AI 中文总结
该研究结合拓扑数据分析与可分形状张量学习，提出双参数滤波方法量化材料微观结构的形状与拓扑，为成像科学及微观结构分析提供新工具。
AI 中文摘要
微观结构分析的挑战之一是，除了平均值比较外，还需对微观结构的形状、尺寸及空间排列进行严格量化。本文阐述了结合拓扑数据分析（Topological Data Analysis, TDA）与曲线间非欧距离的形式化原理，为图像中模式与形状的形式及本质提供新视角。具体而言，提取持久拓扑结构的TDA描述子，与可分形状张量（separable shape tensors, SST）的乘积子流形学习相结合，通过双参数滤波视角，为材料微观结构的电子背散射衍射（electron backscatter diffraction, EBSD）图像提供独特见解。与标准方法不同，该方法凸显了形状距离的不同选择或排列如何导致持久拓扑的不同概念，从而拓宽了TDA的解释范围。所得特征提取可视化结果兼具严谨性与解释性，为现代成像科学提供了应用于材料计量的新工具。更广泛而言，该框架对拓扑与形状精确量化对揭示基础图像模式和特征至关重要的领域具有重大影响，为微观结构分析提供了额外的数据驱动工具。
英文摘要
One of the challenges in microstructure analysis is the rigorous quantification of the shape, size, and spatial arrangement of the microstructure beyond comparison of average values. We expound on formal principles combining Topological Data Analysis (TDA) and non-Euclidean distances between curves to motivate novel perspectives on the form and nature of pattern and shape in images. Specifically, TDA descriptors extracting persistent topological structures are combined with product submanifold learning of separable shape tensors (SST) to offer unique insights about electron backscatter diffraction (EBSD) images of material microstructures through the lens of a dual-parameter filtration. Beyond standard approaches, our methodology highlights how different choices or permutations of shape distances can lead to distinct notions of topological persistence, thereby broadening the interpretive scope of TDA. The resulting visualizations of feature extraction are designed to be both principled and explanatory, offering novel tools for modern imaging science with applications to material metrology. More broadly, this framework has strong potential to impact domains where precise quantification of topology and shape is critical for uncovering fundamental image patterns and features, and enables additional data-driven tools for microstructure analysis.
