Triangle mesh simplification method for graph neural networks

Abstract

The application discloses a triangle mesh graph neural network simplification method, and belongs to the technical field of computer deep learning, which acquires a triangle mesh model to be simplified and constructs a graph structure with vertices as nodes and undirected edges based on the topological relation of triangular facets as graph edges; the nodes are subjected to feature initialization, and geometric, topological and Laplacian position coding information is fused; a multi-scale adjacency relation containing 1st-order adjacent edges and 2nd-order adjacent edges is constructed; the node features and the multi-scale adjacency relation are input into a graph neural network, message passing and feature aggregation are completed in different neighborhoods based on a graph convolution network, and edge importance probability is predicted; the edge geometric structure importance is integrated into a simplification process, and the key feature protection effect is improved; the geometric structure features extracted by deep learning are combined with a classical quadratic error measurement algorithm, and the edge collapse cost is dynamically scaled, so that the number of mesh facets is greatly reduced, and the geometric features and the topological structure of the model are effectively maintained.

Description

Technical Field

[0001] This application relates to the field of computer deep learning technology, and specifically discloses a simplified method for graph neural networks using triangular meshes. Background Technology

[0002] With the rapid development of 3D scanning, photogrammetry, and physical simulation technologies, high-resolution mesh models are widely used in fields such as digital cities, cultural heritage, industrial inspection, and 3D mapping. However, these models typically contain hundreds of thousands or even millions of triangular faces, which puts enormous computational and memory pressure on storage, transmission, and real-time rendering. To meet the needs of interactive applications and resource-constrained scenarios, mesh simplification has become an indispensable core technology. The goal of mesh simplification is to significantly reduce model complexity while preserving the original geometric appearance and structural features as much as possible. Most traditional methods are based on greedy strategies, which define cost functions for vertices or edges and perform simplification operations according to priority. Among them, the quadratic error metric QEM is the most representative. This type of method is widely used because of its computational efficiency and stable implementation.

[0003] 3D mesh simplification is a core problem in computer graphics, aiming to reduce the amount of model data while maintaining its key features, which is crucial for large-scale scene rendering and real-time applications. With the development of geometric deep learning, graph neural networks have provided an effective tool for learning the structural features of 3D meshes. In addition to neighborhood structure design, how to provide global structural information for graph neural networks is also an important research topic.

[0004] Although deep learning has been widely used in model simplification, replacing traditional simplification algorithms with deep models still faces some challenges. Learning-based methods often struggle to control geometric errors as precisely as QEM (Qualitative Mesh Array), and lack stability in high simplification rates or complex topological scenarios. During simplification, the mesh topology continuously evolves, causing guidance information based on initial mesh predictions to gradually become ineffective, resulting in so-called topology drift. Cost functions relying solely on local geometric errors often fail to accurately distinguish between geometrically compressible redundant regions and critical features essential to the overall shape and structure, leading to premature feature edge disappearance, unstable normals, or topological degradation. Therefore, introducing learning-driven structure awareness while maintaining geometric controllability is a key issue in achieving high-fidelity mesh simplification.

[0005] In view of this, the present invention provides a simplified method for graph neural networks using triangular meshes to solve the above-mentioned problems. Summary of the Invention

[0006] This invention proposes a graph neural network simplification method for triangular meshes. This method combines the geometric structure features extracted by deep learning with the classic quadratic error metric algorithm. The core innovation lies in proposing a GNN-guided progressive cost scaling strategy. This strategy learns the importance score of each edge and dynamically scales the cost of edge collapse during the simplification process, thereby effectively maintaining the salient geometric features and topological structure of the mesh while significantly reducing the number of faces.

[0007] To achieve the above objectives, the present invention provides the following basic solution: A simplified method for graph neural networks using triangular meshes includes the following steps: Step S1: Obtain the triangular mesh model to be simplified, and construct the triangular mesh model into a graph structure, wherein the vertices of the triangular mesh are used as nodes, and the undirected edges constructed based on the topological relationship of the triangular facets are used as graph edges; Step S2: Initialize the features of each node in the graph structure to obtain node features. The node features include at least geometric information, topological information, and Laplacian position encoding. Step S3: A multi-scale adjacency relationship is constructed based on the graph structure. The multi-scale adjacency relationship includes at least the first-order adjacency edges corresponding to the original mesh topology and the second-order adjacency edges constructed based on the second-order neighborhood. Step S4: Input the node features and the multi-scale adjacency relationship into the graph neural network model, use the graph convolutional network as the basic message to perform message passing in the first-order neighborhood and the second-order neighborhood respectively, learn the node embedding representation by aggregating neighborhood features, and predict the importance probability of each undirected edge based on this. Step S5: Construct a QEM simplification method based on progressive cost scaling, which incorporates the predicted importance of edge geometry into the simplification process to improve the protection of key features of the mesh model.

[0008] Furthermore, in step S2, the Laplacian position encoding is obtained by performing eigenvalue decomposition on the normalized graph Laplacian matrix, and the eigenvectors corresponding to the smallest non-zero eigenvalues ​​are selected as position encodings; and the eigenvectors are subjected to maximum component sign normalization to eliminate sign uncertainty in the eigenvalue decomposition.

