Measuring congruence of spatial objects

Transcription

1 International Journal of Geographical Information Science Vol. 25, No. 1, January 2011, Measuring congruence of spatial objects Zuoquan Zhao a, *, Roger R. Stough b and Dunjiang Song a a Institute of Policy and Management, Chinese Academy of Sciences, Beijing, China; b School of Public Policy, George Mason University, Fairfax, VA, USA (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 Ò. 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. zhao1@casipm.ac.cn ISSN print/issn online # 2011 Taylor & Francis DOI: /

2 114 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 Ò, 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

3 International Journal of Geographical Information Science 115 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 <-strings (Gudivada 1998, Sciascio et al. 2004). Similarity of spatial relations is also calibrated by combining the individual relations of distance, direction, and topology (Papadias et al. 1999) or by counting gradual changes in these three relations (Bruns and Egenhofer 1996). Similarity of both objects and spatial relations is explored as an additive function of object s and relational matching (Stefanidis et al. 2002, Nedas and Egenhofer 2008). All determinants (or dimensions) of spatial similarity are measured on quantitative scales, and added up to overall

4 116 Z. Zhao et al. similarity after normalizing each determinant (Gudivada 1998, El-Kwae and Kabuka 1999). Like in shape similarity assessment, most algorithms of spatial similarity are invariant to translation, rotation, and scaling (Gudivada and Raghavan 1995, Gudivada 1998, El-Kwae and Kabuka 1999, Stefanidis et al. 2002, Sciascio et al. 2004). This shows that there is something in common between shape and spatial similarities: the relative position, orientation, and size of objects (or scenes) of interest need not be preserved in the evaluation of similarity. Notably, the two aforementioned similarity approaches differ in their perception of similarity. According to the literature on semantic similarity (Schwering 2008), there are two different notions of similarity: one is based on commonalities and differences, suggesting that more commonalities (or fewer differences) lead to increases in similarity between two representations of concepts or objects, while the other is based on distance, indicating that similarity decreases with the semantic distance between two representations of concepts. The shape similarity approach compares two objects in terms of their commonalities (intersection) and differences (non-intersection) with the assistance of geometric matching, while the spatial similarity approach deals with overall scene similarity in a multidimensional framework by examining each dimension (individual object and spatial relation) as a metric. As shown above, spatial similarity is examined as a type of shape similarity (Frontiera et al. 2008) or a derivative of the three spatial relations (Papadias et al. 1999); it is not related directly to the position, orientation, size, and shape of spatial objects under comparison. In the next section, we follow the notion of commonalities-and-differences-based similarity, and present a quantitative framework of spatial congruence by exploring the congruence impacts of the four properties of spatial objects. 3. Spatial congruence and expansion In this section, we first define spatial congruence for two areal objects; we then present the size distance substitution principle in congruence measurement, and use this principle to design an algorithm of spatial congruence based on the expansion transformation. The notion and algorithm of spatial congruence are simplified to differentiate the congruence impacts of the various properties of objects Spatial congruence: definition and function Spatial congruence refers to the extent of geometric congruity of two areal objects in the Euclidean space. It is an increasing function of congruity in the position, orientation, size, and shape of the two objects. Perfect spatial congruence appears when the two objects have the same position, orientation, size, and shape. Here, given two areal objects A and B as in Figure 1, p A, p B refer to their respective positions, represented by their centroids; o A, o B refer to their respective orientations; s A, s B refer to their respective sizes; and h A, h B refer to the respective shapes. Denote the function of spatial congruence I sc : I SC ¼ fðp A ; p B ; o A ; o B ; s A ; s B ; h A ; h B Þ ð1þ According to the definition, I sc would increase if differences between p A and p B, between o A and o B, between s A and s B, and/or between h A and h B reduce, and decrease if these differences become larger.

