High-fidelity reconstruction method for complex surface topography based on VMD and sparse sampling point cloud

Abstract

The application discloses a kind of complex surface topography high-fidelity reconstruction method based on VMD and sparse sampling point cloud.The same part is respectively high-resolution sampling and sparse sampling to obtain high-resolution part surface point cloud and sparse sampling point cloud, and the eigenvalue is obtained by processing two kinds of point clouds using principal component analysis, and the layer data is extracted to obtain the topographic characteristic function and the base shape function along the X-axis direction respectively, the sparse sampling point cloud is expanded in Y-axis direction by interpolation, and the high-fidelity topographic reconstruction data is obtained by weighted topographic reconstruction processing according to topographic characteristic function and linear interpolation base shape function.The application has the characteristics of small sampling data, high reconstruction accuracy and low algorithm complexity, and is suitable for efficient prediction modeling of complex surface part surface.

Description

Technical Field

[0001] This invention relates to a method for reconstructing curved surface parts, specifically a high-fidelity reconstruction modeling method for complex curved surface morphology based on VMD (Variational Mode Decomposition) and measured sparse sampled point clouds. Background Technology

[0002] High-fidelity surface reconstruction is a crucial component of 3D reconstruction. This technology aims to reconstruct the 3D model of curved surface parts while preserving their machining features. Since complex curved surface parts are often used in critical components, such as aircraft turbine engine blades, propellers, and diesel engine crankshafts, their performance is a primary concern. Furthermore, the surface morphology of curved surface parts is often a significant factor influencing their performance; for example, the surface morphology of turbine engine blades determines flow losses.

[0003] Therefore, the current shortcomings are mainly as follows:

[0004] (1) Most of the modeling methods proposed so far are based on ideal surfaces and cannot establish surfaces with real features. Therefore, they cannot perform simulation analysis and calculation of the real shape of the parts. In order to obtain a surface shape that better meets the performance requirements, it is necessary to consider the possible impact of different processing procedures on the surface shape of the parts during the modeling stage.

[0005] (2) To restore a high-fidelity model, a large number of high-resolution data points need to be collected, which is generally impossible.

[0006] To address the above issues, for complex curved surface parts, a high-fidelity modeling method is needed to represent the real surface morphology of the parts while minimizing the amount of data collected. Summary of the Invention

[0007] To address the problems existing in the background technology, the present invention aims to provide a high-fidelity reconstruction method for complex curved surface topography based on VMD and sparse sampling point cloud. This method solves the problem that most current general modeling methods, which are based on ideal surface modeling, cannot create curved surface parts that conform to real topographic features or require a large amount of high-resolution data.

[0008] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0009] Step (1): For the same part, perform high-resolution sampling and sparse sampling respectively to obtain high-resolution part surface point cloud and sparse sampling point cloud;

[0010] Specifically, high- and low-resolution scanners are used to collect point cloud data from the parts.

[0011] The high-resolution sampling has a higher sampling resolution than the sparse sampling, while the sparse sampling has a lower sampling resolution than the high-resolution sampling.

[0012] Step (2): Use Principal Component Analysis (PCA) to process the high-resolution part surface point cloud to obtain feature values. Determine the direction vectors of the X, Y and Z axes based on the feature vectors of the feature values, establish the point cloud XYZ coordinate system, and extract the layer data of each layer in the X direction.

[0013] Step (3): For each layer of the high-resolution part surface point cloud, remove the shape part, and then perform VMD decomposition on the remaining part to obtain k modes. Then, perform sine function fitting on each mode and add them together to obtain the topographic feature function of the layer along the X-axis.

[0014] Step (4): For sparsely sampled point clouds, perform the same processing as in steps (2) and (3) to obtain the layer data of each layer and the basis shape function along the X-axis;

[0015] Step (5): Interpolate and extend the basis shape function of the sparse sampled point cloud along the X-axis direction in the Y-axis direction;

[0016] Step (6): Based on the morphological feature function f(x) obtained in step (3) and the basis shape function obtained by linear interpolation in step (5), perform morphological reconstruction processing to obtain morphological reconstruction data and achieve high-fidelity reconstruction.

[0017] The layer distribution in steps (2) and (4) is the same, and both follow the layering of 3D printing.

