Relations between adjacency trees

Abstract

Adjacency trees can model the nesting structure of spatial regions. In many applications it is necessary to model foreground and background regions which exhibit changes over time such as splitting, where one region divides into two. For example, the qualitative description of the development of wildfires would use the foreground for areas on fire and the background for areas not on fire. Such dynamic behaviour can be modelled by a particular kind of relation between the nodes of two adjacency trees representing the initial and final configurations of the regions at two times. These relations, which we call bipartite, correspond to having an arbitrary relation between the foreground regions at the two times and an arbitrary relation between the background regions at the two times. We show that all bipartite relations between trees arise from sequences of atomic relations between trees. There are just four types of these atomic relations (in addition to one representing no change): inserts, splits, merges and deletes.

Spatial Information Theory

“…This could establish what kinds of spatio-temporal change are possible from particular rules. There will be close connections between the behaviour of the split and merge rules for bigraphs and the splitting and merging studied in [9,15]. There are also many possible application problems for spatiotemporal analysis described in the literature cited in the introduction.…”

“…We can imagine spatio-temporal scenarios between regions, such as one moving to encircle another, or two regions moving further apart to allow a third to pass between them. The most basic scenarios of single regions splitting and merging have been rigorously analysed in [9,15], but it is not clear whether more complex behaviours can be treated in a similar way. In order to study such behaviour it is necessary to have a formal framework that is capable of modelling spatio-temporal change without pre-judging the kinds of higher level events and process that will be significant.…”

Out of Sight, but Not Out of Mind: A Diagrammatic Conversation on Relational Drawing

“…We start with Figure 1, where the main image came from thinking about change over time, as developed in [5,6]. A series of qualitative changes occur and are tracked in the diagram over a series of frames: ring-like darker patches appear; they coalesce into a single patch with two open areas; these two areas merge and enlarge to such an extent that the dark area remains only as a thin ring.…”

stampr: Spatial-Temporal Analysis of Moving Polygons in R

“…Polygons that split or merge through time induce complexities into their distance and directional relationships due to changing the spatial context within which two (or more) polygons are compared. For example, Stell and Worboys (2011) provided recently the theoretical basis for computing relations for splitting and merging polygons using adjacency trees, with focus on the relationship between polygons (in the foreground) and a background region. We will extend the implementation to handle distance and direction between level 4 events in future releases.…”