5 International Journal of Geographical Information Science 117 Figure 1. Two spatial objects with their centroids marked The size distance substitution principle There are two geometrical transformations that can be used to identify whether two areal objects are completely congruent: one is translation and the other is expansion. Given objects A and B as shown in Figure 1, with the same orientation, size, and shape but different positions, translating A and overlapping B with A around their centroids would make A and B perfectly matched; likewise, expanding A and B infinitely at the same rate around their centroids would also lead to a (nearly or quantitatively) complete match between A and B as the distance between their centroids tends to be ignorable with respect to the increased size. Given other conditions constant, the farther apart the two objects, the more they both need to be enlarged to achieve a perfect spatial match; meanwhile, the smaller the two objects, the more the expansion they need to obtain a perfect match. This suggests that the size increase under expansion is equivalent to the decline in distance (or closeness in position) under translation; in other words, the congruence impacts of size and position are relevant and thus substitutable for each other in the examination of spatial congruence. The size distance substitution principle still holds when two objects are not the same shape, size, and/or orientation. Figure 2 shows how size and position are related to each other and how expansion increases the extent of congruence between two objects. With fixed positions under expansion, the square and circle double, triple, and quadruple their sizes (see Figure 2b d), and turn back to their original sizes (see Figure 2e g) as the scale of space Figure 2. Effects of expansion and de-scaling on congruence: (a) no expansion; (b) double expansion; (c) triple expansion; (d) quadruple expansion; (e) 1/2 scaling; (f) 1/3 scaling; (g) 1/4 scaling.

6 118 Z. Zhao et al. contracts one-half, two-thirds, and three-fourths, respectively. The two pairs of objects in Figure 2b and e show identical extents of congruence. It is also the case for the two pairs of objects either in Figure 2c and f or in Figure 2d and g. This shows that expanding two objects is like moving both objects closer, reducing the position effect, and consequently increasing the degree of congruence between the two objects. In short, the position effect is closely related to the size effect: one would increase if the other decreases. This size distance substitution principle suggests that expansion can be used to examine the extent of congruence between areal objects of different positions if the expansionderived size effect can be appropriately deflated or filtered out. As shown above, expansion increases the extent of congruity between two objects with different positions. Meanwhile, expansion inflates the congruence impact of size, and thus the inflated size effect needs to be removed. This suggests that the algorithm of spatial congruence should be inversely related to the magnitude of expansion on the two objects of interest. Given two objects with different sizes, shapes, and orientations, for example, the larger their size difference, the more the smaller object needs to be expanded to attain a best overlap with the larger object, and the more the expansion-led size effect needs to be removed The algorithm of spatial congruence The function of spatial congruence I sc [Equation (1)] is evaluated using the algorithm of spatial congruence as follows: where R I sc ¼ max E x ij R ¼ i ¼ A B A B SA2 S A1 ð2þ ð3þ ð4þ j ¼ SB2 S B1 ð5þ where R is the ratio of overlap between A and B under expansion; i and j are the respective ratios of enlargement of A and B, with the reciprocals of both i and j representing the magnitude of de-scaling (i 1; j 1); A B is the intersection of A and B under expansion; A B is the union of A and B under expansion; S A2 and S B2 are the respective expanded areas of A and B; S A1 and S B1 are the respective areas of A and B before expansion; and E x refers to the expansion transformation relating to I sc. I sc has several properties. First, it is an increasing function of congruity in the position, orientation, size, and shape of the two objects and is sensitive to differences in these four spatial properties of both objects, as defined in Equation (1). Here the overlap ratio R captures primarily the effects of shape and orientation, indicating that the more congruous

