Isotropic grid generation method based on deep learning and grid parameterization

Abstract

The invention belongs to the technical field of computer graphics geometric processing, and particularly relates to an isotropic grid generation method based on deep learning and grid parameterization, which comprises the following steps: acquiring triangular grid data of an object model, and expanding the triangular grid data from a three-dimensional space to a two-dimensional plane by adopting a grid parameterization method; inputting the Voronoi unit region boundary point coordinate set of each vertex into a pre-trained deep neural network to obtain a Voronoi unit region initial sampling point coordinate of each vertex; constructing a CVT energy function and minimizing the CVT energy function to obtain optimized triangular mesh data on a two-dimensional plane; mapping the optimized triangular mesh data from a two-dimensional plane to a three-dimensional space through barycentric coordinates; the method can guarantee the quality of the generated grid and improve the operation efficiency at the same time.

Description

Technical Field

[0001] The present invention belongs to the technical field of computer graphics geometry processing, and in particular relates to an isotropic mesh generation method based on deep learning and mesh parameterization. Background Art

[0002] Triangular mesh optimization is an important task in computer graphics. An isotropic triangular mesh is one whose triangular faces have similar shapes and sizes to equilateral triangles and are evenly distributed. Isotropic triangular meshes are stable in numerical calculations and are mainly used in finite element analysis, physical simulation, rendering, etc. As the industry's requirements for rendering and simulation accuracy and efficiency continue to increase, it is necessary to face increasingly complex geometric shapes with a surge in the number of triangular faces. Therefore, it is particularly important to improve the efficiency and quality of triangular mesh optimization operations.

[0003] In order to optimize the quality of triangular meshes using CVT (Centroidal Voronoi Tessellation) remeshing technology while ensuring time efficiency, the present invention converts triangular meshes in three-dimensional space into two-dimensional planes by mesh parameterization. This method uses Euclidean distance to optimize the mesh, avoiding the time-consuming calculation of geodesic distances in three-dimensional space. Although Euclidean distance is used for calculation, when CVT processes large meshes with hundreds of thousands or even millions of triangular faces, the optimization process needs to be iterated based on the Voronoi unit area between each vertex, which takes a lot of time to calculate.

[0004] As deep learning is gradually mature in the field of geometric processing, some scholars have proposed a method of combining deep learning and 3D reconstruction technology to repair the defective areas of triangular meshes, and some scholars have combined deep learning and surface simplification technology to perform denoising work, but these methods are to optimize the quality of triangular meshes by changing the number of triangular faces of the triangular mesh, which is easy to change the shape characteristics of the triangular mesh itself, and will affect the accuracy of object model rendering and simulation. In response to the above problems, the present invention combines deep learning and mesh parameterization to allow the neural network to learn to maintain the shape characteristics of the triangular mesh while reducing the number of CVT iterations, thereby improving the computing efficiency. Summary of the invention

[0005] To solve the above problems, the present invention provides an isotropic mesh generation method based on deep learning and mesh parameterization, comprising the following steps:

[0006] S1. Obtain triangular mesh data of the object model, and use a mesh parameterization method to expand the triangular mesh data from a three-dimensional space to a two-dimensional plane to obtain triangular mesh data after mesh parameterization;

[0007] S2. Based on the triangular mesh data after mesh parameterization, the Voronoi unit region boundary point coordinate set of each vertex is input into the pre-trained deep neural network to obtain the initial sampling point coordinates of the Voronoi unit region of each vertex;

[0008] S3. construct and minimize the CVT energy function according to the coordinates of the initial sampling points in the Voronoi unit area to obtain optimized triangular mesh data on a two-dimensional plane;

[0009] S4. Map the optimized triangular mesh data from the two-dimensional plane back to the three-dimensional space through the barycentric coordinates.

[0010] Beneficial effects of the present invention:

[0011] Graph convolutional neural networks can process the relationships between graphs and are suitable for processing the topological relationships between vertices of triangular meshes. The mesh parameterization method can reduce the dimensionality of the triangular mesh in three-dimensional space to a two-dimensional plane, avoiding the calculation of geodesic distances during mesh optimization. The use of Euclidean distance can improve computational efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 is a flow chart of the method of the present invention;

[0013] Figure 2 It is a schematic diagram of an embodiment of the present invention;

[0014] Figure 3 Schematic diagram of an isotropic mesh after CVT optimization according to an embodiment of the present invention. DETAILED DESCRIPTION

[0015] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0016] The present invention provides an isotropic mesh generation method based on deep learning and mesh parameterization, such as Figure 1 As shown, the following steps are included:

[0017] S1. Acquire the triangular mesh data of the object model, and use the mesh parameterization method to expand the triangular mesh data from the three-dimensional space to the two-dimensional plane to obtain the triangular mesh data after mesh parameterization.

