Method and system for measuring aspheric mirror surface based on variational autoencoder

Abstract

The application discloses a kind of based on variational autoencoder aspheric mirror surface shape measurement method and its system, belong to optical measurement technical field.Method includes: constructing point cloud training set, and establish the variational autoencoder model including multi-scale encoder, decoder and local correction network.Two-stage training is used: global training stage, encoder and decoder are learned by optimizing four-component loss function containing reconstruction loss and KL divergence loss, and global latent space representation representing overall shape profile is obtained;Local fine-tuning stage, by fixing global network parameters, each sub-region is independently optimized using local correction network to restore high-frequency details.Finally, the measured point cloud is input into the trained model, and high-precision surface reconstruction and measurement are realized.The application solves the problem that macroscopic accuracy and microscopic details are difficult to be considered in discrete point cloud reconstruction through multi-scale feature separation and global-local collaborative optimization, and improves measurement accuracy and adaptability.

Description

A method and system for measuring the shape of aspherical mirrors based on variational autoencoders Technical Field

[0001] This application relates to the field of optical measurement technology, and more specifically to a method and system for measuring the shape of an aspherical mirror based on a variational autoencoder. Background Technology

[0002] Large-aperture aspherical mirrors are core components of high-end optical systems, and their surface shape errors need to be controlled at the sub-micron or even nanometer level. Spectroscopic confocal displacement probe technology, with its advantages of axial nanometer resolution, no need for a reference mirror, and ability to measure high-slope surfaces, has become the mainstream method for acquiring 3D point clouds of mirror surfaces. This technology can rapidly and accurately collect massive amounts of discrete points on the mirror surface, providing a reliable data foundation for subsequent flatness analysis and surface shape evaluation. However, the point cloud data directly acquired by the optical probe is essentially a series of discrete sampling points on the mirror surface, lacking any explicit topological connections or continuous geometric information. Therefore, how to robustly reconstruct a high-fidelity continuous surface model from this disordered, sparse, and potentially noise-laden discrete data has become a key challenge for achieving high-precision measurement of mirror flatness. This challenge not only requires the reconstruction algorithm to accurately restore the macroscopic surface shape features of the mirror but also to effectively preserve key geometric features such as local minute undulations, placing extremely high demands on the accuracy, robustness, and detail preservation capabilities of the reconstruction algorithm.

[0003] In existing technologies, surface reconstruction methods primarily employ polynomial fitting. These methods fit point clouds globally or locally using polynomial functions. While computationally simple, they are prone to oversmoothing or overfitting when handling complex surfaces and noisy data, making it difficult to preserve detailed features. To better preserve geometric structures, methods based on triangulation, such as triangulation, have emerged. These methods construct explicit surfaces by connecting discrete points into triangular patches, but they tend to produce non-manifold structures when point cloud density is uneven or noisy, and their effectiveness in handling sharp features and boundaries remains limited. Building upon this, methods based on implicit functions, such as radial basis functions and Poisson reconstruction, represent surfaces by defining implicit functions, generating smooth, continuous surfaces. However, these methods are sensitive to parameter selection, have high computational complexity, and are inefficient when processing large-scale point clouds. 3D reconstruction methods based on convolutional neural networks have pioneered a new data-driven approach. These methods can learn complex features from large amounts of data and have strong generalization capabilities, but they typically require large amounts of labeled data and high-performance computing resources, and there is still room for improvement in their ability to restore details.

[0004] Therefore, there is an urgent need for a measurement method and system for aspherical mirror shapes based on variational autoencoders to solve the technical problem of inaccurate measurement of large-diameter aspherical mirror shapes in existing technologies. Summary of the Invention

[0005] The purpose of this application is to provide a method and system for measuring the shape of aspherical mirror surfaces based on a variational autoencoder, which can solve at least one of the technical problems mentioned above. The specific solution is as follows:

[0006] According to a specific embodiment of this application, this application provides a method for measuring the shape of an aspherical mirror based on a variational autoencoder, comprising the following steps:

[0007] Obtain the point cloud dataset for model training;

[0008] Construct a model of a multi-scale variational autoencoder;

[0009] A two-stage training strategy is used to train the model to obtain a trained reconstruction model;

[0010] The point cloud of the aspherical mirror surface to be tested is input into the reconstruction model, and the surface coordinates of the aspherical mirror surface to be tested are output.

[0011] The model includes:

[0012] Variational autoencoders are used to perform multi-scale probabilistic encoding of input coordinates in the latent space;

[0013] Variational autodecoder for multi-scale surface reconstruction from latent variables;

[0014] A local correction network is used to refine the details of the variational autodecoder output.