7 International Journal of Geographical Information Science 119 is the shape and orientation of the two objects, the larger is R; and i and j together reveal primarily the effects of both position and size, suggesting that the more congruous are both the size and position of the two objects, the less is the de-scaling, and the larger is R. Relating to the magnitudes of expansion on both objects, the deflation represents a compromise between the effects of position, orientation, size, and shape. Second, I sc varies from 0 to 1. Third, I sc is independent of scale and scaling because it would be constant when the two objects and the underlying plane expand at the same rate. This suggests that the measurement of spatial congruence is not limited by spatial scales. Fourth, symmetry holds for I sc : I sc ða; BÞ ¼ I sc ðb; AÞ. Fifth, transitivity fails to hold for I sc. The extents of congruence among three areal objects all rest upon their closeness in the plane Simplifications I sc [Equation (2)] can be used to assess the extent of congruence between two spatial objects regardless of their position, orientation, size, and shape. Therefore it can be simplified to isolate the effects of position, orientation, size, and shape with the assistance of translation and rotation as shown below: Denote by I soc an increasing function of congruity in the orientation, size, and shape of the two objects A and B; let p A ¼ p B by translating A to B with their centroids coincident. From Equation (2), R I soc ¼ max E X T r j where j is the ratio of enlargement of B and T r refers to the translation transformation relating to I soc. Here we assume B is smaller than A, which need not be enlarged like B. Similarly, denote by I oc an increasing function of congruity in both the size and shape of A and B; let o A ¼ o B by rotating B around its centroid until the lines of orientation of both objects are coincident. From Equation (6), R I oc ¼ max E x T r R o j ð6þ ð7þ where R o refers to the rotation transformation relating to I oc. Likewise, denote by I os an increasing function of congruity (or similarity) in the shape of A and B; expand A around its centroid until both objects achieve best match. From Equation (7), I os ¼ max ðrþ E x T r R o ð8þ where E x refers to the expansion transformation under which the size effect, including the expansion-derived size effect, need not to be removed. We do not use s A ¼ s B as a condition for measuring shape similarity because the shape of spatial objects may be very complex (see Zhao and Stough 2005). The four algorithms of congruence as shown in Equations (2), (6), (7), and (8) show the complexity of spatial congruence between two objects. I sc and I os are the two extremes of the spectrum of spatial congruence: I sc is related to all of the four properties of objects, whereas

8 120 Z. Zhao et al. I os is associated with only a single property - the shape of objects. Both I sc and I soc measure congruity between two objects in the spatial setting, while I oc and I os examine spatial objects as general geometric ones regardless of their position and orientation Effects of position, orientation, size, and shape The four functions of spatial congruence can be ranked as below: Due to the position effect, we have I sc I soc : Due to the orientation effect, we have I soc I oc : Due to the size effect, we have I oc I os : Thus, we have I sc I soc I oc I os : The effects of position, orientation, size, and shape all account for the extent of congruence between two spatial objects, keeping the two objects away from perfect spatial congruence, where the two objects are the same position, orientation, size, and shape. The position effect is identified by subtracting I sc from I soc. The orientation effect is identified by subtracting I soc from I oc. The size effect is identified by subtracting I oc from I os. The shape effect is identified by using I os. 4. Applications In the following two subsections, we first employ three geographical objects the outlines of Texas, Louisiana, and Mississippi to show how the framework of spatial congruence is operated in a GIS environment, including the use of spatial statistics to compute the centroids and orientations of the three states in the plane. We then use three regular geometric objects an equilateral triangle, a square, and a circle to show how the functions of spatial congruence are sensitive to differences or changes in the position, size, and shape of areal objects, and why these algorithms would increase when the spatial differences and changes become small Congruence of geographical objects We choose the outlines of Texas, Louisiana, and Mississippi because the three states not only have meet and disjoint topological relations but also differ remarkably from each other in position, orientation, size, and shape (Figure 3). The boundary data of the three states are collected from the polygon shapefile of the states in ArcGIS9.2 Ò. The projected coordinate system is NAD_1983_Texas_Statewide_Mapping_System predefined in ArcGIS9.2 Ò. Here we acknowledge there is a loss of information for the three states in this global-to-planar projection and leave the spherical measurement of spatial congruence for further research. All operations and procedures in data treatment and computation are implemented in a Visual Basic for Applications (VBA) environment in ArcGIS9.2 Ò Data prehandling To measure the extent of spatial congruence among the three states, we deal with the boundary data as follows. First, the islands of the three states are deleted; second, the