[0018] Step (2) specifically involves:

[0019] (2.1) First, establish an overall point cloud coordinate system for the point cloud, and combine all points in the point cloud into a dataset matrix K:

[0020]

[0021] Where, x n y n z n These represent the X, Y, and Z coordinates of the m-th point in the overall point cloud coordinate system;

[0022] Next, calculate the covariance matrix V of the dataset matrix K:

[0023]

[0024] Where m is the number of points in the point cloud, K T Let K be the transpose of K, and T denote the matrix transpose.

[0025] (2.2) Find the three eigenvalues ​​of the covariance matrix C, arrange the three eigenvalues ​​from largest to smallest and find their corresponding eigenvectors. Then, use the eigenvectors of the three arranged eigenvalues ​​as the direction vectors of the X-axis, Y-axis and Z-axis respectively. Establish the XYZ coordinate system of the point cloud with the centroid of the entire point cloud as the origin.

[0026] (2.3) In the XYZ coordinate system of the point cloud, collect points in the point cloud data at fixed intervals along the X-axis to obtain layer data.

[0027] The layer data is a set of points similar to a curve. Since the point cloud data is collected at fixed intervals, it has a certain directionality.

[0028] This invention employs a hierarchical structure to process data. Specifically, the y-value of a point in the point cloud is taken as the layer number, and the set of intersection points between the XZ plane containing the y-value and the sample data is the layer data corresponding to that y-value. This facilitates calculation and reduces the amount of data.

[0029] The specific process for extracting layer data in step (2.3) is as follows:

[0030] (2.3.1) In the point cloud data, initially determine the initial point P in the current layer. origin Set the iteration starting point P start When the iteration begins, P start =P orighin ;

[0031] (2.3.2) Calculate and determine the ending reference point P according to the following formula. end :

[0032] P end =P start +ΔdV x

[0033] Among them, V x Δd represents the direction vector of the X-axis, and Δd represents the step size preset according to the actual resolution of the sampling.

[0034] (2.3.3) The KD Tree data structure is used to search for the endpoint reference point P in the point cloud data. end The nearest point is taken as the result point P. result Determine the end reference point P. end and result point P result Does the distance d between them exceed the preset distance upper limit threshold?

[0035] If the distance is less than or equal to the upper limit threshold ∈, then the result point P result As the required data points;

[0036] If the first result point P is obtained result and the ending reference point P end If the distance d between them is not less than the upper limit threshold ∈, then stop the iteration and return to the current layer in step (2.3.1) to redetermine the initial point P. origin Process it;

[0037] If it is not the first result point P result and the ending reference point P end If the distance d between them is not less than the upper limit threshold ∈, then stop the iteration;

[0038] (2.3.4) Using the result point P result Replace the starting point P start Returning to step (2.3.2), repeat steps (2.3.2) and (2.3.3) iteratively until the distance d > ∈ [0, 0] stops the iteration, obtaining all result points P. result Finally, all result points P result Extract the layer data that constitutes the current layer.

[0039] Step (3) specifically includes:

[0040] Remove the shape portion from each layer of data extracted in step (2), and perform K-modal VMD decomposition on the remaining portion to obtain K modes, namely imf1, imf2, ..., imf k Then, for each mode, a sinusoidal fitting function φ1, φ2, ..., φ is established. k Then, the fitting functions of all modes are summed to establish the morphological feature function f(x) along the X-axis, as follows:

[0041]

[0042]

[0043] A n =0.5(max(imf n )-min(imf n ))

[0044] C n =average(imf n )

[0045] Where n represents the ordinal number of the mode, A n The parameter ω represents the exclusion of outliers. nThis represents the intermediate frequency value of the mode. C represents the phase parameter of the mode. n The modal average is represented by , max() represents the maximum value function, min() represents the minimum value function, average() represents the average value, k represents the total number of modalities, and x represents the value along the X-axis, which is used as the independent variable.

[0046] The shape portion is the part of the quadratic function obtained after fitting the layer data with a quadratic term.

[0047] This invention takes into account that each mode has fluctuating shape and abstracts the mode into a sine function. This can better match the periodic fluctuation characteristics of the mode, and the calculation error is small, which has a better effect on accurate fitting.

[0048] In step (4), for the sparse sampled point cloud, the same method as in steps (1) and (2) is used to process and obtain the layer data of each layer and the shape feature function along the X-axis direction, wherein the shape feature function of each layer is used as the base shape function of each layer of the sparse sampled point cloud along the X-axis direction.

