1,2…n) are the node coordinates after mirror deformation, and T is the rigid body displacement of the mirror deformation, which is obtained based on the least squares method;

[0096] Multiply the left side of the mirror-distorted node coordinates by the transpose matrix T. -1 The coordinates of the surface shape error nodes after removing the displacement of the mirror cylinder are obtained, as shown in equation (2):

[0097]

[0098] Using Zernike polynomials as basis functions to fit the surface shape error of optical elements, solving for the Zernike polynomial coefficients is equivalent to solving for the surface shape error function W. N The Zernike polynomial linear combination coefficients of (x,y) are shown in equation (3):

[0099]

[0100] Among them, Z i (x,y) is the i-th Zernike polynomial in Cartesian coordinates, C i Let n be the coefficient of the i-th Zernike polynomial, and n be the number of Zernike polynomial terms.

[0101] Substitute the actual measured discrete point displacements into equation (3), solve the Zernike polynomial coefficient matrix using the least squares method, and evaluate the fitting accuracy of the coefficient matrix through the root mean square error. The coefficient matrix obtained after fitting accuracy is the polynomial coefficient that characterizes the surface shape error of the mirror.

[0102] Taking the twin-reflector optical system as the specific research object, its mechanical deformation result can be obtained after tightening with standard preload. The deformation result includes both rigid body displacement and surface error. Based on expressions (1) and (2), matrix operations are performed on the coordinates of the deformed mirror to obtain the surface error node coordinates after removing the rigid body displacement, thus obtaining the surface error cloud map of the primary mirror, such as... Figure 5 As shown. The Zernike polynomial coefficient matrix is ​​solved using expression (3) to achieve surface error fitting. The fitting residual between the sag surface and the mirror deformation obtained from the fitting is shown below. Figure 6 As shown in the figure, the unit is waves = 1e-9m. It can be seen that the fitting residuals are relatively uniformly distributed, and most of the residuals are less than 1e-6mm. The maximum residual is 4.488e-6mm, while the maximum deformation obtained by conventional finite element simulation is 6.1935e-4mm, which is much larger than the fitting residuals. This proves that the fitting error is very small and the surface fitting effect is good.

[0103] S2. Establish a finite element model of the optical system, simulate the tightening process of the bolt connection in the finite element model, and obtain the mirror deformation.

[0104] In practice, a two-mirror optical system is used as the research object, such as... Figure 7 The device mainly consists of a primary mirror, a secondary mirror, a support, and three bolt-connected structures. The bolts fix the primary mirror to the support, and the secondary mirror is installed into the cylindrical hole of the support through a pressure ring.

[0105] In practice, the finite element model of the optical system, including the threaded connection structure, can be created using the finite element preprocessing software Hypermesh 12.0. Furthermore, to simulate the bolt assembly process, fine meshing of the internal and external threads is required. Mesh generation necessitates establishing mathematical expressions for the cross-sectional profiles of the internal and external threads, as follows:

[0106] Based on thread parameters (nominal diameter, minor diameter, pitch, etc.), the thread profile is divided into three distinct parts: the thread root, the thread flank, and the thread crest. Taking external threads as an example... Figure 8 The diagram shows the profile of an external thread with one pitch along the axial section; extending the three sections to a plane yields the thread profile along the transverse section, as shown below. Figure 9 As shown, the root of the thread is selected as the starting point, and θ represents the rotation angle of a point on the thread cross-sectional profile relative to the starting point. The expression for the thread cross-sectional profile is established as shown in equation (4):

[0107]

[0108] in, d represents the nominal diameter, d1 represents the minor diameter of the thread, r represents the distance from the thread profile to the bolt center axis, and ρ represents the radius of the arc at the root of the thread.

[0109] Then, based on the mathematical expressions of the cross-sectional profiles of internal and external threads and the rotational invariance of the thread cross-sectional profiles, a thread meshing method is used to establish the mesh models of internal and external threads; specifically, the thread meshing method proposed by Fukuoka and Nomura can be used.