[0009] Furthermore, the multi-scale adjacency relationship includes first-order adjacency edges and second-order adjacency edges. First-order adjacency edges correspond to the direct connection edges existing in the original mesh topology and are used to characterize the local geometry and surface continuity. The construction method of the second-order adjacency edges is as follows: for any node, search its strict second-order neighbors, and after excluding the node itself and its first-order neighbors, select a preset number of nearest nodes from the strict second-order neighbors according to the Euclidean distance to establish a connection.

[0010] Furthermore, the graph neural network model adopts a dual-branch message passing architecture, including a first-order graph convolution branch and a second-order graph convolution branch; the first-order graph convolution branch is used to aggregate first-order neighbor information, and the second-order graph convolution branch is used to aggregate second-order neighbor information; the node embedding after message passing in each layer is obtained by a linear combination and nonlinear transformation of the first-order graph convolution branch and the second-order graph convolution branch.

[0011] Furthermore, in step S4, the importance probability of each undirected edge is obtained through an edge importance prediction head. The input of the edge importance prediction head includes: the embedding vector representations of the nodes at both ends of the edge, the absolute difference term of the embedding representations of the nodes at both ends, and the explicit geometric edge attributes of the edge. The explicit geometric edge attributes include at least geometric sharpness and normalized edge length.

[0012] Furthermore, step S5 specifically includes: Step S5.1: Obtaining candidate collapsed edges and their optimal target positions after collapse. And obtain the basic geometric error of the candidate collapsed edge based on the two matrix functions; Step S5.2: Construct a progressive cost scaling mechanism, first define the scaling factor as the simplification progresses. The monotonically increasing continuous function is used as the basis for constructing a normalized simplified progress factor and an initial protection threshold. Finally, the edge collapse cost is obtained by predicting the probability and the continuous function.

[0013] Furthermore, in step S5.1, geometric and topological constraints are integrated. If the collapse causes the change in the angle between the normals of adjacent surfaces to exceed a threshold, the edge collapse operation is prohibited.

[0014] Furthermore, a composite objective loss function consisting of structural representation constraints, geometric guidance constraints, and spatial regularization terms is constructed to achieve self-supervised and weakly supervised collaborative training without manual annotation. The loss includes structural contrast loss, geometric perception constraint loss, and local smoothing constraint loss, among which the geometric perception constraint loss includes geometric hinge constraints and pairwise ordering constraints.

[0015] Furthermore, for structural contrastive loss, an interval-based contrastive learning objective is introduced to impose self-supervised constraints on node embedding, enabling the learning to have topology-aware capabilities.

[0016] Furthermore, for geometrically perceptive constraint loss, a unified geometric constraint objective is constructed. By combining absolute constraints and relative ranking constraints, explicit geometric priors are transformed into learnable supervision signals. For locally smooth constraint loss, local prediction noise is suppressed and the continuity of topological evolution is guaranteed.

[0017] The principle and effect of this solution are as follows: 1. Compared with existing technologies, this invention uses a graph neural network model as a structural importance evaluator to predict the geometric importance of edges in a mesh. This information is injected into the classical QEM simplification process in the form of soft modulation. At the same time, a progressive cost scaling strategy is designed to dynamically adjust the edge collapse cost without destroying the geometric optimality of QEM. This makes the simplification process more inclined to protect key geometry and topology in the early stage, and gradually return to the goal of minimizing global error in the later stage. This invention can meet the requirements of multi-resolution display and progressive transmission of high-resolution meshes. The progressive mesh represents the high-resolution triangular mesh model as a base mesh with a very low resolution and a set of continuous progressive records, improving the triangular mesh representation. It is suitable for multi-resolution display and multi-level detail display of models and can be used in applications such as 3D games and real-world maps.

[0018] 2. Compared with existing technologies, this invention proposes a GNN-guided QEM hybrid simplification framework, which organically combines the edge structure importance learned by graph neural networks with the classical quadratic error metric (QEM) algorithm. By predicting the importance score of the edges and injecting it into the simplified process in the form of soft modulation, it significantly improves the protection effect of key features of the mesh model and realizes an interpretable fusion between deep learning methods and classical geometric optimization. At the same time, a composite structure representation network based on spectral geometry and dual-branch architecture is constructed, and Laplace position encoding is introduced to capture global topological information. This invention designs first-order and second-order dual-branch message passing architectures, thereby improving the ability to express complex geometric structures.

[0019] 3. Compared with the prior art, the present invention further designs a progressive cost scaling and phased inference collaboration mechanism, which effectively alleviates the topology drift problem in the simplification process and achieves a balance between computational efficiency and prediction accuracy.