[0049] The aforementioned base shape function is a quadratic function obtained by fitting a quadratic term to each layer of data. The base shape function is used to characterize the basic shape of the surface part layer data.

[0050] In step (5), the specific method is as follows:

[0051] The base shape function of the two adjacent layers obtained in step (4) and The point (x′, y′, z′) to be determined is selected as the starting point for reconstruction. Based on the relationship between the point (x′, y′, z′) to be determined and the basis shape functions of the two adjacent layers obtained in step (4), the following is obtained:

[0052]

[0053] d0+d1=l

[0054] Among them, B y′ (X) represents the basis shape function along the X-axis of the layer containing the point (x′, y′, z′), and d0 is the distance from the reconstruction starting point to an adjacent basis shape function. The shortest distance between them, d1 is the distance from the reconstruction starting point to the shape function of the adjacent basis. The shortest distance between them, where l is the basis shape function. With the base shape function The shortest distance between them, where C represents the volatility parameter;

[0055] This invention calculates By′ When considering the fluctuations in the Y direction (X), a constant C is added for compensation. This ensures that the base shape function is smoothly transitioned and also maintains accuracy.

[0056] The volatility parameter C is calculated as follows:

[0057]

[0058] Among them, B y′ (Y) is the base shape function along the Y-axis of the layer containing the point (x′, y′, z′). They represent and The value that X takes when it takes the value of x′.

[0059] The volatility parameter C is obtained by subtracting the basis shape function at the point y′ to be calculated from the basis shape function after linear interpolation to obtain the offset.

[0060] In step (6), the specific steps are as follows:

[0061] (6.1) The topographic feature function f(x) of each layer obtained in step (3) and the basis shape function of each layer obtained by linear interpolation in step (5) are fused:

[0062] For the original base shape function of each layer after linear interpolation, the morphology feature function of the same layer is applied to the base shape function of the same layer according to the following formula to obtain the preliminary reconstructed data G(X) after refinement in the X-axis direction:

[0063] G i (X)=B yi (X)+f i (x)

[0064] Among them, G i (X) represents the initial reconstructed data G(X) of the i-th layer, B yi (X) represents the base shape function of the i-th layer along the X-axis, f i (x) represents the morphological feature function of the i-th layer;

[0065] The reconstructed data resolution is the same as the original high-precision sampling resolution.

[0066] (6.2) For each layer, the initial reconstruction data of the layer itself is weighted and summed with the initial reconstruction data calculated by the previous layer to obtain the final morphological reconstruction data, as shown in the following formula:

[0067] G i '(X)=c weight G i-1(X)+(1-c weight )G i (X)

[0068] Where G′(x) is the final topography reconstruction data of the i-th layer, G i-1 (X) represents the initial reconstructed data of the (i-1)th layer, c weight This indicates the preset reconstruction weights.

[0069] This invention pre-sets a weight c weight Applying the morphological feature function to the base shape function, and using an iterative reconstruction process that reuses surface morphological features, can significantly improve computational accuracy.

[0070] This invention utilizes high-resolution sampled point clouds of a curved surface, employs PCA to determine the principal direction, and layers the data according to the principal direction. Each layer is decomposed into K modes using the VMD method. A morphological feature function is then used to fit each decomposed mode, identifying a set of morphological feature parameters applicable to the complex curved surface. Sparse sampled point clouds of other regions of the same curved surface are obtained through experimental measurements. After determining the principal direction, a basis shape function is fitted, and the basis shape function is interpolated and expanded. The previously identified surface morphological feature parameters are reused, and through iterative calculations of multiple feature parameters, the surface point cloud is reconstructed and predictively modeled, ultimately achieving high-fidelity surface morphological reconstruction and modeling of the sparsely sampled region.

[0071] The beneficial effects achieved by this invention due to the adoption of the above technical solution are as follows:

[0072] The present invention has high overall reconstruction accuracy, which can better restore the surface morphology of the parts without losing the original shape features.

[0073] The present invention significantly reduces the amount of data collected at each collection point, enabling high-fidelity model reconstruction without requiring a large number of data points, while also ensuring accuracy.

