Persistence for shape comparison

Persistence is a theory for Topological Data Analysis based on analyzing the scale at whichtopological features of a topological space appear and disappear along a filtration of thespace itself. As such, it is particularly suited for handling qualitative rather than quantitativeinformation about the studied space. Moreover, persistence deals with noise consistently, inthat noisy data do not need to be smoothed out in advance. Last but not least, it is modular,meaning that different filtrations give insights from different perspectives on the space understudy.For all these reasons persistence turns out to be a well-suited tool for shape comparison,i.e. the task of assessing similarity between digital shapes.In particular, persistence provides a shape descriptor, the persistence diagram, and adistance between these diagrams, the bottleneck distance. Thus the similarity between twoshapes, represented by spaces endowed by functions, is measured by the bottleneck distancebetween the corresponding persistence diagrams.Persistence diagrams are very concise descriptors, consisting of finitely many points ofthe plane. Moreover, the bottleneck distance between persistence diagrams is stable in thesense that small changes in the filtration imply small changes in the bottleneck distance.Finally, the bottleneck distance between persistence diagrams bounds from below the naturalpseudo-distance between the original shapes.