Open Access

ARTICLE

# Fast Mesh Reconstruction from Single View Based on GCN and Topology Modification

1 Wuxi Research Institute, Nanjing University of Information Science & Technology, Wuxi, 214100, China

2 Engineering Research Center of Digital Forensics, Ministry of Education, Jiangsu Engineering Center of Network Monitoring, School of Computer and Software, Nanjing University of Information Science & Technology, Nanjing, 210044, China

3 Jiangsu Collaborative Innovation Center of Atmospheric Environment and Equipment Technology (CICAEET), Nanjing University of Information Science & Technology, Nanjing, 210044, China

4 School of Automation, Nanjing University of Information Science & Technology, Nanjing, 210044, China

5 Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON, N9B 3P4, Canada

* Corresponding Author: Xiaorui Zhang. Email:

*Computer Systems Science and Engineering* **2023**, *45*(2), 1695-1709. https://doi.org/10.32604/csse.2023.031506

**Received** 19 April 2022; **Accepted** 16 June 2022; **Issue published** 03 November 2022

## Abstract

3D reconstruction based on single view aims to reconstruct the entire 3D shape of an object from one perspective. When existing methods reconstruct the mesh surface of complex objects, the surface details are difficult to predict and the reconstruction visual effect is poor because the mesh representation is not easily integrated into the deep learning framework; the 3D topology is easily limited by predefined templates and inflexible, and unnecessary mesh self-intersections and connections will be generated when reconstructing complex topology, thus destroying the surface details; the training of the reconstruction network is limited by the large amount of information attached to the mesh vertices, and the training time of the reconstructed network is too long. In this paper, we propose a method for fast mesh reconstruction from single view based on Graph Convolutional Network (GCN) and topology modification. We use GCN to ensure the generation of high-quality mesh surfaces and use topology modification to improve the flexibility of the topology. Meanwhile, a feature fusion method is proposed to make full use of the features of each stage of the image hierarchically. We use 3D open dataset ShapeNet to train our network and add a new weight parameter to speed up the training process. Extensive experiments demonstrate that our method can not only reconstruct object meshes on complex topological surfaces, but also has better qualitative and quantitative results.## Keywords

Image-based 3D reconstruction is the process of recovering 3D information from 2D image, with the aim of obtaining a 3D model that matches the 2D image. The advantages of single-view-based reconstruction methods are that they require less input data and usually only need to reconstruct the image of a single perspective of the object as input, so as to restore the whole shape of the object. With the emergence of large-scale 3D data sets such as ShapeNet [1] and Pix3D [2], single-view reconstruction methods integrating deep learning have gradually become the main trend in recent years.

Early single-view 3D reconstruction based on learning represented 3D structures as voxels [3–6] or point clouds [7–11]. Voxels lack much detail due to the limitation of their resolution and expression. Each point in the point cloud is not connected, so it will lack the surface information of the object. In comparison, the representation method of mesh [12–16] has the advantages of light weight and rich shape details. Recent advances in single-view mesh reconstruction can recover more specific object surface details. However, they still have some limitations. For example, due to the fixed connection relationship between the vertices of the initial mesh, most of the current methods only perform well in the reconstruction of objects with approximately predefined templates. When reconstructing complex topologies, unnecessary mesh self-intersections and connections are generated, thus destroying surface details; the flexibility of the mesh topology and the final reconstruction effect cannot be achieved at the same time and increasing the model accuracy can easily reduce its generalization; the mesh representation contains more information, and longer training cycles are required to achieve sufficient accuracy, and the training speed of the model is slower.

