A complex curved surface reconstruction method based on white light interferometry data

Abstract

This invention provides a method for reconstructing complex surfaces based on white light interferometry data: The method acquires surface topography measurement data of the complex surface to be reconstructed using white light interferometry; constructs a sliding local region centered on each point to be estimated within the measurement area; processes the measurement data within each local region to obtain the height value of the corresponding point to be estimated; and outputs the global reconstruction result of the complex surface after traversing all points to be estimated across the entire region. The process of processing the measurement data within each local region includes the following steps: S1, using MM estimation to perform robust pre-estimation of the measurement data within the current local region; S2, using the absolute value of the residual as input, and through mixing... t Distribution clustering adaptively classifies the measurement data to obtain the retained valid measurement data; S3, repeat S1 to S2 until the outlier removal iteration is completed to obtain the final valid measurement dataset; S4, use weighted Jacobi basis functions to perform local regression to obtain the height value of the point to be estimated.

Description

Technical Field

[0001] This invention belongs to the field of precision optical measurement and three-dimensional topography reconstruction technology, specifically involving a method for reconstructing complex curved surfaces based on white light interferometry data. Background Technology

[0002] With the rapid development of precision optical manufacturing technology, high-precision complex optical components are increasingly widely used in aerospace, high-end imaging systems, semiconductor lithography, precision sensing, and advanced manufacturing. These complex optical surfaces typically possess special surface features such as aspherical surfaces, freeform surfaces, or micro / nano-scale arrays. Their surface accuracy directly determines the performance of the final optical system, thus placing increasingly stringent requirements on the high-precision inspection and quality evaluation of surface morphology.

[0003] White light interferometry, with its non-contact, non-destructive, high longitudinal resolution, and full-field 3D topography reconstruction capabilities, has become the mainstream technique for inspecting the surface topography of precision optical components and micro / nano structures. This technique performs a mechanical vertical scan by changing the relative position between the interferometer objective and the surface under test along the optical axis. Interference signal sequences are acquired at different scanning positions, and the height information of the surface is determined by locating the peak position or maximum modulation position of the coherent envelope of the interference signal, thus achieving high-precision characterization of the surface's 3D topography. However, in actual testing, the steep slopes, micro / nano structures, and abrupt changes in the topography of complex optical surfaces can easily lead to decreased contrast of interference fringes, reduced signal-to-noise ratio, and distortion of the coherent envelope in local areas. This can cause peak positioning deviations and height recovery errors, ultimately introducing high-amplitude, non-Gaussian distributed, and high-proportion outliers into the measurement data. These inherent defects, stemming from the measurement principle itself, cannot be completely eliminated through hardware system optimization. They can only be suppressed and corrected through subsequent data processing. The presence of outliers directly leads to severe distortion in the final 3D topography reconstruction results, failing to meet the high-precision detection requirements of complex optical components.

[0004] Existing surface reconstruction methods mostly employ polynomial fitting, moving least squares, and radial basis function interpolation for estimation. These methods achieve good fitting results in ideal scenarios with low data noise levels and Gaussian error distributions. However, in highly contaminated scenarios with a high proportion of outliers in white light interferometry data, they often produce significant reconstruction distortion, failing to restore the true morphology of the measured surface. Furthermore, traditional algorithms rely heavily on manually preset thresholds for local domain selection and outlier identification mechanisms, lacking adaptive adjustment capabilities. This makes it difficult to simultaneously ensure robustness and accuracy in reconstruction, and it cannot meet the high-precision detection requirements of complex optical surfaces with multiple features. Therefore, there is an urgent need for a complex surface reconstruction method specifically suited to the characteristics of white light interferometry data, capable of effectively suppressing the influence of outliers in highly contaminated data environments, balancing reconstruction robustness and accuracy, and fully preserving the fine structural features of complex optical surfaces. Summary of the Invention