[0074] This invention features small sample data volume, high reconstruction accuracy, and low algorithm complexity, making it suitable for efficient prediction and modeling of complex curved surface parts. Attached Figure Description

[0075] Figure 1 This is a flowchart of the method of the present invention;

[0076] Figure 2 A schematic diagram of the point cloud model of a blade curved surface component;

[0077] Figure 3 A schematic diagram of layered data;

[0078] Figure 4 This is a schematic diagram of interpolation extension;

[0079] Figure 5 This is a schematic diagram for reconstructing the surface. Detailed Implementation

[0080] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0081] like Figure 1 As shown, the embodiments and implementation process of the present invention are as follows:

[0082] Step (1): For the same part, perform high-resolution sampling and sparse sampling respectively to obtain high-resolution part surface point cloud and sparse sampling point cloud;

[0083] Step (2): In this example, the blade curved surface part sample is used as the target object. The schematic diagram of the high-resolution part surface point cloud is shown below. Figure 2 As shown, principal component analysis is used to process the high-resolution part surface point cloud to obtain eigenvalues. Based on the eigenvectors of the eigenvalues, the direction vectors of the X, Y, and Z axes are determined, and an XYZ coordinate system for the point cloud is established. Layer data of each layer is extracted from it in the X direction.

[0084] (2.1) First, establish an overall point cloud coordinate system for the point cloud, and combine all points in the point cloud into a dataset matrix K:

[0085]

[0086] Where, x m y m z m These represent the X, Y, and Z coordinates of the m-th point in the overall point cloud coordinate system;

[0087] Next, calculate the covariance matrix V of the dataset matrix K:

[0088]

[0089] Where m is the number of points in the point cloud, K T Let K be the transpose of K, and T denote the matrix transpose.

[0090] (2.2) Find the three eigenvalues ​​of the covariance matrix C, arrange the three eigenvalues ​​from largest to smallest and find their corresponding eigenvectors. Then, the eigenvectors of the three arranged eigenvalues ​​are used as the direction vectors of the X-axis, Y-axis and Z-axis respectively. Establish the XYZ coordinate system of the point cloud with the centroid of the entire point cloud as the origin. The obtained X-axis direction is the layer direction for 3D printing.

[0091] (2.3) In the XYZ coordinate system of the point cloud, collect points in the point cloud data at fixed intervals along the X-axis to obtain layer data.

[0092] (2.3.1) In the point cloud data, initially determine the initial point P in the current layer. origin Set the iteration starting point P start When the iteration begins, P start =P origin ;

[0093] (2.3.2) Calculate and determine the ending reference point P according to the following formula. end :

[0094] P end =P start +ΔdV x Among them, V x Δd represents the direction vector of the X-axis, and Δd represents the step size preset according to the actual resolution of the sampling.

[0095] (2.3.3) The KD Tree data structure is used to search for the endpoint reference point P in the point cloud data. end The nearest point is taken as the result point P. result Determine the end reference point P. end and result point P result Does the distance d between them exceed the preset distance upper limit threshold?

[0096] If the distance is less than or equal to the upper limit threshold ∈, then the result point P result As the required data points;

[0097] If the first result point P is obtained result and the ending reference point P end If the distance d between them is not less than the upper limit threshold ∈, then stop the iteration and return to the current layer in step (2.3.1) to redetermine the initial point P. origin Process it;

[0098] If it is not the first result point P result and the ending reference point P end If the distance d between them is not less than the upper limit threshold ∈, then stop the iteration;

[0099] (2.3.4) Using the result point P result Replace the starting point P start Return to step (2.3.2) to update P. start Repeat steps (2.3.2) and (2.3.3) iteratively until the distance d > ∈ [0, 0], then stop iterating and obtain all result points P. result Finally, all result points P result The layer data constituting the current layer is extracted, and a schematic diagram of the layer data is shown below. Figure 3 As shown, further processing will be performed.

[0100] Step (3): For each layer of the high-resolution part surface point cloud, the shape portion is removed, and the remaining portion is decomposed using VMD to obtain k modes. Then, each mode is fitted with a sine function and the results are summed to obtain the topographic feature function of that layer along the X-axis. In this embodiment, the reference value of k is 6.

