On the Local Behavior of Spaces of Natural Images

Authors:
Gunnar Carlsson;Tigran Ishkhanov;Vin Silva;Afra Zomorodian
Affiliations:
Department of Mathematics, Stanford University, Palo Alto, USA 94305;Department of Mathematics, Stanford University, Palo Alto, USA 94305;Department of Mathematics, Stanford University, Palo Alto, USA 94305;Department of Mathematics, Stanford University, Palo Alto, USA 94305
Venue:
International Journal of Computer Vision
Year:
2008

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Abstract

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 驴. We develop a theoretical model for the high-density 2-dimensional submanifold of 驴 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 驴. We find the best-fitting embedding of the Klein bottle into the ambient space of 驴. Our results are currently being used in developing a compression algorithm based on a Klein bottle dictionary.