9 International Journal of Geographical Information Science 121 Figure 3. The outlines of Texas, Louisiana, and Mississippi (The expanded outlines of Texas and Mississippi show the extent of overlap from which the algorithm of spatial congruence I sc is calculated). boundaries (or outlines) of the three states are generalized and thus smoothened for the simplicity of computation. Using the command generalize in ArcGIS9.2 Ò, we set the Maximum allowable offset to 10 km. The generalization increases the area of Texas from 682,093.6 km 2 to 682,564.8 km 2, that of Louisiana from 117,696.9 km 2 to 117,880.7 km 2, and that of Mississippi from 122,959.3 km 2 to 123,020.5 km 2, leading to a less 0.2% gain of area. Third, the generalized boundary polygons are gridded for the computation of the centroids and orientations of the three states. The grid cell has sizes of 5 km 5 km. Fourth, the centroids and orientations of the three states are estimated using the spatial statistics standard deviational ellipse, whose center and major axis represent the centroid and orientation of the state polygons, respectively (Wong and Lee 2005). Texas, Louisiana, and Mississippi have their centroids at (1,063,097.45, 1,034,840.17), (1,759,149.31, 1,020,334.31), and (1,965,922.12, 1,220,706.33), respectively; their respective orientations are , , and degrees from the north. Here we also acknowledge that the 5 km 5 km gridding results in a loss of boundary information but this gridding has very limited influence on the estimation of the extent of interstate congruence.

10 122 Z. Zhao et al Computation (1) Resize a state polygon by j Given (X k, Y k ) the coordinate vector of the vertices of polygon for each state, resize(x k, Y k, dareamultiplesize), where k refers to the number of vertices and dareamultiplesize is the ratio of expansion in terms of the expanded area divided by the original area of the state polygon. The original area of the state polygon is or A ¼ 1 x 1 x 2 2 y 1 y 2 þ x 2 x 3 y 2 y 3 þ x n x 1 y n y 1 ð9þ A ¼ 1 ð 2 x 1y 2 x 2 y 1 þ x 2 y 3 x 3 y 2 þþx n 1 y n x n y n 1 þ x n y 1 x 1 y n Þ ð10þ Multiply (X k, Y k )byj(j1); then we get an expanded polygon whose area is j 2 times of A. To get the degree of spatial congruence between two state polygons, we expand each polygon 100 times: 50 times for integer-level expansion and 50 times for decimal-level expansion. Set j = 1, 2, 3,..., 50 first, and then set j = m 0.02, m 0.04, m 0.06,..., m 0.50; here j = m (m is an integer) when maximum de-scaled congruence appears. In this case, it is not necessary to run extra expansions on the three state polygons because m 6 (see Table 1 in the following). (2) Translate a state polygon by (dx, dy) Translate(X k, Y k,dx, dy), where dx, dy refer to respective changes in (X k, Y k ) due to translation. (3) Rotate a state polygon by around (x 0, y 0 ) Rotate(X k, Y k, x 0, y 0, dtheta), where x 0, y 0 are the point (or centroid of the state polygon) for rotation, and dtheta is the angle (degrees) of rotation. As shown in Figure 4, point Px ð 1 ; y 1 Þ in the coordinate system xoy will be changed to x 0 1 ; y 0 1 in the rotated coordinate system x 0 o 0 y 0, where there is no translation involved. x 0 1 ¼ x 1 cos þ y 1 sin ð11þ y 0 1 ¼ x 1 sin þ y 1 cos ð12þ Table 1. Degrees of spatial congruence among Texas, Louisiana, and Mississippi (%). State pair I sc I soc I oc I os Texas, Louisiana (i, j) Texas, Mississippi (i, j) Louisiana, Mississippi (i, j) (2.10, 3.90) (1.00, 2.96) (1.00, 3.42) (1.00, 5.76) (4.70, 5.26) (1.00, 3.22) (1.00, 4.50) (1.00, 5.64) (1.50, 3.34) (1.00, 1.00) (1.00, 1.00) (1.00, 1.01) Note: i and j are the ratios of expansion on the corresponding pair of states.

