An interferometric subaperture stitching method based on polynomial fitting and alternating optimization

Abstract

The application discloses an interference sub-aperture splicing method based on polynomial fitting and alternating optimization, and comprises the following steps: eliminating the phase tilt of all sub-apertures by using Zernike polynomials; constructing an alternating iterative optimization model of the sub-aperture phase distribution and the sub-aperture scanning pose error coefficient; calculating the initial value of the sub-aperture scanning pose error coefficient; continuously optimizing the sub-aperture scanning pose error coefficient by using the alternating iterative optimization model to obtain the sub-aperture space pose error corresponding to the measurement; according to the error, correcting the phase of the sub-aperture caused by the rigid body motion; splicing the phase of the overlapping area between the sub-apertures by using the weighted average method; and completing the measurement of the full-aperture phase. The application has good accuracy in correcting the pose error caused by the sub-aperture scanning and fast iterative convergence speed, does not need super-high-precision motion structure hardware, has low demand for hardware, and provides a low-cost and high-precision solution under the condition of poor motion structure precision.

Description

Technical Field

[0001] This invention belongs to the field of sub-aperture phase stitching technology, and in particular, it is an interferometric sub-aperture stitching method based on polynomial fitting and alternating optimization. Background Technology

[0002] With the rapid development of technologies such as astronomical optics and infrared imaging systems, optical systems with large-aperture, complex-surface optical elements are being used more and more widely. Sub-aperture stitching interferometry uses a small-aperture interferometer to achieve high-precision detection of large-aperture optical elements without the addition of auxiliary components, improving measurement resolution and acquiring more mid-to-high frequency information. However, in actual measurement, sub-aperture scanning is an indispensable step. Due to the inaccurate positioning of the mechanical moving structure during sub-aperture scanning, positioning errors are introduced into the measurement, causing the condition of consistency of the overlapping area to no longer be met, thereby reducing the accuracy of full-aperture phase recovery.

[0003] To accurately recover the full-aperture phase from sub-aperture phase maps containing positioning errors, numerous stitching algorithms have been developed. One method directly compensates for positioning errors such as translation, defocusing, and tilting in each sub-aperture phase data. This method is simple and efficient, quickly stitching and restoring the sub-aperture phases. However, it mistakenly identifies the sub-aperture surface distribution as a positioning error, leading to inaccurate measurements. Another method focuses on the consistency of overlapping areas between sub-apertures, using four mechanical error compensation terms—axial translation, x-direction tilt, y-direction tilt, and defocusing—to represent the spatial attitude of sub-apertures with positioning errors. This method offers higher accuracy than the first, but it doesn't adequately represent rigid body motion transformations and cannot decouple positioning errors generated by sub-aperture scanning. This results in an inaccurate description of the true spatial attitude of the sub-apertures. Furthermore, due to the limitations of detector pixels, it fails to represent minute spatial pose changes, leading to a larger error and lower accuracy in the stitched full-aperture phase reconstruction. Summary of the Invention

[0004] The purpose of this invention is to address the problems existing in the prior art by providing an interferometric sub-aperture stitching method based on polynomial fitting and alternating optimization to improve the accuracy of positioning error measurement.

[0005] The technical solution to achieve the purpose of this invention is: an interferometric sub-aperture stitching method based on polynomial fitting and alternating optimization, the method comprising the following steps:

[0006] Step 1, Phase OPD for each sub-aperture nIn the sub-aperture coordinate system, a Zernike polynomial is generated, the Zernike coefficients for each sub-aperture are fitted, and the phase is obtained by interpolation to eliminate the Zernike tilt and translation errors of each sub-aperture.

[0007] Step 2, for sub-aperture phase that eliminates tilt and translation errors. Construct an alternating iterative optimization model for sub-aperture phase distribution and sub-aperture scanning pose error coefficients;

[0008] Step 3: Calculate the initial values ​​of the pose error coefficients for all sub-aperture scans;

[0009] Step 4: Based on the calculated initial value of the sub-aperture scanning pose error coefficient, the sub-aperture scanning pose error coefficient is continuously optimized using an alternating iterative optimization model to obtain the corresponding sub-aperture spatial pose error coefficient during measurement.

[0010] Step 5: Based on the sub-aperture spatial pose error coefficients obtained in Step 4, calculate the sub-aperture phase by correcting the rigid body motion error. Then, the phase Ф(x,y) of the overlapping region is calculated using the weighted average method to restore the phase of the full aperture.