[0110] like Figure 10 As shown, the nominal diameter of the bolt is M3, and the pitch is 0.5mm. Figure 11 The completed finite element mesh model of the bolt-nut connection is shown. All meshes adopt a hexahedral structure with a reasonable sparse transition. The primary mirror, secondary mirror, and support all use free tetrahedral meshes. The final result is as follows. Figure 12 As shown, this significantly improves the efficiency of mesh generation and solution.

[0111] Next, define the model and material parameters:

[0112] 1) Contact Setup: The finite element model is configured with 11 contact pairs. The contact areas include the contact between the support and the secondary mirror, the contact between the support and the primary mirror, the contact between the nut and the primary mirror, the contact between the nut and the support, and internal and external thread contacts. Among them, except for the bonded contact between the support and the secondary mirror, all other contacts are defined as frictional contacts. Each contact pair consists of a contact surface (CONTA173) and a target surface (TARGE170). The coefficient of friction for all contact areas is set to 0.15.

[0113] 2) Material settings: The bracket is made of aluminum alloy (LY12), the primary lens and secondary lens are made of different aluminum alloys (6061-t6), and the bolts and nuts are all made of 5.6 grade structural steel.

[0114] 3) Constraints and loads: Set the area where the main mirror contacts the nut as constrained in the X and Y directions, and set the bracket and the bottom surface of the bolt head as fully constrained; when the bolt is tightened, apply a rotational load to the nut, and constrain all directions except for rotation in the Z direction.

[0115] Finally, to verify whether mesh size and shape affect the calculation results, a sensitivity analysis was conducted on the external thread mesh of a bolted connection structure. Six different meshing methods were used to mesh the bolt finite element model. The mesh type, number of nodes, and number of meshes are detailed in Table 2. All three-dimensional mesh elements used were Solid185, while the mesh type and number of other components remained unchanged. During the simulation, only this bolted connection structure was tightened, and the torque-preload relationship under each finite element model was extracted to obtain the torque-tension relationship curves under different mesh sizes and shapes, as shown below. Figure 13As shown, the preload result under ideal conditions is calculated based on the torsion-tension relationship theory formula. It can be clearly seen that the smaller the mesh size, the closer the result is to the ideal value, and the solution accuracy of the hexahedral mesh is significantly higher than that of the tetrahedral mesh.

[0116] Table 2 Six different bolt head grid division methods

[0117]

[0118] Based on the torsion-tension relationship theory formula, a tightening angle of 3° was applied to the outer surface of the three nuts to simulate the assembly process. The preload force at this time met the preload force requirements for tightening 5.6 grade structural steel bolts, and the static analysis module in Ansysworkbench 19.2 was used to solve the problem. Figure 14 The distortion cloud maps of the primary and secondary mirrors are shown. It can be seen that the distortion of the primary mirror is much greater than that of the secondary mirror, while the distortion of the secondary mirror is negligible.

[0119] S3. Import the mirror deformation into the optomechanical-thermal coupling analysis module and insert Zernike polynomial coefficients to obtain the mirror surface shape error parameters and mirror assembly pose deviation.

[0120] In practice, finite element simulation can obtain the mirror deformation results. The deformation result data is then imported into the optomechanical-thermal coupling analysis tool Sigfit software. By fitting the optical surface deformation with Zernike polynomials, Zernike polynomial coefficient files and zpl sag files containing mirror surface shape error parameters and mirror assembly pose deviations can be obtained.

[0121] S4. Import the mirror surface shape error parameters and mirror assembly pose deviation into the optical simulation module to perform optical path imaging simulation and obtain the energy concentration used to characterize the imaging quality.

