非刚性三维形状分析在模式识别、计算机视觉、生物计算以及医学图像处理等领域 引起了研究人员的广泛关注，其研究任务包括形状分类、检索、对应以及配准等。对于 非刚性形状分析而 ，其核心问题是如何建立有效的形状描述符以及度量方法，实现形 状匹配(shape matching)或形状相似度度量。本文将非刚性三维形状视为黎曼流形，在黎曼 几何框架下提出了一系列基于谱分析的非刚性三维形状匹配方法，面向不同的形状匹配 问题提出对应的形状描述符，基于形状描述符将形状映射到相应的特征空间，建立并求 解度量函数， 终得到形状匹配结果。本文从度量范围(不同类形状和同类形状)以及形状 描述符范围(局部和全局)两个角度综合考虑，提出了四种匹配方法，主要工作和创新点如 下：
1、提出了一种基于波扩散距离分布曲线的非刚性三维形状匹配方法，将非刚性三维 形状匹配转化为波扩散距离分布相似度度量。该方法基于对形状表面的波扩散距离进行 统计构造，提取了距离矩阵的累积分布曲线用于定义非刚性三维形状的低维特征，计算 分布曲线之间的Fr′echet距离度量了不同类形状之间的匹配度。多个实验结果显示该方法 对形状具有全局感知能力，能够简化非刚性三维形状匹配的过程并降低匹配复杂度，该 方法适用于不同类形状之间的全局匹配。
 2、提出了一种基于几何矩谱描述符的非刚性三维形状匹配方法，将谱形状描述符的 矩统计量作为特征用于形状相似度度量。该方法通过分析谱形状描述符在时间域的几何 矩和空间域内的条件矩，对谱形状描述符进行重构并将形状的几何信息和拓扑结构编码 到六个几何矩谱描述符中。随后计算了形状表面上所有采样点的几何矩谱描述符之间的 修正Hausdorff距离度量了不同类形状之间的匹配度。不同的实验结果显示该方法能将参 数选择独立于特定的形状分析任务，增强了非刚性三维形状匹配的稳定性和鲁棒性，适 用于不同类形状之间的局部匹配。
3、提出了一种基于距离特征融合的非刚性三维形状匹配方法，将多元距离特征进 行融合刻画了同类形状的全局差异性。该方法通过对波扩散距离进行奇异值分解进行特 征提取，基于特征融合方法将波扩散距离的奇异值与累积分布曲线进行融合得到距离特 征，计算距离特征之间的欧氏距离度量了同类形状之间的匹配度。插值模型和真实模型实验显示该方法提高了对具有相似外形的同类非刚性三维形状的全局辨别能力，适用于 不同采样精度的同类形状之间的全局高精度匹配。
4、提出了一种基于调和波动核的非刚性三维形状匹配方法，通过特征选择的思想研 究了同类形状的局部匹配度。该方法通过引入波动核签名的双能量参数设计了局部三维 形状描述符，通过泛函映射得到 对三维形状之间的对应点，计算三维形状对应点的调 和波动核签名之间的余弦相似度度量了同类形状之间的匹配度。插值模型和真实模型实 验显示该方法刻画了具有多边界复杂拓扑形状的局部细节，对于近似等距变换鲁棒，适 用于同类非刚性三维形状之间的局部高精度匹配。 
基于上述非刚性三维形状匹配方法，本论文提出了两种应用：（1）三维形状检索， 基于波扩散距离分布曲线和几何矩谱描述符，通过设置合适的检索阈值快速查询并提取 出满足阈值的形状，将其按匹配结果降序排列并进行了评价。（2）颅面关系验证，基于 距离特征和调和波动核描述颅面模型；通过颅面数据库提出了一个数据驱动的框架，该 框架立足于整体和局部几何特征双重角度，结合统计方法中的典型相关分析综合验证了 颅骨与人脸之间的明确关系。 基于谱分析的非刚性三维形状描述符继承了拉普拉斯贝尔特拉米算子的等距（近似 等距）不变性、拓扑鲁棒性、噪声鲁棒性和采样鲁棒性，适用于非刚性三维形状匹配方 法。通过在多个公 非刚性三维形状数据库以及私有颅面数据库的匹配实验可知，本文 提出的三维形状匹配方法能够得到高精度和强鲁棒性的匹配结果。

Non-rigid 3D shape analysis methods have attracted wide attention in many fields, including pattern recognition, computer vision, biological computing, and medical image processing. Shape analysis tasks include shape classification, retrieval, correspondence, and registration, etc. For non-rigid 3D shape analysis, the core problem is how to establish an effective shape descriptor and distance function to achieve shape matching or shape similarity measurement. In this thesis, the non-rigid 3D shape is regarded as a Riemannian manifold, and a series of non-rigid 3D shape matching methods based on spectral analysis are proposed based on the Riemannian geometry theory. For different shape matching problems, the corresponding shape descriptors are proposed. Through effective shape descriptors, 3D shapes are mapped to feature space, and metric functions are established and solved. Finally, the shape matching results are obtained. In this thesis, four matching methods are proposed according to the perspective of the measurement range (the shapes of the different categories and shapes of the same category) and shape descriptor range (local and global descriptors), the contributions and innovations are as follows:
1.A non-rigid 3D shape matching method based on the cumulative distribution curve of wave diffusion distance is proposed, which formulates the 3D shape matching problem as the similarity measurement of wave diffusion distance distribution. Based on the statistical construction of the wave diffusion distance on the shape surface, the cumulative distribution curve of the distance matrix is extracted to define the low dimensional features of the non-rigid 3D shape. The Fr′eChet distance between the distribution curves is calculated to measure the matching degree between different types of shapes. Several experimental results show that this method has the ability of global shape perception, which can simplify the process of non-rigid 3D shape matching and reduce the matching complexity. This method is suitable for global matching between different categories of shapes.