[0011] Furthermore, the specific process of fitting the Zernike coefficients for each sub-aperture in step 1 includes:

[0012] Step 1-1: Measure the phase OPD of each sub-aperture. n for:

[0013]

[0014] In the formula, n is the sub-aperture index, n = 1, 2, 3, ..., N, N is the total number of sub-apertures, m is the total number of terms in the Zernike polynomial, and ρ and θ are the normalized polar coordinates of the pixel. Represents the nth sub-aperture Pth aperture n The Zernike coefficient of the term, Represents the nth sub-aperture Pth aperture n Zernike polynomial of the term;

[0015] Steps 1-2: Phase OPD of each sub-aperture n Rewritten as:

[0016]

[0017] Then, the least squares method is used, utilizing the known OPD. n and Calculate The value;

[0018] Steps 1-3: Phase OPD of each sub-aperture n Rewritten as:

[0019]

[0020] In the formula, and These are the first three Zernike coefficients of the nth sub-aperture. and The first three Zernike terms of the nth sub-aperture represent the translation wavefront relative to the z-axis, the tilt wavefront relative to the x-axis, and the tilt wavefront relative to the y-axis, respectively. It is the nth sub-aperture phase obtained by eliminating tilt and translation errors during actual testing.

[0021] Furthermore, the sub-aperture phase described in step 2 for eliminating tilt and translation errors... An iterative optimization model is constructed for the sub-aperture phase distribution and sub-aperture scanning pose error coefficients, specifically including:

[0022] Step 2-1, for sub-aperture phase that eliminates tilt and translation errors. Sub-aperture phase The expression for the pose error of the six-dimensional sub-aperture scanning is:

[0023]

[0024] In the formula, x n y n Let x and y be the x and y coordinates of all pixels at the nth sub-aperture under the full aperture. This represents the nth sub-aperture phase obtained after eliminating tilt and translation errors during actual testing. Δx represents the nth sub-aperture phase under ideal conditions where there are no sub-aperture scanning pose errors. n Δy represents the rigid body motion pose error coefficient for the translation along the x-axis of the nth sub-aperture. n Δz represents the rigid body motion pose error coefficient for the translation along the y-axis of the nth sub-aperture. n Δα represents the rigid body motion pose error coefficient for the translation along the z-axis of the nth sub-aperture. n Δβ represents the rigid body motion pose error coefficient for the nth sub-aperture rotating along the x-axis. n Δγ represents the rigid body motion pose error coefficient for the nth sub-aperture rotating along the y-axis. n The rigid body motion pose error coefficient representing the rotation along the z-axis direction of the nth sub-aperture;

[0025] Step 2-2, phase the sub-aperture In the expression for pose error of six-dimensional sub-aperture scanning and Approximately and The expression can then be rewritten as:

[0026]

[0027] In the formula, Represents the sub-aperture scanning pose error coefficient, where and Let represent the sub-aperture scanning pose error polynomial, where

[0028]

[0029]

[0030]

[0031] Steps 2-3 combine the sub-aperture scanning pose error coefficient after the kth iteration. and The phase of the nth sub-aperture after eliminating the sub-aperture scanning pose error after the kth iteration, calculated through step 2-2. By finding the overlapping regions between pairs of sub-apertures and calculating the difference between them and the polynomial of the scanning pose error of the overlapping sub-apertures, a system of equations is constructed:

[0032]

[0033] In the formula, i, j, and n represent the sub-aperture numbers, and N is the total number of sub-apertures; x i y i Let x and y be the x and y coordinates of all pixels at the full aperture for the i-th sub-aperture. This represents the i-th sub-aperture phase after eliminating the pose error of the sub-aperture scan after the k-th iteration;

[0034] The sub-aperture scanning pose error coefficient for the (k+1)th iteration is calculated using the least squares method. and

[0035] Step 2-4: Repeat steps 2-1 to 2-3, performing alternating iterative optimization between sub-aperture phase distribution and sub-aperture scanning pose error coefficients until Δx... n Δy n Δzn ,Δα n Δβ n and Δγ n Convergence has been achieved. The convergence calculation formula is as follows:

[0036]

[0037]

[0038]

[0039] In the formula, ε1, ε2, ε3, ε4, ε5 and ε6 are the convergence thresholds of two consecutive iterations, and k represents the number of iterations.

[0040] Furthermore, the initial value of the pose error coefficient for all sub-aperture scanning described in step 3 includes the following specific steps:

[0041] Sub-aperture phase for eliminating tilt and translation errors Search The overlapping regions between each pair of sub-apertures are calculated, and the difference between the overlapping regions of the two sub-apertures is taken. Then, the initial values ​​of the sub-aperture scanning pose error coefficients are fitted using the least squares method.

