Spatial data spectral domain decomposition method and apparatus applying laplace operator

Abstract

The application discloses a spatial data spectral domain decomposition method and device applying Laplace operator, and belongs to the technical field of three-dimensional spatial perception data. In view of the three-dimensional space field model construction widely existing in the coal mine scene, the space model coordinate transformation has the invariance representation, the space model is not dependent on the specific coordinate and is decomposed, and the decomposition quantity is used for coding and decoding; the Laplace operator calculation based on cotangent and Hodge dual area is carried out on the original characteristic spectrum, so that the calculation of the characteristic value and the characteristic equation of the original model is realized, and the original model is smoothed and spectrally decomposed and visualized by using the characteristic value and the characteristic equation. The space occupation is low, the model calculation time is low, the fine part is convenient to express, the number of features k can be selected according to the selected precision requirement, the original model is restored, and details of the original model are selected and reserved according to the number of k.

Description

Technical Field

[0001] This invention relates to a method and apparatus for spatial data spectral domain decomposition using the Laplacian operator, belonging to the field of digital reconstruction technology of maps, building interior space, and underground space scenes. Background Technology

[0002] The Laplace operator is a widely used second-order differential operator, with important applications in numerous studies, including damped-spring systems, graphical learning systems, and finite element systems. For three-dimensional spatial models, there are two types of Laplace operators: discrete and continuous. The continuous form is helpful for analyzing the properties of functions acting on smooth, continuous surfaces. However, in practical applications, three-dimensional spatial models always exist in discretized data form; therefore, the discretized Laplace operator is more widely used in practical applications.

[0003] 3D spatial perception data is a necessary data format in many practical applications, serving as prior maps for positioning and navigation, and for the design of smart buildings, smart factories, underground spaces, and mine working faces using 3D digital representations. In practical applications, 3D perception data takes various representations, including point cloud maps, patch-like maps, vector-based subway maps, probabilistic grid maps, and implicit maps. Existing technologies such as point cloud and patch-like implicit maps have the advantage of representing 3D perception models with arbitrary precision, but they suffer from the drawback of large data storage requirements and are highly dependent on a unified coordinate system when reconstructing 3D models from the data. While vector and probabilistic grid maps occupy less storage space, they are not suitable for representing finer details. Summary of the Invention

[0004] To address the shortcomings of existing technologies, a spatial data spectral domain decomposition method and apparatus using the Laplacian operator are provided for model reconstruction, restoration, and smoothing. In other words, the spectral domain decomposition method is used to decompose the model and the decomposed data is used for encoding and decoding. Compared with the original patch model, it has advantages such as controllable accuracy in approximating the original model and smaller data storage scale.

[0005] To achieve the above technical objectives, this invention provides a spatial data spectral domain decomposition method using the Laplace operator. This method constructs a representation of the spatial model that is invariant to coordinate transformations, addressing the widespread existence of three-dimensional spatial field models in building interiors, underground spaces, and coal mine working faces. It decomposes the spatial model independently of specific coordinates and uses the decomposition quantities for encoding and decoding. By performing Laplace operator calculations based on cotangent and Hodge dual areas on the original feature spectrum, the eigenvalues ​​and characteristic equations of the original model are calculated. The eigenvalues ​​and characteristic equations are then used to smooth the original model and visualize the spectral domain decomposition.

[0006] The specific steps are as follows:

[0007] Use a camera to build a mesh, and import the mesh into a computer to form a spatial 3D model that needs to be decomposed.

[0008] A Laplacian operator is constructed on a 3D spatial model to represent the local information of the 3D spatial model. The solution of the local information of the 3D spatial model is to first calculate the neighborhood area of ​​the input 3D spatial model, analyze the cotangent angle of all connected edges in the vertex neighborhood, and finally calculate the Laplacian operator matrix.

[0009] Solve for the characteristic equation and eigenvalues ​​of the Laplacian operator matrix;

[0010] The Laplacian operator matrix is ​​decomposed into eigenvalue sequences and characteristic equations.