[0020] 4. Compared with existing technologies, the method of this invention effectively alleviates the resource allocation limitations of classic QEM in complex geometric scenes. Especially in models with rich high-frequency details, the proposed method achieves a more stable and controllable trade-off between geometric accuracy, normal consistency and feature preservation, providing a practical solution for the deep integration of learning methods and traditional geometric simplification algorithms. Attached Figure Description

[0021] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0022] Figure 1This paper illustrates a grid simplification framework diagram of a graph neural network simplification method for triangular grids proposed in an embodiment of this application. Figure 2 The diagram illustrates the composition and multi-scale graph of the node feature vectors in a graph neural network simplification method for triangular meshes proposed in this application embodiment. Figure 3 A schematic diagram of the GNN network structure of a graph neural network simplification method for triangular meshes proposed in an embodiment of this application is shown. Figure 4 The diagram illustrates a QEM simplification mechanism based on progressive cost scaling for a triangular mesh graph neural network simplification method proposed in this application, wherein (a) is a graph showing the change in simplification progress and (b) is a trend graph of the continuous penalty function. Figure 5 The diagram shows the error field of the mesh simplification result of a graph neural network simplification method for triangular meshes proposed in this application, wherein (a) is the error field of the mesh simplification result of the cat0 model, and (b) is the error field of the mesh simplification result of the dog3 model. Figure 6 A heatmap comparing the Win-rate of a triangular mesh graph neural network simplification method proposed in this application with that of QEM / FQMS is shown, where (a) is the Win-rate comparison map of QEM and (b) is the Win-rate comparison map of FQMS. Figure 7 A flowchart of a simplified graph neural network method for triangular meshes proposed in an embodiment of this application is shown. Detailed Implementation

[0023] To further illustrate the technical means and effects of the present invention in achieving its intended purpose, the following detailed description of the specific implementation methods, structures, features, and effects of the present invention, in conjunction with the accompanying drawings and preferred embodiments, is provided below.

[0024] Implementation, for example Figures 1-7 As shown: A method for simplifying a triangular mesh into a graph neural network includes the following steps: Step S1: Obtain the triangular mesh model to be simplified and construct the triangular mesh model into a graph structure, wherein the vertices of the triangular mesh are used as nodes and the undirected edges constructed based on the topological relationship of the triangular facets are used as graph edges.

[0025] Please refer to the details. Figure 1 , input 3D triangular mesh Model as a graph structure ,in For the set of vertices, For triangular facets In step S2, the undirected edge set is constructed. Step S2: the features of each node in the graph structure are initialized to obtain node features. The node features include at least geometric information, topological information and Laplacian position encoding.

[0026] In detail: To capture local geometric details and global topological context, each vertex It is initialized as a high-dimensional feature vector , Representing the input feature dimension, this vector is composed of the following three parts: Geometric information: three-dimensional coordinates of the vertices. And vertex normal vectors based on the summation and normalization of adjacent face normal vectors Used to explicitly describe surface morphology; topological information: includes normalized vertex degrees and a binarized boundary indicator. , used to distinguish between points inside a manifold and points on open boundaries; Laplace position encoding: Standard message passing neural networks mainly rely on local neighborhood information for feature aggregation under the constraint of node permutation invariance. However, limited by the theoretical upper limit of the 1-Weisfeiler-Lehman (1-WL) test, this framework is not sensitive to the absolute position of nodes and has difficulty distinguishing symmetric structures with local isomorphic features.

[0027] To overcome the aforementioned expression bottleneck, this method introduces a positional encoding based on spectral geometry for the processing of the nodes, and uses the eigenvalue decomposition of the graph Laplacian operator to capture global topological information.

[0028] Specifically, the Laplacian positional encoding is obtained by performing eigenvalue decomposition on the normalized graph Laplacian matrix, and selecting the value corresponding to the smallest value. The feature vectors with non-zero eigenvalues ​​are used as position codes. The feature vectors are then subjected to sign normalization. These low-frequency feature vectors characterize the smooth geometric modes of the grid surface and provide a long-distance global position reference.

[0029] Furthermore, to address the inherent sign uncertainty in eigenvalue decomposition, this method employs a maximum component sign normalization strategy, forcibly aligning the direction of the component with the largest absolute value, thereby ensuring the determinism and consistency of the input features. This normalized graph Laplacian matrix serves as the initial node embedding for the graph neural network. The normalized graph Laplacian matrix is ​​as follows: Where A is the adjacency matrix, D is the degree matrix, and I is the identity matrix.

[0030] To facilitate overall computation in the graph neural network, for each vertex in the grid, a corresponding feature vector is first constructed based on its geometric properties, topological descriptor, and Laplacian position code. Then, all feature vectors are stacked row by row in vertex index order to form the overall input feature matrix. Each row of this matrix corresponds to the feature representation of a vertex, and each column corresponds to a feature dimension, thus mapping the discrete grid structure to a unified tensor representation space, where H... (0) The initial node embedding of the graph neural network is directly assigned by the input feature matrix and serves as the input for subsequent multi-layer message passing. This definition conforms to the input format of a standard message passing neural network. In subsequent layers, it is gradually updated to a higher-level structural representation through neighborhood aggregation. ; Where V is the set of vertices. Represents the total number of vertices. Let be the input feature vector of vertex vi, and Din be the input feature dimension. It is the input feature matrix composed of all vertices. Indicates the first Layer node embedding matrix.