[0042]

[0043] In the formula, i and j represent the sub-aperture numbers, N is the total number of sub-apertures, and k i and k j These represent the scanning pose error coefficients or sub-aperture scanning pose error polynomials for the i-th and j-th sub-apertures, respectively.

[0044] Furthermore, in step 5, based on the sub-aperture spatial pose error coefficient obtained in step 4, the rigid body motion error is calculated to obtain the sub-aperture phase. Then, the phase Ф(x,y) of the overlapping region is calculated using a weighted average method to restore the phase of the full aperture. Specifically, this includes:

[0045] Step 5-1, based on the sub-aperture spatial pose error coefficient Δx obtained in step 4 n Δy n Δz n ,Δα n Δβ n and Δγ n ,calculate

[0046]

[0047] Step 5-2: Calculate the phase Ф(x,y) of the overlapping region using the weighted average method to restore the phase of the full aperture. The calculation formula is as follows:

[0048]

[0049] In the formula, Ф(x,y) represents the phase of the overlapping region calculated using the weighted average method, and x and y represent the horizontal and vertical coordinates of the pixel at full aperture. n y n L represents the x and y coordinates of the pixel at the nth sub-aperture. n R is the distance from the data point in the overlapping region of the nth sub-aperture to the center of that sub-aperture, and R is the radius of a single sub-aperture.

[0050] Compared with the prior art, the significant advantages of this invention are:

[0051] 1) Six rigid body motion pose error polynomials are used to represent the positioning error caused by sub-aperture scanning, which can more completely express the rigid body sub-aperture scanning transformation and improve the accuracy of full-aperture phase recovery.

[0052] 2) An alternating iterative optimization model for sub-aperture phase distribution and sub-aperture scanning pose error coefficients was established, which avoids the problem of getting trapped in local optima and improves the robustness of full-aperture phase recovery.

[0053] 3) By using interpolation amplification, small positioning errors can be identified and corrected, and even with small mechanical positioning errors, the full-aperture phase recovery accuracy is good.

[0054] The present invention will now be described in further detail with reference to the accompanying drawings. Attached Figure Description

[0055] Figure 1 This is a flowchart of the interferometric sub-aperture stitching method based on polynomial fitting and alternating optimization of the present invention.

[0056] Figure 2 Here are nine sub-aperture phase maps used for calculation in one embodiment, wherein Figure 2 In the diagram, (a) represents the phase distribution obtained during the actual measurement of the first sub-aperture, i.e., OPD1. Figure 2 (b) in the figure represents the phase distribution obtained during the actual measurement of the second sub-aperture, i.e., OPD2. Figure 2 (c) in the figure represents the phase distribution obtained during the actual measurement of the third sub-aperture, i.e., OPD3. Figure 2 In the diagram, (d) represents the phase distribution obtained during the actual measurement of the fourth sub-aperture, i.e., OPD4. Figure 2 In the diagram, (e) represents the phase distribution obtained during the actual measurement of the fifth sub-aperture, i.e., OPD5. Figure 2In the diagram, (f) represents the phase distribution obtained during the actual measurement of the sixth sub-aperture, i.e., OPD6. Figure 2 In the diagram, (g) represents the phase distribution obtained during the actual measurement of the seventh sub-aperture, i.e., OPD7. Figure 2 In this context, (h) represents the phase distribution obtained during the actual measurement of the eighth sub-aperture, i.e., OPD8. Figure 2 In the diagram, (i) represents the phase distribution obtained during the actual measurement of the th sub-aperture, i.e., OPD9.

[0057] Figure 3 This is a schematic diagram of the restored full-aperture phase distribution in one embodiment.

[0058] Figure 4 This is a schematic diagram of the full-aperture phase distribution obtained using the conventional four-step phase shifting method in one embodiment.

[0059] Figure 5 This is a schematic diagram of the phase residual distribution obtained by using the method of the present invention and synchronous phase shift measurement in one embodiment. Detailed Implementation

[0060] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0061] In one embodiment, combined Figure 1 This paper provides a method for stitching interferometric sub-apertures based on polynomial fitting and alternating optimization, characterized by the following steps:

[0062] Step 1, Phase OPD for each sub-aperture n In the sub-aperture coordinate system, a Zernike polynomial is generated, the Zernike coefficients for each sub-aperture are fitted, and the phase is obtained by interpolation to eliminate the Zernike tilt and translation errors of each sub-aperture.

