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Timpf Sabine

Hierarchical Structures in Map Series
by Sabine Timpf

Among the different operations in Geographic Information Systems (GIS), changing scale is crucial for the handling and analysis of multi-scale data. The term multi-scale is used to indicate that there are data of more than one level of detail in the database. Different levels of detail are needed when dealing with spatial data. Current scaling functions blow up and shrink the objects to display or change to a radically different representation of the display.

In this research we investigate how to link map objects at different levels of detail that represent the same real-world entity. We observe the behavior of map objects at four levels of detail in eight case studies. We apply an observation method derived from the method of abduction. It relies on careful and controlled observation of phenomena, similar to the method of reverse engineering in computer science. We define four types of map objects to be observed:
a trans-hydro network, that is a combination of the street network, the railroad network, and the hydrographic network,
containers, that represent the regions between the links of the trans-hydro network,
areas, that represent the land-use areas and are contained in the containers, and
map elements, that are contained in the areas, representing buildings, symbols, and dead-end streets and railroads.

We propose a hierarchical data structure that supports and describes the behavior of map objects over four levels of detail. Hierarchical data structures are best represented by trees or forests. The data structure of the map cube model is a combination of three different trees, representing three different types of hierarchies.

Hierarchies are defined according to the functions associated with the edges of their trees: aggregate, generalize, and filter. The aggregation hierarchy is built by aggregating objects. The generalization hierarchy defines how classes relate to more generic super classes. The filter hierarchy filters objects according to a filter criterion.

The map cube model can be imagined as a three dimensional composition. Each map is a level in a three-dimensional cube, the vertical axis represents the level of detail. The hierarchical data structure for the map series is a combination of the trans-hydro network, a tree of containers including the information on areas and elements, an area graph, and an element graph. The tree of containers stores which container includes which areas and elements. It does not store how these areas and elements are related to areas and elements on a lower or higher level of detail. This information is encoded in the area graph or element graph, respectively. We formalize the map cube model in a functional programming language with an object-oriented paradigm.

In our model the trans-hydro network is represented by a filter hierarchy, the filter criterion is the relative importance of the link in the network. The container tree is an aggregation hierarchy - the more abstract the representation, the more containers are aggregated. The area graph is a collection of area trees; each tree represents a generalization hierarchy. The element graph is a collection of trees; each tree is a constructed by several functions. Those functions are ‘omit’, ‘typify’, ‘merge’, and ‘continue’. The first three functions correspond to the cartographic functions with the same name. Thus, the model incorporates all three types of hierarchies.

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