[0005] To address the shortcomings and deficiencies of existing technologies, this invention provides a method for reconstructing complex surfaces based on white light interferometry measurement data. It aims to solve the core problems in existing white light interferometry complex surface detection methods, such as severe reconstruction distortion in highly polluted data environments due to the presence of high-amplitude, high-proportion non-Gaussian distributed outliers; reliance on manually preset outlier detection thresholds; inability to balance reconstruction robustness and accuracy; and difficulty in preserving the fine structure of complex surfaces. This invention employs a sliding local domain processing architecture. For the measurement data within each local domain, robust pre-estimation against outlier interference is first performed using MM estimation, and the corresponding residuals are calculated. Then, using the absolute value of the residuals as input, hybrid t-distribution clustering is used to achieve adaptive classification of the measurement data, accurately identifying and removing outliers. After iterative closed-loop preprocessing to gradually suppress outlier interference, a weighted Jacobi basis function is used to perform local regression on the retained effective measurement data to calculate the height value of the points to be estimated. After traversing all points to be estimated across the entire region, a high-precision global reconstruction result of the complex surface is output. This invention requires no manual intervention for debugging and maintains excellent reconstruction robustness in highly contaminated data environments with a high proportion of outliers, while fully preserving the steep edges and fine features of complex optical surfaces, such as micro / nano-scale arrays. Further spherical parameter estimation of the reconstructed surface and evaluation of the reconstruction results using the deviation between the estimated and calibrated radii demonstrate that the method of this invention achieves a small correlation deviation under conditions of high outlier contamination, thereby improving the reconstruction accuracy and robustness of complex optical surfaces. Therefore, it is widely applicable to white-light interferometric 3D topography detection scenarios for various complex optical components, including aspherical surfaces, freeform surfaces, and micro / nano-array structures.

[0006] The specific technical solution adopted by this invention to solve its technical problem is as follows:

[0007] A method for reconstructing complex surfaces based on white light interferometry data is proposed. This method acquires surface topography measurement data of the complex surface to be reconstructed using white light interferometry. A sliding local region is constructed centered on each point to be estimated within the measurement area. The measurement data within each local region is processed to obtain the height value of the corresponding point to be estimated. After traversing all points to be estimated across the entire region, the global reconstruction result of the complex surface is output. The process of processing the measurement data within each local region includes the following steps:

[0008] S1. Use MM estimation to make a robust pre-estimation of the measurement data in the current local domain to resist the interference of non-Gaussian outliers in white light interferometry, and calculate the residual between the measurement data and the local pre-estimation model.

[0009] S2. Using the absolute value of the residual as input, adaptively classify the measurement data through mixed t-distribution clustering, identify and remove outliers, and obtain the retained valid measurement data.

[0010] S3. Using the currently retained valid measurement data as input, repeat S1 to S2 until the outlier removal iteration is completed, and obtain the final valid measurement dataset.

[0011] S4. Use the weighted Jacobi basis function to perform local regression on the final effective measurement dataset to calculate the height value of the point to be estimated.

[0012] Furthermore, the MM estimation is achieved by cascading high-crash-point S-estimation and high-efficiency M-estimation: first, the initial estimation parameters and scale parameters for resisting outliers are obtained through S-estimation, and then the final local pre-estimation model is obtained by iterative optimization through M-estimation based on the initial estimation results.

[0013] Furthermore, the mixed t-distribution clustering uses the t-distribution as the probability distribution model of each mixed component, takes the absolute value of the residual corresponding to the local estimation model as the input of the clustering sample, estimates the parameters of each mixed component through the expectation-maximization algorithm, and performs adaptive classification based on the posterior probability corresponding to each sample.

[0014] Furthermore, in step S3, the termination condition for the outlier removal iteration is reaching the preset maximum number of iterations.

[0015] Furthermore, the weighted Jacobian basis function is constructed using orthogonal Jacobian polynomials, and local weighted regression is achieved by assigning higher weights to measurement data that are closer to the point to be estimated.

[0016] Furthermore, the weights are calculated using a compactly supported weight function, the expression of which is:

[0017]

[0018] in, Let be the normalized distance from the i-th measurement data point to the center of the local domain. The attenuation coefficient of the weighting function.

[0019] Furthermore, the radius of the sliding local region is preset based on the morphological characteristics of the measured surface and the sampling interval of the measurement data.

[0020] And, a complex surface reconstruction system based on white light interferometry, comprising:

[0021] The data acquisition module is used to acquire surface topography measurement data of the complex surface to be reconstructed obtained by white light interferometry.

[0022] The local domain construction module is used to construct a sliding local domain centered on each point to be estimated within the measurement area, and to obtain the measurement data within the corresponding local domain.

[0023] The robust prediction module is used to perform the operation of step S1 above to obtain the local prediction model and the corresponding residuals.

[0024] The adaptive outlier handling module is used to perform the operations of steps S2 and S3 above to obtain the final valid measurement dataset;

[0025] The local weighted regression module is used to perform the operation of step S4 above and calculate the height value of the point to be estimated.

[0026] The global reconstruction module is used to traverse all points to be estimated in the entire region, integrate the height values ​​of each point to be estimated, and output the global reconstruction result of the complex surface.

[0027] Furthermore, the system incorporates an embedded processing unit of a white light interferometric 3D topography measurement system. After inputting the raw measurement data collected by the white light interferometric measurement device, it automatically outputs the global reconstruction results and surface shape detection data of the complex surface.