[0122] In implementation, Zemax software was selected for optical imaging simulation. When using Zemax for optical imaging simulation, it is first necessary to establish the corresponding reflecting surfaces of the primary and secondary mirrors. The primary mirror is a paraboloid of revolution with a 16.2mm aperture at its center, and the secondary mirror is an even-order hyperboloid. The distance between the primary and secondary mirrors is 38mm, the thickness tolerance of the primary and secondary mirrors is less than 0.01mm, the surface shape error (PV) is less than 0.3μm, the RMS value is less than 0.05μm, and the concentricity is controlled at approximately 10μm. Specific design parameters are detailed in Table 3. Figure 15The ideal optical path is shown. This optical system has a focal length of 120mm, a field of view of ±12.2°, and an F-number of 1.4. The incident light wavelength is set to 3.7-5.3μm, and the entrance pupil diameter is set to 84mm. Imaging quality is then evaluated using MTF (Modulation Transfer Function) and energy concentration. The MTF curve is shown below. Figure 16 As shown, the MTF across the entire field of view is greater than 0.9 at 9.6 lp / mm and still greater than 0.8 at 19.2 lp / mm, approaching the diffraction limit. The energy concentration curve is shown below. Figure 17 As shown, over 80% of the incident beam energy is concentrated within the Airy disk with a radius of 8.021 μm. Based on these two evaluation metrics, it can be concluded that the designed optical system possesses excellent imaging quality without considering mirror distortion, pose deviation, or an ideal optical path.

[0123] Next, the fitted zpl elevation file is loaded using a Zemax macro file, and the mirror assembly pose deviation is superimposed with the reflecting surface to generate a composite surface, thereby fitting the mirror deformation to the mirror surface; the mirror surface shape error is set and adjusted by inserting mirror coordinate discontinuities to achieve the setting and adjustment of the eccentricity and tilt of the primary and secondary mirrors.

[0124] Table 3 Design parameters for primary and secondary mirrors

[0125]

[0126] The optical system designed in this invention is mainly used for point target imaging and detection in engineering practice. Energy concentration determines the level of energy collected from the point target by a single pixel, directly affecting the image signal-to-noise ratio of the point target detection system. Compared to the modulation transfer function (MTF), energy concentration is more advantageous in evaluating beam quality and energy distribution. Therefore, from an energy perspective, energy concentration is chosen as a quantitative evaluation index for the imaging quality of the optical system. It should be noted that energy concentration is related to the detector pixel size. In the optical system designed in this invention, the detector pixel size is exactly equal to the Airy disk diameter, i.e., the detector pixel size is 16μm × 16μm. At this point, the energy concentration of this ideal optical path is 83.8%.

[0127] S5. A dataset is constructed based on the mirror surface shape error parameters, mirror assembly pose deviation, and energy concentration, and an SVR surrogate model containing local and global hybrid kernel functions is established.

[0128] In practice, the assembly pose errors of the primary and secondary mirrors, in the Cartesian coordinate system, include eccentricity errors along the X, Y, and Z directions, as well as tilt errors about these three coordinate axes. The Z-axis is the incident direction parallel to the light path. In actual assembly, the relative distance between the primary and secondary mirrors in the Z-axis direction is easily adjusted to achieve optimal imaging quality of the optical system. Since the primary and secondary mirrors have a circularly symmetrical structure, rotational errors in the Z-axis direction have a very small impact on image quality. Figure 18 The study demonstrates the change in energy concentration of the primary mirror under different rotation angles, even with mirror surface shape errors. The deviation in energy concentration of the optical system is less than 1% when rotating 0°, 90°, and 180° along the Z-axis. Therefore, the pose errors of the primary and secondary mirrors are mainly considered as eccentricity errors along the X and Y directions and tilt errors around the X and Y directions.

[0129] To address the mirror surface shape error, different tightening angles (2° to 6°) were applied to the bolts during simulation to generate varying preload. Then, a Zernike polynomial was used to fit the mirror surface shape error, yielding the first 45 Zernike polynomial coefficients, as follows: Figure 19 As shown, it can be seen that as the bolt preload increases, the changes in the 5th (defocus), 6th (first-order astigmatism - Y-axis), and 10th (first-order cloverleaf aberration - Y-axis) terms of the Zernike polynomial coefficients are much greater than those of the other terms. This indicates that these three Zernike polynomial coefficients have a significant impact on the principal mirror shape error. Therefore, the Zernike polynomial coefficients of the 5th, 6th, and 10th terms are selected as sensitive parameters for the mirror shape.

