Hierarchies Relating Topology and Geometry

Abstract

Cognitive Vision has to represent, reason and learn about objects in its environment it has to manipulate and react to. There are deformable objects like humans which cannot be described easily in simple geometric terms. In many cases they are composed of several pieces forming a “structured subset” of ℝⁿ or ℤⁿ. We introduce the potential topological representations for structured objects: plane graphs, combinatorial and generalized maps. They capture abstract spatial relations derived from geometry and enable reconstructions through attributing the relations by e.g. coordinates. In addition they offer the possibility to combine both topology and geometry in a hierarchical framework: irregular pyramids. The basic operations to construct these hierarchies are edge contraction and edge removal. We show preliminary results in using them to hold a whole set of segmentations of an image that enable reasoning and planning actions at various levels of detail down to a single pixel in a homogeneous way. We further speculate that the higher levels map the inherent structure of objects and can be used to integrate (and “learn”) the specific object properties over time by up-projecting individual measurements. The construction of the hierarchies follows the philosophy to reduce the data amount at each higher level of the hierarchy by a reduction factor > 1 while preserving important topological properties like connectivity and inclusion.