[0011] The characteristic equations are sorted in ascending order of their corresponding eigenvalues, and in a one-to-one correspondence between eigenvalues ​​and characteristic equations, the numerical values ​​of the characteristic equations in different regions corresponding to different eigenvalues ​​are related to the local shape and geometric information of the 3D model. Since the 3D model used is not completely symmetrical, the frequency domain of the 3D model as a whole also presents an incompletely symmetrical form, and the characteristic equations corresponding to the same spectrum change relatively smoothly.

[0012] To verify that the decomposed representation can effectively represent the original model, the characteristic equations of the Laplacian operator are sorted in ascending order of their corresponding eigenvalues. Using the characteristic equations corresponding to the first k smallest eigenvalues ​​of the Laplacian operator matrix, an approximate model of the original 3D model is recovered and calculated based on the local information of the neighborhood near the vertices that has been encoded. The value of k is selected according to different accuracy requirements. The larger the value of k, the closer the model recovered by applying the characteristic equations is to the original model.

[0013] Furthermore, the continuous form of the Laplace operator is defined as follows:

[0014]

[0015] in Indicates the gradient sign. The divergence symbol represents the divergence of the gradient of the function u; although the Laplace operator is formally dependent on coordinates, it is actually independent of coordinate representation.

[0016] Furthermore, the discrete form of the Laplacian operator is calculated based on the continuous form: the locality information of the discrete Laplacian operator is calculated using the discrete form 3D perception model composed of triangular facets. The neighborhood of any vertex i in the discrete form 3D perception model, i.e., all triangular facets containing vertex i, is considered. Since this invention aims to decompose the 3D model for invariance to spatial rigid body motion, the locality characteristics of the 3D model itself are considered, i.e., the neighborhood nodes of vertex i are considered, represented as follows: When calculating the area of ​​a vertex's neighborhood, consider Hodge duality, and apply the centroid of the triangle as the dual representation of vertex i, i.e., append... Figure 1 If the centroids of the triangles neighboring vertex i are connected sequentially, the area region formed by connecting the centroids of the triangles corresponding to vertex i is A. i Using positive values ​​of the area field to represent upward convexity and negative values ​​to represent downward concavity, the discrete form of the Laplace operator and the formula for calculating the Laplace operator matrix are defined as follows:

[0017]

[0018]

[0019]

[0020] In the formula, α and β represent the relative angles of the edges ij in the Laplace operator; express The value in the i-th row and k-th column of the matrix is ​​calculated by first calculating the matrix... The values ​​of the off-diagonal lines are calculated last, and the value when i = j is calculated at the end; L represents the Laplace operator matrix; The elements of the matrix are vertex i and its adjacent vertices. The sum of the cotangents of the two angles corresponding to the edges is half of the sum of the diagonal elements, and for the diagonal elements it is the negative of the sum of the other non-zero terms in the row. This shows that the Laplacian matrix has the properties of being real symmetric and self-adjoint. Since the Laplacian matrix is ​​a weighted sum of the matrix representing the geometric angles and the matrix representing the area of ​​the vertex neighborhood, the Laplacian operator of the same model has similar values ​​for 3D perception models of different resolutions.

[0021] Furthermore, the characteristic equation and eigenvalues ​​are solved using a matrix eigenvalue decomposition library: Since the Laplacian operator matrix is ​​a sparse matrix, the factors are efficiently calculated using the sparse matrix solver provided by the Eigen library, where λ represents the eigenvalue.

[0022] Lφ=λφ (5)

[0023] According to the spectral domain theorem, its characteristic equation {φ iThe function space of the corresponding dimension of Zhang Cheng's 3D perception model, that is, any function u acting on the n-vertex 3D perception model, can necessarily be uniquely decomposed into: Where w i φ represents the weighting coefficients for the decomposition. i This is the i-th characteristic equation;

[0024] After performing eigenvalue decomposition on the Laplacian operator matrix, the eigenvalue sequence {λ} is obtained. i} and the sequence of characteristic equations {φ i Since the Laplace operator matrix is ​​positive semidefinite, its eigenvalues ​​must satisfy:

[0025] λ i ≥0 (6)

[0026] For a 3D model with n vertices, use A vertex array of dimension is used to store the data, where each row represents the x / y / z components of the coordinates. An array containing the row number of all vertices in the vertex array is used as the index of each vertex. This array, of dimension , is then used to construct the vertex connectivity relationships within the patch. Each row represents the vertex index of a triangular facet, that is, the position of the three vertices of the facet in the vertex array. The calculated Laplacian matrix has dimensions of... It is a positive semi-definite real symmetric matrix with n eigenvalues ​​and eigenvectors.

