Symmetric functions for fast image retrieval with persistent homology

Abstract

Persistence diagrams, combining geometry and topology for an effective shape description used in pattern recognition, have already proven to be an effective tool for shape representation with respect to a certainfiltering function. Comparing the persistence diagram of a query with those of a database allows automatic classification or retrieval, but unfortunately, the standard method for comparing persistence diagrams, the bottleneck distance, has a high computational cost. A possible algebraic solution to this problem is to switch to comparisons between the complex polynomials whose roots are the cornerpoints of the persistence diagrams. This strategy allows to reduce the computational cost in a significant way, thereby making persistent homology based applications suitable for large scale databases. The definition of new distances in the polynomial frame-work poses some interesting problems, both of theoretical and practical nature. In this paper, these questions have been addressed by considering possible transformations of the half-plane where the persistence diagrams lie onto the complex plane, and by considering a certain re-normalisation the symmetric functions associated to the polynomial roots of the resulting transformed polynomial. The encouraging numerical results, obtained in a dermatology application test, suggest that the proposed method may even improve the achievements obtained by the standard methods using persistence diagrams and the bottleneck distance.