The topology of boundaries

Abstract

High-level representations used in reasoning distinguish a special set of boundary locations, at which function values can change abruptly and across which adjacent regions may not be connected. Standard models of space and time, based on segmenting $\text{R}$ⁿ, do not allow these possibilities because they have the wrong topological structure at boundaries. This mismatch has made it difficult to develop formal mathematical models for high-level reasoning algorithms. This paper shows how to modify an $\text{R}$ⁿ model so as to have an appropriate topological structure. It then illustrates how the new models support standard reasoning algorithms, provide simple models for previously difficult situations, and suggest interesting new analyses based on change or non-change in scene topology.