International Journal of Geographical Information Science Vol. 25, No. 1, January 2011, 113–130
Measuring congruence of spatial objects Zuoquan Zhaoa,*, Roger R. Stoughb and Dunjiang Songa a
Institute of Policy and Management, Chinese Academy of Sciences, Beijing, China; bSchool of Public Policy, George Mason University, Fairfax, VA, USA
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(Received 2 September 2009; final version received 16 February 2010) This article develops and tests an algorithm of spatial congruence based on geometric congruity of two spatial areal objects in the Euclidean plane. Spatial congruence is defined and thus evaluated as an increasing continuous function of congruity in the position, orientation, size, and shape of spatial objects, dependent upon scaling, translation, and rotation. Expansion-based geometric matching is used to seek the best match between the two objects of interest for the examination and differentiation of the congruence effects of their spatial and geometric properties, while the expansion-inflated size effect is deflated or filtered out accordingly. The use of both expansion and deflation not only allows for a trade-off between size and position, both of which are found substitutable for each other in congruence measurement, but also enables the congruence algorithm to be highly sensitive to differences or changes in these properties. Three geographical objects (the states of Texas, Mississippi, and Louisiana) are used to show how trade-offs among the four properties are manipulated by the congruence algorithm in a geographic information system (GIS) environment, ArcGIS. In addition, three regular geometric objects are used to demonstrate how the congruence algorithm is sensitive even to small changes in each of the four properties of objects. The results show that the proposed congruence algorithm is capable of quantifying the extent of congruity between two spatial objects regardless of how they are related as described in topological relations. Keywords: spatial congruence; spatial relations; spatial similarity; expansion; spatial matching
1.
Introduction
Quantifying spatial congruence is widely used in applications such as inferring spatial relations in geographical information science (Egenhofer 1989, Egenhofer and Franzosa 1991, Bruns and Egenhofer 1996, Frontiera et al. 2008), finding spatial change in a spatiotemporal process (Egenhofer and Al-Taha 1992, Galton 2000, Jiang and Worboys 2009), detecting alterations to spatial databases (Hagedoorn and Veltkamp 1999), and calculating pattern (or shape) similarity in image processing (Endreny and Wood 2003, Frontiera et al. 2008). Existing methods for measuring spatial congruence are established based on one or two of the four spatial and geometric properties – position, orientation, size, and shape of spatial objects (Egenhofer et al. 1998, Cristani 1999, Chen et al. 2001, Godoy and Rodriguez 2004). For example, the overlap index is used to measure the degree of shape similarity between two spatial objects (Zhao and Stough 2005); distance relations are used to evaluate the extent of incongruence in the position of spatial objects (Bruns and Egenhofer *Corresponding author. Email:
[email protected] ISSN 1365-8816 print/ISSN 1362-3087 online # 2011 Taylor & Francis DOI: 10.1080/13658811003766928 http://www.informaworld.com
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Z. Zhao et al.
1996, Papadias et al. 1999, Stefanidis et al. 2002). With the rapid development of spatial databases over the past decades, there is an increasing need for a measure of spatial congruence that can incorporate the four properties into a quantitative framework. However, it is difficult to simultaneously measure the four properties of spatial objects in quantitative terms, though it is easy to do so in qualitative terms, e.g. topological relations (Egenhofer 1989, Egenhofer and Al-Taha 1992). The difficulty in building such a quantitative measure of spatial congruence lies in the way in which the effects of these spatial and geometric properties on congruence measurement are traded off and integrated, and in the extent to which the measure is sensitive to small changes or differences in the four properties. To overcome this difficulty, this article develops and tests an algorithm of spatial congruence based on geometric congruity of two spatial areal objects (e.g., regions) in the Euclidean plane. Spatial congruence is defined and thus quantified as an increasing continuous function of congruity in the position, orientation, size, and shape of two spatial objects, dependent upon scaling, translation, and rotation. An expansion-based geometric matching technique is used to seek the best match between the two objects of interest for the examination and differentiation of the congruence effects of their spatial and geometric properties, while the expansion-inflated size effect is deflated or filtered out accordingly. The use of both expansion and deflation not only allows for a trade-off between size and position, both of which are found substitutable for each other by observing that congruity between two identical objects at different positions can be equally examined using expansion and translation, but also ensures that the congruence algorithm is highly sensitive to differences (even small ones) in the four properties. Three geographical objects (the states of Texas, Mississippi, and Louisiana) are used to show how the congruence algorithm computes tradeoffs among the four spatial and geometric properties in a geographic information system (GIS) environment, ArcGIS, where spatial statistical (or centrographic) methods are employed to