[0063] Step 2, for sub-aperture phase that eliminates tilt and translation errors. Construct an alternating iterative optimization model for sub-aperture phase distribution and sub-aperture scanning pose error coefficients;

[0064] Step 3: Calculate the initial values ​​of the pose error coefficients for all sub-aperture scans;

[0065] Step 4: Based on the calculated initial value of the sub-aperture scanning pose error coefficient, the sub-aperture scanning pose error coefficient is continuously optimized using an alternating iterative optimization model to obtain the corresponding sub-aperture spatial pose error coefficient during measurement.

[0066] Step 5: Based on the sub-aperture spatial pose error coefficients obtained in Step 4, calculate the sub-aperture phase by correcting the rigid body motion error. Then, the phase Ф(x,y) of the overlapping region is calculated using the weighted average method to restore the phase of the full aperture.

[0067] Furthermore, in one embodiment, the process of fitting the Zernike coefficient for each sub-aperture in step 1 specifically includes:

[0068] Step 1-1: Measure the phase OPD of each sub-aperture. n for:

[0069]

[0070] In the formula, n is the sub-aperture index, n = 1, 2, 3, ..., N, N is the total number of sub-apertures, m is the total number of terms in the Zernike polynomial, and ρ and θ are the normalized polar coordinates of the pixel. Represents the nth sub-aperture Pth aperture n The Zernike coefficient of the term, Represents the nth sub-aperture Pth aperture n Zernike polynomial of the term;

[0071] Steps 1-2: Phase OPD of each sub-aperture n Rewritten as:

[0072]

[0073] Then, the least squares method is used, utilizing the known OPD. n and Calculate The value;

[0074] Steps 1-3: Phase OPD of each sub-aperture n Rewritten as:

[0075]

[0076] In the formula, and These are the first three Zernike coefficients of the nth sub-aperture. and The first three Zernike terms of the nth sub-aperture represent the translation wavefront relative to the z-axis, the tilt wavefront relative to the x-axis, and the tilt wavefront relative to the y-axis, respectively. It is the nth sub-aperture phase obtained by eliminating tilt and translation errors during actual testing.

[0077] Furthermore, in one embodiment, step 2 refers to the sub-aperture phase for eliminating tilt and translation errors. An iterative optimization model is constructed for the sub-aperture phase distribution and sub-aperture scanning pose error coefficients, specifically including:

[0078] Step 2-1, for sub-aperture phase that eliminates tilt and translation errors. Sub-aperture phase The expression for the pose error of the six-dimensional sub-aperture scanning is:

[0079]

[0080] In the formula, x n y n Let x and y be the x and y coordinates of all pixels at the nth sub-aperture under the full aperture. This represents the nth sub-aperture phase obtained after eliminating tilt and translation errors during actual testing. Δx represents the nth sub-aperture phase under ideal conditions where there are no sub-aperture scanning pose errors. n Δy represents the rigid body motion pose error coefficient for the translation along the x-axis of the nth sub-aperture. n Δz represents the rigid body motion pose error coefficient for the translation along the y-axis of the nth sub-aperture. n Δα represents the rigid body motion pose error coefficient for the translation along the z-axis of the nth sub-aperture. n Δβ represents the rigid body motion pose error coefficient for the nth sub-aperture rotating along the x-axis. n Δγ represents the rigid body motion pose error coefficient for the nth sub-aperture rotating along the y-axis. n The rigid body motion pose error coefficient representing the rotation along the z-axis direction of the nth sub-aperture;

[0081] Step 2-2, phase the sub-aperture In the expression for pose error of six-dimensional sub-aperture scanning and Approximately and The expression can then be rewritten as:

[0082]

[0083] In the formula, C in Represents the sub-aperture scanning pose error coefficient, where and Let represent the sub-aperture scanning pose error polynomial, where

[0084] Steps 2-3 combine the sub-aperture scanning pose error coefficient after the kth iteration. and The nth sub-aperture phase obtained by approximation in step 2-2 after eliminating the sub-aperture scanning pose error after the kth iteration. By finding the overlapping regions between pairs of sub-apertures and calculating the difference between them and the polynomial of the scanning pose error of the overlapping sub-apertures, a system of equations is constructed:

[0085]

[0086] In the formula, i, j, and n represent the sub-aperture numbers, and N is the total number of sub-apertures; x i y i Let x and y be the x and y coordinates of all pixels at the full aperture for the i-th sub-aperture. This represents the i-th sub-aperture phase after eliminating the pose error of the sub-aperture scan after the k-th iteration;

[0087] The sub-aperture scanning pose error coefficient for the (k+1)th iteration is calculated using the least squares method. and

[0088] Step 2-4: Repeat steps 2-1 to 2-3, performing alternating iterative optimization between sub-aperture phase distribution and sub-aperture scanning pose error coefficients until Δx... n Δy n Δz n ,Δα n Δβ n and Δγ n Convergence has been achieved. The convergence calculation formula is as follows:

[0089]

[0090]

[0091]

[0092] In the formula, ε1, ε2, ε3, ε4, ε5 and ε6 are the convergence thresholds of two consecutive iterations, and k represents the number of iterations.