[0031] As mentioned above, limited by the theoretical upper bound of the 1-Weisfeiler-Lehman (1-WL) test, in order to overcome the limitation that single-layer graph convolution can only aggregate first-order neighborhood information, this application constructs multi-scale adjacency relationships based on the graph structure and defines two types of edge indices: first-order adjacency edges ( ): Corresponds to the direct connecting edges existing in the original mesh topology, i.e., 1-hop adjacency relationships, used to characterize local geometry and surface continuity; 2-hop adjacency edges ( This invention introduces sparse connectivity based on 2-hop neighborhood construction, for each vertex Search for its strict second-order neighbors, i.e., exclude itself and its 1-hop neighbors, and select the nearest neighbors according to Euclidean distance. The nodes are connected. Based on experience, when K is 12, the network is allowed to cross local small undulations when aggregating features, and can capture a wider range of geometric patterns.

[0032] like Figure 2 As shown, the three components of a node feature vector and the construction method of a multi-scale graph are described in [reference needed]. Figure 2 Specifically: first-order adjacent edges are represented by solid lines, and second-order adjacent edges are represented by dashed lines.

[0033] The node features and the multi-scale adjacency relationship are input into the graph neural network model (GCN). A graph convolutional network is used as the basic message to pass messages in the first-order and second-order neighborhoods respectively. The node embedding representation is learned by aggregating neighborhood features, and the importance probability of each undirected edge is predicted based on this.

[0034] Please refer to Figure 3 , Figure 3 This is a schematic diagram of a GNN network structure. The left image shows a dual-branch GCN layer, used to aggregate features from 1-hop and 2-hop neighbors respectively. The right image shows an edge importance prediction head, which embeds and combines nodes to predict the geometric importance of edges. Since traditional graph convolutional networks cannot effectively learn the feature mixing relationships between neighbors of different orders, this invention designs a dual-branch message-passing architecture. This architecture significantly enhances the graph neural network model's ability to represent complex mesh geometries. This dual-branch message-passing architecture consists of… The system consists of graph convolutional layers, which process graph topologies of different scales in parallel at each layer.

[0035] The dual-branch message passing architecture includes a first-order graph convolution branch and a second-order graph convolution branch. Specifically: the first-order graph convolution branch ( : Aggregates first-order neighbor information, focusing on capturing local geometric features of the manifold surface, and uses second-order graph convolution branches ( ): Aggregates information on second-order topological neighborhoods, enabling the model to capture larger-scale geometric patterns and topological associations.

[0036] The node feature update process is achieved through linear combination and nonlinear transformation of multi-scale features. The transformation includes layer normalization, activation function and Dropout regularization.

[0037] ; in, Representative after the first Node embedding representation after layer message passing Indicates the first The feature dimensions of the layer It is a non-linear activation function, specifically the ReLU function, with hyperparameters... This is used to balance the relative contributions of first-order neighborhood information and second-order neighborhood context information in the feature fusion process. and They share the standard graph convolution mathematical framework, but each has its own independent weight matrix, which is responsible for extracting spatial features of different ranges, forming a complementary dual-branch message passing architecture.

[0038] In step S4, the importance probability of each undirected edge is obtained through the edge importance prediction head. First, as mentioned above, after... The final node embedding matrix is ​​obtained after layer message passing. , of which The embedding vector of each node is denoted as . , corresponding to matrix The Line; for any undirected edge Construct its edge feature vector The feature vector The edge consists of the embedding vectors of the nodes at both ends, the absolute difference term represented by the embeddings of the nodes at both ends, and the explicit geometric edge properties. The edges are cascaded together and then predicted using a two-layer perceptron with a hidden layer to obtain the predicted probabilities. Among them, the two-layer perceptron is a feedforward neural network structure used to perform nonlinear mapping processing on input features. The structure includes two fully connected transformation units connected in sequence. The hidden layer corresponds to the first layer, where the input features are linearly transformed and mapped through a nonlinear activation function to obtain an intermediate feature representation. The second layer further performs a linear transformation based on the intermediate feature representation and outputs the target result.

[0039] Regarding edge feature vectors : .

[0040] Regarding the probability of prediction : .

[0041] in, It consists of geometric sharpness and normalized edge length. Geometric sharpness is calculated from normalized dihedral angles, and the topological boundary edges are explicitly enhanced and their maximum values ​​are taken, thus characterizing the geometric saliency of internal sharp edges and open boundaries on a unified numerical scale. This represents the embedding vector of node u. The embedding vector of node v, and the absolute difference term. This enables the model to sensitively detect drastic changes in node embedding when crossing feature boundaries such as sharp edges, thereby giving higher retention weights to geometrically salient regions. Φ() represents the two-layer perceptron mapping function.