[0130] Based on the analysis of mirror surface shape error parameters and assembly pose error parameters, this paper analyzes and discusses the most significant factors affecting image quality during assembly, introducing the concept of a dot plot: For actual optical systems, light rays are not ideal straight lines, and the image is a diffuse light spot. The resulting light spot image is called a dot plot, and the distribution of points in the dot plot approximately represents the energy distribution of the image points. According to the definition of energy concentration, if all points in the dot plot fall within the Airy disk, the energy concentration reaches its maximum value. Figure 20 The diagram shows the imaging results of the optical system under different conditions. Figure a corresponds to the imaging result under an ideal optical path; figures b and c correspond to the imaging results under the individual effects of mirror surface shape error and assembly pose error, respectively; and figure d shows the imaging result under the combined effect of both, with energy concentrations of 83.8%, 83.8%, 52.4%, and 47.5%, respectively. It is clear that, compared to mirror surface shape error, assembly pose error has a more significant impact on the image quality of the optical system.

[0131] In summary, the mirror surface shape error parameters were selected from the Zernike polynomial coefficients, including defocus, first-order astigmatism-Y-axis, and first-order cloverleaf aberration-Y-axis. The mirror assembly pose deviation was selected from the eccentricity error of the primary and secondary mirrors along the X and Y directions and the tilt error around the X and Y directions, totaling 11 sensitive parameters as characteristic variables, with energy concentration as the target variable.

[0132] To eliminate the influence of different units, the feature variables and target variables are normalized so that their values ​​fall within the range of [0, 1]. The normalized feature variables and target variables are then divided into training and testing sets according to a set ratio, which can be 8:2.

[0133] In practice, the torque method can be used to assemble bolt fasteners. The preload deviation is approximately ±30%. With a consistent friction coefficient, the bolts are tightened at nine different angles, generating nine sets of bolt preload forces to simulate the actual assembly error of the bolts. The eccentricity error of the primary and secondary mirrors along the X and Y axes is ±0.01mm, and the tilt error around the X and Y axes is ±0.0167mm. For each set of bolt preload forces, 128 sets of eccentricity and tilt errors are randomly generated within the error range, resulting in a total of 1152 data sets. For each set of assembly errors, the energy concentration index of the optical system is calculated based on finite element assembly simulation, Zernike polynomial surface fitting method, and optical system imaging simulation, thus constructing a dataset.

[0134] Support Vector Machine (SVM) is a supervised learning method based on the Vapnik-Chervonenkis Dimension (VC Dimension) and structural risk minimization theory of statistical learning, primarily used for classification. For linearly separable problems, SVM is typically abstracted as a line or a plane, serving as a boundary to distinguish different categories. For linearly inseparable problems, kernel functions can be used to map low-dimensional data to a high-dimensional space, transforming the linearly inseparable problem in the low-dimensional space into a linearly separable problem in the high-dimensional space, which is then solved and mapped back to the low-dimensional space.

[0135] Support Vector Regression (SVR) is an extension of SVM, primarily used for regression problems. It uses SVM to fit curves to achieve regression analysis of data. The regression principle can be applied using... Figure 21 The ε-pipeline model shown illustrates this: Suppose we describe the function y(x) with precision ε, and predict y(x) using another function f(x) such that y(x) falls within the range of f(x) ± ε. Figure 21As shown, within the ε-elastic pipe, the support vectors at its edges prevent the pipe from expanding further, while the presence of relaxation vectors allows some training points to be outside the pipe, but the impact of these points on the regression results needs to be measured by the penalty coefficient.

