中国科学院自动化研究所机构知识库

三维网格由于其高效性，简洁性和灵活性，是计算机中构建三维几何形状的重要媒介。相比于二维图像，三维网格包含了更加丰富而真实的三维外形信息，更接近人类对真实世界的实际感知。随着三维模型获取和分析技术的不断提高和不断普及，三维网格的表示方法成为计算机图形学近年来着重关注的一个研究热点。

三维网格的表示方法是指有效表示网格几何形状和网格形状特征的方法。鲁棒性是指表示方法具有适应复杂对象或复杂场景的能力。论文围绕如何设计鲁棒的表示方法，以更好地对三维网格进行呈现和分析。论文重点研究复杂情况下三维网格表示方法的鲁棒性问题，并关注三维网格的三个方面：即三维网格的优化表示方法、特征表示方法和表示学习方法。其中优化表示方法是特征表示方法的基础，而特征表示方法又是表示学习方法的基础。论文的主要工作和贡献如下：

1. 提出了一种三维网格复杂几何形状的优化表示方法

实际应用中，许多三维网格具有复杂的几何形状，例如具有尖锐特征或者曲率高度变化的形状。虽然三维网格的优化表示已有大量的工作，但是以往的方法很难对这种具有复杂几何形状的网格进行有效优化。如何针对具有复杂几何形状的三维网格进行有效的优化表示，对实际应用具有重要意义。为此，针对复杂几何形状，本文提出了一种鲁棒的优化表示方法。该方法设计了保持几何特征的局部大小角度优化操作。通过迭代地应用角度优化、顶点平滑和连接度优化操作，该算法将输入网格重新网格化为高质量的网格表示。实验结果表明，该方法在保持高效性的基础上，可以在具有尖锐特征或者曲率高度变化的网格上鲁棒地优化角度质量，并控制角度的上下界。不仅如此，该方法还可以为后续三维网格的鲁棒分析提供网格连接关系可控的数据生成方法。

2. 提出了一种三维网格不同形状结构的特征表示方法

三维网格由于获取设备和方法的不同，通常具有不同的形状结构，即不同的尺度变换、刚性变换和分辨率。传统的特征表示方法很难在保持特征判别性的基础上，生成对形状结构变化鲁棒的特征。因此，如何满足特征表示方法的鲁棒性要求是一个具有挑战的研究问题。针对这一问题，本文提出了一种鲁棒的特征表示方法。该方法利用坐标信息分别在频域和小波域中重建狄利克雷能量，通过相应的能量分配方法来生成局部点特征和小波能量分解特征，以实现将与尺度变化和刚性变换相关的坐标信息直接转化为对尺度变化和刚性变换理论上不变的信息。实验结果表明，该方法提取的特征具有尺度与刚性变换的不变性，并且对网格分辨率变化不敏感，实现了特征表示效率和判别性的提升。

3. 提出了一种三维网格不同离散化的表示学习方法

在图形学中，同一个形状往往有多种离散化（分辨率和连接关系）方式。传统的表示学习方法很少考虑三维网格的离散化问题，生成的描述子很难对离散化的变化鲁棒。基于测地圆盘规则采样的预处理方法避免了深度神经网络考虑网格离散化的问题，但是该预处理方法的性能和鲁棒性都无法满足要求。本文首先提出了基于测地圆盘规则采样的改进方法。该方法以紧凑的方式将每个顶点周围的多尺度局部频域特征编码为局部的顶点频域图像，并使用三元组（Triplet）损失函数高效地训练卷积神经网络，以生成鲁棒的描述子。但是，基于采样的预处理方法会造成信息损失，描述子的性能无法进一步提高。为此，本文提出了一种鲁棒的表示学习方法。该方法将小波嵌入到图卷积神经网络当中，通过设计对离散化不敏感的图卷积算子，以直接在原始三维网格上提取形状描述子。实验结果表明，该方法可以进一步提升描述子形状匹配的质量，并且使形状描述子不敏感于不同网格离散化的结果。

Due to efficiency, simplicity, and flexibility, the 3D mesh is an important medium for modeling 3D geometric shapes in computers. Compared with 2D images, 3D meshes contain richer and more real information of shape, which is more in line with human perception of the real world. With the continuous improvement and popularization of 3D model acquisition and analysis technology, the representation method of 3D meshes has become a research hotspot of computer graphics in recent years.