[0027] Furthermore, using the Laplacian operator matrix calculation formula recorded in formulas (2)-(4), it can be seen that it has encoded the local information of the neighborhood near the vertex. The original three-dimensional model is recovered based on the local information. Since the Laplacian operator matrix cannot distinguish between convex and concave, the recovered three-dimensional model uses the neighborhood area with a sign to distinguish between convex and concave, so as to avoid inconsistency between the recovered model and the original model.

[0028] Finally, the original 3D model is recovered based on the first k characteristic equations of the Laplacian operator matrix, and the recovered 3D model is smoothed to verify the effectiveness of the decomposition method.

[0029] The formula for generating a 3D model using the first k characteristic equations of the Laplacian operator matrix is ​​as follows:

[0030]

[0031] in Let P represent the matrix consisting of the first k characteristic equations, with dimensions n×k, where k is the number of characteristic equations. Let P represent the vertex matrix of the original model, with dimensions n×3. This is the vertex matrix of the new 3D model.

[0032] A spatial data spectral domain decomposition device using the Laplacian operator, comprising:

[0033] The 3D spatial model generation unit uses a camera to construct a mesh and convert it into a 3D spatial model within the computer.

[0034] The Laplacian operator constructs units to represent local information of a 3D spatial model and to solve for local information of the 3D spatial model.

[0035] The characteristic equation and eigenvalue calculation unit of the Laplacian operator matrix is ​​used to solve for the characteristic equation and eigenvalue of the Laplacian operator matrix;

[0036] The characteristic equation sorting unit sorts the characteristic equations of the calculated Laplacian operator matrix from bottom to top according to their corresponding eigenvalues.

[0037] The restoration verification unit is used to restore the original 3D model based on the set feature values ​​and the corresponding feature equations, using the local information of the neighborhood near the vertices that has been encoded.

[0038] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-described spatial data spectral domain decomposition method using the Laplace operator.

[0039] Beneficial effects:

[0040] 1) For spatial 3D models, a Laplacian operator based on Hodge duality is designed for local information representation; 2) Based on the Laplacian operator, spectral domain eigenvalue decomposition is performed to represent the 3D model independently of its spatial coordinates, avoiding inconsistencies in the 3D model representation caused by rigid body transformation. Compared with point clouds, probabilistic meshes, etc., the storage requirements are significantly smaller. Selecting the k characteristic equations with smaller eigenvalues ​​after decomposition as the representation of the original model can effectively reduce space storage consumption; 3) The eigenvalues ​​of the Laplacian operator are color-mapped to facilitate intuitive visualization of the characteristic equations, thereby intuitively visualizing the represented local information; 4) Based on the decomposed representation, this invention designs a method to recover the original 3D model with arbitrary precision.

[0041] Technical advantages:

[0042] Compared with existing 3D model representation methods, the representation method of this invention can effectively reduce space occupation and model calculation time; the representation after spectral domain decomposition proposed in this invention can realize the model's invariance to spatial rigid body transformations, which is convenient for analysis and processing and for expressing refined parts; the method of this invention can recover the original model by selecting the number of features k according to the accuracy requirements, thereby selecting and retaining the details of the original model according to the number of k. Attached Figure Description

[0043] Figure 1 This is a flowchart illustrating the spatial data spectral domain decomposition method using the Laplace operator as described in this invention.