[0136] Taking linear regression as an example, the goal is to define a linear function as the objective function. The equation is:

[0137] f(x) = ω·x + b

[0138] In the formula, ω is the normal vector, b is the bias term, and x is the input sample. Assuming that all training data can be fitted with a linear function at accuracy ε, the problem of finding the minimum ω can be expressed as a convex optimization problem, i.e., minimizing ω. Simultaneously satisfying the constraints. Secondly, considering that regression errors exceeding the accuracy ε may exist in practice, two slack variables ξ are introduced. i ,ξ i * ≥0, according to the structural risk minimization criterion of statistical learning theory, the regression estimation problem is transformed into,

[0139]

[0140] In the formula, C is the penalty coefficient. The constrained optimization problem is solved using duality theory, transforming it into a quadratic programming problem, and the Lagrangian function L is constructed.

[0141]

[0142] In the formula, the Lagrange multiplier a i a i * η i η i * All should be greater than or equal to 0, and the parameters w, b, and ξ should also be greater than or equal to 0 i ξ i * Since all partial derivatives are zero, the following dual optimization form can be obtained.

[0143]

[0144]

[0145] For nonlinear problems, kernel functions are introduced to map them to a higher-dimensional space, transforming the linearly separable dividing line f(x) into a hyperplane. The location is determined by solving the optimal solution under the dual optimization problem. Commonly used kernel functions include linear kernel functions, polynomial kernel functions, Gaussian kernel functions, and sigmoid kernel functions. Kernel functions are divided into global kernel functions and local kernel functions. The former can extract global features of samples well, with strong generalization ability but weak learning ability; a typical example is the polynomial kernel function. The latter can reflect local information of samples well, with strong learning ability but weak generalization ability; a typical example is the Gaussian kernel function.

[0146] In practice, a Gaussian kernel function is selected as the local kernel function, and a polynomial kernel function is selected as the global kernel function to construct a hybrid kernel function. By adjusting the weights of the global kernel function and the local kernel function, it can simultaneously possess the advantages of both global and local kernel functions, as shown in equation (5).

[0147] K mix =λK local +(1-λ)K global (5)

[0148] Among them, K local Refers to the local kernel function, K global The global kernel function is used, and λ is a scaling factor used to adjust the ratio between the local and global kernel functions.

[0149] S6. Use the dataset to train the SVR surrogate model to obtain the optical system imaging quality prediction model.

[0150] In practice, the SVR agent model is trained using training set data, and the trained SVR agent model is tested using test set data to calculate the coefficient of determination R. 2 The root mean square error (RMSE) and the mean relative error (MRE) are shown in equations (6), (7), and (8), respectively:

[0151]

[0152]

[0153]

[0154] Where n is the number of samples in the test set, y i Let y be the true value of energy concentration, and f(x) be the average of the true values ​​of energy concentration. i ) represents the predicted energy concentration, and m represents the number of model training cycles;

[0155] S62. Based on the calculated coefficient of determination, root mean square error, and average relative error, determine whether the prediction accuracy and generalization ability of the SVR surrogate model meet expectations. If they meet expectations, end the training. If they do not meet expectations, adjust the model parameters and execute S63.

[0156] S63. Re-divide the model into training and testing sets according to the set ratio and return to execute S61. Combine the prediction accuracy of single-cycle prediction with the prediction fluctuation of multi-cycle prediction to evaluate the prediction accuracy and generalization ability of the model. The regularization parameter C, the fault tolerance parameter ε, the polynomial kernel order, the scaling factor and the Gaussian kernel width are optimized multiple times to obtain the optical system imaging quality prediction model that meets the expectations.

[0157] In practice, for the 230 sets of test data constructed above, in order to intuitively demonstrate the predictive capabilities of the constructed proxy model, Figure 22 A scatter plot showing the predicted and actual values ​​from the surrogate model is presented. The plot shows that for most data points, the predicted values ​​coincide with or are very close to the actual values, with only a few data points showing some deviation. The coefficient of determination and root mean square error (RMSE) of the predicted values ​​in the test set are calculated using the formula, where R0... 2 The value is 0.95, and the corresponding RMSE value is 0.05, which indicates that the model has high reliability and accuracy in predicting image quality. The average relative error of the predicted values ​​is calculated again, and its value is 6.51%, that is, the average prediction accuracy of the constructed surrogate model reaches 93.49%.