[0015] The two-stage training includes:

[0016] The first stage involves global training, which enables the variational autoencoder and the variational autodecoder to learn the global latent spatial distribution and overall shape contour of the aspherical mirror shape.

[0017] The second stage involves local training, where the local correction network is used to optimize the high-frequency details of the output of the variational autodecoder.

[0018] Furthermore, the variational autoencoder is a three-layer network structure, including a low-frequency layer, a mid-frequency layer, and a high-frequency layer, which sequentially extract and encode the low-frequency features, mid-frequency features, and high-frequency features of the input coordinates into the latent space;

[0019] The variational autoencoder outputs latent spatial distribution parameters to characterize the input coordinate features, and generates latent variables based on the latent spatial distribution parameters as input to the variational autodecoder.

[0020] Furthermore, the variational self-decoder has a three-layer network structure, including: a low-frequency layer, a mid-frequency layer, and a high-frequency layer;

[0021] Matching the three-layer structure of the variational autoencoder, the low-frequency features, mid-frequency features and high-frequency features are reconstructed sequentially according to the latent variables, and the preliminary surface coordinates are output.

[0022] Furthermore, the local correction network includes:

[0023] The preliminary surface coordinates output by the variational autodecoder and the latent variables are received as inputs.

[0024] The input is processed through at least one fully connected layer to output a displacement vector for correcting high-frequency details;

[0025] The displacement vector is added to the initial surface coordinates to obtain the surface coordinates after detail optimization.

[0026] Furthermore, the global training includes:

[0027] The point cloud dataset is input into the low-frequency, mid-frequency, and high-frequency layers of the variational autoencoder to extract the overall trend, mesoscale fluctuations, and local basic details, respectively.

[0028] Based on the potential spatial distribution of the high-frequency layer output, the potential variables shared across the entire surface are established;

[0029] The latent variables are input into the variational autodecoder for multi-scale surface reconstruction.

[0030] The reconstruction results and the point cloud dataset are jointly optimized to obtain a global latent space representation.

[0031] Furthermore, the joint optimization is as follows:

[0032] Optimization is performed using a four-component loss function;

[0033] The four-component loss function includes: reconstruction loss function, KL divergence loss function, local correction loss function, and smoothness loss function.

[0034] Furthermore, the local training includes:

[0035] The latent variables are input into the variational autodecoder to generate preliminary reconstruction results for the corresponding intervals;

[0036] The preliminary reconstruction results and the data of the corresponding intervals in the point cloud dataset are input into the local correction network;

[0037] The high-frequency details are supplemented and corrected by the local correction network to obtain the final reconstruction result of the interval.

[0038] Furthermore, during the local training process, the parameters of the variational autoencoder and the variational autodecoder remain fixed, while the parameters of the local correction network are optimized.

[0039] Further, obtaining the point cloud dataset for model training includes:

[0040] Collect raw 3D point cloud data of the aspherical mirror surface;

[0041] In the space where the original three-dimensional point cloud data is located, a sampling interval is defined, and regular grid points are defined as principal points;

[0042] A local sampling window is defined with each of the principal points as the center, and sampling is performed within each of the windows to obtain a local point cloud set;

[0043] The local point cloud set is denoised using a Gaussian filtering method to obtain the point cloud dataset.

[0044] This application also provides a measurement system for aspherical mirror shape based on a variational autoencoder, comprising:

[0045] The data acquisition and preprocessing module is used to acquire and process the point cloud dataset for model training and the point cloud of the aspherical mirror surface to be tested.

[0046] The model building module is used to build a multi-scale variational autoencoder model that includes a variational autoencoder, a variational autodecoder, and a local correction network.

[0047] The model training module is used to train the model using a two-stage training strategy to obtain a trained reconstructed model.

[0048] The surface reconstruction module is used to input the point cloud of the aspherical mirror surface to be tested into the trained reconstruction model and output the surface coordinates of the surface to be tested.

[0049] Compared with the prior art, the above-described solutions of this application have at least the following beneficial effects:

[0050] 1. This application discloses a measurement method and system for aspherical mirror shape based on variational autoencoder. It constructs a multi-scale variational autoencoder model, and designs both the encoder and decoder as three layers: low frequency, medium frequency, and high frequency. It explicitly separates and learns the surface characteristics of different spatial frequencies. The high-frequency layer is optimized independently and is no longer overwhelmed by low-frequency energy. The model simultaneously reconstructs large-scale trends and minute details, avoiding the oversmoothing of single-scale models.