To solve the above challenges, we propose a method for fast mesh reconstruction from single view based on Graph Convolutional Network (GCN) and Topology Modification. First, we use GCN with residual connection, namely G-ResNet [13,14], to predict the position offset of the mesh vertices and the surrounding shape features, so as to generate 3D meshes with good surface details; at the same time, the meshes are processed after each deformation. After each deformation, the mesh topology is modified, the error surface is trimmed, and the original fixed structure is broken, to improve the flexibility of the topology structure. Secondly, we propose a new feature fusion method to make full use of the features of different stages of the image in a hierarchical input manner, to meet the input requirements of different modules and improve the compatibility of data structures between modules. Finally, we use 3D supervision to constrain the formation of the meshes and add a weight parameter to the corresponding loss function, so that the entire network can pay more attention to the backbone during training, thereby increasing the training speed and reducing resource consumption.

The contributions of this paper are mainly as follows:

• An end-to-end learning framework is proposed, which combines GCN and topology modification for the first time. It is used to reconstruct the 3D mesh model from a single image, considering the details of the reconstructed mesh surface and the flexibility of the mesh topology, and it can be applied to the reconstruction of complex structures with good generalization ability.

• Different from the previous method that simply vectorizes the image, we propose a new feature fusion method that uses the features of the image at different stages multiple times to meet the input requirements of different modules and make each module compatible with each other. This module can be integrated into other learning frameworks.

• A weight parameter related to the 3D loss function is proposed, which can give priority to the location of key points during training process, so as to achieve the purpose of improving the training speed. We use this to optimize our training methods and improve the stability of the network.

Human beings are good at using prior knowledge to make inferences and predictions [17]. They can infer the approximate size and approximate geometric shape of objects with just one eye. For neural networks, the same is true for reconstructing objects in a 2D image from a single perspective. This chapter will introduce in detail the work of our algorithm in related fields from three aspects: single-view-based 3D reconstruction technology, GCN and topology modification.

2.1 Single Image 3D Reconstruction

According to different representations, the single-view-based 3D reconstruction technology is mainly divided into three directions: voxel, point cloud and mesh. Voxels discretize an object into a 3D voxel grid. Its advantage is that it is easy to integrate deep learning frameworks (such as 3D convolution and max pooling). Choy et al. proposed 3D-R2N2 based on the voxel representation and used the 3DLSTM network structure to establish a mapping from 2D graphics to 3D voxel models, which completed single-view or multi-view 3D reconstruction based on voxels [3]. Deep Marching Cubes [5] is an end-to-end trainable network, which can predict an explicit surface representation of any topology. OctNet [6] recursively subdivides the 3D voxel grid into eight quadrants based on an octree, thereby reducing the computational complexity of 3D convolution. However, 3D voxel representation lacks geometric details, and the increase in accuracy requires an increase in resolution. The increase in resolution will greatly increase the calculation time and occupy expensive memory resources. Only a few technologies can achieve subvoxel accuracy.

In comparison, the point cloud is a simple, unified and easy-to-learn structure. Since the connectivity between vertices does not need to be updated, the point cloud is easier to manipulate during geometric transformation and deformation. Point Set Generation Network (PSGN) proposed by Fan et al. solves the problem of loss when training a point cloud network [7]. Li et al. introduced the Generative Adversarial Network (GAN) to deal with the disturbance problem in the point cloud generation process [8]. DeformNet proposed by Kurenkov et al. can find the most similar shape template in advance, and then generate a 3D dense point cloud through the deformed template [11]. Algorithms based on point cloud representation can process 3D objects of arbitrary topology, but the point cloud reconstruction lacks connectivity, so the surface information of the object will be lacking, and the surface will be uneven after reconstruction.

Polygon mesh is composed of vertices and triangular faces. It has the characteristics of scalability and curved surface, as well as light weight and rich shape details. The most important parts of mesh are the connections between adjacent points. N3MR [12], as the first 3D reconstruction method using mesh representation, can reconstruct some low-precision models with a small number of vertices. Pixel2Mesh is a coarse-to-fine network architecture that adds mesh vertices through the graphical pool layer to refine the mesh surface details [13]. Pan et al. proposed a topology modification network, which is characterized by its ability to prune the mesh topology in multiple stages [15]. Image2Mesh combines a rough topology map structure according to the image characteristics, and then uses the Free Form Deformation(FFD) to restore the dense 3D mesh model according to the estimated deformation [16].