estimate the centroid (representing position) and orientation of geographic objects, and GIS operations are used to compute the area of overlap between objects during the object expansion process. Meanwhile, three regular geometric objects are used to demonstrate how the congruence algorithm is sensitive to small changes or differences in each of the four properties of objects. The results show that the proposed congruence algorithm is capable of quantifying the extent of congruity between two spatial objects regardless of their position, orientation, size, and shape or how they are related (e.g. far, near, touching, or overlapped) as described in topological relations. It is necessary to mention several caveats about the scope of the proposed research. First, the term spatial congruence used in this article differs considerably from the concept of ‘spatial similarity’ or ‘spatial congruence’ widely used in the literature on spatial or image databases. The former refers to congruity of spatial objects in the same metric space, while the latter deals with congruity of objects between different databases without preserving the relative position of the objects (Cristani 1999, El-Kwae and Kabuka 1999, Petrakis 2002, Sciascio et al. 2004); our notion of spatial congruence is dependent on the position, orientation, size, and shape of spatial objects and thus on geometrical transformations such as scaling, translation, and rotation, while the concept of spatial similarity is not directly relevant to the position, orientation, size, and shape of spatial objects and therefore is independent of the three geometrical transformations (El-Kwae and Kabuka 1999, Stefanidis et al. 2002, Sciascio et al. 2004). Second, shape similarity is a special and the simplest case of spatial congruence, with the effects of position, orientation, and size to be removed in the measurement of the former (Zhao and Stough 2005) but to be kept in the assessment of the latter. Third, the notion of spatial congruence is limited to geometrical areal objects; it is not related to line or network features (Endreny and Wood 2003), point
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International Journal of Geographical Information Science
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objects (Kendall 1989) or nongeometric (e.g., attribute and density) similarities (Robinson and Bryson 1957, Tobler 1965, Cliff 1970). Fourth, the concept of a spatial object as used in this article refers to a two-dimensional (2-D) bounded object with specified position, orientation, size, and shape in Euclidean space. Fifth, orientation refers to the longest direction or axis along which an areal object extends, measured by an angle of the direction from north. Sixth, expansion and the other two types of geometrical transformations of translation and rotation are independently operated or manipulated around the centroids of the two objects of interest (see Zhao and Stough 2005). This article is structured around five sections following this introduction. Section 2 contains a literature review on spatial (and shape) similarity measurement. The function of spatial congruence is defined and the size–distance substitution principle presented in Section 3. They are then used to create a spatial matching algorithm. Examination of the performance of the algorithm is accomplished in Section 4 using three geographic and three geometric objects. This shows how the algorithm can be used in a GIS environment and spatial setting. The last section provides conclusions and a discussion of the need for future related research. 2. Literature review In the following, a brief literature review is presented with focus on geometric approaches to assessing similarity (or congruence) of two areal objects (or patterns) in scenes and images. For the purpose of comparison, this review is limited to the notion and algorithm of similarity as well as methods being used to compute the extent of similarity. Currently, two approaches have been widely used: the shape similarity approach and the spatial similarity approach. The shape similarity approach deals mainly with the shape correspondence of two single objects (Wentz 2000, Zhao and Stough 2005). Similarity is defined as a function of shape matching regardless of the position, orientation, and size of objects under comparison. With the assistance of geometrical transformations (translation, rotation, and scaling), geometric overlapping is used to obtain a best match between both objects. The best match may be achieved unconditionally (Wentz 2000) or under the condition that the two objects’ centroids are coincided for removing the effects of position, orientation, and size on shape, and the related algorithms of similarity are invariant to the three geometric transformations (Zhao and Stough 2005). The spatial similarity approach compares objects and their spatial relations across different spatial databases for the purpose of image retrieval (Sciascio et al. 2004) or sometimes of information consistency assessment (Abdelmoty and El-Geresy 2000). Spatial similarity is perceived as a function of correspondence of objects, spatial relations of objects, or both, across spatial scenes or databases (Frontiera et al. 2008, Nedas and Egenhofer 2008). Similarity of objects is examined in terms of best matching between their geometric approximations, e.g. the minimum bounding boxes, the convex hulls (Frontiera et al. 2008), and the sketched outlines (Stefanidis et al. 2002). Similarity of spatial relations is calculated by matching their geometric representations, e.g. the spatial orientation graphs (Gudivada and Raghavan 1995, El-Kwae and Kabuka 1999) and