[0044] Figure 2 This is a schematic diagram of the local calculation of the discrete Laplace operator of the present invention;

[0045] Figure 3 This is a schematic diagram of an example spatial model according to an embodiment of the present invention;

[0046] Figure 4 It is aimed at Figure 3 A schematic diagram of the characteristic equations corresponding to the first 6 spectra of the three-dimensional spatial model in the Chinese embodiment;

[0047] Figure 5 It is aimed at Figure 3 A schematic diagram of the three-dimensional spatial model in the embodiment, reconstructed using the spectrum k=5, k=10, k=20, k=30, k=50, k=80. Detailed Implementation

[0048] The present invention will be further described below with reference to the accompanying drawings. The following embodiments are only used to illustrate the technical solution of the present invention more clearly, and should not be used to limit the scope of protection of the present invention.

[0049] like Figure 1 As shown, this invention provides a spatial data spectral domain decomposition method using the Laplace operator. This method constructs a representation of the spatial model that is invariant to coordinate transformations, addressing the widespread existence of three-dimensional spatial field models in building interiors, underground spaces, and coal mine working faces. It decomposes the spatial model independently of specific coordinates and uses the decomposition quantities for encoding and decoding. By performing Laplace operator calculations based on cotangent and Hodge dual areas on the original feature spectrum, the eigenvalues ​​and characteristic equations of the original model are calculated. The eigenvalues ​​and characteristic equations are then used to smooth the original model and visualize the spectral domain decomposition.

[0050] The specific steps are as follows:

[0051] Use a camera to build a mesh, import the mesh into a computer to form a spatial 3D model that needs to be decomposed, such as... Figure 3 As shown, the three-dimensional spatial model includes a spatial map, the interior space of buildings, underground space, and the underground working face space of a coal mine;

[0052] A Laplacian operator is constructed on a 3D spatial model to represent the local information of the 3D spatial model. The solution of the local information of the 3D spatial model is to first calculate the neighborhood area of ​​the input 3D spatial model, analyze the cotangent angle of all connected edges in the vertex neighborhood, and finally calculate the Laplacian operator matrix.

[0053] Solve for the characteristic equation and eigenvalues ​​of the Laplacian operator matrix;

[0054] The Laplacian operator matrix is ​​decomposed into eigenvalue sequences and characteristic equations.

[0055] The characteristic equations are sorted in ascending order of their corresponding eigenvalues, and in a one-to-one correspondence between eigenvalues ​​and characteristic equations, the numerical values ​​of the characteristic equations in different regions corresponding to different eigenvalues ​​are related to the local shape and geometric information of the 3D model. Since the 3D model used is not completely symmetrical, the frequency domain of the 3D model as a whole also presents an incompletely symmetrical form, and the characteristic equations corresponding to the same spectrum change relatively smoothly.

[0056] To verify that the decomposed representation can effectively represent the original model, the characteristic equations of the Laplacian operator are sorted in ascending order of their corresponding eigenvalues. Using the characteristic equations corresponding to the first k smallest eigenvalues ​​of the Laplacian operator matrix, an approximate model of the original 3D model is reconstructed based on the local information of the vertex neighborhoods that have already been encoded. In practice, the value of k can be determined according to different accuracy requirements; the larger the value of k, the closer the model reconstructed using the characteristic equations is to the original model.

[0057] Three-dimensional spatial perception data has wide applications, including spatial field reconstruction in coal mine scenes and spatial models in the construction industry. However, spatial models have unique properties compared to other types of perception data or models:

[0058] 1) Generally speaking, it has invariance to rigid body motion transformations;

[0059] 2) The same perceptual model object can have different resolutions and different discretization forms;

[0060] 3) The discretized primitives can be triangular patches, quadrilaterals, other polygons, or hybrid forms;

[0061] 4) Spatial models obtained using techniques such as Poisson reconstruction and SLAM often exhibit fragmented forms and face problems such as holes. These properties pose significant challenges to processing 3D perceptual data, but also provide new processing approaches, namely focusing on the locality of 3D data rather than information such as the number of point pairs, surface patterns, and relative spatial positions of the whole data.