[0028] And a computer device including a memory, a processor, and a computer program stored in the memory, wherein the processor executes the computer program to implement the method described above.

[0029] A non-transitory computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method described above.

[0030] Compared to existing technologies, this invention and its preferred solution address the core shortcomings of current methods for reconstructing complex surfaces in white light interferometry, namely insufficient resistance to outliers and reliance on manually preset thresholds. This invention employs iterative closed-loop processing of MM estimation and mixed t-distribution clustering to adaptively identify and remove outliers from measurement data without requiring manual adjustment of threshold parameters. It maintains excellent reconstruction robustness even in highly contaminated data environments with a high proportion of outliers, effectively avoiding reconstruction distortion caused by outlier interference in traditional methods. Furthermore, addressing the difficulty of existing methods in balancing reconstruction accuracy with the preservation of fine structures in complex surfaces, this invention utilizes a weighted Jacobi basis function. By performing local regression, while ensuring the smoothness of global reconstruction, it can effectively approximate the center of the interval with high precision, and completely preserve the fine surface features such as the steep edge structure and micro-nano scale array of the measured surface, thus achieving high-precision restoration of complex optical surfaces. The overall solution of this invention has strong adaptability and can be widely applied to white light interferometric topography detection scenarios of various complex optical components without the need for significant adjustments for different measured components. At the same time, the solution process is clear and controllable and can be directly integrated into the processing unit of the white light interferometric three-dimensional topography measurement system to realize automated, high-precision reconstruction and detection of complex surfaces, and has good engineering application value. Attached Figure Description

[0031] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0032] Figure 1 This is a flowchart illustrating the principle of the complex surface reconstruction method based on white light interferometry data according to an embodiment of the present invention.

[0033] Figure 2 This is a schematic diagram of the white light interferometer and the sample being tested used in the verification experiment of this invention.

[0034] Figure 3 The figure shows a comparison of the reconstructed surfaces of the existing moving least squares method and the method of the present invention under different outlier ratios in the embodiments of the present invention. In the figure, (a1) is the first group of original measurement data to be tested, (a2) is the second group of original measurement data to be tested, (a3) ​​is the third group of original measurement data to be tested, (a4) is the fourth group of original measurement data to be tested, (b1~b4) are the surface results after reconstruction of the corresponding original measurement data by the moving least squares method, and (c1~c4) are the surface results after reconstruction of the corresponding original measurement data by the method of the present invention. Detailed Implementation

[0035] To make the features and advantages of the present invention more apparent and understandable, specific embodiments are described below in detail:

[0036] It should be noted that the following detailed descriptions are exemplary and intended to provide further explanation of this application. Unless otherwise specified, all technical and scientific terms used in this specification have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0037] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments according to this application. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0038] To address the shortcomings and deficiencies of existing technologies, this invention proposes a method for reconstructing complex surfaces using white light interferometry data. This method is a novel and robust surface reconstruction method that eliminates the need for manually setting outlier detection thresholds. Simply input the measurement data into the method to obtain a high-precision and robust reconstructed surface.

[0039] This invention combines MM estimation with mixed t-distribution clustering to construct a data preprocessing mechanism. This mechanism, along with weighted Jacobi basis functions, is then introduced into the local domain reconstruction framework, thereby achieving robust reconstruction of complex surfaces. The principle diagram of this invention is shown below. Figure 1 As shown, the overall processing flow is as follows: Using the original surface topography measurement data obtained by white light interferometry as input, local processing units are first constructed centered on each point to be estimated within the entire region through a sliding local domain approach. For each local domain, robust pre-estimation of the measurement data is first performed using MM estimation, and the corresponding residuals are calculated. Then, using the absolute value of the residuals as input, outliers are adaptively identified and iteratively removed through mixed t-distribution clustering to complete robust preprocessing of the data. Finally, for the effective measurement data retained after preprocessing, local regression is performed using a weighted Jacobi basis function to calculate the height value of the point to be estimated. After completing the above processing by traversing all points to be estimated across the entire region, the high-precision global reconstruction result of the complex optical surface is output. The following is a detailed description of each step of the method of this invention:

[0040] First, select the point to be estimated, and then determine the measurement data within the local region based on the location of the point to be estimated and the local region radius.

[0041] Secondly, MM estimation is used to pre-estimate the measurement data within the local domain, establishing a local estimation model. The absolute value of the residual between this model and the measurement data within the local domain is then used as the input for mixed t-distribution clustering. For the clustering results, the measurement data corresponding to the component with the largest mean absolute residual is identified as outliers and removed. To remove outliers as much as possible and suppress their interference, this step needs to be repeated for the measurement data retained after each removal until the maximum number of outer iterations is reached.

