Persistent homology and partial similarity of shapes

The ability to perform shape retrieval based not only on full similarity, but also partial similarity is a key property for any content-based search engine. We prove that persistence diagrams can reveal a partial similarity between two shapes by showing a common subset of points. This can be explained using the MayerVietoris formulas that we develop for ordinary, relative and extended persistent homology. An experiment outlines the potential of persistence diagrams as shape descriptors in retrieval tasks based on both full and partial similarity.