[0062] The Laplacian operator can be used to effectively handle local information because it is independent of the coordinate system, and its continuous form is defined as follows:

[0063]

[0064] in Indicates the gradient sign. The divergence sign is used to represent the divergence of the gradient of the function u. Although the Laplace operator formally depends on coordinates, it is actually independent of coordinate representation. This can be demonstrated by proving the Laplace transform under the rigid body transformation (R, t) and utilizing the properties of the special orthogonal group R. T R = RR T =I can prove it.

[0065] In the discrete form, since calculating partial differentials becomes less intuitive, it is necessary to define a Laplacian operator that satisfies the aforementioned locality and coordinate independence requirements. This invention considers a discrete 3D perception model composed of triangular facets. For such a model, for any vertex i, its corresponding neighborhood is as follows: Figure 2 As shown in the shaded area, when considering locality, i.e., considering the neighboring nodes of vertex i, it is represented as... Considering Hodge duality, vertex i corresponds to the area region A formed by connecting the centroids of the triangles in sequence. i Then the discrete form of the Laplace operator is defined as:

[0066]

[0067]

[0068]

[0069] In the formula The elements of the matrix are vertex i and its adjacent vertices. The Laplacian matrix is ​​half the sum of the cotangents of the two angles corresponding to the edges, and for diagonal elements, it is the negative of the sum of the other non-zero terms in that row. This property ensures that the calculated Laplacian matrix has good locality and possesses real symmetry and self-adjoint properties, which are also required properties of the Laplacian matrix. Since the Laplacian matrix is ​​derived by weighting a matrix representing geometric angles with a matrix representing the area of ​​the vertex neighborhood, the Laplacian operator of the same 3D perception model has similar values ​​for different resolutions.

[0070] After obtaining the Laplacian operator matrix, its characteristic equation and eigenvalues ​​are solved. This can be achieved using common matrix eigenvalue decomposition libraries, such as the Eigen library used in this invention. Since the Laplacian operator matrix is ​​sparse, the factors can be decomposed using the sparse matrix solver provided by the Eigen library for more efficient computation. That is, to find:

[0071] Lφ=λφ (5)

[0072] Due to the favorable properties of the Laplace operator matrix, according to the spectral domain theorem, its characteristic equation {φ} i The function space spanned by the corresponding dimension of the 3D perception model can be defined, meaning that any function u acting on the n-vertex 3D perception model can be uniquely decomposed into the following functions: Where w i These are the weighting coefficients for the decomposition.

[0073] After performing eigenvalue decomposition on the Laplacian operator matrix, the eigenvalue sequence {λ} is obtained. i} and the sequence of characteristic equations {φ i Since the Laplace operator matrix is ​​positive semidefinite, its eigenvalues ​​must satisfy:

[0074] λ i ≥0 (6)

[0075] In the calculation, for a 3D model with n vertices, this invention uses A vertex array of dimension is used to store the data, where each row represents the x / y / z components of the coordinates. An array containing the row number of all vertices in the vertex array is used as the index of each vertex. This array, of dimension , is then used to construct the vertex connectivity relationships within the patch. Each row represents the vertex index of a triangular facet (that is, the position of the three vertices of the facet in the vertex array). The calculated Laplacian matrix has dimensions of... It is a positive semi-definite real symmetric matrix, therefore it has n eigenvalues ​​and eigenvectors.

[0076] Sort them in ascending order, and according to the one-to-one correspondence between eigenvalues ​​and characteristic equations, for the appended... Figure 1 The 3D model in the image is used to visually represent the decomposed characteristic equations. The values ​​of the first six characteristic equations (k=6) are converted into color maps, and the results are shown in the appendix. Figure 4 As shown. According to the appendix Figure 4 It can be seen that the characteristic equations corresponding to different eigenvalues ​​have different numerical values ​​in different regions. This is closely related to the local shape and geometric information of the three-dimensional model. Furthermore, since the three-dimensional model used is not completely symmetrical, the overall frequency domain also exhibits an incompletely symmetrical form, and the characteristic equations corresponding to the same spectrum change relatively smoothly.