[0042] Finally, the weighted Jacobi basis function is used to estimate the measurement data that are finally retained in the local domain, and the height value of the point to be estimated in the local domain is calculated by the local estimation model.

[0043] The implementation of this solution will be demonstrated and described in more detail below through more specific embodiments:

[0044] First, for each local domain, assume the measurement data is... The error model is established as follows:

[0045] (1)

[0046] Where β is the parameter to be estimated, and ε i This is the random error term.

[0047] Then, based on this error model, least squares estimation is performed on the measurement data to obtain initial values ​​for the estimated parameters:

[0048] (2)

[0049] in,

[0050] (3)

[0051] Where m represents the order of the local estimation model.

[0052] Then use the initial parameter β (0) Calculate the estimated value:

[0053] (4)

[0054] And obtain the initial residual:

[0055] (5)

[0056] Next, the S-estimation of high crash points is used to update the parameters to be estimated. The specific steps are as follows:

[0057] (1) Initial scaling estimation: The initial residuals are standardized. In the first iteration, scaling estimation based on the median of absolute deviations is used:

[0058] (6)

[0059] The constant 0.6745 is the correction factor between the median and standard deviation of the absolute deviation under the assumption of normal distribution.

[0060] (2) Residual Standardization: The residuals are standardized using the scaling estimate, i.e.:

[0061] (7)

[0062] Where h represents the number of iterations.

[0063] (3) Weight calculation: Calculate the weight w based on the standardized residuals. i (h) The first iteration uses the Tukey double-weight function form, namely:

[0064] (8)

[0065] Among them, c s =1.547 is the cutoff constant for the S estimation stage.

[0066] (4) Parameter update: Construct a diagonal weight matrix, i.e.:

[0067] (9)

[0068] Update the estimated parameters, i.e.:

[0069] (10)

[0070] Then recalculate the estimated value and the residual, i.e.:

[0071] (11)

[0072] (5) Scale update: After the first iteration, the scale is no longer updated using the median absolute deviation, but instead using a weighted form, i.e.:

[0073] (12)

[0074] Where C is the consistency correction constant, used to correct the scaling estimation bias of the weighted sum of squared residuals.

[0075] (6) Convergence determination: Repeat steps (2) to (5) above until the parameters meet the convergence condition:

[0076] (13)

[0077] Where ε is a preset, sufficiently small convergence threshold. After convergence, the estimated S parameter β is obtained. s and estimation scale σ s Since the S-estimation has a high resistance to outliers, it can be used as the initial estimate in the M-estimation stage.

[0078] Next, the parameters to be estimated are updated using the efficient M-estimation. The specific steps are as follows:

[0079] (1) Initialization of the stage: The estimated parameter β obtained by estimating S. s and estimation scale σ s As the initial value, the residual is recalculated, i.e.:

[0080] (14)

[0081] Reconstruct the standardized residuals, that is:

[0082] (15)

[0083] (2) Weight calculation: The Tukey double weight function is still used in the M estimation stage, but a larger cutoff constant c is used. m =4.685, to improve estimation efficiency, i.e.:

[0084] (16)

[0085] Compared to S-estimation, M-estimation assigns higher weights to normal measurement data while still suppressing outliers.

[0086] (3) Parameter update: Construct a diagonal weight matrix, i.e.:

[0087] (17)

[0088] Solving for the estimated parameters, i.e.

[0089] (18)

[0090] After updating, recalculate the residuals and estimate the scale, and repeat the iteration.

[0091] (4) Convergence determination: Repeat steps (1) to (3) above until the convergence condition is met:

[0092] (19)

[0093] Finally, the MM estimation parameter β is obtained. MM = β m The corresponding estimated value is y i = f (x i , βMM The residuals between the measured data and the estimated model are calculated as d = (d1, d2, d3, …, d). n This serves as the input for subsequent mixed t-distribution clustering to identify outliers.

[0094] In white-light interferometry, outliers typically account for over 20%, and even more in extreme scenarios. Furthermore, the errors do not conform to a Gaussian distribution, making traditional least-squares estimation ineffective against outliers and prone to prediction distortion. This proposed solution employs MM estimation, which cascades S-estimation with high-collapse-point outlier collapse points. Leveraging robust statistical laws, it can still obtain a local prediction model that closely resembles the actual surface shape even under high-proportion non-Gaussian outlier interference. This provides reliable residual data for accurate subsequent outlier identification, specifically addressing the prediction failure problem in highly contaminated white-light interferometry data scenarios.