[0093] Furthermore, in one embodiment, the initial value of calculating the pose error coefficients of all sub-aperture scanning in step 3 specifically includes:

[0094] Sub-aperture phase for eliminating tilt and translation errors Search The overlapping regions between each pair of sub-apertures are calculated, and the difference between the overlapping regions of the two sub-apertures is taken. Then, the initial values ​​of the sub-aperture scanning pose error coefficients are fitted using the least squares method.

[0095]

[0096] In the formula, i and j represent the sub-aperture numbers, N is the total number of sub-apertures, and k i and k j These represent the scanning pose error coefficients or sub-aperture scanning pose error polynomials for the i-th and j-th sub-apertures, respectively.

[0097] Furthermore, in one embodiment, step 5 involves calculating the sub-aperture phase based on the sub-aperture spatial pose error coefficients obtained in step 4 to correct rigid body motion errors. Then, the phase Ф(x,y) of the overlapping region is calculated using a weighted average method to restore the phase of the full aperture. Specifically, this includes:

[0098] Step 5-1, based on the sub-aperture spatial pose error coefficient Δx obtained in step 4 n Δy n Δz n ,Δα n Δβ n and Δγ n ,calculate

[0099]

[0100] Step 5-2: Calculate the phase Ф(x,y) of the overlapping region using the weighted average method to restore the phase of the full aperture. The calculation formula is as follows:

[0101]

[0102] In the formula, Ф(x,y) represents the phase of the overlapping region calculated using the weighted average method, and x and y represent the horizontal and vertical coordinates of the pixel at full aperture. n y n L represents the x and y coordinates of the pixel at the nth sub-aperture. n R is the distance from the data point in the overlapping region of the nth sub-aperture to the center of that sub-aperture, and R is the radius of a single sub-aperture.

[0103] In one embodiment, an interferometric sub-aperture stitching system based on polynomial fitting and alternating optimization is provided, characterized in that the system comprises:

[0104] The first module is used for phase OPD for each sub-aperture. nIn the sub-aperture coordinate system, a Zernike polynomial is generated, the Zernike coefficients for each sub-aperture are fitted, and the phase is obtained by interpolation to eliminate the Zernike tilt and translation errors of each sub-aperture.

[0105] The second module is used to address the sub-aperture phase, which eliminates tilt and translation errors. Construct an alternating iterative optimization model for sub-aperture phase distribution and sub-aperture scanning pose error coefficients;

[0106] The third module is used to calculate the initial values ​​of the pose error coefficients for all sub-aperture scanning.

[0107] The fourth module is used to continuously optimize the sub-aperture scanning pose error coefficient based on the calculated initial value of the sub-aperture scanning pose error coefficient using an alternating iterative optimization model, so as to obtain the sub-aperture spatial pose error coefficient corresponding to the measurement.

[0108] The fifth module is used to calculate the sub-aperture phase by correcting rigid body motion errors based on the sub-aperture spatial pose error coefficients obtained above. Then, the phase Ф(x,y) of the overlapping region is calculated using the weighted average method to restore the phase of the full aperture.

[0109] Specific limitations regarding the high-fidelity image reconstruction system for super-resolution structured light illumination microscopy can be found in the above description of the limitations on the high-fidelity image reconstruction method for super-resolution structured light illumination microscopy, and will not be repeated here. Each module in the aforementioned high-fidelity image reconstruction system for super-resolution structured light illumination microscopy can be implemented entirely or partially through software, hardware, or a combination thereof. These modules can be embedded in or independent of the processor in a computer device, or stored in the memory of a computer device as software, so that the processor can call and execute the corresponding operations of each module.

[0110] In one embodiment, a computer device is provided, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to perform the following steps:

[0111] Step 1, Phase OPD for each sub-aperture n In the sub-aperture coordinate system, a Zernike polynomial is generated, the Zernike coefficients for each sub-aperture are fitted, and the phase is obtained by interpolation to eliminate the Zernike tilt and translation errors of each sub-aperture.