[0018] Specifically, the present invention uses BFF (Boundary First Flattening) to expand the three-dimensional space grid into a two-dimensional plane, and constructs a linear conformal mapping through the Cherrier formula, such as Figure 2 As shown, BFF can map the triangular mesh in three-dimensional space to a two-dimensional plane with free boundaries or to a two-dimensional plane with fixed shape boundaries.

[0019] Step S1 uses a mesh parameterization method to expand the triangular mesh data from a three-dimensional space to a two-dimensional plane, including:

[0020] S11. Consider the vertex of each triangle face in the triangle mesh data as a boundary vertex;

[0021] S12. Determine the Dirichlet Poisson problem based on the boundary vertices as follows

[0022]

[0023] The boundary discrete Newman number is obtained by solving the problem, which is expressed as

[0024]

[0025] Among them, A II Indicates that only the Laplacian matrix of the internal vertices is calculated, A IB Represents the calculation of the Laplacian matrix of internal vertices and boundary vertices, A BB Indicates that only the Laplacian matrix of the boundary vertices is calculated, α I represents the internal vertex coordinates, α B represents the coordinates of the boundary vertices, represents the internal vertex divergence value, represents the divergence value of the boundary vertex, g represents the Dirichlet number, and h represents the boundary discrete Newman number;

[0026] If the discrete Gaussian curvature of the boundary is 0:

[0027]

[0028] Where k represents the current discrete geodesic curvature, represents the target discrete geodesic curvature;

[0029] S13. Based on the Newman-Poisson problem, given the target discrete geodesic curvature Find the scaling factor u:

[0030] g=u

[0031] S14. Use the least squares conformal energy function to find the coordinate values ​​of the vertices inside the mesh:

[0032]

[0033] Among them, a is the boundary vertex coordinate, b is the internal vertex coordinate, and then the grid vertex coordinates in the three-dimensional space are expanded into the vertex coordinates of the two-dimensional plane.

[0034] S2. Based on the triangular mesh data after mesh parameterization, the coordinate set of the boundary points of the Voronoi unit area of ​​each vertex is input into the pre-trained deep neural network to obtain the initial sampling point coordinates of the Voronoi unit area of ​​each vertex.

[0035] Specifically, the pre-training process of the deep neural network model in step S2 includes:

[0036] S21. Acquire an isotropic triangular mesh data set, wherein each isotropic triangular mesh data set includes original mesh data and isotropic mesh data;

[0037] S22. Use the Boundary First Flattening method to map the isotropic triangular mesh data set from three-dimensional space to a two-dimensional plane;

[0038] S23. Take the set of boundary points of the Voronoi unit region of the vertex as input and the coordinates of the initial sampling points of the Voronoi unit region of the vertex as output, and train the graph convolutional neural network (GCN), in which the mean square error loss function is used to calculate the loss, expressed as

[0039]

[0040] h i =GCN(p i )

[0041] Where N is the number of vertices, p i ={p i,1 ,p i,2 ,…,p i,m} represents the coordinate set of the Voronoi cell region boundary point of the i-th vertex in the original grid data, m represents the number of Voronoi cell region boundary points; h i Indicates p i The predicted coordinates of the initial sampling points in the Voronoi unit area obtained by the graph convolutional neural network; v i Represents the coordinate of the i-th vertex in the isotropic mesh data, which is used as the true value of the loss function. The GCN is trained by minimizing the loss function L to obtain a pre-trained deep neural network.

[0042] Specifically, obtaining the coordinate set of the boundary points of the Voronoi cell region of the vertex includes:

[0043] Determine all 1-neighborhood triangles of the vertex, take the center of the circumscribed circle of each 1-neighborhood triangle as a boundary point of a Voronoi cell area and obtain the coordinates.

[0044] S3. Construct and minimize the CVT energy function based on the initial sampling point coordinate set of the Voronoi unit area to obtain the optimized triangular mesh data on the two-dimensional plane.

