Multi-scale qualitative location: A direction-based model
Introduction
Spatial locations are fundamental in geo-applications such as spatial data analysis and location-based service. They have three components: spatial objects which need to be located, a reference frame in which the locations of objects can be described, and the relations between the objects and the reference frame. In GIS, spatial locations are traditionally related to the Cartesian or spherical reference frame. However, people tend to locate objects by relating them to a reference frame based on qualitative relations. For example, it is common to say that John is inside Beijing instead of at 39.93°N, 116.26°E. In this example, John is the object to be located, inside refers to a qualitative relation, and Beijing is the reference object.
The reference frames in existing qualitative location models can be either (1) a single reference object, (2) a set of discrete objects, or (3) regional partitions. For the first case, a qualitative location is defined by a qualitative relation with respect to another object. An example for this case can be found in Yao and Thill (2006). For the second case, the reference frames are composed of discrete objects, which in most cases are points. For example, in route direction, some salient or well-known features are chosen as landmarks to guide travelers (Winter et al., 2008); and in qualitative positional information (Clementini et al., 1997), a set of points is regarded as the reference objects on which the orientation relations and qualitative distances are used to locate objects. In the first two frames, the reference objects do not cover the whole space, thus the spatial objects can be located in the empty space without reference objects. For the third case, a regional partition is regarded as the reference frame and the locations of objects are represented by relating them to each region in the partitions using qualitative relations (Bittner and Stell, 1998, Bittner and Stell, 2002).
Four kinds of qualitative relations can be used in qualitative location models: direction relations (Haar, 1976, Frank, 1996, Goyal, 2000, Skiadopoulos and Koubarakis, 2004), topological relations (Egenhofer and Herring, 1991, Cohn et al., 1997), qualitative distances (Clementini et al., 1997), and nearness relations (Worboys, 2001). For qualitative location models that regard a single object or a set of objects as reference frames, these relations can be used jointly, such as the use of direction relations and qualitative distances for defining and reasoning qualitative locations with respect to a single object (Frank, 1992) or a set of discrete points (Clementini et al., 1997). They can also be used individually, such as the compositions of topological relations (Egenhofer, 1994) and direction relations (Skiadopoulos and Koubarakis, 2004) for a pair of objects. For qualitative location models that rely on regional partitions, so far only single type of relation, i.e., topological relation was explored (Bittner and Stell, 1998, Bittner and Stell, 2002, Bittner and Stell, 2003); Stell, 1999, Stell, 2003 used graphs to represent static and dynamic entities at multi-granularity. These studies, however, do not consider the direction relations and their changes across two scales.
A recent study in cognitive science confirmed that the assumption “topology matters, metric refines” in Naive Geography (Egenhofer and Mark, 1995) is not always true; in some cases metrics (directions and sizes) are more important (Klippel et al., 2013). A qualitative location model based on direction thus deserves attention. As discussed in the previous paragraph, existing works incorporating direction relations are all based on either a single object or a set of objects, especially points. They thus lack consideration of spatial scales embedded in regional partitions. This study develops a qualitative location model that emphasizes the changes in the representations of spatial objects and regional partitions, as well as direction relations between objects and that between objects and regional partitions induced by map generalization operators.
Termed direction-based multi-scale qualitative location (DMQL), the proposed model is built on the maximum bounding rectangle (MBR) of each region, i.e., a partition we called cell. It is based mainly on direction relations. To accommodate direction operations in a reference frame, spatial objects are linked to a regional partition by direction relations, which is equal to locating objects in the cell partition. Therefore, the cell representation is a form of qualitative locations concerned with directions. Based on the cell representations of objects in multiple levels of partitions, approaches are presented to identify direction changes between different reference frames, between objects and regions, and between objects.
Compared to similar partition-based models (Bittner and Stell, 1998, Bittner and Stell, 2002) that rely on topological relations, the direction relations in the DMQL model can provide support to hybrid topological and direction operations and more importantly, a more precise means to locate objects. From the practical perspective, the DMQL model considers the changes in geometrical representations of objects, direction relations and locations across multiple scales, and thus can play an important role in evaluating the consistency about multi-scale spatial data (Sheeren et al., 2004, Delis and Hadzilacos, 1997), querying multi-resolution maps (Podestà et al., 2007), and analyzing image (Bittner and Winter, 1999).
This paper is organized as follows. Section 2 introduces qualitative locations. Section 3 formalizes the DMQL model. Section 4 develops the approaches to derive direction changes in the DMQL model by considering two generalization operators, i.e., the reduction of frames and the merging of spatial objects. From both locational and directional perspectives, Section 5 presents the methods to evaluate consistencies using the DMQL model. Finally, suggestions for further research are given in Section 6. The notations used in the following sections are summarized in Table 1.
Section snippets
Background
Multi-scale regional partitions can be either strict or weak. The former imposes stricter constraints on regions in different details than the latter. Both types of multi-scale regional partitions are presented in this section as they are suitable to model different geographical phenomena.
Importance and essential direction types
Incorporating directions into qualitative location models brings multiple benefits in handing locations. In addition to supporting direction queries and reasoning among spatial objects, the direction information can improve the accuracy of approximating the location of a spatial object. For example, the spatial object a in Fig. 1a is approximated by p2 and p4 but only a small portion. By integrating the topological relations α(a, p2) = α(a, p4) = overlap and the direction relation β(a, p6) = {NE6}, the
Direction changes in DMQLs
In multi-scale locations, a detailed reference frame is reduced to a coarse one, resulting in the potential changes in direction relations. This section addresses how direction changes with the reduction, which can help to check the consistencies of DMQLs.
Applications in checking the consistencies of multi-scale spatial data
The DMQL model can locate spatial objects in multi-scale regional partitions by relating objects to the regions in partitions with direction relations. Both objects and regions are represented as sets of cells. Due to the generalization of spatial objects and regional partitions, the multiple representations of the same objects may have different cell representations, and thus inconsistencies. In this context, evaluating whether the cell representations of the same objects at multiple scales
Conclusions and future directions
This study presented a direction-based multi-scale qualitative location (DMQL) model which locates spatial objects in regional partitions at different scales by direction relations. Related operations based on DMQL model were also investigated. Existing qualitative location models locate spatial objects in regional partitions by topological relations, which approximate spatial objects using regions in regional partitions. In contrast to these topology-based models, the spatial objects in the
Acknowledgements
The work presented in this paper was supported by the grants from National Natural Science Foundation of China (No. 41171297) and Program for New Century Excellent Talents in University (NCET-10-0189). The work of the second author is supported by National University of Singapore Academic Research Fund (R-109-000-112-112).
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