11 International Journal of Geographical Information Science 123 y y y 1 P d y 1 e θ a b c x 1 x Figure 4. O O Rotation of coordinate systems. f (4) Get the ratio of overlap R for two state polygons GetIntersectionArea(X k1, Y k1, X k2, Y k2 )/GetUnionArea(X k1, Y k1, X k2, Y k2 ), where (X k1, Y k1 ) and (X k2, Y k2 ) are the coordinate vectors of the vertices of state polygons 1 and 2. (5) Get the index of spatial congruence between two state polygons SpatialCongruence(X k1, Y k1, X k2, Y k2 ) Remove the expansion-led size effect wherever necessary Plot the surface of spatial congruence Find the peak of the congruence surface A 3-D surface is constructed to demonstrate the variation of spatial congruence with the magnitudes of expansion on two state polygons. The surface of spatial congruence consists of 2500 values since each polygon is expanded 50 times. The peak of the congruence surface corresponds to the degree of spatial congruence we seek. θ x 1 x Results Table 1 and Figure 5 show the extents of spatial congruence among the states of Texas, Louisiana, and Mississippi. The results confirm that spatial congruence is an increasing function of geometric congruity between two areal objects. For each pair of the three states, the degree of congruence increases all the way from I sc, I soc, I oc,toi os. I os represents the shape effect alone, showing the maximal congruence that can be reached for any pair of the three states under geometrical transformations. The results demonstrate how shape, size, orientation, and position make different contributions to the extent of spatial congruence. The shape effect demonstrates a high congruence among the three states, while the maximal shape effect is found between Texas and Mississippi; The maximal size effect is found between Texas and Mississippi, reducing the extent of congruence between the two states by The maximal effects of position and orientation are found between Louisiana and Mississippi, decreasing the extent of congruence between the two states

12 124 Z. Zhao et al. Figure 5. Congruence among the outlines of Texas, Louisiana, and Mississippi. by and 0.113, respectively. In relative terms, it is between Texas and Mississippi that position, orientation, and size all make the largest contributions to congruence: a 709.9% increase from I sc to I soc, a 29.5% increase from I soc to I oc, and a 125.6% increase from I oc to I os. The results also show how the three states differ from one another in the dimension of spatial congruence. For example, Mississippi and Louisiana have the largest extent of spatial congruence but the least degree of shape similarity, while Mississippi and Texas have the least extent of spatial congruence but the largest degree of shape similarity. Figure 6 shows how the algorithms of congruence vary explicitly with the magnitude of expansion on each pair of the three states. The surface of congruence reveals the rough landscape including peaks of congruence when any pair of the three states of interest enlarge their sizes. As shown in Figure 6 (top left), for example, expansion makes Texas and Mississippi more congruent by going through several topological relations : disjoint (far, near), meet, and overlap. At the same time, the expansion-derived size effect is filtered out by dividing the magnitude of expansion on the two states. Figure 7 demonstrates how the algorithm of spatial congruence I sc changes more explicitly (than those shown in Figure 6) with the magnitude of expansion on the states of Texas and Mississippi. The surface of congruence displays a detailed and high-resolution landscape, including the peak of congruence between the two states when each is expanded around the magnitude 5. It is peaked at (4.70, 5.26) with a higher value of congruence than when peaked at (5, 5) as shown in Figure 6 (top left). It shows that decimal-level expansion is necessary to find the maximal value of the spatial congruence between two objects under comparison.

13 International Journal of Geographical Information Science 125 Figure 6. Surfaces of spatial congruence among Texas, Louisiana and Mississippi. The red to blue gradient displays a gradual decline in Spatial congruence Congruence of geometric objects The three regular geometric objects - an equilateral triangle, a square, and a circle - are used to show the extent of sensitivity of the functions of spatial congruence to changes in the position, size, and shape of areal objects. Of the nine pairs of spatial objects as shown in Figures 8 and 9, each differs in position. Some pairs have identical shapes and sizes (see Figure 8), while others are different in size, shape, and/or both (Figure 9). In Figure 8, three pairs of identical objects are disjoint from each other as if a third counterpart is in between each pair (Figure 8a c), while a pair of circles touch each other (Figure 8d). In Figure 9a c, the small square, one-fifth the size of the large square in segment length, meets the large

