Topology of Shape and Data in Material Microstructures
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.