visual boundary (a) a vector format region (b) a raster format region region b region a Boundary Interior of region a Interior of region b

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1 A Spatial Algebra for Content-based Retrieval Mohan S. Kankanhalli Jiang Xunda Jiankang Wu Real World Computing Partnership, Novel Function ISS Laboratory Institute of Systems Science, National University of Singapore Kent Ridge, Singapore ( Abstract We present a scheme for retrieval by spatial reasoning in a spatial information system such asage- ographic Information System. We base this scheme on a spatial algebra consisting of topological (point, line and region), metric and morphological operators. We detail the operations available in this algebra and show howcontext can be built in for exible retrieval of spatial data. 1 Introduction We have developed a spatial algebra for the purpose of exible retrieval of spatial information. Traditionally, spatial algebra work has been been conned mainly to topological operations [] or to metric operations [1]. Also, all this work has been limited to mainly -D point sets (areal objects). Wepresent a comprehensive spatial algebra which can combine topological, metric and morphological operations. The topological operators allow for lineal and point objects also. These are distinguishing features of our spatial algebra 1. Data Structure This will use the dual rastervector data structure [6] and hence will be able to exibly handle both vector as well as raster representation of data. Due to the concept of the visual boundary which ensures a zero-thickness boundary in the raster domain, many operations will be very easy in this domain.. Contexts Since most real world operations have an implicit context built, we provide a mechanism for explicit building of contexts by the user/application of this spatial algebra.. Eciency All operations are performed in the raster domain for the sake of eciency. For this purpose we have developed an ecient vector-toraster conversion algorithm.. Comprehensive We have developed a wide suite of operators which can be used as primitive spatial operators for a very large range of spatial queries. In particular, we have developed (a) Topological Operators These operations are invariant under any topological homeomorphism. They are not sensitive to metric information. The spatial algebra we have developed can handle point objects and lineal objects. They can be exibly combined with areal operators. (b) Metric Operators These operators depend on the underlying metric space. (c) Morphological Operators These are shape transforming operators. Data Representation We consider the dual raster-vector data structure (DRVDS) [6] as the basis for all operations. We emphasize this aspect because most of the previous work has neglected practical implementation aspects of spatial algebras. We believe that the practicality consideration is vital and hence wehave designed the algebra for easy implementation. Basically, DRVDS represents the continuous data (i.e. the vector data) in the cartesian coordinate system having innite precision. The discrete information (raster data) is represented in a digital grid system having nite precision. An example of vector and raster data is shown in gure 1. There is a problem having data in two very dierent representations { a zero-thickness vector line boundary has a one-pixel thick analog in the raster domain. This makes the spatial algebra very complicated requiring additional neighborhood information []. It also introduces ambiguity in situations such asgure. We avoid this problem by using the concept of a visual boundary which is a zero-thickness line representation which separates two areal regions (shown in two dierent colors). So fundamentally, aline boundary corresponds to the raster visual boundary between two areal objects. See gure for an example. This representation scheme has three classes of objects 1. Class 1 These are point objects which are objects that represent positional information e.g. control points, ight paths etc.. Class These are lineal objects which are objects whose one or more dimensions are neglectable e.g. a cable.. Class These are areal objects which are objects with two dimensions e.g. areas, buildings etc. We use the vector representation for class 1 and class objects while for the class objects, we use the native data format i.e. either vector or raster. For example,