2.2 Graph Convolutional Network (GCN)

In reality, many important data are stored in the form of graphs, such as social network information, knowledge graphs, protein networks, the World Wide Web, and so on. Since it is difficult for CNN to choose a fixed convolution kernel to adapt to the irregularities of the entire graph [18–20], its translation invariance is often not applicable to graph topology. As a network structure designed for graphs [21,22], GCN’s essential purpose is to extract the spatial characteristics of topological graphs. Its core idea is to use edge in formation to aggregate node information to generate new node representations. The extraction of graph features by GCN mainly relies on the adjacency matrix and Laplacian matrix [23]. Li et al. proposed an adaptive graph convolutional network (AGCN), which learns unknown hidden structural relationships through the adjacency matrix of the graph [24].

3D mesh can also be regarded as a graph topology, and there are corresponding connections between the vertices and edges of the mesh. Wang et al. proposed for the first time that GCN is applied to the grid 3D reconstruction of a single image [13], which has greatly improved the training speed and model accuracy compared with previous methods. We follow their ideas, and the purpose is to perform controllable operations on high-quality 3D representation.

However, the uncertainty in the design of the corresponding convolution parameters for the mesh is large, and the mesh representation is no suitable for conventional 3D convolution operations. Therefore, in this work, we use polygon mesh as the 3D format and introduce graph convolutional neural network (GCN) to control this structure and solve the problem of incompatibility between mesh representation and neural network.

The graph topology is controlled by the connection of vertices and edges, so a structure containing a large number of vertices like a 3D mesh is not easy to update, and it consumes more resources when performing convolution operations. Topology modification, as a technology to update the topology in real time, has been used in the design of broadband antenna structures [25]. The most common method for designing compact broadband antennas is to make various topological modifications to the antenna radiator, feeder, or ground plane, and reduce the area occupied by the antenna by finding a balance point in structural flexibility and material adaptability.

Pan et al. applied topology modification to the 3D reconstruction of the mesh, which solved the problem that the deformation of the mesh is limited by the predefined shape template and can adapt to the reconstruction of the surface of complex objects [15]. We introduce topology modification to process the output results of GCN, trims the surface predicted by GCN that produces large errors, and strikes a balance between the high-quality mesh surface and the flexibility of the topology.

As an end-to-end network structure, when we are given an input image, the system outputs a 3D mesh model. An overview of the framework of this article is shown in Fig. 1. The entire network architecture is based on the “Encoder-Decoder” structure. The “encoder” is the image coding layer used to extract the features of the input image, while the “decoder” consists of two identical subnets. Each subnet contains a mesh deform block, a topology modification module and a boundary optimization module.

The encoding layer is used to extract 2D image features hierarchically, convert the input image into feature maps and feature vectors, and input them to the deform block and the topology modification module, respectively. The deform block modifies the predefined sphere mesh. By manipulating the feature vector attached to the vertices, the vertices can be deformed, so that the sphere mesh gradually tends to the object described by the input image. The topology modification module dynamically trims the mesh surface after each deformation, so that the mesh topology is no longer limited to a predefined template. After each deformation of the vertices and modification of the topology, we use a boundary optimization loss function to trim the zigzag boundary and smooth the model surface. In order to make the network produce stable deformation and generate accurate meshes, we combine the commonly used 3D loss function with boundary refinement loss to train our network.

Next, we will introduce the encoding layer, deform block, topology modification module and the 3D loss function used one by one.

The encoding layer uses VGG-19 as the main architecture of the network. First, input the image and extract it into feature maps of different layers and 1000-dimensional feature vectors, as shown in Fig. 2. Due to the use of the same size convolution kernel, the VGG-19 architecture has a small number of hyperparameters, which is simple and convenient in image encoding. Compared with the more commonly used two-dimensional image feature network VGG-16, VGG-19 has three additional convolutional layers, which can extract some deeper image features [26].

