An isotropic mesh generation method based on deep learning and mesh parameterization

Abstract

This invention belongs to the field of computer graphics geometry processing technology, specifically relating to an isotropic mesh generation method based on deep learning and mesh parameterization. The method includes acquiring triangular mesh data of an object model; unfolding the triangular mesh data from three-dimensional space to a two-dimensional plane using a mesh parameterization method; inputting the set of Voronoi unit region boundary point coordinates of each vertex into a pre-trained deep neural network to obtain the initial sampling point coordinates of the Voronoi unit region for each vertex; constructing and minimizing a CVT energy function to obtain optimized triangular mesh data on the two-dimensional plane; and mapping the optimized triangular mesh data back from the two-dimensional plane to three-dimensional space using barycentric coordinates. This invention can ensure the quality of the generated mesh while improving computational efficiency.

Description

Technical Field

[0001] This invention belongs to the field of computer graphics geometry processing technology, specifically relating to an isotropic mesh generation method based on deep learning and mesh parameterization. Background Technology

[0002] Triangle mesh optimization is a crucial task in computer graphics. Isotropic triangle meshes are characterized by triangular faces that are similar in shape and size to equilateral triangles and exhibit uniform anisotropy. Isotropic triangle meshes demonstrate stability in numerical computations and are primarily used in finite element analysis, physical simulations, and rendering. As industry demands for accuracy and efficiency in rendering and simulation increase, the need to handle increasingly complex geometries with a rapidly growing number of triangular faces necessitates improving the computational efficiency and quality of triangle mesh optimization.

[0003] To optimize triangular mesh quality while maintaining time efficiency using CVT (Centroidal Voronoi Tessellation) remeshing technology, this invention converts the triangular mesh in three-dimensional space into a two-dimensional plane through mesh parameterization. This method utilizes Euclidean distance for mesh optimization, avoiding the time-consuming calculation of geodesic distances in three-dimensional space. Although Euclidean distance is used for calculation, CVT requires significant computation time for iterating over Voronoi cell regions between each vertex when dealing with massive meshes with hundreds of thousands or even millions of triangular faces.

[0004] As the application of deep learning in geometric processing matures, some scholars have proposed methods combining deep learning and 3D reconstruction techniques to repair defective areas in triangular meshes. Others have combined deep learning with surface simplification techniques for denoising. However, these methods optimize the quality of triangular meshes by changing the number of triangle faces, which can easily alter the shape characteristics of the mesh itself and affect the accuracy of object model rendering and simulation. This invention addresses these problems by combining 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 computational efficiency. Summary of the Invention

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

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

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

[0008] S3. Construct the CVT energy function based on the initial sampling point coordinates of the Voronoi unit region and minimize it to obtain the optimized triangular mesh data on the two-dimensional plane;

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

[0010] The beneficial effects of this invention are:

[0011] Graph convolutional neural networks can handle relationships between graphs and are suitable for handling topological relationships between vertices of triangular meshes. Mesh parameterization methods can reduce the dimensionality of triangular meshes in three-dimensional space to a two-dimensional plane, avoiding the calculation of geodesic distances during mesh optimization. Using Euclidean distance can improve computational efficiency. Attached Figure Description

[0012] Figure 1 This is a flowchart of the method of the present invention;

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

[0014] Figure 3 This is a schematic diagram of an isotropic mesh after CVT optimization according to an embodiment of the present invention. Detailed Implementation

[0015] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0016] This invention provides an isotropic mesh generation method based on deep learning and mesh parameterization, such as... Figure 1 As shown, it includes the following steps:

[0017] S1. Obtain the triangular mesh data of the object model, and use the mesh parameterization method to unfold the triangular mesh data from three-dimensional space to a two-dimensional plane to obtain the parameterized triangular mesh data.