[0095] In this invention, the calculation process for mixed t-distribution clustering is as follows:

[0096] This step uses the absolute value of the residual e from the MM estimate output. i = |d i Using | as input, the optimal parameters of the K-component mixture t-distribution model are solved through the EM algorithm to achieve adaptive classification of normal measurement data and outliers. Its core probability density function is:

[0097] (20)

[0098] Where, π j Let be the mixing ratio coefficient of the j-th component, satisfying Let µ represent the t-distribution probability density function corresponding to the j-th t-component. j σ represents the position parameter. j 2 Represents the scale parameter, v j This represents the degree of freedom parameter.

[0099] The specific calculation steps are as follows:

[0100] (1) Model initialization: For the absolute residual sample e i Perform the initial partitioning, given the number of mixed components K, and initialize the parameter π. j (0) , µ j (0) , σ j 2(0) , v j (0) At the same time, the convergence condition ε1 is set;

[0101] (2) E-step - Calculate latent variables and posterior probabilities: Two latent variables are introduced to realize the EM solution of the mixed t-distribution model. One is the label variable z. ij , indicating sample e i Whether it belongs to the j-th component; secondly, the scaling variable u. ij When z is given ij When =1, u ij They are independent and follow a Gamma distribution, satisfying u ij | z ij = 1~Γ(v j / 2, v j / 2). The label variable z ij For sample category determination, specifically after subsequent iterations converge, the sample category is determined based on the maximum a posteriori probability, with the scale variable u. ij Used to mitigate the impact of outliers on model parameter updates.

[0102] In the t-th iteration, based on the current model parameters, two calculations are performed:

[0103] Calculate sample e under the current parameters i The posterior probability of belonging to the j-th component is:

[0104] (twenty one)

[0105] Given the current parameters, the scaling variable u of the i-th sample in the j-th component ij Conditional expectation:

[0106] (twenty two)

[0107] In one-dimensional residual clustering, p=1, and ρ represents the standardized squared distance between the sample and the component center, i.e.:

[0108] (twenty three)

[0109] This weight reflects the degree of standardized deviation of the sample from the current component center. The greater the deviation, the higher the value of u. ij The smaller the sample size, the weaker its impact on parameter updates.

[0110] (3) M-step - Update all parameters of the model: Based on the posterior probability and latent variable expectation obtained in step (2), update all parameters of the mixture t-distribution in sequence:

[0111] Based on the posterior probability obtained in the current iteration, update the mixing coefficients of each component, i.e.:

[0112] (twenty four)

[0113] Based on the posterior probability τ ij (t) and scale variable u ij (t) Given the conditional expectation, update the mean parameter of the j-th component, i.e.:

[0114] (25)

[0115] Update the scale parameter of the j-th component, i.e.:

[0116] (26)

[0117] (4) Convergence judgment and iterative control:

[0118] If the parameter changes in two consecutive iterations satisfy the convergence condition: and If ε1 is a sufficiently small convergence threshold, then convergence is considered complete; otherwise, return to step (2) above to continue iterating.

[0119] Iterative convergence, i.e., after clustering is complete, determines the sample category based on the maximum posterior probability:

[0120] (27)

[0121] The components with larger means correspond to outliers in the measurement data. The measurement data corresponding to the outliers are removed in the local domain, and the remaining measurement data is used as the input for the next MM estimation. The above MM estimation-mixed t-distribution clustering process is iterated repeatedly to gradually remove outliers in the local domain, thereby achieving the goal of effectively resisting the influence of outliers.

[0122] This invention addresses the issue of outliers in white light interferometric measurement data. Traditional methods such as polynomial fitting, moving least squares, and radial basis function interpolation exhibit poor data adaptability and are prone to reconstruction distortion. Based on locally robust pre-estimation, this scheme further employs mixed t-distribution clustering of the residual absolute values. Since the t-distribution has good characterization capabilities for heavy-tailed data, mixed t-distribution clustering can classify samples with different residual amplitudes, thus adaptively identifying outliers in measurement data of different optical components. This solves the technical problems of traditional methods requiring manual threshold adjustment and poor versatility, further enhancing the method's adaptability to various complex optical surface detection scenarios.

