Symmetric functions for fast image retrieval with persistent homology

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 certain filtering 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\u2010scale databases. The definition of new distances in the polynomial framework 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\u2010plane where the persistence diagrams lie onto the complex plane, and by considering a certain re\u2010normalisation the symmetric functions associated with 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

Shortened persistent homology for a biomedical retrieval system with relevance feedback

This is the report of a preliminary study, in which a new coding of persistence diagrams and two relevance feedback methods, designed for use with persistent homology, are combined. The coding consists in substituting persistence diagrams with complex polynomials; these are "shortened", in the sense that only the first few coefficients are used. The relevance feedback methods play on the user's feedback for changing the impact of the diferent filtering functions in determining the output