[0112] Step 2, for sub-aperture phase that eliminates tilt and translation errors. Construct an alternating iterative optimization model for sub-aperture phase distribution and sub-aperture scanning pose error coefficients;

[0113] Step 3: Calculate the initial values ​​of the pose error coefficients for all sub-aperture scans;

[0114] Step 4: Based on the calculated initial value of the sub-aperture scanning pose error coefficient, the sub-aperture scanning pose error coefficient is continuously optimized using an alternating iterative optimization model to obtain the corresponding sub-aperture spatial pose error coefficient during measurement.

[0115] Step 5: Based on the sub-aperture spatial pose error coefficients obtained in Step 4, calculate the sub-aperture phase by correcting the rigid body motion error. Then, the phase Ф(x,y) of the overlapping region is calculated using the weighted average method to restore the phase of the full aperture.

[0116] For specific limitations on each step, please refer to the limitations on the interferometric sub-aperture stitching method based on polynomial fitting and alternating optimization mentioned above, which will not be repeated here.

[0117] In one embodiment, a computer-readable storage medium is provided having a computer program stored thereon, the computer program performing the following steps when executed by a processor:

[0118] Step 1, Phase OPD for each sub-aperture n In the sub-aperture coordinate system, a Zernike polynomial is generated, the Zernike coefficients for each sub-aperture are fitted, and the phase is obtained by interpolation to eliminate the Zernike tilt and translation errors of each sub-aperture.

[0119] Step 2, for sub-aperture phase that eliminates tilt and translation errors. Construct an alternating iterative optimization model for sub-aperture phase distribution and sub-aperture scanning pose error coefficients;

[0120] Step 3: Calculate the initial values ​​of the pose error coefficients for all sub-aperture scans;

[0121] Step 4: Based on the calculated initial value of the sub-aperture scanning pose error coefficient, the sub-aperture scanning pose error coefficient is continuously optimized using an alternating iterative optimization model to obtain the corresponding sub-aperture spatial pose error coefficient during measurement.

[0122] Step 5: Based on the sub-aperture spatial pose error coefficients obtained in Step 4, calculate the sub-aperture phase by correcting the rigid body motion error. Then, the phase Ф(x,y) of the overlapping region is calculated using the weighted average method to restore the phase of the full aperture.

[0123] For specific limitations on each step, please refer to the limitations on the interferometric sub-aperture stitching method based on polynomial fitting and alternating optimization mentioned above, which will not be repeated here.

[0124] As a specific example, in one embodiment, the interferometric sub-aperture stitching method based on polynomial fitting and alternating optimization of the present invention is verified and illustrated, specifically including:

[0125] In this embodiment, a 4D dynamic interferometer is used to acquire interferograms of a plane mirror with an aperture of 975.6 mm, a PV of 1.0646λ, and an RMS of 0.1382λ. Sub-aperture scanning is performed according to a pre-planned path, acquiring nine sub-aperture interferograms. These interferograms are 504×504 pixels in size and are then analyzed. Figure 2 As shown, the nine sub-aperture phases are stitched together and restored using the following steps:

[0126] Step 1, Phase OPD for 9 sub-apertures n In the sub-aperture coordinate system, a Zernike polynomial is generated, the Zernike coefficients for each sub-aperture are fitted, and interpolation is used to obtain the phase that eliminates the Zernike tilt and translation errors for each sub-aperture.

[0127] Step 2, targeting the nine sub-aperture phases to eliminate tilt and translation errors. An alternating iterative optimization model for the sub-aperture phase distribution and sub-aperture scanning pose error coefficients is constructed, and the convergence value is set to ε1 = 10 in steps 2-4. -9 ε2=10 -9 ε3=10 -9 ε4=10 -4 ε5=10 -4 and ε6 = 10 -5 The interpolation factor is set to 2;

[0128] Step 3: Calculate the initial values ​​of the pose error coefficients for all sub-aperture scanning.

[0129] Step 4: Based on the calculated initial values ​​of the rigid body motion pose error coefficients, the sub-aperture scanning pose error coefficients are continuously optimized using an alternating iterative optimization model to obtain the corresponding sub-aperture spatial pose error coefficient Δx during measurement. n Δy n Δz n ,Δα n Δβ n and Δγ n ;

[0130] Step 5, based on the sub-aperture spatial pose error coefficient Δx n Δyn Δz n ,Δα n Δβ n and Δγ n The phase of the sub-aperture calculated by correcting rigid body motion errors is The phase of the full aperture is reconstructed by using the weighted average method to calculate the phase Ф(x,y) of the overlapping region, such as... Figure 3 As shown.

