Two-dimensional micrographs are a common data format in several important machine learning applications. One example is histopathological classification: the detection of disease-related abnormalities in 2D scans of biological tissue. Another, from materials science, is the classification of polycrystalline materials and the prediction of their physical properties based on images of their microstructures. In both applications, the local topology and geometry — namely the shape and arrangement of cells/grains — are thought to be essential. However, information about these features may be lost in traditional machine learning pipelines such as those involving convolutional neural nets (CNNs). In this talk, I will discuss two methods to represent the geometry of micrographs in formats amenable to machine learning. The first augments images of biological tissue with additional fields representing the persistent homology of local windows. The second represents the grain structure of a polycrystal as a metric measure space of local configurations.