[0045] Specifically, step S3 includes:

[0046] S31. Calculate the CVT energy function E(Z), expressed as

[0047]

[0048] Where N is the number of vertices, q i The coordinate set of the boundary points of the Voronoi cell region representing the i-th vertex; z i Indicates q i The initial sampling point coordinates of the Voronoi unit area obtained by the graph convolutional neural network; Ω i represents the i-th Voronoi unit area, each vertex corresponds to a Voronoi unit area, and ρ represents the density function;

[0049] S32. Minimize the energy function E(Z) to obtain the gradient Expressed as

[0050]

[0051] Among them, c i represents the target centroid of the ith Voronoi cell area, q i,j represents the jth Voronoi unit region boundary point of the i-th vertex, and m represents the number of Voronoi unit region boundary points;

[0052] S33. Update the coordinates of the initial sampling points in the Voronoi cell area according to the gradient so that they are close to the target centroid position, that is,

[0053]

[0054] Among them, α represents the gradient descent rate;

[0055] S34. Update the vertex coordinates to the updated values ​​of the initial sampling point coordinates of the Voronoi unit region, and return to step S31 until the energy function reaches the threshold.

[0056] S4. Map the optimized triangular mesh data from the two-dimensional plane back to the three-dimensional space through the barycentric coordinates.

[0057] Specifically, step S4 includes:

[0058] S41. In the triangular mesh data, each vertex corresponds to an index, and the vertex index remains unchanged in subsequent processing;

[0059] S42. For each vertex v in the optimized triangle mesh data on the two-dimensional plane c , find the vertex v in the triangle mesh data after mesh parameterization c Vertices v with the same index b ; Similarly, find the vertex v in the triangle mesh data c Vertices v with the same index o ;

[0060] S43. Get the triangle mesh data containing the vertex v in the mesh parameterized triangle mesh data. b triangular face B; get the vertex v in the triangular mesh data o The triangular face O;

[0061] S44. Calculate the area of ​​triangle B, and calculate the area of ​​the three vertices of triangle B after combining them in pairs and adding them to v c The area of ​​the new triangle is calculated based on the area weight, expressed as

[0062]

[0063] Among them, B1, B2, and B3 are the three vertices of triangle face B. represents the area of ​​triangle B, Represents vertices B2, B3 and vertex v c The area of ​​the new triangle is Represents vertices B1, B3 and vertex v c The area of ​​the new triangle is Represents vertices B1, B2 and vertex v c The area of ​​the new triangle. δ, β, γ represent weights.

[0064] S45. Get vertex v according to triangle O and weight c The vertex coordinates v mapped back to space new , expressed as

[0065] v new =δ×v o1 +β×v o2 +γ×v o3

[0066] Among them, v o1 、v o2 、v o3Respectively represent the coordinates of the three vertices of the triangular surface O;

[0067] S46. Repeat steps S42-S45 to obtain the result of mapping the optimized triangle mesh data from the two-dimensional plane back to the three-dimensional space. Figure 3 As shown, the left image is converted into an isotropic triangular mesh on the right after CVT optimization.

[0068] In the present invention, unless otherwise clearly stipulated and limited, the terms such as "installation", "setting", "connection", "fixation" and "rotation" should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral one; it can be a mechanical connection or an electrical connection; it can be directly connected or indirectly connected through an intermediate medium; it can be the internal connection of two elements or the interaction relationship between two elements. Unless otherwise clearly defined, ordinary technicians in this field can understand the specific meanings of the above terms in the present invention according to the specific circumstances.

[0069] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. An isotropic mesh generation method based on deep learning and mesh parameterization, characterized in that: The following steps are involved: S1. Obtain triangular mesh data of the object model, and use a mesh parameterization method to expand the triangular mesh data from a three-dimensional space to a two-dimensional plane to obtain triangular mesh data after mesh parameterization; S2. Based on the triangular mesh data after mesh parameterization, the Voronoi unit region boundary point coordinate set of each vertex is input into the pre-trained deep neural network to obtain the initial sampling point coordinates of the Voronoi unit region of each vertex; S3. construct and minimize the CVT energy function according to the coordinates of the initial sampling points in the Voronoi unit area to obtain optimized triangular mesh data on a two-dimensional plane; S4. Map the optimized triangular mesh data from the two-dimensional plane back to the three-dimensional space through the barycentric coordinates.