[0131] The phase distribution of the entire aperture was calculated by using a 4D dynamic interferometer for synchronous phase shifting measurement, and the resulting phase distribution is as follows: Figure 4 As shown, the method of this invention has a high degree of consistency with the calculation results of synchronous phase shift measurement using a 4D dynamic interferometer; the residual phase of the two methods is as follows: Figure 5 As shown, the phase residuals PV and RMS are very small, with PV being 1.3318e-5λ and RMS being 7.8678e-7λ. Therefore, the method of this invention can effectively stitch together and restore the phase of the entire aperture. For cases with small rigid body motion errors, it can effectively correct the pose errors of rigid body motion and has a relatively fast iterative convergence speed.

[0132] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the claimed invention.

Claims

1. A method for stitching interferometric sub-apertures based on polynomial fitting and alternating optimization, characterized in that, The method includes the following steps: Step 1, Phase OPD for each sub-aperture n In the sub-aperture coordinate system, a Zernike polynomial is generated, the Zernike coefficients for each sub-aperture are fitted, and the phase is obtained by interpolation to eliminate the Zernike tilt and translation errors of each sub-aperture. ; Step 2, for sub-aperture phase that eliminates tilt and translation errors. An iterative optimization model is constructed for the sub-aperture phase distribution and sub-aperture scanning pose error coefficients; specifically including: Step 2-1, for sub-aperture phase that eliminates tilt and translation errors. Sub-aperture phase The expression for the pose error of the six-dimensional sub-aperture scanning is: In the formula, x n y n Let x and y be the x and y coordinates of all pixels at the nth sub-aperture under the full aperture. This represents the nth sub-aperture phase obtained after eliminating tilt and translation errors during actual testing. Δx represents the nth sub-aperture phase under ideal conditions where there are no sub-aperture scanning pose errors. n Δy represents the rigid body motion pose error coefficient for the translation along the x-axis of the nth sub-aperture. n Δz represents the rigid body motion pose error coefficient for the translation along the y-axis of the nth sub-aperture. n Δα represents the rigid body motion pose error coefficient for the translation along the z-axis of the nth sub-aperture. n Indicates the edge of the nth sub-aperture The pose error coefficient for rigid body motion with rotation along the axis, Δβ n Δγ represents the rigid body motion pose error coefficient for the nth sub-aperture rotating along the y-axis. n The rigid body motion pose error coefficient representing the rotation along the z-axis direction of the nth sub-aperture; Step 2-2, phase the sub-aperture In the expression for pose error of six-dimensional sub-aperture scanning and Approximately and The expression is then rewritten as: In the formula, Represents the sub-aperture scanning pose error coefficient, where , , , , and , Let represent the sub-aperture scanning pose error polynomial, where , , , , , ; Steps 2-3 combine the sub-aperture scanning pose error coefficient after the kth iteration. , , , , and The phase of the nth sub-aperture after eliminating the sub-aperture scanning pose error after the kth iteration, obtained through step 2-2. By finding the overlapping regions between pairs of sub-apertures and calculating the difference between them and the polynomial of the scanning pose error of the overlapping sub-apertures, a system of equations is constructed: In the formula, i, j, and n represent the sub-aperture numbers, and N is the total number of sub-apertures; x i y i Let x and y be the x and y coordinates of all pixels at the full aperture for the i-th sub-aperture. This represents the i-th sub-aperture phase after eliminating the pose error of the sub-aperture scan after the k-th iteration; The sub-aperture scanning pose error coefficient for the (k+1)th iteration is calculated using the least squares method. , , , , and ; Step 2-4: Repeat steps 2-1 to 2-3, performing alternating iterative optimization between sub-aperture phase distribution and sub-aperture scanning pose error coefficients until Δx... n Δy n Δz n ,Δα n Δβ n and Δγ n Convergence has been achieved. The convergence calculation formula is as follows: , , , In the formula, , , , , and Both are convergence thresholds between two consecutive iterations, and k represents the number of iterations; Step 3: Calculate the initial values ​​of the pose error coefficients for all sub-aperture scans; Step 4: Based on the calculated initial value of the sub-aperture scanning pose error coefficient, the sub-aperture scanning pose error coefficient is continuously optimized using an alternating iterative optimization model to obtain the corresponding sub-aperture spatial pose error coefficient during measurement. Step 5: Based on the sub-aperture spatial pose error coefficients obtained in Step 4, calculate the sub-aperture phase by correcting the rigid body motion error. Then, the phase Ф(x,y) of the overlapping region is calculated using the weighted average method to restore the phase of the full aperture.

