In this study we concentrate on qualitative topological analysis of the local behavior of the space of natural images. To this end, we use a space of 3 by 3 high-contrast patches M. We develop a theoretical model for the highdensity 2-dimensional submanifold of M showing that it has the topology of the Klein bottle. Using our topological software package PLEX we experimentally verify our theoretical conclusions. We use polynomial representation to give coordinatization to various subspaces of M. We find the best-fitting embedding of the Klein bottle into the ambient space of M. Our results are currently being used in developing a compression algorithm based on a Klein bottle dictionary.

Abstract. We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverseimage persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence and their stability can be described in this framework. This includes the classical case of sublevelset persistent homology. We introduce a distinction between 'soft' and 'hard' stability theorems. While our treatment is direct and elementary, the approach can be explained abstractly in terms of monoidal functors.