2. The isotropic mesh generation method based on deep learning and mesh parameterization according to claim 1, characterized in that: Step S1 uses the Boundary First Flattening method to map the triangular mesh data from the three-dimensional space to the two-dimensional plane.

3. The isotropic mesh generation method based on deep learning and mesh parameterization according to claim 1, characterized in that: Step S2: The pre-training process of the deep neural network model includes: S21. Acquire an isotropic triangular mesh data set, wherein each isotropic triangular mesh data set includes original mesh data and isotropic mesh data; S22. Use the Boundary First Flattening method to map the isotropic triangular mesh data set from three-dimensional space to a two-dimensional plane; S23. Take the coordinate set of the boundary points of the Voronoi unit area of ​​the vertex as input and the coordinates of the initial sampling points of the Voronoi unit area of ​​the vertex as output to train the graph convolutional neural network, in which the mean square error loss function is used to calculate the loss, expressed as h i =GCN(p i ) Where N is the number of vertices, p i Represents the coordinate set of the Voronoi cell region boundary point of the i-th vertex in the original grid data, h i Indicates p i The predicted coordinates of the initial sampling points in the Voronoi unit area obtained by the graph convolutional neural network, v i Represents the coordinates of the i-th vertex in the isotropic mesh data.

4. The isotropic mesh generation method based on deep learning and mesh parameterization according to any one of claims 1 or 3, characterized in that: The acquisition of the coordinate set of the boundary points of the Voronoi cell region of the vertex includes: Determine all 1-neighborhood triangles of the vertex, take the center of the circumscribed circle of each 1-neighborhood triangle as a boundary point of a Voronoi cell area and obtain the coordinates.

5. The isotropic mesh generation method based on deep learning and mesh parameterization according to claim 1, characterized in that: Step S3 specifically includes: S31. Calculate the CVT energy function E(Z), expressed as Where N is the number of vertices, q i The coordinate set of the boundary points of the Voronoi cell region representing the i-th vertex; z i Indicates q i The initial sampling point coordinates of the Voronoi unit area obtained by the graph convolutional neural network; Ω i represents the i-th Voronoi unit area, each vertex corresponds to a Voronoi unit area, and ρ represents the density function; S32. Minimize the energy function E(Z) to obtain the gradient Expressed as Among them, c i represents the target centroid of the i-th Voronoi unit area, q i,j represents the jth Voronoi unit region boundary point of the i-th vertex, and m represents the number of Voronoi unit region boundary points; S33. Update the coordinates of the initial sampling points in the Voronoi cell region according to the gradient, Among them, α represents the gradient descent rate; S34. Update the vertex coordinates to the updated values ​​of the initial sampling point coordinates of the Voronoi unit region, and return to step S31 until the energy function reaches the threshold.

6. The isotropic mesh generation method based on deep learning and mesh parameterization according to claim 1, characterized in that: Step S4 specifically includes: S41. In the triangular mesh data, each vertex corresponds to an index, and the vertex index remains unchanged in subsequent processing; S42. For each vertex v in the optimized triangle mesh data on the two-dimensional plane c , find the vertex v in the triangle mesh data after mesh parameterization c Vertices v with the same index b ; Similarly, find the vertex v in the triangle mesh data c Vertices v with the same index o ; S43. Get the triangle mesh data containing the vertex v in the mesh parameterized triangle mesh data. b triangular face B; get the vertex v in the triangular mesh data o The triangular face O; S44. Calculate the area of ​​triangle B, and calculate the area of ​​the three vertices of triangle B after combining them in pairs and adding them to v c The area of ​​the new triangle is calculated based on the area weight, expressed as Among them, B1, B2, and B3 are the three vertices of triangle face B. represents the area of ​​triangle B, Represents vertices B2, B3 and vertex v c The area of ​​the new triangle is Represents vertices B1, B3 and vertex v c The area of ​​the new triangle is Represents vertices B1, B2 and vertex v c The area of ​​the new triangular face; (δ, β, γ) represents the barycentric coordinates; S45. Get vertex v based on triangular surface O and barycentric coordinates c The vertex coordinates v mapped back to space new , expressed as in new =δ×v o1 +β×v o2 +γ×v o3 Among them, v o1 、v o2 、v o3 Respectively represent the coordinates of the three vertices of the triangular surface O; S46. Repeat steps S42-S45 to obtain the result of mapping the optimized triangular mesh data from the two-dimensional plane back to the three-dimensional space.

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