The whole process of image encoding can be reused. On the one hand, the three layers of feature maps, conv3_4, conv4_4 and conv5_4, are stitched together in series, and given any vertex in the 3D grid, its projection point on the input image is found according to the camera parameters, and the bilinear difference method finds and fuses the four pixels adjacent to the corresponding point of that vertex on the feature map as the feature vector for manipulating the deformation of that vertex in the grid deformation module. On the other hand, the 1024-dimensional vector formed by the whole VGG-19 network is an important benchmark to guide the topology modification. In this way, one feature extraction of the image satisfies the input requirements of both the mesh deformation and topology modification modules.

Define the mesh structure as

where

However, mesh only predicted by G-ResNet is prone to obvious self-intersection, so it is necessary to trim the topology to achieve a suitable visual effect.

In order to reduce the calculation of the deformation process and generate a more realistic 3D model, a topology modification process is added after each deform block to dynamically modify the topological relationship between the vertices and the surface of the mesh. A topology correction network is used to update the topological structure of the reconstructed grid by trimming the surfaces that clearly deviate from the ground truth.

Randomly sample points on the surface of the predicted grid topology M and connect the copied shape feature vector with the matrix containing all the sample points. Multilayer Perceptron (MLP) takes the spliced feature matrix as input and predicts the error distance of each vertex to the ground truth value. Calculate the average value of the prediction errors of all sampling points on the triangular surface of the grid and obtain the final error of each triangular surface.

We apply a threshold strategy to delete those faces whose errors exceed the predefined threshold, thereby updating the mesh topology. The threshold τ needs to be adjusted according to the actual situation to reach the most suitable grid structure for pruning. If the threshold τ is too high, it will reduce the trimming part and increase the reconstruction error; if the threshold τ is too low, it will delete too many triangular surfaces and destroy the topological structure of the mesh. Therefore, a coarse-to-fine method is adopted. First, a higher τ is given in the first module, and then τ is sequentially reduced in the subsequent modules to gradually refine the area to be trimmed.

Since the parameters of the network have not been trained at the initial stage, the 3D model after a round of mesh deformation and topology modification cannot achieve sufficient accuracy, so we use the corresponding 3D loss function to repeat the process many times until the generated model error is within the expected range.

In this paper, the network is trained by 3D ground truth to constrain the deformation results of the mesh. The loss function is mainly based on Chamfer Distance

Chamfer loss. Chamfer Distance, as the most common constraint function in the field of 3D reconstruction, was originally used in the point cloud collection to represent the difference between the predicted vertex and the ground truth. Its main function is to limit the position of the vertex , gradually approaching the ground truth. If the loss is large, the difference between the two sets of vertices is large; if it is small, the reconstruction effect is better. The Chamfer loss can be defined as:

Earth Mover’s loss. Earth Mover’s Distance is defined as the minimum sum of the distances between a point in one set and a point in another set on all possible corresponding arrangements. Earth Mover’s loss can be defined as:

Through Chamfer loss and Earth Mover’s loss, the vertices can be gradually returned to the appropriate position, but it is not enough to produce a well-structured and stable mesh. Inspired by the work of Pixel2Mesh [13] and parameters compressing [28], we added a weight vector parameter

Here,

Therefore, the Chamfer loss and Earth Mover’s loss can be further defined as:

Boundary regularize. Sine the topological trimming of the mesh model will leave a jagged edge, which greatly destroys the visual appearance of the reconstructed mesh. In order to further improve the visual quality of the reconstructed mesh, we incorporate a boundary regularization term in the original loss, and penalize zigzag by forcing the boundary curve to remain smooth and consistent:

Here, x is the boundary point of the prediction mesh,

Therefore, the final training goal of the model can be defined as:

Here,

Figures and tables should be inserted in the text of the manuscript.

Next, we will introduce our experimental setup and details.