[0101] The shape portion is removed from each layer of data extracted in step (2). The shape portion is the quadratic function obtained after fitting the layer data with a quadratic term. The remaining portion after removing the shape portion is subjected to VMD decomposition to obtain K modes, namely imf1, imf2, ..., imf k Then, for each mode, a sinusoidal fitting function φ1, φ2, ..., φ is established. k Then, the fitting functions of all modes are summed to establish the morphological feature function f(x) along the X-axis, as follows:

[0102]

[0103]

[0104] A n =0.5(max(imf n )-min(imf n ))

[0105] C n =average(imf n )

[0106] Where n represents the ordinal number of the mode, A n The parameter ω represents the exclusion of outliers. n This represents the intermediate frequency value of the mode. C represents the phase parameter of the mode. n The modal average is represented by , max() is the maximum value function, min() is the maximum value function, average() is the average value, k is the total number of modalities, and x is the x-value along the X-axis as the independent variable.

[0107] In practice, the methods for determining each parameter are as follows:

[0108] The value of n is in the range of [1, k].

[0109] A n The maximum and minimum values ​​in this mode can be used to determine, i.e., A. n =0.5(max(imf n )-min(imf nHowever, when determining the maximum and minimum values, the influence of outliers needs to be excluded. If the maximum value is max(imf n or minimum value min(imf) n If a value is clearly an outlier, it can be removed manually.

[0110] ω n The center frequency of each mode can be used to determine the frequency, which is the midpoint frequency value within a bandwidth. The center frequency of each mode can be obtained during VMD decomposition, in the following order: but

[0111] It is difficult to determine because the phase of each mode is not uniform everywhere; specifically, let

[0112] C n The modal average value, C, can be used to determine this. n =average(imf n ).

[0113] Step (4): For sparse sampling point clouds, the same method as in steps (1) and (2) is used to process and obtain the layer data of each layer and the shape feature function along the X-axis direction. The shape feature function of each layer is used as the basis shape function of each layer of the sparse sampling point cloud along the X-axis direction. That is, a coordinate system is established to extract the layer data in the X direction, and then the basis shape function is fitted to each layer of data.

[0114] Step (5): Extend the sparse sampled point cloud by interpolation along the Y-axis direction using the basis shape function along the X-axis direction. The extension diagram is shown below. Figure 4 As shown.

[0115] The base shape function of the two adjacent layers obtained in step (4) and In any layer where linear interpolation is required, the point to be determined (x′, y′, z′) is selected as the starting point for reconstruction. The sparse sampled point cloud is processed in the same way as steps (2) and (3) in the layer containing the point to be determined (x′, y′, z′) to obtain the basis shape function B along the Y-axis. y′ (Y) is obtained based on the relationship between the point to be determined (x′, y′, z′) and the base shape functions of the two adjacent layers obtained in step (4):

[0116]

[0117] d0+d1=l

[0118] Among them, B y′(X) represents the basis shape function along the X-axis of the layer containing the point (x′, y′, z′), and d0 is the distance from the reconstruction starting point to an adjacent basis shape function. The shortest distance between them, d1 is the distance from the reconstruction starting point to the shape function of the adjacent basis. The shortest distance between them, where l is the basis shape function. With the base shape function The shortest distance between them, where C represents the volatility parameter, specifically taken as a constant;

[0119] The volatility parameter C is calculated as follows:

[0120]

[0121] Among them, B y′ (Y) represents the base shape function along the Y-axis of the layer containing the point (x′, y′, z′), serving as the basic shape representation in the Y-direction and also as the starting point data in the X-direction. They represent and The value that X takes when it takes the value of x′.

[0122] Step (6): A high-fidelity reconstruction algorithm that reuses surface features is adopted to improve reconstruction accuracy.

[0123] (6.1) Perform the following on the topographic feature function f(x) of each layer obtained in step (3) and the base shape function of each layer obtained by linear interpolation in step (5):

[0124] For the original base shape function of each layer after linear interpolation, the morphology feature function of the same layer is applied to the base shape function of the same layer according to the following formula to obtain the preliminary reconstructed data G(X) after refinement in the X-axis direction:

[0125] G i (X)=B yi (X)+f i (x)

[0126] Among them, G i (X) represents the initial reconstructed data G(X) of the i-th layer, B yi (X) represents the base shape function of the i-th layer along the X-axis, f i (x) represents the morphological feature function of the i-th layer;

[0127] (6.2) For each layer, the initial reconstruction data of the layer itself is weighted and summed with the initial reconstruction data calculated by the previous layer to obtain the final morphological reconstruction data, as shown in the following formula:

[0128] G i ′(X)=cweight G i-1 (X)+(1-c weight )G i (X)

[0129] Where G′(x) is the final topography reconstruction data of the i-th layer, G i-1 (X) represents the initial reconstructed data of the (i-1)th layer, c weight This indicates the preset reconstruction weights.

