Simplification and Hierarchical Voronoi Graph Construction

Abstract

In the previous chapter, we proposed the HAGVG as an exhaustive environmental model for a mobile robot. As stated, our general aim is that the graph representation be learned incrementally from local information. Sensor information or locally computed metric representations used to compute a local AGVG will be discarded immediately.

In this chapter, we will start by focusing on the problem of automatically deriving coarser AGVG representations and constructing complete HAGVG representations from a given AGVG. For this purpose, we will introduce relevance measures for Voronoi nodes and the regions accessible via their edges. These measures serve two aims: being able to deal with the stability problems described in Sect. 3.7 by identifying and removing unstable parts of an AGVG, and assessing the significance of Voronoi nodes in an AGVG as a decision point for navigation with the goal of deciding which nodes should be retained on higher levels of abstraction.

As we will show, the two notions of stability and relevance for navigation are very much related and thus can be covered by the same measures. The process of constructing a coarser AGVG based on these measures will be referred to as simplification. Simplification can be used for deriving more abstract levels of representation as well as for removing unstable parts caused by noise. The work described in this chapter is related to work on shape representation and matching approaches developed in the vision community that also employ Voronoi diagrams or skeletons (e.g., Mayya & Rajan, 1996; Ogniewicz & Kübler, 1995; Siddiqi & Kimia, 1996). However, the basic conditions in these approaches are different as here a complete description of the shape’s boundary can be used to simplify the Voronoi structures. In contrast, our incremental construction in which no global representation of the shape of the environment is maintained requires a different approach, in which the simplification is purely based on the information stored in the AGVG.