[0018] Specifically, this invention uses BFF (Boundary First Flattening) to unfold a three-dimensional mesh into a two-dimensional plane, and constructs a linear conformal mapping using the Cherrier formula, such as... Figure 2 As shown, BFF can map a 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 unfold the triangular mesh data from three-dimensional space to a two-dimensional plane, including:

[0020] S11. Treat each vertex of a 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, and is expressed as:

[0024]

[0025] Among them, A II A represents the Laplacian matrix calculated only for the internal vertices. IB Let A represent the Laplacian matrix used to compute the interior vertices and boundary vertices. BB This indicates that only the Laplacian matrix of the boundary vertices is calculated, α I Represents the coordinates of the interior vertices, α B Represents the coordinates of the boundary vertices. Represents the divergence value of the internal vertices. denoted by , g represents the divergence value of the boundary vertex, 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 discrete geodesic curvature of the target;

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

[0030] g = u

[0031] S14. Use the least squares conformal energy function to find the vertex coordinates inside the mesh:

[0032]

[0033] Where a represents the coordinates of the boundary vertices and b represents the coordinates of the interior vertices, the grid vertex coordinates in three-dimensional space are then expanded into vertex coordinates in two-dimensional plane.

[0034] S2. Based on the parameterized triangular mesh data, input the set of Voronoi cell region boundary point coordinates of each vertex into a pre-trained deep neural network to obtain the initial sampling point coordinates of the Voronoi cell region of each vertex.

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

[0036] S21. Obtain an isotropic triangular mesh dataset, wherein each isotropic triangular mesh dataset includes the original mesh data and isotropic mesh data;

[0037] S22. The Boundary First Flattening method is used to map the isotropic triangular mesh dataset from three-dimensional space to a two-dimensional plane;

[0038] S23. Using the set of boundary points of the Voronoi unit regions of the vertices as input and the coordinates of the initial sampling points of the Voronoi unit regions of the vertices as output, train a graphical convolutional neural network (GCN), where the mean squared error loss function is used to calculate the loss, expressed as follows:

[0039]

[0040] h i =GCN(p i )

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

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

[0043] Determine all 1-neighbor triangles of a vertex, and take the circumcenter of the circumcircle of each 1-neighbor triangle as a boundary point of a Voronoi cell region and obtain its coordinates.

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

[0045] Specifically, step S3 includes:

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

[0047]

[0048] Where N represents the number of vertices, q i z represents the set of coordinates of the boundary points of the Voronoi cell region at the i-th vertex; i q i The initial sampling point coordinates of the Voronoi cell region obtained by the graph convolutional neural network; Ω i Let represent the i-th Voronoi cell region, with each vertex corresponding to a Voronoi cell region, and ρ represent the density function;

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

[0050]

[0051] Among them, c i Let q represent the target centroid of the i-th Voronoi unit region. i,j Let m represent the boundary point of the j-th Voronoi cell region at the i-th vertex, and m represent the number of boundary points of the Voronoi cell region.

[0052] S33. Update the initial sampling point coordinates of the Voronoi cell region according to the gradient, making them approximate the target centroid position, i.e., let

[0053]

[0054] Where α represents the gradient descent rate;

[0055] S34. Update the vertex coordinates to the updated initial sampling point coordinates of the Voronoi cell 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 using the centroid 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 triangular mesh data on the two-dimensional plane c Searching for the vertex v in the parameterized triangle mesh data c Vertices v with the same index b Similarly, in the triangular mesh data, find the vertex v. c Vertices v with the same index o ;

[0060] S43. Obtain the triangle mesh containing vertex v from the parameterized mesh data. b The triangle B; obtain the triangle containing vertex v from the triangle mesh data. o The triangular face O;

[0061] S44. Calculate the area of ​​triangle B. Calculate the area of ​​triangle B by combining each of its three vertices in pairs with v. c The area of ​​the newly formed triangle is weighted based on the area and expressed as:

[0062]

[0063] Where B1, B2, and B3 are the three vertices of triangle B. This represents the area of ​​triangle B. Representing vertices B2, B3 and vertex v c The area of ​​the newly formed triangle, Representing vertices B1, B3 and vertex v c The area of ​​the newly formed triangle, Representing vertices B1, B2 and vertex v c The area of ​​the newly formed triangle; δ, β, and γ represent the weights;

[0064] S45. Obtain vertex v based on triangle O and its weights. c Vertex coordinates v mapped back to space new , represented as

[0065] v new =δ×v o1 +β×v o2 +γ×v o3

[0066] Among them, v o1 v o2 v o3These represent the coordinates of the three vertices of triangle O;

[0067] S46. Repeat steps S42-S45 to obtain the optimized triangular mesh data mapped back to three-dimensional space from the two-dimensional plane. For example... Figure 3 As shown, the left image, after CVT optimization, is transformed into a triangular mesh that tends to be isotropic, as shown in the right image.