[0042] Step S5 specifically includes: Step S5.1: Obtaining candidate collapsed edges and their optimal target positions after collapse. And obtain the basic geometric error of the candidate collapsed edge based on the two matrix functions; Step S5.2: Construct a progressive cost scaling mechanism, first define the scaling factor as the simplification progresses. Monotonically increasing continuous function Secondly, a normalized simplified progress factor and initial protection threshold are constructed, and the prediction probability is used in conjunction with a continuous function. This results in the final cost of edge collapse.

[0043] Specifically, for candidate collapsed edges and its optimal target position after collapse Basic geometric error C QEM Determined by the cumulative quadratic matrix Q: ;in, and These represent the quadratic error matrices obtained by accumulating the plane equations of adjacent triangular faces at the corresponding vertices.

[0044] Furthermore, to prevent degenerate meshes from forming during the simplification process, this invention integrates geometric and topological constraints. If the collapse causes the angle between the normals of adjacent surfaces to change beyond a threshold... If the operation is prohibited, it will be penalized or prohibited; if it results in the generation of extremely narrow triangles (low aspect ratio), it will be prohibited; edge collapse operations are only allowed on 2-manifold edges to preserve the topology.

[0045] To adjust the change in feature protection strength with the simplification progress, this invention proposes a dynamic cost modulation strategy. This strategy introduces a relaxation threshold mechanism that dynamically evolves with the simplification progress, thereby balancing geometric accuracy and simplification rate during topology evolution.

[0046] Figure 4 This is a schematic diagram of a QEM simplification mechanism based on progressive cost scaling. First, we define the scaling mechanism as the simplification progresses. Monotonically increasing continuous function This function is used to assess the importance of geometry in the cost of soft-modulated edge collapse. Essentially, it constructs a progressive constraint release mechanism, its idea stemming from the continuation method and annealing optimization strategy. Changes with the progress of simplification, such as Figure 4 As shown in (a), strong constraints are maintained on edges with high geometric saliency in the early stage of simplification, while feature protection requirements are gradually relaxed in the later stage of simplification. This ensures the reachability of the target number of faces and the stability of global error control without compromising the geometric optimality of QEM.

[0047] Continuous functions : ,in, The normalized simplified progress factor (N is the current number of patches). As the initial protection threshold, in this application, It is 0.2. To control the power exponent of the relaxation rate ( ).

[0048] The final cost of edge collapse Perform progressive scaling as follows: Edge collapse cost ;in, The continuously scaling penalty term is defined by incorporating the geometrical importance prediction probability of the GNN. Compared with the current threshold Continuous penalty function The trend is as follows Figure 4 As shown in (b), when the importance of an edge is higher than the current dynamic threshold, its collapse cost will be amplified, thereby reducing the collapse priority of the edge in the priority queue, and thus suppressing important geometric features from being prematurely collapsed in the early stages of simplification, resulting in:

[0049] in, To ensure the stability of the denominator value, This is a penalty intensity coefficient used to enhance the priority protection of highly important edges in the priority queue.

[0050] Regarding the loss, this invention constructs a composite objective loss function consisting of structural representation constraints, geometric guidance constraints, and a spatial regularization term. It enables collaborative training of self-supervised and weakly supervised methods without the need for manual annotation.

[0051] The losses include structural contrast loss, geometric perception constraint loss, and local smoothing constraint loss. Among them, geometric perception constraint loss includes geometric hinge constraints and pairwise ordering constraints.

[0052] Regarding structural contrast loss To learn potential representations with topology awareness, this invention introduces an interval-based contrastive learning objective to self-supervise node embeddings. This loss encourages topologically adjacent nodes to maintain high similarity in the embedding space while suppressing excessively high similarity between non-adjacent nodes, thereby constructing a structured feature representation space.

[0053] ;

[0054] in, This indicates that nodes u and v are connected by an edge. This indicates that nodes u and v are not connected by an edge. For cosine similarity operators, This represents the vertex embeddings learned by the GNN. The similarity interval parameter (as set in this invention) This objective encourages the explicit encoding of mesh adjacency relationships in the embedded space and makes the feature representations of disconnected nodes tend to be orthogonal, providing a stable structural representation for subsequent geometric importance prediction.

[0055] Regarding geometrically perceptual constraint loss To transform explicit geometric priors into learnable supervisory signals, this invention constructs a unified geometric constraint objective. By combining absolute constraints and relative ranking constraints, it guides the network to learn an edge importance distribution that conforms to geometric consistency, i.e.:

[0056] .

[0057] To balance the weighting coefficients of the two terms, Geometric hinge constraints obtained through neural network learning For a set of sharp feature edges defined by a high dihedral angle threshold or boundary conditions Apply hinge constraints to ensure that its importance score is not lower than the empirical threshold. ,have to: ; in, The minimum importance empirical threshold, Set it to 0.6.

[0058] Regarding pairwise ordering constraints To reinforce the importance difference between feature regions and flat regions, a pairwise ranking constraint is introduced, which requires ranking from the set of sharp features. Set with flat regions Extract edge pairs And ensure that their predictive importance scores differ by at least one significant interval. : ; The sorting interval parameter is set to =0.1, This represents a smooth geometric region automatically identified by a low dihedral angle threshold. This constraint is explicitly injected into the geometric prior in the form of a relative ranking, prompting the network to learn that the importance score of sharp edges is significantly higher than that of smooth edges, thus prioritizing the protection of significant geometric structures during the simplification process.