[0051] 2. The method and system for measuring aspherical mirror shape based on variational autoencoder of this application adopts a two-stage collaborative training. The first stage is global training, which focuses on learning the overall shape contour and low-frequency trends to establish a robust global latent space. The second stage is local fine-tuning, which optimizes high-frequency details in each interval by using a lightweight correction network on the basis of a fixed global model. This can ensure the accurate reconstruction of large-scale surface shape, such as off-axis amount and overall curvature, while effectively restoring key micro-features such as fine processing marks and local undulations.

[0052] 3. This application discloses a method and system for measuring aspherical mirror shapes based on variational autoencoders. The four-component joint optimization loss and reconstruction loss ensure accurate fitting of the sampling points; KL divergence loss ensures that the latent space is regular and continuous, which is convenient for generation; and smoothness loss further constrains the physical rationality of the generated surface. It can robustly infer and reconstruct a complete, continuous and smooth surface from a limited number of discrete sampling points, effectively overcoming the technical problems caused by data sparsity and noise, and providing a complete data foundation for high-confidence surface shape evaluation. Attached Figure Description

[0053] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this application and, together with the description, serve to explain the principles of this application. It is obvious that the drawings described below are merely some embodiments of this application, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort. In the drawings:

[0054] Figure 1 is a flowchart illustrating a method for measuring the shape of an aspherical mirror based on a variational autoencoder, as provided in an embodiment of this application.

[0055] Figure 2 is a schematic diagram of the structure of the multi-scale variational autoencoder model provided in the embodiment of this application. Detailed Implementation

[0056] To make the objectives, technical solutions, and advantages of this application clearer, the application will be further described in detail below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0057] It should also be noted that the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a product or device that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a product or device. Without further limitation, an element defined by the phrase "comprising one" does not exclude the presence of other identical elements in the product or device that includes that element.

[0058] The embodiments of this application are described in detail below with reference to the accompanying drawings.

[0059] This application provides a method and system for measuring the shape of aspherical mirrors based on a variational autoencoder. The data acquisition hardware employs a spectral confocal measurement system to acquire point cloud data. The communication scheme is RS232 communication, and an FPGA chip with customized logic ensures the data transmission of point cloud data to a host computer. The host computer implements the technical solution for the method and system for measuring the shape of aspherical mirrors based on a variational autoencoder.

[0060] Example 1:

[0061] As shown in Figure 1, this application provides a method for measuring the shape of an aspherical mirror based on a variational autoencoder, including the following steps:

[0062] S1. Obtain the point cloud dataset for model training, including:

[0063] S101. Acquire raw 3D point cloud data of the aspherical mirror surface. Use a spectral confocal measurement system to acquire and measure the surface of the aspherical mirror to obtain raw 3D point cloud data.

[0064] S102. Define a sampling interval within the space containing the original 3D point cloud data, and define regular grid points as principal points. Analyze and process the acquired data. First, define a sampling interval in 3D space to determine the boundaries of the point cloud acquisition. Then, define regular grid points as principal points within this interval to systematically divide the sampling area.

[0065] S103. Define a local sampling window centered on each principal point, and perform sampling within each window to obtain a local point cloud set. Define a local sampling window centered on each regular grid principal point. Perform sampling within the local sampling window of each principal point to obtain the position of the point and the surface information of its neighborhood. Each principal point and the sampling points around it together constitute the preliminary local point cloud data. In this embodiment, the size of the local sampling window is 10μm×10μm.

[0066] S104. Use Gaussian filtering to denoise the local point cloud set and obtain a point cloud dataset. For the obtained local point cloud data, use Gaussian filtering to remove abnormal measurement noise, completing data cleaning. After this processing, denoised and normalized point cloud data is obtained, which serves as the final point cloud dataset for model learning.

[0067] S2. Construct the model of the multi-scale variational autoencoder, as shown in Figure 2. The multi-scale variational autoencoder model consists of three parts: a variational autoencoder, a variational autodecoder, and a local correction network. The variational autoencoder is used to perform multi-scale probabilistic encoding of the input coordinates in the latent space. The variational autodecoder is used to reconstruct the surface from the latent variables at multiple scales. The local correction network is used to refine the details of the variational autodecoder output.

[0068] This application provides a preferred technical solution where a variational autoencoder maps input data to a probability distribution in the latent space, rather than a fixed vector representation. The variational autoencoder is a three-layer fully connected network structure: a low-frequency layer, a mid-frequency layer, and a high-frequency layer, corresponding to macroscopic trend features, moderate fluctuation features, and basic detail features, respectively. In this embodiment, all three fully connected network layers use ReLU as the activation function. In this embodiment, the latent space is a low-dimensional continuous vector space learned by the variational autoencoder through training.