[0158] To test the robustness and reliability of the surrogate model and reduce random errors caused by random factors, the dataset was randomly divided again in an 8:2 ratio (repeated 10 times), and the corresponding relative error data was calculated, such as... Figure 23 As shown, the maximum relative error is 7.33%, the minimum relative error is 5.85%, and the average is 6.51%. This indicates that under repeated experimental conditions, the prediction accuracy of the SVR surrogate model proposed in this invention exceeds 93%. Therefore, the surrogate model demonstrates high accuracy and consistency in predicting the imaging quality of optical systems.

[0159] Comparison of prediction accuracy after selecting different kernel functions

[0160] Three commonly used single kernel functions—linear, polynomial, and Gaussian—were compared, as well as the case without a kernel function. Similarly, to reduce test error and mitigate the influence of random factors, the dataset was randomly divided into training and test sets in an 8:2 ratio, and the experiment was repeated 10 times. The average prediction error of the test set was calculated for each of the 10 trials. Figure 24As shown in the figure, the average errors of both the kernel-free and linear kernel functions are greater than 30%, averaging around 40%, indicating poor optical image quality prediction performance. Using the Gaussian kernel function, the prediction errors in all three sets of experiments were greater than 15%, with an average prediction error of 10%. Using the polynomial kernel function, the prediction error in one set of experiments exceeded 13%, with an average error of 8%. In contrast, using the hybrid kernel function proposed in this invention, the test errors in all ten sets of experiments were less than 8%, with an average error of only 6.51%. Therefore, compared to the single kernel function and the kernel-free case, the proposed hybrid kernel function exhibits the best optical image quality prediction performance.

[0161] Comparison of prediction accuracy after selecting other surrogate models

[0162] A prediction model for a two-reflector optical system was constructed using an SVR regression model and a hybrid kernel function, demonstrating excellent prediction accuracy. Several other common surrogate models were selected for comparison, including Random Forest Regression (RFR), Multilayer Perceptron (MLP), Decision Tree Regression (DTR), and Generalized Additive Model (GAM). The training and test sets were partitioned using the same method, and the experiment was repeated 10 times. To achieve optimal prediction accuracy for these four models, fine-tuning of the parameters was performed. The prediction errors of different surrogate models are shown below. Figure 25 As shown in the figure, the decision tree regression model exhibits the worst prediction accuracy, with an average prediction error of 13.15%, followed by the multilayer perceptron model with an average prediction error of 11.38%. The random forest regression model and the generalized additive model have comparable prediction accuracy, with average prediction errors of 10.38% and 10.37%, respectively. However, both are lower than the proposed SVR model with a mixed kernel function (6.51%). Therefore, compared to other surrogate models, the SVR regression model and mixed kernel function proposed in this invention demonstrate the best optical imaging quality prediction performance.

[0163] This invention employs Zernike polynomials to accurately fit the mirror surface shape caused by assembly stress, and based on this, establishes an optomechanical integration simulation method for a two-reflector optical system considering assembly errors. It proposes an SVR surrogate model based on a mixture of local and global kernel functions to extract sensitive mirror surface shape errors and assembly pose error parameters, constructing an assembly error and imaging quality dataset. Through model training, high-precision prediction of imaging quality under different assembly error conditions is achieved. The following conclusions are drawn:

[0164] 1) By combining finite element simulation of the assembly process, Zernike polynomial mirror surface fitting, and Zemax optical path simulation, an effective approach is provided for imaging simulation and quality assessment of optical systems considering different assembly errors. An energy concentration index is established, and quantitative evaluation of the imaging quality of optical systems is realized.