[0059] Regarding the local smoothing constraint loss To suppress local prediction noise and ensure the continuity of topological evolution, this application introduces a local smoothing constraint loss. This loss requires edges sharing the same vertex to have similar importance scores, thereby avoiding disordered collapse sequences in local regions. ;in, Represents the vertex The average importance of all connected edges. The value represents the average importance of all edges connected to vertex v. This regularization term improves the robustness of the simplification process in terms of topological evolution by penalizing discontinuous changes in local scores.

[0060] Based on the above, we obtain the total loss function, which is the composite objective loss function. The final training objective is defined as a linearly weighted combination of the loss terms: ;in, The weighting coefficients for each loss are used to achieve a controllable trade-off between structural consistency, geometric preservation, and prediction smoothness.

[0061] The specific implementation is as follows: The present invention uses the benchmark TOSCA dataset containing 80 high-resolution grids to train the proposed method, and adopts QEM and FQMS as baseline methods.

[0062] QEM: QEM serves as a theoretical benchmark for minimizing geometric errors. Based on the standard implementation provided by Open3D, it explicitly preserves the original mesh boundaries. Therefore, it will produce non-zero values ​​in TOSCA models containing open boundaries, such as cat (cat0) and dog (dog3) models. This phenomenon reflects the topological characteristics of the data itself, rather than algorithmic degradation. For models like the Centaur watertight model, this implementation correctly maintains topological integrity and achieves 0%. Compared to implementations that partially ignore boundaries or implicitly fill holes, Open3D's strategy provides a more conservative and consistent evaluation standard.

[0063] Among them, FQMS: This invention selects FQMS as the baseline method of engineering-level feature-first strategy to compare the performance differences between learning-based methods and engineering-level simplification strategies. FQMS is based on the QEM framework and introduces heuristic boundary penalties and topological compactness constraints at the implementation level, making it more inclined to preserve visual contours and boundary structures under high simplification rates. This method usually performs better in feature preservation metrics, but may sacrifice some geometric positional accuracy. Therefore, including FQMS in the comparison helps to objectively evaluate the trade-off between geometric accuracy and visual feature preservation of different methods. The results are shown in Table 1 below:

[0064]

[0065] Table 1 shows the performance comparison results (Ratio) of various methods on the TOSCA dataset.

[0066] To comprehensively evaluate the geometry preservation capability of the proposed method in mesh simplification tasks, this invention compares the performance of three methods—QEM, FQMS, and the proposed method—on multiple geometric error metrics under different simplification rates.

[0067] Table 1 shows the results of each method under different simplification rates (0.05, 0.1, 0.2, 0.5, and 0.7). , , and The statistical results for the indicators are listed, including the mean and median of each indicator. The overall results show that as the simplification rate gradually increases, the effectiveness of each method improves. , and All error indicators showed a gradual downward trend.

[0068] The index is used to measure the sealing performance of the mesh model; a higher value indicates a higher proportion of incompletely sealed edges. In low simplification scenarios (Ratio=0.05, 0.1), the proposed method... The mean values ​​were 0.3699 and 0.2621, respectively, which were higher than those of QEM (0.1232, 0.1094) and FQMS (0.1686, 0.1500). However, as the simplification rate increased to 0.5 and 0.7, the differences between the methods gradually decreased, indicating that under high retention rate conditions, the effect of simplification on mesh sealing tended to be consistent.

[0069] The metric measures the geometric deviation between the original and simplified models. The proposed method performs close to QEM overall across all simplification rates and outperforms FQMS. For example, at a very low simplification rate (0.05), the proposed method... The mean value is 2.4662, slightly lower than QEM (2.4773) and FQMS (2.7078). In the medium simplification range, the differences among the three methods further converge, but the proposed method still maintains a low error, indicating that it can achieve a level comparable to or even slightly better than the classical QEM method in terms of overall geometry preservation.

[0070] The index is used to reflect the consistency of the normal direction. The proposed method shows a trend of being comparable to QEM and generally superior to FQMS under different simplification rates. Taking a simplification rate of 0.05 as an example, its... The mean value is 0.1724, lower than QEM (0.1746) and FQMS (0.1878); this advantage is maintained at simplification rates of 0.1 and 0.2, indicating that the proposed method can better preserve the normal information of the model surface during the simplification process, thereby reducing surface orientation distortion caused by the simplification operation. The indicators are used to assess the retention of the overall structure.

[0071] The methods show little difference in this metric; for example, at a simplification rate of 0.1, The mean values ​​are all approximately 0.273. Under high simplification rates (0.5–0.7), the performance of each method remains similar. This shows that the proposed method improves the local geometry preservation capability without significantly affecting the global mesh structure, thus achieving balanced structural stability.