[0069] Two-dimensional coordinates of a point The input is fed into the low-frequency layer, which outputs 256 neurons, or a 256-dimensional feature vector. The output of the low-frequency layer is then fed into the mid-frequency layer, which outputs 128 neurons, or a 128-dimensional feature vector. Finally, the output of the mid-frequency layer is fed into the high-frequency layer, which outputs the latent space distribution parameters representing the input features. In this embodiment, these parameters are the mean and logarithmic standard deviation of the latent variables in the latent space. The latent spatial distribution of the variational autoencoder output is a Gaussian distribution, expressed as:

[0070]

[0071] in, The coordinates represent the two-dimensional coordinates of the input to the variational autoencoder; c represents the latent variable; N represents the normal distribution. The mean of the latent variables in the latent space; The variance term represents the distribution of the latent space, which is a measure of the uncertainty of the latent variables in each dimension; I represents the identity matrix, which reflects that in the latent space, the dimensions are assumed to be independent of each other and have the same variance structure. and It is calculated by a variational autoencoder.

[0072] Latent variables are generated through reparameterization techniques, enabling the model to undergo backpropagation training. The introduction of reparameterization allows the network to effectively compute gradients and perform backpropagation optimization, representing low-frequency, mid-frequency, and high-frequency features. The latent variables are expressed as:

[0073]

[0074] in, The standard deviation vector representing the potential spatial distribution; This represents the noise term sampled from the standard normal distribution.

[0075] The technical solution of this application embodiment uses a variational autoencoder with a three-layer fully connected network structure: a low-frequency layer, a mid-frequency layer, and a high-frequency layer. The low-frequency layer transforms and abstracts the original, specific coordinate values ​​through the fully connected layers. The low-frequency layer learns and outputs 256 abstract features that best represent the macroscopic, large-scale trend of the location. The mid-frequency layer further refines and compresses these features. The mid-frequency layer focuses on separating and extracting medium-scale surface undulations and fluctuations from the macroscopic features. The output dimension is reduced from 256 to 128, reflecting a more refined and advanced feature representation. The high-frequency layer is the final output layer, receiving the refined features containing low- and mid-frequency information and performing probabilistic modeling accordingly. The high-frequency layer output is no longer a deterministic feature vector, but rather two parameters used to define a Gaussian probability distribution, i.e., outputting distribution parameters describing the latent space.

[0076] This application provides a preferred technical solution where a variational autodecoder maps latent space variables back to the input space, and the output is the reconstructed coordinates. In this embodiment, a three-layer fully connected network structure is used, including a low-frequency layer, a mid-frequency layer, and a high-frequency layer, corresponding to a variational autoencoder. The variational autodecoder recovers the low-dimensional representation of the latent space into high-dimensional coordinates through a layer-by-layer fully connected network. The input is the latent variables of the latent space, and the three-layer network sequentially reconstructs the low-frequency, mid-frequency, and high-frequency features, outputting preliminary surface coordinates. The decoding process of the variational autodecoder is expressed as:

[0077]

[0078] in, This represents the initial surface coordinates.

[0079] The technical solution of this application embodiment uses a variational autodecoder as a three-layer fully connected network structure corresponding to the variational autoencoder, consisting of a high-frequency layer, a mid-frequency layer, and a low-frequency layer. The variational autodecoder takes a latent variable *c* as input. Its high-frequency layer first maps the latent variable into a feature vector representing high-level details; the mid-frequency layer then expands and concretizes this feature to recover medium-scale surface undulation information; and the low-frequency layer finally maps the fused multi-scale features into preliminarily reconstructed surface coordinates. The variational autodecoder and variational autoencoder together realize a complete multi-scale modeling process from concrete coordinates to probabilistic latent representations, and then back to high-fidelity reconstructed coordinates.

[0080] This application provides a preferred technical solution where the local correction network is a lightweight two-layer fully connected network. Its inputs are the initial surface coordinates output from the variational autodecoder and latent variables, and its output is a displacement vector used to correct high-frequency details. This displacement vector is added to the initial coordinates to obtain the surface coordinates after detail optimization.

[0081] S3. A two-stage training strategy is used to train the model and obtain a trained reconstructed model; the two-stage training includes:

[0082] S301. The first stage involves global training, enabling the variational autoencoder and variational autodecoder to learn the global latent spatial distribution and overall shape contour of the aspherical mirror shape.

[0083] S302. The second stage involves local training, using a local correction network to optimize high-frequency details in the output of the variational autodecoder. Global and local training work together to ensure that the surface reconstruction maintains high accuracy in both macroscopic trends and microscopic details.

[0084] This application provides a preferred technical solution, the global training phase including the following steps:

[0085] S301-1. Input the point cloud dataset into the low-frequency, mid-frequency, and high-frequency layers of the variational autoencoder to extract the overall trend, mesoscale undulations, and local basic details, respectively.