[0130] As can be seen from the above, in addition to sparsely sampling point clouds, the method of the present invention also performs a measurement of actual data at regular intervals. However, only one layer of data is needed for the actual measurement, which greatly reduces the amount of data collected. The weight values ​​are also determined to be within the range of [0,1]. Through experimental testing, the weight values ​​should be dynamically changed. As the number of iterations increases, the weights should gradually decrease to ensure accuracy.

[0131] The weight values ​​are dynamically determined. In this embodiment, the weight value is set to decrease uniformly from 0.9 to 0 in the first five calculations and remains unchanged in subsequent calculations.

[0132] The final schematic diagram of the reconstructed surface is as follows: Figure 5 As shown in the figure. After comparative calculation, the average error between the reconstructed surface and the original surface is within 2µm, and the standard deviation is also within 2µm.

Claims

1. A high-fidelity reconstruction method for complex surface topography based on VMD and sparsely sampled point clouds, characterized in that, The method includes the following steps: Step (1): For the same part, perform high-resolution sampling and sparse sampling respectively to obtain high-resolution part surface point cloud and sparse sampling point cloud; Step (2): Use principal component analysis to process the point cloud of the high-resolution part surface to obtain feature values. Determine the direction vectors of the X, Y and Z axes based on the feature vectors of the feature values, establish the point cloud XYZ coordinate system, and extract the layer data of each layer in the X direction. Step (3): For each layer of the high-resolution part surface point cloud, remove the shape part, and then perform VMD decomposition on the remaining part to obtain k modes. Then, fit each mode with a sine function and add them together to obtain the topographic feature function of the layer along the X-axis. Step (4): For sparse sampled point clouds, perform the same processing as in steps (2) and (3) to obtain the layer data of each layer and the basis shape function along the X-axis; Step (5): Interpolate and extend the basis shape function of the sparse sampled point cloud along the X-axis direction in the Y-axis direction; In step (5), the specific method is as follows: The base shape function B of the two adjacent layers obtained in step (4) beforehand y0 (X) and B y1 (X) determines the point (x',y',z') to be determined as the starting point for reconstruction. Based on the relationship between the point (x',y',z') to be determined and the basis shape functions of the two adjacent layers obtained in step (4), the following is obtained: Among them, B y' (X) represents the basis shape function along the X-axis of the layer containing the point (x', y', z'), and d0 is the distance from the reconstruction starting point to an adjacent basis shape function B. y0 The shortest distance between (X), where d1 is the distance from the reconstruction starting point to the adjacent basis shape function B. y1 The shortest distance between (X), where l is the basis shape function B. y0 (X) and the basis shape function B y1 The shortest distance between (X), where C represents the volatility parameter; The volatility parameter C is calculated as follows: Among them, B y' (Y) is the base shape function along the Y-axis of the layer containing the point (x', y', z'), B y0 (X'), B y1 (X') represent B respectively y0 (X) and B y1 (X) The value of X when X takes the value of x'; Step (6): Based on the morphological feature function f(x) obtained in step (3) and the basis shape function obtained by linear interpolation in step (5), perform morphological reconstruction processing to obtain morphological reconstruction data.

2. The method for high-fidelity reconstruction of complex surface topography based on VMD and sparsely sampled point clouds as described in claim 1, characterized in that: Step (2) specifically involves: (2.1) First, establish an overall point cloud coordinate system for the point cloud, and combine all points in the point cloud into a dataset matrix K: Where, x n y n z n These represent the X, Y, and Z coordinates of the m-th point in the overall point cloud coordinate system; Next, calculate the covariance matrix V of the dataset matrix K: V=(KK T ) / m Where m is the number of points in the point cloud, K T Let K be the transpose of K, and T denote the matrix transpose. (2.2) Find the three eigenvalues ​​of the covariance matrix C, arrange the three eigenvalues ​​from largest to smallest and find their corresponding eigenvectors. Then, use the eigenvectors of the three arranged eigenvalues ​​as the direction vectors of the X-axis, Y-axis and Z-axis respectively. Establish the XYZ coordinate system of the point cloud with the centroid of the entire point cloud as the origin. (2.3) In the XYZ coordinate system of the point cloud, collect points in the point cloud data at fixed intervals along the X-axis to obtain layer data.