The dataset ShapeNet is used for training, which contains 13 different object categories and corresponding 50,000 model images. We divide the dataset into a training set and a testing set. On the testing set, we can determine when to stop training by tracking the loss size of the method and all benchmarks.

On the basis of following the standard 3D shape reconstruction evaluation method, we use two different numerical indicators to evaluate the performance of the model and compare with the existing advanced technology. The Chamfer Distance (CD) and Earth Mover’s Distance (EMD) can be used both in training and testing. They are able to measure the error of the vertices between the predicted meshes and ground truth. When the two results are smaller, the experimental effect is better.

We compare the proposed method with some existing 3D reconstruction techniques. Specifically, such as Deep Marching Cubes and PSGN, which are the more influential methods in volume reconstruction and point cloud reconstruction, respectively. In addition, we also compare Pixel2Mesh and TMNet in mesh reconstruction.

The input image size is set to 224*224. First, we pre-train the network structure shown in Fig. 1. In the pre-training, we only train the deform block, eliminating topology modification and boundary refinement. Then train Subnet-1 and Subnet-2 respectively. The training period of pre-training and Subnet-1 is set to 420, the training period of Subnet-2 is set to 120, and the learning rate is set to 0.001. The trained model uniformly contains 2562 vertices. The value of the hyperparameter mentioned in Eq. (8) is set as

4.2 Comparison to State of the Art

As shown in Fig. 3, we show the reconstructed visual effects of some chairs and compare the visual effects with the current state-of-the-art methods. Pixel2Mesh can construct the approximate outline and surface details of the object, but when dealing with the reconstruction of non-approximately spherical objects, it is easy to produce wrong surface connections. TMNet has made improvements on this point. It has independent topology modification capabilities, so it can reconstruct some complex topologies. For example, the back of a chair and its seat are clearly separated, and unnecessary connections are not generated at the legs of the chair. For example, there is a clear separation between the backrest of the chair and the seat of the chair, and there is no unnecessary connection at the legs of the chair. However, its reconstruction details are not good, and the reconstruction will take longer. Relatively speaking, our method combines the advantages of the two. Although there is still a certain gap with the ground truth, it can produce a shape topological structure with clear outlines and rich details.

We uniformly sample 1000 points on the surface of the generated model, and measure CD and EMD between them and the real point cloud of ground truth. Since PSG only generates the point cloud of the target, the ball-pivoting algorithm [30] is used to estimate the grid structure before sampling. The iterative closest point algorithm (ICP) [31] is used for the measurement results, so that it can be better compared with the ground truth value. The final results are recorded in Tabs. 1 and 2. It can be observed that most of the results of PSGN and Deep Marching Cubes are not ideal, because they use point cloud or voxel representation methods, so the surface reconstruction results are not good. Since Pixel2Mesh uses mesh representation and a coarse-to-fine reconstruction process, its error rate is lower than the previous two. TMNet is close to the results of this method, but it uses MLP to perform affine transformation on objects to achieve the purpose of deformation, which consumes more resources in the training process, and the error rate in the initial training period will be higher. In comparison, our method is superior to the latest methods in most results, especially when reconstructing the surface of objects with complex topologies, such as tables and chairs with thin structures; at the same time, because we use the deformation module with the G-ResNet with residual connection and the corresponding weight parameter added to the loss function, our model consumes less resources during training and converges faster.

Now we conduct an ablation experiment to analyze the importance of each component in the entire model. Fig. 4 lists the reconstruction results of the 3D model when the corresponding components are missing, and each column corresponds to a reduction of one component. We found that after reducing the corresponding components, the quality of the reconstruction is reduced to varying degrees, and some of them cannot even generate suitable recognizable 3D shapes (for example, the column (b) in Fig. 4). After removing the specific components, we perform the sampling point analysis experiment on the training results again and measure the error with the ground truth. The quantitative results of the ablation experiment are recorded in Tab. 3.