[0077] As can be seen from the calculation formula of the Laplacian operator matrix, it has encoded the local information of the neighborhood near the vertex, so the original model can be recovered from it. It is worth noting that since the Laplacian operator matrix cannot distinguish between convex and concave surfaces, the recovered 3D model may show some inconsistencies with the original model.

[0078] The formula for calculating the new 3D model using the first k characteristic equations of the Laplacian operator matrix is ​​as follows:

[0079]

[0080] in Let P represent the matrix composed of the first k characteristic equations, with dimensions n×k, and let P represent the vertex matrix of the original model, with dimensions n×3. This represents the vertex matrix of the new 3D model. From this, the original model can be reconstructed based on the first k characteristic equations, and smoothed to some extent.

[0081] As attached Figure 5 As shown, the three-dimensional spatial models generated according to the above method when the number of characteristic equations used is k=5, k=10, k=20, k=30, k=50, and k=80.

[0082] A spatial data spectral domain decomposition device using the Laplacian operator, comprising:

[0083] The 3D spatial model generation unit uses a camera to construct a mesh and convert it into a 3D spatial model within the computer.

[0084] The Laplacian operator constructs units to represent local information of a 3D spatial model and to solve for local information of the 3D spatial model.

[0085] The characteristic equation and eigenvalue calculation unit of the Laplacian operator matrix is ​​used to solve for the characteristic equation and eigenvalue of the Laplacian operator matrix;

[0086] The characteristic equation sorting unit sorts the characteristic equations of the calculated Laplacian operator matrix from bottom to top according to their corresponding eigenvalues.

[0087] The restoration verification unit is used to restore the original 3D model based on the set feature values ​​and the corresponding feature equations, using the local information of the neighborhood near the vertices that has been encoded.

[0088] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-described spatial data spectral domain decomposition method using the Laplace operator.

Claims

1. A spatial data spectral domain decomposition method using the Laplacian operator, characterized in that: This paper addresses the construction of 3D spatial field models widely found in maps, building spaces, and underground spaces. These models possess coordinate transformation invariance and are decomposed without relying on specific coordinates. The decomposition quantities are then used for encoding and decoding. By calculating the original feature spectrum using the Laplacian operator based on the cotangent and Hodge dual area, the eigenvalues ​​and characteristic equations of the original model are calculated. Finally, the eigenvalues ​​and characteristic equations are used to smooth the original model and visualize the spectral domain decomposition. The specific steps are as follows: Use a camera to build a mesh, and import the mesh into a computer to form a spatial 3D model that needs to be decomposed. A Laplacian operator is constructed on a 3D spatial model to represent the local information of the 3D spatial model. The solution of the local information of the 3D spatial model is to first calculate the neighborhood area of ​​the input 3D spatial model, analyze the cotangent angle of all connected edges in the vertex neighborhood, and finally calculate the Laplacian operator matrix. This implementation uses a matrix eigenvalue decomposition library to solve for its characteristic equation and eigenvalues: Since the Laplacian operator matrix is ​​a sparse matrix, the factors are efficiently calculated using the sparse matrix solver provided by the Eigen library for eigenvalue decomposition. λ represents the eigenvalue. ， According to the spectral domain theorem Its characteristic equation Zhang Cheng's corresponding dimension of the three-dimensional perception model's function space, that is, any action on... Functions on a vertex 3D perception model It can necessarily be uniquely decomposed into ,in These are the weighting coefficients for the decomposition. For the first One characteristic equation; The Laplacian operator matrix is ​​decomposed into eigenvalue sequences. and characteristic equation sequence Because of the positive semidefiniteness of the Laplace operator matrix, its eigenvalues ​​must satisfy: ， For containing A 3D model with vertices, using The vertices are stored in a dimensional array, where each row represents the coordinates. The components are used, and an array of vertex connections in a patch is constructed using a row number containing all vertices in the vertex array as the index of each vertex. The array has dimensions of 1. Each row represents the vertex index of a triangular facet, that is, the position of the three vertices of the facet in the vertex array. The calculated Laplacian matrix has dimensions of . It is a positive semi-definite real symmetric matrix with a total of Each eigenvalue and eigenvector; The characteristic equations are sorted in ascending order of their corresponding eigenvalues, and then arranged in a one-to-one correspondence between eigenvalues ​​and characteristic equations. To verify that the decomposed representation can effectively represent the original model, the characteristic equations of the Laplacian operator are sorted in ascending order of their corresponding eigenvalues. Using the characteristic equations corresponding to the first k smallest eigenvalues ​​of the Laplacian operator matrix, an approximate model of the original 3D model is recovered and calculated based on the local information of the neighborhood near the vertices that has been encoded. The value of k is selected according to different accuracy requirements. The larger the value of k, the closer the model recovered by applying the characteristic equations is to the original model.