[0086] Point cloud dataset As input, where Represents a point cloud dataset; Let represent the X, Y, and Z coordinates of the point cloud dataset, respectively; ... The data are sequentially input into the three-layer fully connected network structure of the variational autoencoder, where multi-scale features of overall trend, mesoscale fluctuations and local basic details are extracted from the low-frequency layer, mid-frequency layer and high-frequency layer respectively.

[0087] S301-2. Based on the potential spatial distribution of the high-frequency layer output, establish potential variables shared across the entire surface.

[0088] Based on the distribution parameters of the high-frequency layer output, specific latent variables are generated using reparameterization techniques. During training, the model optimizes the objective to cause the distribution of latent variables corresponding to all data points to converge towards a common, compact region. The center and morphology of this convergence region constitute the prior understanding of the latent variables shared across the entire surface, representing the overall shaping characteristics and statistical regularities of low-frequency morphology drift of this type of aspherical mirror.

[0089] The technical solution of this application embodiment forces all data points to learn a shared latent variable distribution during the global training phase, which constrains all reconstruction results to conform to a unified and physically reasonable overall surface shape prior.

[0090] S301-3. Input the latent variables into the variational self-decoder to perform multi-scale surface reconstruction.

[0091] The sampled latent variables are input into the variational autodecoder (VAD). The VAD also performs the inverse operation using a three-layer fully connected network structure. The high-frequency layer of the VAD maps the latent variables into a 128-dimensional feature vector, initiating feature initialization for the reconstruction process. The mid-frequency layer of the VAD expands the 128-dimensional features into 256-dimensional features, restoring mesoscale details. Finally, the low-frequency layer of the VAD maps the 256-dimensional features back to two-dimensional coordinate space, outputting the preliminary surface coordinates of the point. The expression is:

[0092]

[0093] in, Represents the initial surface coordinates; This represents the reconstruction function, which is the base neural network of a fully connected network; Represents the latent variable at point i.

[0094] S301-4. Jointly optimize the reconstruction results and the point cloud dataset to obtain a global latent space representation. The joint optimization method is a four-component loss function, including: reconstruction loss function, KL divergence loss function, local correction loss function, and smoothness loss function.

[0095] This stage drives the joint optimization of the parameters of the variational autoencoder and variational autodecoder by minimizing a four-component loss function. The mathematical objective of the optimization is to minimize the global expectation loss. In this embodiment, the expression for the weighted sum of the four components of the four-component loss function is as follows:

[0096]

[0097] in, Represents the four-component loss function; Represents the reconstruction loss function; Represents the KL divergence loss function; This represents the locally corrected loss function; Represents the smoothness loss function; These represent the weighting coefficients of the locally modified loss function; These represent the weighting coefficients of the smoothness loss function.

[0098] Reconstruct the loss function In variational autoencoders, mean squared error (MSE) or binary cross-entropy is typically used to measure the similarity between generated and real data. In the embodiments of this application, MSE is used as the reconstruction loss function, which is suitable for regression tasks and can effectively measure errors in surface reconstruction. The mean squared error (MSE) between the initial surface coordinates p' and the true two-dimensional coordinates p of all points is calculated, forcing the model to learn an accurate reconstruction mapping. The expression is:

[0099]

[0100] Where n represents the total number of samples involved in the reconstruction loss calculation; Represents the actual two-dimensional coordinates of point i; This represents the initial surface coordinates of point i.

[0101] The reconstruction loss function in this embodiment drives the model to continuously optimize the performance of the variational autodecoder during training, ensuring that the generated data is as close as possible to the real data.

[0102] KL divergence loss function This is used to measure the latent space distribution of the variational autoencoder output and its KL divergence with the standard normal prior, constraining the latent space c to be regular, continuous, and possessing good generative properties. The expression is:

[0103]

[0104] in, The logarithmic form representing the variance of the latent variables; This represents the mean value generated by the encoder in the variational autoencoder. This represents the variance generated by the encoder in the variational autoencoder.

[0105] The KL divergence loss function in this embodiment minimizes the KL divergence, bringing the latent space distribution output by the variational autoencoder closer to the standard normal distribution N(0,I), thus enabling the latent space to have good generation capabilities. The KL divergence loss function forces the latent distribution learned by the variational autoencoder to approximate a continuous and regular prior distribution, namely the standard normal distribution, making the entire latent space continuous, smooth, and structured. This ensures that at any point in the space, including locations not directly sampled, a reasonable and more realistic shape can be decoded during sampling.