We first remove the deform blocks in the two subnets, and directly perform topology modification and boundary refinement on the initial 3D sphere. It can be observed that the undeformed sphere lacks GCN’s control over the topology, and a large number of error surfaces are predicted. Therefore, the topology modification trims most of the surfaces and destroys the original mesh topology, leaving only some Remaining grid fragments. Since the training result contains only a few vertices and mesh faces, we cannot perform sampling point analysis on them, as shown in Tab. 3. This is obviously inconsistent with the results we expected.

Second, we remove the topology modification modules in the two subnets and re-train the network. The generated model has a specific 3D shape, but there are some self-intersecting connections between the error surface and the grid. In particular, unnecessary connections exist in the thinner parts such as the chair legs and armrests. The reason is the lack of error prediction and surface trimming for topology modification, which only maintains the basic posture of the reconstructed object; at the same time, GCN will not break the constraints of spherical topology to form such a “hollow” surface.

After clarifying the indispensability of these two modules to the model, we also conduct ablation experiments and analysis on the number of modules. After training Subnet-1, we remove the deform blocks in both subnets and topology modification modules in Subnet-2. As shown in the detailed results in Fig. 5, the lack of deform blocks (Fig. 5d) can generate a specific shape, but the necessary part of the chair “legs” is over-trimmed, thus destroying the original shape. The error is quite different from the ground truth. We speculate that this is caused by the smaller threshold τ when predicting the error surface in Subnet-2. The lack of one topological modification (as shown in Fig. 5e) does not seem to have a significant impact on the surface details, and its error is relatively small, but there is still a certain gap between the final reconstruction effect. In addition, the verification based on the necessity of each regularization has been completed in Pixel2Mesh, so we only conduct ablation experiments and analysis for boundary regularization. As shown in Fig. 5f, although the lack of boundary regularization is not much different from the complete model in terms of error, its qualitative results have obvious jagged surfaces, especially near the trimmed thin structures.

Finally, we find that the discontinuous surface with more complex structure (such as the office chair in Fig. 4) is more affected in the ablation experiment analysis than the continuous surface approaching a sphere (the sofa in Fig. 4). It further illustrates that the method in this paper has good adaptability to the reconstruction of complex structures. At the same time, we conclude that among all other possible module combinations, the result after the complete subnet training of the two components is relatively the most appropriate and optimal.

Based on GCN and topology modification technology, we propose an improved end-to-end network architecture that can quickly generate 3D mesh models with complex topologies from a single perspective. Through the iterative use of GCN and topology modification, the problem that the high-quality surface reconstruction effect and the high flexibility of the topological structure cannot be achieved is solved. At the same time, the feature fusion method we propose uses hierarchical input to make full use of the various stages of the image and solve the problem of data input incompatibility between modules; in addition, the proposed weight parameters can help the network pay attention to the backbone position during training and reduce training consumption. A large number of experiments and measurement results show that the method in this paper can have a good reconstruction effect on common categories (especially categories with complex topological structures).

For future work, we will test our algorithm on other 3D data sets, such as Pix3D with pixel-level 2D-3D correspondence and Pascal3D + [32] with 3D graphics annotations. We will adjust our network structure based on the test results to improve the generalization ability of the model. At the same time, we will integrate the work done by existing reconstruction methods on surface detail optimization, such as color reconstruction, background detection [33–35], digital watermarking reconstruction [36] and texture fitting [37], to further improve the accuracy and authenticity of target reconstruction.

Acknowledgement: Thanks to the supervisor for writing guidance and other colleagues in the laboratory for their help.

Funding Statement: This work was supported, in part, by the Natural Science Foundation of Jiangsu Province under Grant Numbers BK20201136, BK20191401; in part, by the National Nature Science Foundation of China under Grant Numbers 61502240, 61502096, 61304205, 61773219; in part, by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) fund.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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*Computer Systems Science and Engineering*,

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*Comput. Syst. Sci. Eng.*, vol. 45, no. 2, pp. 1695-1709. 2023. https://doi.org/10.32604/csse.2023.031506