[0072] In summary, under different simplification rates, this method demonstrates stability on indices reflecting local geometry and normal consistency, such as PCD and PNE, and outperforms FQMS in most cases, while maintaining an error level comparable to the classical QEM method; simultaneously, in In terms of indicators reflecting the overall structural characteristics, the proposed method shows little difference from the baseline method, indicating that the proposed method improves the local geometry preservation capability without compromising the overall structural stability of the model, thus achieving a relatively balanced mesh simplification effect.

[0073] To further analyze the spatial error distribution characteristics of different simplification methods, this invention visualizes the error field of the results at a simplification rate of 0.1, as shown below. Figure 5 As shown, (a) Figure: Visualization of the error field of the mesh simplification result of the cat0 model at a simplification ratio of 0.1; (b) Figure: Visualization of the error field of the mesh simplification result of the dog3 model at a simplification ratio of 0.1. Figure 5 The first row displays the geometric distance error, and the second row displays the normal error. The color gradually transitions from white to dark purple, indicating the change in error from low to high.

[0074] From the perspective of the spatial structure of error distribution, there are obvious differences between different methods. FQMS produces relatively concentrated high error regions in some high curvature areas. For example, in the dog3 model, near the limb joints and facial structure, many dark purple areas can be observed, indicating that the local geometric deviation is relatively obvious. This error concentration phenomenon is more obvious in both the distance error and normal error maps. In contrast, the QEM method has a more uniform overall error distribution, but there is still some local error accumulation in some detailed areas.

[0075] The method of this invention, while maintaining an overall error level essentially consistent with QEM, further reduces the area of ​​high-error regions, resulting in a smoother and more continuous error distribution. For example, in the dog3 model, the method of this invention will... The FQMS value decreased from 0.461 to 0.354, and a significant reduction in high-error regions can be observed in the error plot. Figure 5 It can be seen that, while maintaining the overall simplification quality, this method can distribute geometric errors more evenly, thereby reducing the phenomenon of error concentration in local areas. This indicates that the proposed geometry-aware strategy can more effectively preserve the geometric features of key structural regions during the mesh simplification process.

[0076] To quantitatively evaluate the performance advantages of the proposed method, this invention calculates the win-rate of the proposed method relative to two baseline methods, QEM and FQMS, under different simplification rates and various geometric error indices.

[0077] Regarding Win-rate: The proportion of times the proposed method achieves a smaller error on a certain metric across all tested models. This metric directly reflects the overall advantage of the proposed method over the baseline method.

[0078] like Figure 6 As shown, Figure 6 This is a comparative heatmap, specifically a heatmap comparing the Win-rate (%) of this method with that of QEM or FQMS. Figure 6 Figure (a) shows the win-rate comparison between this method and QEM; Figure 6 Figure (b) is a comparison of the Win-rate of the proposed method and FQMS.

[0079] heat Figure 6 The paper demonstrates the superiority of the proposed method over the baseline method under different simplification rates and geometric error metrics. A value higher than 50% indicates that the proposed method outperforms the corresponding baseline method on that metric. Figure 6 As can be seen, under most simplification rates and evaluation metrics, this method achieved a high win rate, particularly in... and The advantages are particularly evident in two metrics: at lower simplification rates (0.05–0.2), the win-rate compared to QEM is significantly higher. and The indicators generally exceed 80%, indicating that this method can effectively maintain the consistency of local geometry and normals.

[0080] As the simplification rate increases to 0.5 and 0.7, the model's geometric loss becomes more severe, but this method still maintains a high win-rate on several metrics, such as... and In terms of metrics, it still shows a significant advantage over FQMS, indicating that the proposed method can effectively maintain the geometric features of the model even at a high compression ratio.

[0081] The proposed method achieves a high average win-rate across all evaluation metrics. Compared to QEM, the average win-rate for most metrics is around 70% or higher; while in comparison with FQMS, The win rate of the indicator is close to 100%, indicating that the proposed method has a significant advantage on this indicator. Overall, the proposed method performs relatively well across different evaluation indicators, demonstrating good comprehensive performance.

[0082] The global Win-rate results show that the proposed method... and The advantages are most significant in terms of metrics, while in PWA and The performance advantage in terms of metrics is relatively mild, a phenomenon consistent with the design goal of this method. That is, this method predicts edge importance through graph neural networks and applies dynamic penalties to highly important edges during QEM simplification, thereby prioritizing the protection of geometrically significant structures, such as sharp edges and high-curvature regions. and This primarily reflects the preservation of local geometry and normal consistency; therefore, this strategy can significantly improve the performance of these metrics. In contrast, More emphasis is placed on maintaining the overall area distribution, while These metrics, reflecting the global structural characteristics of the mesh, rely more on global geometric distribution than on local feature preservation. Therefore, our method has a relatively greater advantage in these metrics. However, experimental results show that, compared to QEM, and The average win-rates still reached 76.3% and 78.1% respectively, indicating that the proposed method did not have a significant negative impact on the overall structural characteristics while maintaining the local geometric structure.

[0083] The results show that the proposed method not only outperforms the classical QEM simplification method in terms of local geometry preservation, but also shows a significant advantage over FQMS. This method can effectively maintain the geometric features of the model while ensuring a high simplification rate, thereby achieving high-quality mesh simplification.