14 126 Z. Zhao et al. Figure 7. Surface of spatial congruence between Texas and Mississippi. square at three different positions. From Figure 9d and e, the circle doubles its size, and the square triples its size but with their positions unchanged. The degree of congruence (I sc ) for the nine pairs of spatial objects is reported in Table 2. The results demonstrate that, of the three pairs of identical objects, the two squares are the most congruent, and the two equilateral triangles are the least congruent. The shape effect contributes to the difference in I sc among the first three pairs. In addition, the position effect is obvious for each pair with identical shapes, sizes, and orientations. Meanwhile, the two adjacent circles are more congruent than the two disjoint ones due to the position effect (Figure 8c and d). The results also demonstrate that the degree of congruence between two spatial objects increases as the position effect declines (Figure 9). For the two squares of different sizes as shown in Figure 9a c, I sc rises from 2.8 through 3.8 to 5.7% as the small square moves up from the bottom along the right side of the large square. The increases in congruence result from the decline of the position effect. Here, the extent of congruence between the two squares is attributed to the combined effects of position and size because both objects have different positions and sizes but identical orientations and shapes. In Figure 9d and e, I sc increases remarkably from 1.6 to 9.5% as the circle and square increase their sizes. The increase in I sc results from the increasing size effect and decreasing position effect. Here, the Table 2. Degrees of congruence (I sc ) among circles, squares, and equilateral triangles (%). Object pair in figure no. 8a 8b 8c 8d 9a 9b 9c 9d 9e I sc

15 International Journal of Geographical Information Science 127 Figure 8. (a d) Four pairs of geometrical objects with identical shapes and sizes (dotted objects are set as the references of the relative positions of these objects). integrated effects of shape, size, and position account for the magnitude of congruence between the square and circle. The three geometric objects are also used to show how the proposed function of object congruence [Equation (7)] works. Circles and the squares and triangles that are inscribed in the circles are used to assess the extent of object congruence with the involvement of translation and rotation besides expansion (Figure 10). The resulting degrees of object congruence (I oc ) are shown in Table 3 together with the degrees of object (or shape) similarity I os, which were first reported in Zhao and Stough (2005). It is shown that, for each pair of objects, the degree of shape similarity is larger than the corresponding degree of object congruence because of the size effect. Moreover, the inscribed square is more congruent to the circle than the inscribed equilateral triangle, which is more congruent to the square than to the circle (Figure 10a and b). Both the shape and size effects account for the variation in congruence among the three objects (Figure 10a c). (a) (b) (c) (d) (e) Figure 9. (a e) Five pairs of geometrical objects with different sizes (dotted objects are set as the size references of the relative positions and sizes of these objects).

16 128 Z. Zhao et al. (a) (b) (c) (d) Figure 10. (a d) Five pairs of geometrical objects with identical positions but different sizes (dotted objects are set as the size references of objects). Table 3. Comparison between object congruence I oc and object similarity I os (%). Object pair Circle-triangle (Figure 10a) Circle-square (Figure 10b) Square-triangle (Figure 10c) Circle-circle (Figure 10d) I oc I os Conclusion In this article, we presented a quantitative geometric framework of spatial congruence for evaluating the extent of congruity between two areal objects in a Euclidean plane. Spatial congruence was defined as an increasing function of geometric congruity between two areal objects under comparison. An algorithm of spatial congruence was introduced to estimate the congruence effects of four spatial properties of objects using an expansion-based geometric matching approach. The expansion transformation allows for examining and differentiating the congruence effects of position, orientation, size, and shape, while the expansion-derived size effect is deflated. Using three geographical objects (the states of Texas, Louisiana, and Mississippi) and three regular geometric objects (an equilateral triangle, a square, and a circle), this article demonstrated that spatial congruence is an increasing function of geometric congruity between two areal objects, and thus is capable of quantifying relations of two objects from a congruence perspective. Several directions should be addressed in future research. Theoretically, the proposed functions of spatial congruence are extended to examine congruity of areal objects on the spherical surface and in 3-D space, and to explore the extent of congruence among distributions of attributes, e.g. population density and housing prices. On the spherical surface, for example, operations and procedures for expansion, rotation, and translation are very different from and more difficult than those on the planar surface; for assessing congruence between two non-uniform attribute distributions, appropriate methods need to be developed to count the value of attribute when there is a process of expansion on attribute distributions. The proposed framework of spatial congruence is also used to examine more complex spatial objects in real-world situations than the geographical and geometric objects used in this article. Theoretically, the spatial congruence approach is capable of handling perforated and fragmented objects because the shape effect includes the effects of both perforation and fragmentation (Zhao and Stough 2005). Therefore, the proposed approach could be used to evaluate numerous real-world spatial relations in spatial analyses and spatial databases. In examining such spatial processes as urban sprawl, where a series of spatial objects appear at different periods of time, the four indices of spatial congruence can be used to investigate how an object changes its position, orientation, size, and shape compared to itself at an earlier time. In spatial databases in which the spatial relations among numerous objects need