2. The interferometric sub-aperture stitching method based on polynomial fitting and alternating optimization according to claim 1, characterized in that, The specific process of fitting the Zernike coefficients for each sub-aperture in step 1 includes: Step 1-1: Measure the phase OPD of each sub-aperture. n for: In the formula, n is the sub-aperture number. N is the total number of sub-apertures, m is the total number of terms in the Zernike polynomial, and ρ and θ are the normalized polar coordinates of the pixel. Represents the nth sub-aperture Pth aperture n The Zernike coefficient of the term, Represents the nth sub-aperture Pth aperture n Zernike polynomial of the term; Steps 1-2: Phase OPD of each sub-aperture n Rewritten as: Then, the least squares method is used, utilizing the known OPD. n and Calculate The value; Steps 1-3: Phase OPD of each sub-aperture n Rewritten as: In the formula, , and These are the first three Zernike coefficients of the nth sub-aperture. , and The first three Zernike terms of the nth sub-aperture represent the translation wavefront relative to the z-axis, the tilt wavefront relative to the x-axis, and the tilt wavefront relative to the y-axis, respectively. It is the nth sub-aperture phase obtained by eliminating tilt and translation errors during actual testing.

3. The interferometric sub-aperture stitching method based on polynomial fitting and alternating optimization according to claim 2, characterized in that, Step 3, which involves calculating the initial values ​​of the pose error coefficients for all sub-aperture scans, specifically includes the following steps: Sub-aperture phase for eliminating tilt and translation errors Search The overlapping regions between each pair of sub-apertures are calculated, and the difference between the overlapping regions of the two sub-apertures is taken. Then, the initial values ​​of the sub-aperture scanning pose error coefficients are fitted using the least squares method. In the formula, i and j represent the sub-aperture numbers, N is the total number of sub-apertures, and k i and k j These represent the scanning pose error coefficients or sub-aperture scanning pose error polynomials for the i-th and j-th sub-apertures, respectively.

4. The interferometric sub-aperture stitching method based on polynomial fitting and alternating optimization according to claim 3, characterized in that, Step 5 involves calculating the sub-aperture phase based on the sub-aperture spatial pose error coefficients obtained in Step 4, and then performing rigid body motion error correction. Then, the phase Ф(x,y) of the overlapping region is calculated using a weighted average method to restore the phase of the full aperture. Specifically, this includes: Step 5-1, based on the sub-aperture spatial pose error coefficient Δx obtained in step 4 n Δy n Δz n ,Δα n Δβ n and Δγ n ,calculate : Step 5-2: Calculate the phase Ф(x,y) of the overlapping region using the weighted average method to restore the phase of the full aperture. The calculation formula is as follows: In the formula, Ф(x,y) represents the phase of the overlapping region calculated using the weighted average method, and x and y represent the horizontal and vertical coordinates of the pixel at full aperture. n y n This represents the x and y coordinates of the pixel at the nth sub-aperture. R is the distance from the data point in the overlapping region of the nth sub-aperture to the center of that sub-aperture, and R is the radius of a single sub-aperture.

5. An interferometric sub-aperture stitching system based on polynomial fitting and alternating optimization according to any one of claims 1 to 4, characterized in that, The system includes: The first module is used for phase OPD for each sub-aperture. n In the sub-aperture coordinate system, a Zernike polynomial is generated, the Zernike coefficients for each sub-aperture are fitted, and the phase is obtained by interpolation to eliminate the Zernike tilt and translation errors of each sub-aperture. ; The second module is used to address the sub-aperture phase, which eliminates tilt and translation errors. An iterative optimization model for sub-aperture phase distribution and sub-aperture scanning pose error coefficients is constructed. The third module is used to calculate the initial values ​​of the pose error coefficients for all sub-aperture scanning. The fourth module is used to continuously optimize the sub-aperture scanning pose error coefficient based on the calculated initial value of the sub-aperture scanning pose error coefficient using an alternating iterative optimization model, so as to obtain the sub-aperture spatial pose error coefficient corresponding to the measurement. The fifth module is used to calculate the sub-aperture phase by correcting rigid body motion errors based on the sub-aperture spatial pose error coefficients obtained above. Then, the phase Ф(x,y) of the overlapping region is calculated using the weighted average method to restore the phase of the full aperture.

6. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1 to 4.

7. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the steps of the method according to any one of claims 1 to 4.

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