[0084] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A simplified method for graph neural networks using triangular meshes, characterized in that, Includes the following steps: Step S1: Obtain the triangular mesh model to be simplified, and construct the triangular mesh model into a graph structure, wherein the vertices of the triangular mesh are used as nodes, and the undirected edges constructed based on the topological relationship of the triangular facets are used as graph edges; Step S2: Initialize the features of each node in the graph structure to obtain node features. The node features include at least geometric information, topological information, and Laplacian position encoding. Step S3: A multi-scale adjacency relationship is constructed based on the graph structure. The multi-scale adjacency relationship includes at least the first-order adjacency edges corresponding to the original mesh topology and the second-order adjacency edges constructed based on the second-order neighborhood. Step S4: Input the node features and the multi-scale adjacency relationship into the graph neural network model, use the graph convolutional network as the basic message to perform message passing in the first-order neighborhood and the second-order neighborhood respectively, learn the node embedding representation by aggregating neighborhood features, and predict the importance probability of each undirected edge based on this. Step S5: Construct a QEM simplification method based on progressive cost scaling, which incorporates the predicted importance of edge geometry into the simplification process to improve the protection of key features of the mesh model.

2. The method for simplifying a triangular mesh graph neural network according to claim 1, characterized in that, In step S2, the Laplacian position encoding is obtained by performing eigenvalue decomposition on the normalized graph Laplacian matrix, and the eigenvectors corresponding to the smallest non-zero eigenvalues ​​are selected as position encodings; and the eigenvectors are subjected to maximum component sign normalization to eliminate sign uncertainty in the eigenvalue decomposition.

3. The method for simplifying a triangular mesh graph neural network according to claim 2, characterized in that, The multi-scale adjacency relationship includes first-order adjacency edges and second-order adjacency edges. First-order adjacency edges correspond to the direct connection edges existing in the original mesh topology and are used to characterize the local geometry and surface continuity. The construction method of the second-order adjacency edges is as follows: for any node, search its strict second-order neighbors, and after excluding the node itself and its first-order neighbors, select a preset number of nearest nodes from the strict second-order neighbors according to the Euclidean distance to establish a connection.

4. The method for simplifying a triangular mesh graph neural network according to claim 3, characterized in that, The graph neural network model adopts a dual-branch message passing architecture, including a first-order graph convolution branch and a second-order graph convolution branch. The first-order graph convolution branch is used to aggregate first-order neighbor information, and the second-order graph convolution branch is used to aggregate second-order neighbor information. The node embedding after message passing in each layer is obtained by a linear combination and nonlinear transformation of the first-order graph convolution branch and the second-order graph convolution branch.

5. The method for simplifying a triangular mesh graph neural network according to claim 4, characterized in that, In step S4, the importance probability of each undirected edge is obtained through an edge importance prediction head. The input of the edge importance prediction head includes: the embedding vector representations of the nodes at both ends of the edge, the absolute difference term of the embedding representations of the nodes at both ends, and the explicit geometric edge attributes of the edge. The explicit geometric edge attributes include at least geometric sharpness and normalized edge length.

6. The method for simplifying a triangular mesh graph neural network according to claim 5, characterized in that, Step S5 specifically includes: Step S5.1: Obtain candidate collapsed edges and their optimal target positions after collapse. And based on the two matrix functions, the basic geometric error of the candidate collapsed edge is obtained; Step S5.2: Construct a progressive cost scaling mechanism. First, define the scaling mechanism as the simplification progresses. The monotonically increasing continuous function is used as the basis for constructing a normalized simplified progress factor and an initial protection threshold. Finally, the edge collapse cost is obtained by predicting the probability and the continuous function.

7. The method for simplifying a triangular mesh graph neural network according to claim 6, characterized in that, In step S5.1, geometric and topological constraints are integrated. If the collapse causes the change in the angle between the normals of adjacent surfaces to exceed a threshold, the edge collapse operation is prohibited.

8. The method for simplifying a triangular mesh graph neural network according to claim 6, characterized in that, A composite objective loss function consisting of structural representation constraints, geometric guidance constraints, and spatial regularization terms is constructed to achieve self-supervised and weakly supervised collaborative training without manual annotation. The loss includes structural contrast loss, geometric perception constraint loss, and local smoothing constraint loss, among which the geometric perception constraint loss includes geometric hinge constraints and pairwise ordering constraints.

9. The method for simplifying a triangular mesh graph neural network according to claim 8, characterized in that, For structural contrastive loss, an interval-based contrastive learning objective is introduced to impose self-supervised constraints on node embedding, enabling the learning to have topology-aware capabilities.

10. The method for simplifying a triangular mesh graph neural network according to claim 8, characterized in that, For geometrically perceptive constraint loss, a unified geometric constraint objective is constructed. By combining absolute constraints and relative ranking constraints, explicit geometric priors are transformed into learnable supervision signals. For locally smooth constraint loss, local prediction noise is suppressed and the continuity of topological evolution is guaranteed.

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