17 International Journal of Geographical Information Science 129 to be identified, the spatial congruence approach can be used to compare each pair of objects and to display the related results through a matrix. In some situations, the extents of spatial congruence among several sample objects may sufficiently represent the ones among a number of objects in a spatial image because the spatial congruence framework can provide as many as four congruence indicators for any pair of the sample objects. Acknowledgments The authors would like to thank David Wong and Adam Lederer for their comments and contributions to the manuscript during its infancy stages, and the editor and two anonymous reviewers for very valuable comments on earlier versions of the manuscript. This research is funded in part by the National Natural Science Foundation of China under grant number References Abdelmoty, A. and El-Geresy, B., Assessing spatial similarity in geographic databases. In: P. Atkinson and D. Martin, eds. GIS and geocomputation, innovations in GIS (7). London: Taylor & Francis, Bruns, H.T. and Egenhofer, M.J., Similarity of spatial scenes. In: Seventh international symposium on spatial data handling. London: Taylor & Francis, 4A.31 4A.42. Chen, J., et al., AVoronoi-based 9-intersection model for spatial relations. International Journal of Geographical Information Science, 15, Cliff, A.D., Computing the spatial correspondence between geographical patterns. Transactions of the Institute of British Geographers, 50, Cristani, M., The complexity of reasoning about spatial congruence. Journal of Artificial Intelligence Research, 11, Egenhofer, M.J., A formal definition of binary topological relationships. Lecture Notes in Computer Science, 367, Egenhofer, M.J. and Al-Taha, K.K., Reasoning about gradual changes of topological relations. Lecture Notes in Computer Science, 639, Egenhofer, M.J. and Franzosa, R., Point-set topological spatial relations. International Journal of Geographical Information Systems, 5, Egenhofer, M.J., Shariff, A., and Rashid, B.M., Metric details for natural-language spatial relations. ACM Transactions on Information Systems, 16, El-Kwae, E.A. and Kabuka, M.R., A robust framework for content-based retrieval by spatial similarity in image databases. ACM Transactions on Information Systems, 17, Endreny, T.A. and Wood, E.F., Maximizing spatial congruence of observed and DEM-delineated overland flow networks. International Journal of Geographical Information Science, 17, Frontiera, P., Larson, R., and Radke, J., A comparison of geometric approaches to assessing spatial similarity for GIR. International Journal of Geographical Information Science, 22, Galton, A., Qualitative spatial change. Oxford, UK: Oxford University Press. Godoy, F. and Rodriguez, A., Defining and comparing content measures of topological relations. GeoInformatica, 8, Gudivada, V.N., <-string: a geometry-based representation for efficient and effective retrieval of image by spatial similarity. IEEE Transactions on Knowledge and Data Engineering, 10, Gudivada, V.N. and Raghavan, V.V., Design and evaluation of functions for image retrieval by spatial similarity. ACM Transactions on Information Systems, 13, Hagedoorn, M. and Veltkamp, R., Reliable and efficient pattern matching using an affine invariant metric. International Journal of Computer Vision, 31, Jiang, J. and Worboys, M., Event-based topology for dynamic planar areal objects. International Journal of Geographical Information Science, 23, Kendall, D.G., A survey of the statistical theory of shape. Statistical Science, 4,

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