[0123] After the above iterative process is completed, the weighted Jacobi basis function is used to regress the remaining measurement data within the local domain. First, the coordinates of the remaining measurement data within the local domain are normalized:

[0124] (28)

[0125] in, Let be the center of the local domain, and δ be the radius of the local domain. The measured data are approximated using a Jacobian polynomial over the normalized interval [-1, 1], where the Jacobian polynomial is defined as:

[0126] (29)

[0127] in:

[0128] (30)

[0129] Taking α1=α2=1, the following system of equations can be derived:

[0130] (31)

[0131] Jacobi basis functions can be expressed as:

[0132] (32)

[0133] This can be viewed as a system of linear equations:

[0134] (33)

[0135] Where Φ is the Jacobian basis function matrix, its form is shown in equations (34) and (35):

[0136] (34)

[0137] Right now:

[0138] (35)

[0139] Where n is the number of measurement data, θ is the column vector of the estimated parameters with m+1 rows, and y is the column vector of the measurement data values ​​with n rows.

[0140] Next, a weight w is introduced for each measurement data point within the local domain. i The weighting function uses a compactly supported weighting function, that is:

[0141] (36)

[0142] Here, λ is the attenuation coefficient of the weight function. This compactly supported weight function assigns higher weights to measurement data closer to the point to be estimated, thus improving the accuracy of local weighted regression.

[0143] Therefore, the parameter θ can be estimated by the following formula:

[0144] (37)

[0145] This leads to the estimation model f(ξ) = Φ within the local domain. (ξ) Substituting the local normalized coordinates ξ=0 corresponding to the point to be estimated into the estimation model, we obtain the value of the point to be estimated, f(0)= Φ. (0) θ. By moving the local domain across the entire measurement area and repeating the above process, the global reconstruction result can be obtained.

[0146] The core objective of white-light interferometry is to achieve high-precision characterization of the fine surface features of optical components, especially the steep edges and micro-nano scale arrays of complex optical surfaces or microlens arrays. Traditional polynomial fitting is prone to Runge effect at interval boundaries, and radial basis function fitting is prone to overfitting, failing to accurately reproduce the fine structure of complex surfaces. This scheme utilizes the orthogonality and high approximation of the interval center of Jacobi polynomials, combined with compactly supported weight functions to assign higher weights to measurement points closer to the point to be estimated. While ensuring the smoothness of the fitting, it accurately preserves the local fine features of complex optical surfaces, specifically solving the technical problem of reconstruction distortion at the edges of optical surfaces and micro-nano structures, thus achieving high-precision reproduction of complex optical surfaces.

[0147] By performing the above steps, a method for reconstructing complex surfaces for white light interferometry data can be realized.

[0148] The robust reconstruction method for complex surfaces in white light interferometric measurement data provided in the embodiments of the present invention achieves adaptive suppression and high-precision reconstruction of outliers in complex surface measurement data by combining MM estimation, mixed t-distribution clustering, and weighted Jacobian basis function estimation. Compared with existing measurement data processing methods, the present invention does not require manual setting of outlier detection thresholds, can maintain high reconstruction accuracy and robustness for measurement data in highly polluted scenarios, and effectively preserves the geometric features of complex surfaces. It has strong robustness and versatility and can be widely applied to data processing and complex optical surface quality evaluation in white light interferometric three-dimensional topography measurement systems, including aspherical optical element inspection, freeform surface optical processing quality evaluation, micro / nano structure surface topography analysis, and online inspection in precision optical manufacturing.

[0149] This method can be used as the core algorithm of the data processing unit of the white light interferometry three-dimensional topography measurement system. It can be directly written into the embedded processing system of the measurement equipment. The output of the method can be directly used as the final measurement data of the equipment. Based on this design and the above-mentioned solution provided by the present invention, the surface reconstruction can be automatically performed by simply inputting the data obtained by the white light interferometry measurement system.

[0150] The experimental instruments and test samples involved in implementing this scheme are shown in the following diagram. Figure 2 As shown.

[0151] To further verify the reconstruction accuracy and robustness of the proposed method in white light interferometry data processing, a comparative experiment was conducted with the moving least squares method. Experimental data are provided by... Figure 2 The white light interferometer shown in the figure shows that the sample under test is a standard sphere with a calibration radius of 14.402 mm. Four sets of measurement data with progressively increasing outlier ratios were then selected, and the surface was reconstructed for each set of measurement data using the moving least squares method and the algorithm proposed in this invention, respectively. Each set of data consists of 60×60 grid points, with x, y, and z coordinate ranges of [-30 μm, 30 μm], [-30 μm, 30 μm], and [0, 0.2 μm], respectively.

[0152] Reconstruction results of different experimental data as follows Figure 3 As shown.