[0165] 2) Sensitive factor analysis revealed that the key assembly parameters affecting the imaging quality of the optical system include the 5th, 6th and 10th terms of the Zernike polynomial fitting coefficients of the mirror surface shape, as well as the eccentricity and tilt error parameters of the primary and secondary mirrors along the X and Y axes (the Z axis is the incident direction parallel to the light path).

[0166] 3) An SVR surrogate model based on a mixture of local and global kernel functions is proposed, which can achieve high-precision prediction of energy concentration under different assembly error conditions. Compared with the SVR model with a single kernel function / no kernel function and other regression models, the average prediction error is only 6.51%, which shows excellent imaging quality prediction performance.

[0167] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention are included within the scope of protection of the present invention.

Claims

1. A method for constructing an optical system imaging quality prediction model considering both face shape error and pose error, characterized in that, The method comprises the following steps: S1, fitting the mirror surface shape in the optical system by using Zernike polynomials to obtain Zernike polynomial coefficients for characterizing the mirror surface shape error; S2, establishing a finite element model of the optical system, simulating the tightening process of the bolt connection in the finite element model, and obtaining the mirror deformation condition; S3, importing the mirror deformation condition into the optical-mechanical-thermal coupling analysis module and inserting the Zernike polynomial coefficients to obtain the mirror surface shape error parameters and the mirror assembly pose deviation; S4, importing the mirror surface shape error parameters and the mirror assembly pose deviation into the optical simulation module to perform optical path imaging simulation, and obtaining the energy concentration degree for characterizing the imaging quality; S5, constructing a data set based on the mirror surface shape error parameters, the mirror assembly pose deviation and the energy concentration degree, and establishing an SVR proxy model containing a local and global hybrid kernel function; S6, training the SVR proxy model by using the data set to obtain an optical system imaging quality prediction model; The process of establishing the finite element model of the optical system in S2 comprises: Establishing a finite element model of the optical system containing a threaded connection structure, and establishing a free tetrahedral mesh of the primary mirror, the secondary mirror and the support; Establishing mathematical expressions of the inner thread and outer thread cross-sectional profiles, the process being as follows: According to the thread parameters, the thread profile is divided into three different parts: thread root part, thread side part and thread top part; the three divided parts are extended to a plane to obtain the thread profile along the transverse section, the thread root part is selected as the starting point, θ represents the rotation angle of the point on the thread cross-sectional profile relative to the starting point, and the expression of the thread cross-sectional profile is established, as shown in formula (4): （4） wherein , , , , d denotes the nominal diameter, di denotes the minor diameter, r denotes the distance of the thread profile to the bolt center axis, p denotes the radius of the thread root circular arc; Based on the mathematical expressions of the inner thread and outer thread cross-sectional profiles and the rotational invariance of the thread cross-sectional profile, a thread mesh division method is used to establish the mesh model of the inner thread and the outer thread; The process of constructing a data set based on the mirror surface shape error parameters, the mirror assembly pose deviation and the energy concentration degree in S5 comprises: The mirror surface shape error parameters select the defocus, the first-order astigmatism-Y axis and the first-order trefoil-Y axis in the Zernike polynomial coefficients, the mirror assembly pose deviation selects the eccentric error and the tilt error of the primary mirror and the secondary mirror along the X and Y directions, and a total of 11 sensitive parameters are selected as characteristic variables, and the energy concentration degree is selected as a target variable; The characteristic variables and the target variable are normalized, and the normalized characteristic variables and the target variable are divided into a training set and a test set according to a set proportion.