[0106] Locally modified loss function This loss function is used to measure the difference between the local correction term and the true data. During the global training phase, this loss function guides the model to reserve optimization space for subsequent local correction networks. Its calculation is based on a simplified local consistency constraint, expressed as:

[0107]

[0108] in, This represents the correction value predicted by the locally modified network; It represents the true value, that is, the true data of the high-frequency part of the surface.

[0109] In this embodiment, the local correction network is optimized by minimizing the local correction loss function, thereby making the surface generated by the model more refined.

[0110] Smoothness loss function The second-order difference is calculated on the reconstructed surface coordinates as an approximation of curvature, and its norm is minimized. A smoothness loss function is used to ensure that the reconstructed macroscopic surface shape is smooth and natural, conforming to the physical properties of an optical mirror; its expression is:

[0111]

[0112] in, This represents the second derivative of the reconstructed surface coordinates, which is the curvature.

[0113] The technical solution of this application embodiment minimizes the four-component loss function through iterative optimization using the backpropagation algorithm. Once training converges, the variational autoencoder learns to map any input coordinates to a well-structured global latent space, while the variational autodecoder learns to reconstruct an accurate macroscopic surface shape from this space. The model parameters and latent space characteristics obtained at this point constitute the global latent space representation. In other words, using all training data, a robust global latent space representation is learned that can characterize the overall shape and low-frequency trends of the entire aspherical mirror surface.

[0114] The model after the global training phase can accurately describe the overall trend, but in actual measurement, there are subtle differences in each local area, such as high-frequency processing textures, local deformation, probe measurement noise, and small random shifts caused by environmental disturbances. The model after the global training phase cannot fully capture these differences, so it is necessary to fine-tune for each sub-interval, which is to enter the local training phase.

[0115] This application provides a preferred technical solution, in which the local training phase includes the following steps:

[0116] S302-1. Input the latent variables into the variational self-decoder to generate preliminary reconstruction results for the corresponding intervals.

[0117] For any specific local interval defined in the point cloud dataset, the shared latent variables determined during the global training phase are input into the variational autodecoder with frozen parameters. The variational autodecoder propagates forward, generating preliminary reconstruction results for all coordinate points within that interval. The preliminary reconstruction results already possess accurate macroscopic and mesoscopic geometric features.

[0118] The expression for a specific local interval is:

[0119]

[0120] in, This represents the subset of data used in the k-th local fine-tuning interval; The coordinates represent the coordinates in the data subset; j represents the sample index, i.e., the j-th point in the local dataset; M represents the total number of sampling points in this local interval.

[0121] S302-2. Input the preliminary reconstruction results and the corresponding intervals of the point cloud dataset into the local correction network.

[0122] The preliminary reconstruction results obtained in the previous step, along with the corresponding real point cloud data for this local interval, are fed into the local correction network as input. Simultaneously, the shared latent variables determined during the global training phase are also used as conditional input to ensure that the local correction does not deviate from the global constraints. The expression for the local correction network is:

[0123]

[0124] in, This indicates the amount of correction made by the local correction network. Let x represent the k-th local correction network; x and y represent the two-dimensional spatial coordinates of the current point; and c represent the latent variables.

[0125] S302-3. High-frequency details are supplemented and corrected through a local correction network to obtain the final reconstruction result of the interval.

[0126] In this embodiment, the local correction network is a lightweight fully connected network that analyzes the differences between the preliminary reconstruction results and the actual data in that local area, especially the differences in high-frequency components, and outputs a correction amount for high-frequency details. Add this correction amount to the initial surface coordinates to obtain the final high-precision reconstructed coordinates of that point, which is the final reconstruction result. The expression is:

[0127]

[0128] in, This indicates the final reconstruction result.

[0129] During the local training phase, all parameters of the variational autoencoder and variational autodecoder are fixed, and only the parameters of the local refinement network are trained. The optimization objective is straightforward: minimize the difference between the final reconstruction and the true local data. By performing this fine-tuning sequentially on all intervals, the model ultimately gains the ability to recover details for different regions, thus achieving high-fidelity reconstruction of aspherical mirrors across the entire shape range.

[0130] In the local training phase, also known as the local fine-tuning phase, the shared global latent variables are used as conditional inputs to each independent local correction network. This ensures that the detailed optimization of each region is a personalized adjustment under the strong constraint of global consistency. It can both adaptively correct details according to the characteristics of different regions, such as different processing errors, and ensure that all locally corrected surfaces are smoothly spliced ​​together to form a harmonious and consistent complete surface shape, thereby enhancing the modeling ability of complex and diverse aspherical mirror surfaces.

[0131] The model is trained using the two-stage training strategy described above to obtain a well-trained reconstructed model.