[0068] In this invention, unless otherwise explicitly specified and limited, the terms "installation," "setting," "connection," "fixing," "rotation," etc., should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral part; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal connection of two components or the interaction between two components. Unless otherwise explicitly limited, those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

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

Claims

1. A method for generating isotropic meshes based on deep learning and mesh parameterization, characterized in that, Includes the following steps: S1. Obtain the triangular mesh data of the object model, and use the mesh parameterization method to unfold the triangular mesh data from three-dimensional space to a two-dimensional plane to obtain the parameterized triangular mesh data; S2. Based on the parameterized triangular mesh data, input the set of Voronoi cell region boundary point coordinates of each vertex into a pre-trained deep neural network to obtain the initial sampling point coordinates of the Voronoi cell region of each vertex; S3. Construct the CVT energy function based on the initial sampling point coordinates of the Voronoi unit region and minimize it to obtain the optimized triangular mesh data on the two-dimensional plane; Step S3 specifically includes: S31. Calculate the CVT energy function E(Z), expressed as: Where N represents the number of vertices, q i z represents the set of coordinates of the boundary points of the Voronoi cell region at the i-th vertex; i q i The initial sampling point coordinates of the Voronoi cell region obtained by the graph convolutional neural network; Ω i Let represent the i-th Voronoi cell region, with each vertex corresponding to a Voronoi cell region, and ρ represent the density function; S32. Minimize the energy function E(Z) to obtain the gradient. Represented as Among them, c i Let q represent the target centroid of the i-th Voronoi unit region. i,j Let m represent the boundary point of the j-th Voronoi cell region at the i-th vertex, and m represent the number of boundary points of the Voronoi cell region. S33. Update the initial sampling point coordinates of the Voronoi cell region according to the gradient, let Where α represents the gradient descent rate; S34. Update the vertex coordinates to the updated initial sampling point coordinates of the Voronoi cell region, and return to step S31 until the energy function reaches the threshold. S4. Map the optimized triangular mesh data from the two-dimensional plane back to the three-dimensional space using the centroid 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 three-dimensional space to a 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. Obtain an isotropic triangular mesh dataset, wherein each isotropic triangular mesh dataset includes the original mesh data and the isotropic mesh data; S22. The Boundary First Flattening method is used to map the isotropic triangular mesh dataset from three-dimensional space to a two-dimensional plane; S23. Using the set of boundary point coordinates of the Voronoi unit region of the vertex as input and the initial sampling point coordinates of the Voronoi unit region of the vertex as output, train a graph convolutional neural network, where the mean squared error loss function is used to calculate the loss, expressed as: h i =GCN(p i ) Where N represents the number of vertices, p i h represents the set of coordinates of the boundary points of the Voronoi cell region at the i-th vertex in the original mesh data. i p i The predicted coordinates of the initial sampling points in the Voronoi cell region obtained through the graph convolutional neural network, v i This 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, Obtaining the set of coordinates of the boundary points of the Voronoi cell region of the vertex includes: Determine all 1-neighbor triangles of a vertex, and take the circumcenter of the circumcircle of each 1-neighbor triangle as a boundary point of a Voronoi cell region and obtain its coordinates.

5. 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 triangular mesh data on the two-dimensional plane c Searching for the vertex v in the parameterized triangle mesh data c Vertices v with the same index b Similarly, in the triangular mesh data, find the vertex v. c Vertices v with the same index o ; S43. Obtain the triangle mesh containing vertex v from the parameterized mesh data. b The triangle B; obtain the triangle containing vertex v from the triangle mesh data. o The triangular face O; S44. Calculate the area of ​​triangle B. Calculate the area of ​​triangle B by combining each of its three vertices in pairs with v. c The area of ​​the newly formed triangle is weighted based on the area and expressed as: Where B1, B2, and B3 are the three vertices of triangle B. This represents the area of ​​triangle B. Representing vertices B2, B3 and vertex v c The area of ​​the newly formed triangle, Representing vertices B1, B3 and vertex v c The area of ​​the newly formed triangle, Representing vertices B1, B2 and vertex v c The area of ​​the newly formed triangle; (δ,β,γ) represents the coordinates of the barycenter; S45. Obtain vertex v based on the coordinates of triangle O and the centroid. c Vertex coordinates v mapped back to space new , represented as in new =δ×v o1 +β×v o2 +γ×v o3 Among them, v o1 v o2 v o3 These represent the coordinates of the three vertices of triangle 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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