2 the map information is usually in vector format while satellite information is usually in a raster format. For all mixed operations, we use the raster representation for eciency. A vector-to-raster conversion algorithm can be used for all the vector objects. We have developed an ecient algorithm for this conversion which has a provably nite bounded loss in accuracy. Topological Operators In this section we consider the pure topological operations i.e. they are invariant under all topological transformations (which means they are homeomorphic). We assume a point set topological model [] and our work is similar to that of []..1 Areal-Areal Object Operators We assume that all topological operators are for \spatial regions" which are dened to be -D points sets homeomorphic to a -disk []. For an object A (corresponding to a spatial region), the interior of A, denoted by A 0, is dened as the union of all open sets in A. The closure of A, denoted by A, istheintersec- tion of all closed sets of A. The exterior, denoted by A ;, is the complement ofa, which are all the points belonging to R not contained in A. The boundary of A, denoted is the intersection of the closure of A and the closure of A ;. Assume that 1. The continuous domain spans RR and the discrete domain spans ZZ.. In the discrete domain, the interior and the boundary are 8-connected.. In the discrete domain, the exterior is - connected. Assume that we have two objects A and B having interiors A 0, B 0 exteriors A ;, B ; Now, unlike the Egenhofer and Sharma [] approach, we have no problems in translating ideas in RR []tothezz domain. The reason for this is that boundaries in RR correspond to visual boundaries in ZZ which alsohave no thickness. So, essentially at the expense of slight accuracy loss (boundary conversion is bounded by an error of 0 pixel []), we have removed the neighborhood problems of going from continuous to discrete. Note that both vector and raster representations can be subsumed under the point set topological basis. We now dene the following intersection matrix, R(A B), which will be used to dene the topological \ \ B \ B ; A 0 A 0 \ B 0 A 0 \ B ; A ; A ; \ B 0 A ; \ B ; We will now dene the various operators in terms of these matrices. 1. disjoint. overlap. meet. inside. equal 6. contains 7. covers 8. covered by Note that our areal object is like shown in Figure 1. Also note that we obtain 9 cases unlike 16 cases obtained in []. The reduced number of cases is due to the DRVDS representation which reduces ambiguity due to zero-thickness boundaries in the raster domain. Moreover, unlike [], we do not assume connectedness conditions to prevent special cases arising out of degeneracy. All these factors make the implementation a lot more ecient.

3 . Lineal-Lineal Object Operators Till now, we have considered operations only on -D point sets (areal objects). However, the real world consist of 0-D (point objects) and 1-D (lineal objects) point sets as well. While many of the earlier operators can be extended for the point and lineal objects, some operations are unique for for the point/lineal objects. We now detail such operations. Again we assume a point set representation. An areal object A has an interior A 0, a and an exterior A ;. A lineal object L has only a and an exterior L ;. A point object P also has only a and an exterior P ;. We dene the intersection matrix for two lineal objects L 1 and L R(L 1 L ) \ 1 \ L ; L ; 1 L ; 1 \ L; This has ve possibilities (out of the total possible 16) 1. on. equal R(L 1 L ) = R(L 1 L ) =. Lineal-Areal Object Operators We dene the intersection matrix for these \ \ A \ A ; L ; L ; \ A 0 L ; \ A ; There are four possibilities here (out of the 6 total possible) 1. disjoint. touches. goes thru. within. disjoint R(L 1 L ) =. covered by. overlap. covers R(L 1 L ) = R(L 1 L ) =. Point-Point Object Operators We dene the intersection matrix for these objects R(P 1 P ) \ 1 \ P ; P ; 1 P ; 1 \ P ; This has two possibilities (out of the total possible 16) 1. equal. disjoint R(P 1 P ) = R(P 1 P ) =. Point-Lineal Object Operators We dene the intersection matrix for these \ \ L ; P ; P ; \ L ; This has two possibilities (out of the total possible 16) 1. on. disjoint.6 Point-Areal Object Operators We dene the intersection matrix for these \ \ A \ A ; P ; P ; \ A 0 P ; \ A ; This has three possibilities (out of the total possible 6 )

