Before revision: Topological descriptors  have been shown to be useful for summarizing and differentiating shapes . Related work uses persistence diagrams and Euler characteristic curves to differentiate between shapes and quantifies the number of descriptorsnecessary for shape reconstruction, given certain assumptions such as minimum curvature .
After revision: Topological descriptors have been shown to be useful for summarizing and differentiating shapes , both empirically and theoretically. The theoretical justification for their use is based on the fact that shapes can be fully reconstructed using just the descriptors .