2. The method of claim 1, wherein the method further comprises: determining a plurality of sets of surface error and pose error; and determining a plurality of sets of surface error and pose error for each set of surface error and pose error. S1, fitting the mirror surface shape in the optical system by using Zernike polynomials to obtain Zernike polynomial coefficients for characterizing the mirror surface shape error, specifically comprising: The translation amounts of the ideal position X, Y and Z axes after the mirror deformation are set as u x , u y and u z , the rotation amounts around the X, Y and Z axes are θx, θy and θz respectively, and cosθx=cosθy=cosθz=1, sinθx=θx, sinθy=θy, sinθz=θz; according to the homogeneous coordinate transformation method, the coordinate transformation before and after the mirror deformation is shown in formula (1): （1） wherein X i Y i Z i are the node coordinates before mirror surface deformation, X i ’ Y i ’ Z i ’ are the node coordinates after mirror surface deformation, and T is the rigid body displacement of mirror surface deformation obtained based on the least square method. Multiply the node coordinates after mirror surface deformation by the left side of the transpose matrix T -1 to obtain the node coordinates of the surface shape error after removing the displacement of the mirror surface cylinder, as shown in equation (2): （2） The Zernike polynomial is used as a base function to fit the surface shape error of the optical element, and solving the Zernike polynomial coefficient is to solve the surface shape error function W N The Zernike polynomial linear combination coefficient of (x, y) is shown as formula (3): （3） wherein Z i (x,y) is the Zernike polynomial of the i-th item in the Cartesian coordinate system, C i is the Zernike polynomial coefficient of the i-th item, and n is the number of Zernike polynomial items. The actual measured discrete point displacement is brought into formula (3), the least square method is used to solve the Zernike polynomial coefficient matrix, and the fitting accuracy of the coefficient matrix is evaluated by the root mean square error, and the coefficient matrix obtained after the fitting accuracy is the polynomial coefficient for characterizing the mirror surface shape error.

3. The method of claim 1, wherein the method further comprises: determining a plurality of sets of surface error and pose error; and determining a plurality of sets of surface error and pose error for each set of surface error and pose error. The process of S4, importing the mirror surface shape error parameters and the mirror assembly pose deviation into the optical simulation module to perform optical path imaging simulation, comprises: The corresponding reflection surfaces of the primary mirror and the secondary mirror are established, wherein the primary mirror is a rotating parabolic surface, the secondary mirror is an even hyperboloid, and the primary mirror center position aperture size, the primary and secondary mirror spacing distance, the primary and secondary mirror thickness tolerance, the surface shape error, and the concentricity are set; The mirror surface assembly pose deviation is superimposed with the reflection surface to generate a composite surface, so as to deform and fit the mirror surface to the mirror surface; The mirror surface shape error is set by inserting the mirror surface coordinate discontinuous surface to realize the eccentricity and tilt of the primary mirror and the secondary mirror.

4. The method of claim 1, wherein the process of establishing the SVR surrogate model containing the mixed kernel function of the local and global in S5 comprises: selecting a Gaussian kernel function as the local kernel function, selecting a polynomial kernel function as the global kernel function, and constructing the mixed kernel function as shown in formula (5): Setting the parameters of the SVR surrogate model, including the regularization parameter C, the fault tolerance parameter ε, the polynomial kernel order, the scale factor, and the Gaussian kernel width. （5） wherein, denotes a local kernel function, denotes a global kernel function, and λ is a scaling factor to adjust the proportion between the local kernel function and the global kernel function.

5. The method of claim 4, wherein the process of training the SVR surrogate model to obtain the optical system imaging quality prediction model using the data set in S6 comprises: S62, determining whether the prediction accuracy and the generalization ability of the SVR surrogate model meet the expectation based on the calculated determination coefficient, the root mean square error, and the average relative error, ending the training if the expectation is met, and adjusting the parameters of the model and executing S63 if the expectation is not met; S61. Train the SVR agent model using the training set data, test the trained SVR agent model using the test set data, and calculate the coefficient of determination. The root mean square error (RMSE) and the mean relative error (MRE) are shown in equations (6), (7), and (8), respectively: （6） （7） （8） wherein n is the number of test set samples, y i is the true value of the energy concentration, is the average value of the true value of the energy concentration, f(x i is the predicted value of the energy concentration, and m is the number of model training rounds. S63, re-dividing the training set and the test set according to the set proportion and returning to execute S61. ​

Images