[0153] Figure 3 This is a comparison diagram of the reconstructed surfaces of the existing moving least squares method and the method of the present invention under different outlier ratios in the embodiments of the present invention. In the diagram, (a1) is the first group of original measurement data to be tested, (a2) is the second group of original measurement data to be tested, (a3) ​​is the third group of original measurement data to be tested, and (a4) is the fourth group of original measurement data to be tested; (b1), (b2), (b3), and (b4) are the surface results after reconstruction of the original measurement data corresponding to (a1), (a2), (a3), and (a4) by the existing moving least squares method, respectively; (c1), (c2), (c3), and (c4) are the surface results after reconstruction of the original measurement data corresponding to (a1), (a2), (a3), and (a4) by the method of the present invention, respectively.

[0154] from Figure 3 The comparison results show that as the proportion of outliers in the test data gradually increases, the reconstructed surface obtained by the existing moving least squares method exhibits obvious morphological distortion. Compared with the reference morphology of the tested sphere, the local distortion phenomenon gradually increases. However, the method of the present invention can effectively suppress the interference of outliers on the reconstruction results in four sets of test data with different outlier proportions. The reconstructed surface closely matches the reference morphology of the tested sphere, with a smooth surface and no distortion, fully demonstrating the excellent outlier resistance and reconstruction robustness of the method of the present invention.

[0155] To quantitatively evaluate the performance of the reconstruction algorithm, the estimated radius of the reconstructed surface is compared with the calibration radius. After obtaining the reconstructed surface, to evaluate the accuracy of the reconstruction results of the method of this invention, a spherical parameter optimization method based on simulated annealing is further used to determine the estimated radius, and its objective function is as follows:

[0156] (38)

[0157] Where, p i = (x i , y i , z i Let c = (x0, y0, z0) be a point on the reconstructed surface, c = (x0, y0, z0) be the estimated center of the sphere, and r be the estimated radius. Both the center c and the radius r are parameters to be estimated. Reconstruction performance is quantitatively evaluated by comparing the estimated radius with the calibration radius. Table 1 shows the estimated radii obtained using the moving least squares method and the method of this invention, where r1 = |r... 移动最小二乘法 -r 标定 |,r2=|r 本发明方法 -r 标定 The relative improvement is calculated as follows: (|r1 - r2|) / r1 × 100%

[0158] Table 1. Estimated radius of the reconstructed surface [mm]

[0159]

[0160] As shown in Table 1, the estimated radius obtained by the method of the present invention is generally closer to the calibrated radius of 14.402 mm. Compared with the moving least squares method, the method of the present invention shows a significantly greater relative improvement on data with a high proportion of outliers, indicating that the method has better resistance to outliers and better robustness in surface reconstruction in white light interferometry data processing.

[0161] From the quantitative results, the reconstructed surfaces obtained by the method of this invention and the moving least squares method were optimized by simulated annealing of the spherical parameters, and the deviation between the estimated radius and the calibrated radius was used as a quantitative evaluation index. Based on this evaluation method, the deviation of the estimated radius obtained by the method of this invention after reconstructing the third and fourth sets of measurement data was significantly reduced. The corresponding estimated radius deviations decreased from 239 μm and 165 μm to 34 μm and 2 μm, respectively, with relative improvements of 85.77% and 98.79%, respectively. This indicates that the method can effectively reduce the reconstruction error under high proportion of outlier contamination conditions, improve the reconstruction accuracy and robustness of complex optical surfaces, and solve the industry pain point that existing methods cannot achieve reliable detection of ultra-precision optical components in high-contamination data environments.

[0162] Based on the same inventive concept, the present invention also provides a computer device comprising: one or more processors and a memory for storing one or more computer programs; the computer programs include program instructions, and the processor is used to execute the program instructions to implement the complex surface reconstruction method based on white light interferometry data described in the above embodiments.

[0163] The processor may be a central processing unit (CPU), a digital signal processor (DSP), a field-programmable gate array (FPGA), or other programmable logic devices. A general-purpose processor may be a microprocessor or any conventional processor.

[0164] Based on the same inventive concept, this invention also provides a computer-readable storage medium storing a computer program. When the computer program is executed by a processor, it performs the complex surface reconstruction method based on white light interferometry data described in the above embodiments. The computer-readable storage medium can be an electrical, magnetic, optical, or semiconductor system, device, or instrument, such as a portable computer disk, hard disk, random access memory (RAM), read-only memory (ROM), flash memory, fiber optic medium, portable compact disk read-only memory (CD-ROM), etc., and is applicable to various tangible storage media required for patent examination.

[0165] It should be noted that, unless otherwise defined, the technical or scientific terms used in this invention should have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "first," "second," and similar terms used in this invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Terms such as "comprising" or "including" mean that the element or object preceding the word encompasses the elements or objects listed following the word and their equivalents, without excluding other elements or objects. Terms such as "connected" or "linked" are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. Terms such as "upper," "lower," "left," and "right" are used only to indicate relative positional relationships; when the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0166] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention in any other way. Any person skilled in the art may make changes or modifications to the above-disclosed technical content to create equivalent embodiments. However, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the protection scope of the present invention.