[0132] S4. Input the point cloud of the aspherical mirror surface to be measured into the reconstruction model, and output the surface coordinates of the aspherical mirror surface to be measured.

[0133] For a given aspherical mirror under test, the same data acquisition and preprocessing procedures used during model training are employed to obtain point cloud data of the mirror's surface. This point cloud data is then input into the trained reconstruction model. Utilizing its learned knowledge, the model outputs high-precision, complete surface coordinates of the aspherical mirror, thereby achieving high-precision measurement of its surface shape.

[0134] Example 2:

[0135] This application provides a measurement system for aspherical mirror shape based on a variational autoencoder, comprising:

[0136] The data acquisition and preprocessing module is used to acquire and process the point cloud dataset for model training and the point cloud of the aspherical mirror surface to be tested. In this embodiment, the aspherical mirror is scanned by a spectral confocal measurement system, reliable data acquisition is achieved by an FPGA chip, interval division, local window sampling, and Gaussian filtering for noise reduction are completed in the host computer software, and finally a standardized point cloud dataset or the point cloud to be tested is output.

[0137] The model building module is used to construct a multi-scale variational autoencoder model that includes a variational autoencoder, a variational autodecoder, and a local correction network. In this embodiment, the multi-scale variational autoencoder model is constructed within a deep learning framework. The multi-scale variational autoencoder model includes a variational autoencoder with a defined three-layer structure, a variational autodecoder, and a lightweight local correction network.

[0138] The model training module is used to train the model using a two-stage training strategy to obtain a trained reconstructed model. In this embodiment, the global training submodule calls the point cloud dataset, runs the training algorithm, and drives the encoder and decoder to learn the global latent space representation by optimizing the four-component loss function. The local fine-tuning submodule takes over the training process after the global training is completed. For different data subsets, it fixes the variational autoencoder and variational autodecoder and only optimizes the parameters of the local correction network to learn the local detail correction strategy.

[0139] The surface reconstruction module is used to input the point cloud to be tested into the trained reconstruction model and output the surface coordinates of the surface to be tested. In this embodiment, the surface reconstruction module loads the trained model parameters, provides an interface to receive the point cloud data of the aspherical mirror to be tested, performs forward propagation calculation, and outputs the final reconstructed surface coordinates.

[0140] This application's embodiments verify the effectiveness of its technical solution through comparative experiments, comparing four methods: the classic point cloud processing network method PointNet, the improved method PointNet++ based on PointNet, and the dynamic graph convolutional network DGCNN. The core evaluation metrics are root mean square error (RMSE), mean absolute error (MAE), peak-to-valley ratio (PV), curvature error, number of model parameters, and reconstruction time. Specific data for the four methods are shown in Table 1.

[0141] Table 1 Comparison of Experimental Results

[0142]

[0143] Comparative experiments show that RMSE and MAE are the core indicators for measuring the overall deviation between the reconstructed surface and the real surface. The proposed method achieves optimal values ​​of 2.51 μm and 2.11 μm for RMSE and MAE, respectively, demonstrating that its average reconstruction accuracy surpasses other methods globally and achieves high-precision measurement of the overall reconstruction. The PV value is 11.8 μm, slightly higher than DGCNN's 11.1 μm; the curvature error value is 3.64 × 10⁻³, which is optimal and significantly superior to the proposed method. The PV value reflects the maximum possible local distortion in the reconstructed surface. While slightly inferior to DGCNN, it outperforms the others, indicating good performance in controlling the worst-case error. The lowest curvature error indicates that the reconstructed surface is the smoothest, most natural, and best conforms to the physical characteristics of an optical mirror, proving the effectiveness of the smoothness loss function and multi-scale probabilistic modeling in the proposed method, solving the problem of surface unevenness in traditional methods. The model parameter count is 9.4M, the lowest among all methods. With only 42% of the parameters of DGCNN and 73% of PointNet++, the multi-scale variational autoencoder model and local correction network architecture of this application are designed to be efficient and lightweight, reducing computational resource requirements and improving deployment flexibility. The reconstruction time is 17.3ms, which is at a moderate level and meets the industrial detection requirements for real-time or near-real-time processing.

[0144] In summary, the technical solution of this application, by constructing a multi-scale variational autoencoder model and performing two-stage training on the model, successfully achieves high-precision reconstruction of macroscopic surface shapes and high-fidelity preservation of microscopic curvature, resolving the contradiction between accuracy and detail in existing technologies. Through sharing latent variables and lightweight local corrections, it effectively combines efficient and rapid modeling, solving the technical problem of reconstructing high-fidelity continuous surfaces from discrete point clouds.

