Multiscale structural analysis of defective graphene in transmission electron microscopy images using persistent homology

We developed a persistent-homology-based strategy that converts transmission electron microscopy images of defective graphene into quantitative, multiscale descriptors spanning local carbon-atom polygons and their global connectivity. Zero-dimensional persistence diagrams for grayscale images extracted carbon-ring interiors as local structures, allowing identification of defect structures. Roundness as a measure of the distortion of carbon rings and polygon number were quantitatively calculated. It was found that electron irradiation increased both the fraction of non-hexagonal polygons and distortion of carbon-atom polygons in graphene. One-dimensional persistence diagrams for polygon-center point clouds after downsampling the center of non-hexagonal polygons yielded large rings that reliably enclosed connected defect clusters ranging from simple defects to complex clusters. These rings also enclosed non-crystalline regions composed of hexagons, which were difficult to capture considering only polygon number. The large rings enable automated counting of the number of each type of polygon within each defect cluster. The number of polygons within defect clusters thereby quantified the connectivity of carbon-atom polygons and revealed that electron-beam irradiation induced the growth of defect clusters in an interconnected manner.