2.A non-rigid 3D shape matching method based on geometric moment spectral descriptors is proposed and the statistics moment of spectral shape descriptor are defined as features to measure shape similarity. This method reconstructs the spectral shape descriptor by analyzing the geometric moments of the spectral shape descriptor in the time domain and the conditional moments in the spatial domain and encodes the geometric information and topological structure of the shape into six geometric moment spectral descriptors. The modified Hausdorff distance between the geometric moment spectral descriptors of all sampling points on the shape surface is calculated to measure the matching degree between different types of shapes. Different experimental results show that the method makes the parameter selection independent of the specific shape analysis tasks, enhances the stability and robustness of non-rigid 3D shape matching, and is suitable for local matching between different categories of shapes.
3.A non-rigid 3D shape matching method based on the fusion distance feature is proposed and the distance features are fused to describe the global difference of similar shapes. Based on the feature fusion method, the singular value of the wave diffusion distance is fused with the cumulative distribution curve of the wave diffusion distance to obtain the distance feature. The matching degree between the two shapes is measured by the Euclidean distance between the distance feature fusion. Experiment results of linear interpolation models and real models show that this method improves the global discrimination ability of similar non-rigid 3D shapes with similar appearances, and is suitable for global high-precision matching between shapes of the same category with different sampling accuracy.
4.A non-rigid 3D shape matching method based on harmonic wave kernel is proposed and the local matching degree of similar shapes is studied by feature selection. In this method, the local 3D shape descriptor is designed by introducing the two energy parameters of the wave kernel signature. The corresponding points set between a pair of 3D shapes is obtained based on the functional maps. The cosine distance between the harmonic wave kernel signatures of the corresponding points set of the 3D shapes is calculated and the matching degree between the two shapes is obtained. Experiment results of linear interpolation models and real models show that the proposed method depicts the local details of complex topological shapes with multiple boundaries. This method is robust to approximate isometric transformation and is suitable for local high-precision matching between the non-rigid 3D shapes of the same category.
Based on the above-mentioned non-rigid 3D shape matching method, this thesis proposes two applications: (1) 3D shape retrieval: Based on the wave diffusion distance distribution curve and the geometric moment spectral descriptor as the shape descriptors. This method can quickly query and extract the shapes according to the appropriate retrieval threshold. The correspondence results are arranged and evaluated in descending order. (2) Craniofacial relationship verification: A data-driven framework is proposed through the craniofacial database. To comprehensively verify the clear relationship between the skull and the face, this framework is based on the dual perspectives of global and local geometric features and combined with the canonical correlation analysis in statistical methods.
The non-rigid 3D shape descriptors based on spectral analysis inherit the isometric (approximately isometric) invariance, topological robustness, noise robustness, and sampling robustness of the Laplace-Beltrami operator, which are suitable for non-rigid 3D shape matching methods. This thesis tests the matching method on the multiple public non-rigid 3D shape databases and private craniofacial database and shows the method can obtain more accurate and robust results.