2. The spatial data spectral domain decomposition method using the Laplacian operator according to claim 1, characterized in that, The continuous form of the Laplace operator is defined as follows: ， in Indicates the gradient sign. The divergence sign is represented by its function. The divergence of the gradient, x i This indicates taking the partial derivative with respect to the coordinate components.

3. The spatial data spectral domain decomposition method using the Laplacian operator according to claim 2, characterized in that, Calculating the discrete form of the Laplacian operator based on the continuous form: Locality information of the discrete Laplacian operator is calculated using a discrete 3D perception model composed of triangular facets, where any vertex of the discrete 3D perception model... The corresponding neighborhood, i.e., the vertex Considering all triangles within the triangle, the vertices are taken into account. Neighboring nodes, denoted as When calculating the area of ​​a vertex's neighborhood, Hodge duality is considered, and the centroid of the triangle is used as the vertex. The dual representation of , vertex Connecting the centroids of the neighborhood triangles sequentially, the vertices... The area corresponding to the triangle formed by connecting its centroids in sequence is Using positive values ​​of the area field to represent upward convexity and negative values ​​to represent downward concavity, the discrete form of the Laplace operator and the formula for calculating the Laplace operator matrix are defined as follows: ， ， ， In the formula, Describes the edges in the discrete Laplace operator. Relative angle; express The first in the matrix Line number The values ​​of the columns are calculated by first calculating the matrix. The values ​​of the off-diagonal lines are calculated last. The value of time, Represents the Laplacian operator matrix; The elements of the matrix are those derived from the vertices. and adjacent vertices The sum of the cotangents of the two angles corresponding to the sides is half of the sum of the two angles, and for diagonal elements, it is the negative of the sum of the other non-zero terms in that row.

4. The spatial data spectral domain decomposition method using the Laplacian operator according to claim 2, characterized in that, The original 3D model is recovered based on local information. The recovered 3D model is then analyzed using the area of ​​the neighborhood with signs, with positive and negative signs used to distinguish between convex and concave areas. Finally, the original 3D model is recovered based on the first k characteristic equations of the Laplacian operator matrix, and the recovered 3D model is smoothed to verify the effectiveness of the decomposition method. The formula for generating a 3D model using the first k characteristic equations of the Laplacian operator matrix is ​​as follows: ， in This represents a matrix composed of the first k characteristic equations, with dimension . k is the number of characteristic equations. This represents the vertex matrix of the original model, with dimension . , This is the vertex matrix of the new 3D model.

5. A spatial data spectral domain decomposition apparatus using the Laplacian operator as described in claim 1, characterized in that, include: The 3D spatial model generation unit is used to construct a mesh using a camera and convert it into a 3D spatial model within the computer. The Laplacian operator constructs units to represent local information of a 3D spatial model and to solve for local information of the 3D spatial model. The characteristic equation and eigenvalue calculation unit of the Laplacian operator matrix is ​​used to solve for the characteristic equation and eigenvalue of the Laplacian operator matrix; The characteristic equation sorting unit sorts the characteristic equations of the calculated Laplacian operator matrix in ascending order according to their corresponding eigenvalues. The restoration verification unit is used to restore the original 3D model based on the set feature values ​​and the corresponding feature equations, using the local information of the neighborhood near the vertices that has been encoded.

6. A 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 spatial data spectral domain decomposition method using the Laplacian operator as described in any one of claims 1-4.

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