[0145] Finally, it should be noted that the various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the systems or apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the descriptions are relatively simple, and relevant parts can be referred to the method section.

[0146] The above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application.

Claims

1. A method for measuring the shape of an aspherical mirror surface based on a variational autoencoder, characterized in that, The method includes the following steps: acquiring a point cloud dataset for model training; constructing a multi-scale variational autoencoder model; training the model using a two-stage training strategy to obtain a trained reconstruction model; inputting the point cloud of the aspherical mirror surface to be tested into the reconstruction model, and outputting the surface coordinates of the aspherical mirror surface to be tested; wherein, the model includes: a variational autoencoder for multi-scale probabilistic encoding of the input coordinates in the latent space; a variational autodecoder for multi-scale surface reconstruction from the latent variables; and a local correction network for detail correction of the variational autodecoder output; wherein, the two-stage training includes: a first stage of global training, enabling the variational autoencoder and the variational autodecoder to learn the global latent space distribution and overall shape contour of the aspherical mirror surface; and a second stage of local training, using the local correction network to perform high-frequency detail optimization on the output of the variational autodecoder.

2. The measurement method according to claim 1, characterized in that, The variational autoencoder is a three-layer network structure, including a low-frequency layer, a mid-frequency layer, and a high-frequency layer. It sequentially extracts and encodes the low-frequency features, mid-frequency features, and high-frequency features of the input coordinates into the latent space. The variational autoencoder outputs latent space distribution parameters to characterize the features of the input coordinates, and generates latent variables based on the latent space distribution parameters as input to the variational autodecoder.

3. The measurement method according to claim 2, characterized in that, The variational autodecoder has a three-layer network structure, including a low-frequency layer, a mid-frequency layer, and a high-frequency layer. Matching the three-layer structure of the variational autoencoder, it reconstructs the low-frequency features, mid-frequency features, and high-frequency features sequentially according to the latent variables, and outputs preliminary surface coordinates.

4. The measurement method according to claim 3, characterized in that, The local correction network includes: receiving the initial surface coordinates output by the variational autodecoder and the latent variables as input; processing the input through at least one fully connected layer to output a displacement vector for correcting high-frequency details; and adding the displacement vector to the initial surface coordinates to obtain the surface coordinates after detail optimization.

5. The measurement method according to claim 1, characterized in that, The global training includes: inputting the point cloud dataset into the low-frequency, mid-frequency, and high-frequency layers of the variational autoencoder to extract the overall trend, mesoscale undulations, and local basic details, respectively; establishing the latent variables shared across the entire surface based on the latent spatial distribution output by the high-frequency layer; inputting the latent variables into the variational autodecoder for multi-scale surface reconstruction; and jointly optimizing the reconstruction results with the point cloud dataset to obtain a global latent spatial representation.

6. The measurement method according to claim 5, characterized in that, The joint optimization is performed using a four-component loss function; wherein the four-component loss function includes: a reconstruction loss function, a KL divergence loss function, a local correction loss function, and a smoothness loss function.

7. The measurement method according to claim 5, characterized in that, The local training includes: inputting the latent variables into the variational autodecoder to generate preliminary reconstruction results for the corresponding intervals; inputting the preliminary reconstruction results and the data of the corresponding intervals in the point cloud dataset into the local correction network; and supplementing and correcting high-frequency details through the local correction network to obtain the final reconstruction results for the intervals.

8. The measurement method according to claim 7, characterized in that, During the local training process, the parameters of the variational autoencoder and the variational autodecoder remain fixed, while the parameters of the local correction network are optimized.

9. The measurement method according to claim 1, characterized in that, The step of acquiring the point cloud dataset for model training includes: collecting raw 3D point cloud data of the aspherical mirror surface; defining a sampling interval in the space where the raw 3D point cloud data is located, and defining regular grid points as principal points; defining a local sampling window centered on each principal point, and sampling within each window to obtain a local point cloud set; and using a Gaussian filtering method to denoise the local point cloud set to obtain the point cloud dataset.

10. A measurement system for an aspherical mirror shape based on a variational autoencoder for implementing the measurement method as described in any one of claims 1-9, characterized in that, include: The data acquisition and preprocessing module is used to acquire and process the point cloud dataset for model training and the point cloud of the aspherical mirror surface to be tested. The model building module is used to build a multi-scale variational autoencoder model that includes a variational autoencoder, a variational autodecoder, and a local correction network; the model training module is used to train the model using a two-stage training strategy to obtain a trained reconstruction model; the surface reconstruction module is used to input the point cloud of the aspherical mirror surface to be tested into the trained reconstruction model and output the surface coordinates of the surface to be tested.

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