3. The method for high-fidelity reconstruction of complex surface topography based on VMD and sparsely sampled point clouds as described in claim 2, characterized in that: The specific process for extracting layer data in step (2.3) is as follows: (2.3.1) In the point cloud data, initially determine the initial point P in the current layer. origin Set the iteration starting point P start When the iteration begins, P start =P origin ; (2.3.2) Calculate and determine the ending reference point P according to the following formula. end : P end = P start +ΔdV x Among them, V x Δd represents the direction vector of the X-axis, and Δd represents the step size preset according to the actual resolution of the sampling. (2.3.3) Use the KD Tree data structure to search for the endpoint reference point P in the point cloud data. end The nearest point is taken as the result point P. result Determine the end reference point P. end and result point P result Does the distance d between them exceed the preset distance upper limit threshold ϵ? If the distance is less than or equal to the upper limit threshold ϵ, then the result point P result As the required data points; If the first result point P is obtained result and the ending reference point P end If the distance d between them is not less than the upper limit threshold ϵ, then stop the iteration and return to step (2.3.1) to redetermine the initial point P in the current layer. origin Process it; If it is not the first result point P result and the ending reference point P end If the distance between them is not less than the upper limit threshold, then stop the iteration; (2.3.4) Using the result point P result Replace the starting point P start Returning to step (2.3.2), repeatedly iterate through steps (2.3.2) and (2.3.3) until the distance d > ϵ, at which point the iteration stops and all result points P are obtained. result Finally, all result points P result Extract the layer data that constitutes the current layer.

4. The method for high-fidelity reconstruction of complex surface topography based on VMD and sparsely sampled point clouds as described in claim 1, characterized in that: Step (3) specifically includes: Remove the shape portion from each layer of data extracted in step (2), and perform K-modal VMD decomposition on the remaining portion to obtain K modes, namely imf1, imf2, ..., imf k Then, for each mode, a sinusoidal fitting function ϕ1,ϕ2,…,ϕ is established. k Then, the fitting functions of all modes are summed to establish the morphological feature function f(x) along the X-axis, as follows: Where n represents the ordinal number of the mode, A n The parameter ω represents the exclusion of outliers. n φ represents the intermediate frequency value of the mode. n The phase parameter of the mode, C n The modal average is represented by max(), min(), average(), k, and x, which represents the value along the X-axis and is used as the independent variable.

5. The method for high-fidelity reconstruction of complex surface topography based on VMD and sparsely sampled point clouds as described in claim 4, characterized in that: The shape portion is the part of the quadratic function obtained after fitting the layer data with a quadratic term.

6. The method for high-fidelity reconstruction of complex surface topography based on VMD and sparsely sampled point clouds as described in claim 1, characterized in that, In step (4), for the sparse sampled point cloud, the same method as in steps (1) and (2) is used to process and obtain the layer data of each layer and the shape feature function along the X-axis direction, wherein the shape feature function of each layer is used as the base shape function of each layer of the sparse sampled point cloud along the X-axis direction.

7. The method for high-fidelity reconstruction of complex surface topography based on VMD and sparsely sampled point clouds as described in claim 1, characterized in that: In step (6), the specific steps are as follows: (6.1) The topographic feature function f(x) of each layer obtained in step (3) and the basis shape function of each layer obtained by linear interpolation in step (5) are fused: For the original base shape function of each layer after linear interpolation, the morphology feature function of the same layer is applied to the base shape function of the same layer according to the following formula to obtain the preliminary reconstructed data G(X) after refinement in the X-axis direction: G i (X)=B yi (X)+ f i (X) Among them, G i (X) represents the initial reconstructed data G(X) of the i-th layer, B yi (X) represents the base shape function of the i-th layer along the X-axis, f i (X) represents the morphological feature function of the i-th layer; (6.2) For each layer, the initial reconstruction data of the layer itself is weighted and summed with the initial reconstruction data calculated by the previous layer to obtain the final morphological reconstruction data, as shown in the following formula: G'(x)= c weight G i-1 (X)+(1-c weight )G i (X) Where G'(x) is the final topographic reconstruction data of the i-th layer, G i-1 (X) represents the initial reconstructed data of the (i-1)th layer, c weight This indicates the preset reconstruction weights.

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