[0167] This invention is not limited to the preferred embodiment described above. Anyone inspired by this invention can derive other forms of complex surface reconstruction methods based on white light interferometry data. All equivalent variations and modifications made within the scope of the claims of this invention should be included within the scope of this invention.

Claims

1. A method for reconstructing complex surfaces based on white light interferometry data, characterized in that, The surface topography measurement data of the complex surface to be reconstructed is obtained by white light interferometry. A sliding local domain is constructed with each point to be estimated within the measurement area as the center. The measurement data in each local domain is processed to obtain the height value of the corresponding point to be estimated. After traversing all points to be estimated in the entire region, the global reconstruction result of the complex surface is output. The process of processing the measurement data in each local domain includes the following steps: S1. Use MM estimation to make a robust pre-estimation of the measurement data in the current local domain to resist the non-Gaussian outlier interference introduced in white light interferometry, and calculate the residual between the measurement data and the local pre-estimation model; S2. Using the absolute value of the residual as input, adaptively classify the measurement data through mixed t-distribution clustering, identify and remove outliers, and obtain the retained valid measurement data. S3. Using the currently retained valid measurement data as input, repeat S1 to S2 until the outlier removal iteration is completed, and obtain the final valid measurement dataset. S4. Use the weighted Jacobi basis function to perform local regression on the final effective measurement dataset to calculate the height value of the point to be estimated.

2. The method for reconstructing complex surfaces based on white light interferometry data according to claim 1, characterized in that: The MM estimation is achieved by cascading high-crash-point S-estimation and high-efficiency M-estimation: first, the initial estimation parameters and scale parameters for robustness to outliers are obtained through S-estimation, and then the final local pre-estimation model is obtained by iterative optimization through M-estimation based on the initial estimation results.

3. The method for reconstructing complex surfaces based on white light interferometry data according to claim 1, characterized in that: The mixed t-distribution clustering uses the t-distribution as the probability distribution model for each mixed component, takes the absolute value of the residual corresponding to the local estimation model as the input for clustering, estimates the parameters of each mixed component through the expectation-maximization algorithm, and performs adaptive classification based on the posterior probability corresponding to each sample.

4. The method for reconstructing complex surfaces based on white light interferometry data according to claim 1, characterized in that: In step S3, the termination condition for the outlier removal iteration is reaching the preset maximum number of iterations.

5. The method for reconstructing complex surfaces based on white light interferometry data according to claim 1, characterized in that: The weighted Jacobian basis function is constructed using orthogonal Jacobian polynomials, and local weighted regression is achieved by assigning higher weights to measurement data that are closer to the point to be estimated.

6. The method for reconstructing complex surfaces based on white light interferometry data according to claim 5, characterized in that: The weights are calculated using a compactly supported weight function, the expression of which is: in, Let be the normalized distance from the i-th measurement data point to the center of the local domain. The attenuation coefficient of the weighting function.

7. The method for reconstructing complex surfaces based on white light interferometry data according to claim 1, characterized in that: The radius of the sliding local region is preset based on the morphological characteristics of the measured surface and the sampling interval of the measurement data.

8. A complex surface reconstruction system based on white light interferometry data, characterized in that, include: The data acquisition module is used to acquire surface topography measurement data of the complex surface to be reconstructed obtained by white light interferometry. The local domain construction module is used to construct a sliding local domain centered on each point to be estimated within the measurement area, and to obtain the measurement data within the corresponding local domain. A robust pre-estimation module is used to perform the operation of step S1 in claim 1 to obtain a local pre-estimation model and the corresponding residuals; An adaptive outlier processing module is used to perform the operations of steps S2 and S3 in claim 1 to obtain the final valid measurement dataset. The local weighted regression module is used to perform the operation of step S4 in claim 1 and calculate the height value of the point to be estimated. The global reconstruction module is used to traverse all points to be estimated in the entire region, integrate the height values ​​of each point to be estimated, and output the global reconstruction result of the complex surface.

9. A complex surface reconstruction system based on white light interferometry data according to claim 8, characterized in that, The system incorporates an embedded processing unit of a white light interferometry 3D topography measurement system. After inputting the raw measurement data collected by the white light interferometry measurement device, it automatically outputs the global reconstruction results and surface shape detection data of complex curved surfaces.

10. A non-transitory computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the complex surface reconstruction method based on white light interferometry data as described in any one of claims 1 to 7.

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