The representation method of 3D meshes refers to a method that effectively represents the geometric shape and shape feature of meshes. The robustness refers to the ability of the representation method to adapt to complex objects or scenarios. This thesis revolves around how to design robust representation methods to better present and analyze 3D meshes. It focuses on the robustness of the 3D mesh representation method in complex situations. The thesis pays attention to three aspects of 3D meshes: optimization representation method, feature representation method, and representation learning method of 3D meshes. Optimization representation method is the basis of feature representation method, and feature representation method is the basis of representation learning method. The main contributions of this thesis are summarized as follows:

1. Optimization representation method for complex geometric shapes of 3D meshes

In practice, many 3D meshes have complex geometric shapes, such as shapes with sharp features or highly variable curvatures. Although there has been a lot of works on the optimization of 3D meshes, it is difficult to effectively optimize such meshes with complex geometries in the past. How to effectively optimize the representation of three-dimensional meshes with complex geometric shapes is of great significance to practical applications. For this reason, this thesis proposes a robust optimization representation method. This method designs a local large and small angle optimization operation to maintain the geometric feature. By iteratively applying angle optimization, vertex smoothing, and valence optimization operations, the algorithm remesh the input mesh into a high-quality mesh representation. Experimental results show that, while maintaining high efficiency, the proposed method robustly optimizes the angle quality on a mesh with sharp features or a highly variable curvature, and controls the upper and lower bounds of the angle. Not only that, this method also provides a data generation method with controllable mesh connectivity for the subsequent robust analysis of 3D meshes.

2. Feature representation method for different shape structures of 3D meshes

Due to different acquisition equipment and methods, 3D meshes usually have different shape structures, such as different resolutions, scale transformations, and rigid transformations. Traditional feature representation methods fail to generate features that are robust to the change of shape structures while maintaining the discriminative nature of features. Therefore, how to meet the robustness requirements of feature representation methods is a challenging research problem. To address this issue, this thesis proposes a robust feature representation method. This method uses coordinate information to reconstruct Dirichlet energy in the frequency domain and wavelet domain respectively, and generates local point features and wavelet energy decomposition features through corresponding energy distribution methods, so as to directly transform the coordinate information related to scale changes and rigid transformations into information that is theoretically invariant to scale changes and rigid transformations. Experimental results show that the features extracted by this method are invariant to scale and rigid transformations, not sensitive to the change of mesh resolutions, and more effective and discriminative.

3. Representation learning method for different discretizations of 3D meshes

In graphics, there are often multiple discretizations (resolutions and connectivity) for the same shape. Traditional representation learning methods rarely consider the discretization of 3d meshes, and the generated descriptors are difficult to be robust to the change of discretization. The pre-processing method based on the regular sampling of geodesic disks avoids the problem of mesh discretization in deep neural networks, but its performance and robustness cannot meet the requirements. This thesis first proposes an improved method based on the regular sampling of geodesic disks. This method encodes the multiscale local spectral features around each vertex into a local vertex spectral image in a compact manner and uses the triplet loss function to efficiently train the convolutional neural network to generate robust descriptors. However, the sampling-based preprocessing method will cause information loss, and the performance of the descriptor cannot be further improved. For this reason, this paper proposes a robust representation learning method. This method embeds the wavelet into the graph convolutional neural network and designs a graph convolution operator that is not sensitive to discretization to directly extract shape descriptors from the original 3D meshes. Experimental results show that this method further improves the quality of descriptor for shape matching, and makes the shape descriptor insensitive to the discretization results of meshes.