4 1. within. on. outside We havedeveloped in detail the composition of the various topological operators []. However, due to space constraints we cannot present themin this paper. Metric Operators In this section, we will concentrate on the spatial operations which depend on the metric properties of the underlying domain. Hence they are not invariant under topological transformations. 1. Near All metric operations have to depend on the underlying metric space. Certain operations also depend on context. For example, to dene the `near' operator. acontext has to be provided to dene the concept of nearness. This can vary from person to person. Hence it is important that a user-denable context be allowed. We assume a crisp threshold which separates the near and the non-near sets. However, this concept can easily be made more exible by employing a fuzzy settheoretic underpinning. Assume that we havetwo objects A and B { we can assume the point set model introduced earlier. Also assume that we have a context function T dened as T (x) = (x < k) which means T (x) = 1ifx < k and T (x) = 0 if x k. We denote the near operator having context k by k. Then the operator A k B (A near B), returns a boolean value of either 1 or 0 A k B = T (H(A B)) We use the Hausdor distance to compute the k operator. Given two nite point sets A and B A = fa 1 a a m g B = fb 1 b b n g Then the Hausdor distance is dened as where H(A B) = max(h(a B) h(b A)) h(a B) = max min jja ; bjj aa bb where jj jj is some norm over the metric space e.g. L (Euclidean norm). Note that this can be computed for all points on the raster boundary for eciency, in case of raster encoding. Foravector encoding, this needs to be computed only for the vertex points.. Within This is a unary operator and it also has acontext k. We denote this operator by k. For an object A, k (A) returns all objects within a distance k from A. The Hausdor metric (dened earlier) is used to compute the distance.. Above This operator is denoted by.. Below This operator is denoted by.. Left This operator is denoted by. 6. Right This operator is denoted by. For all the above four operators we assume that there are two objects A and B, with their associated isothetic minimum bounding rectangle. Suppose that the bounding rectangle of A has the following coordinates A xmin, A xmax, A ymin and A ymax and B has B xmin, B xmax, B ymin and B ymax. Then we dene A B (A ymax > B ymax ) ^ (A ymin > B ymin ) A B (A ymax < B ymax ) ^ (A ymin < B ymin ) A B (A xmin < B xmin ) ^ (A xmax < B xmax ) A B (A xmin > B xmin ) ^ (A xmin > B xmin ) Note that like in the case of topological operators, we can have compositions of the metric operators [] but we do not present them here. Morphological Operators In this section we describe a set of morphological operators useful for a spatial information system { some of these operators are standard and some are new. 1. dilation For an object A and a structuring element B, the dilation of A by B is dened as A B = fcjc = a + b for some aa and bbg. erosion For an object A and a structuring element B, the dilation of A by B is dened as A B = fxj(x + b)a for every bbg. opening The opening of an object A by a structuring element K, is dened as A B = (A K) K

5 . closing The closing of an object A by a structuring element K, is dened as A B = (A K) K Boundary Interior. non-uniform dilation This operator is a new operator with a context dened. Suppose that there is a real-valued function F dened over the pointset constituting the domain. This function is assumed to be dened over all the points. For example, this function could be a population density function or the income distribution function or perhaps the disposable income distribution function over a spatial domain. Then suppose there exists a context function G i.e. (a) a vector format region (b) a raster format region Figure 1 A spatial region in vector and raster form region b region a G(x) = (x > k) which means that G returns a boolean value i.e. G(x) = 1 if x >= k and G(x) = 0 if x < k. Then we dene the non-uniform dilation of an object A by a structuring element B as A ^B = fcj(c = a + b for some aa and bb) ^ (G(F (c) = 1))g This operator non-uniformly dilates A based on the density function F and the context function G. 6. non-uniform erosion Assume a real-valued function F and a context function G. The nonuniform erosion of A by a structuring element B is dened as A ^ B = fxj((x + b)a for every bb) 6 Conclusions ^(G(F (c) =1))g We have presented a comprehensive spatial algebra which consists of topological, metric and morphological operators. The operators can be composed to deduce relations between objects. This algebra is easy to implement, is ecient and does not suer from the neighborhood problems when applied in the raster domain. This can be used for spatial reasoning for content-based retrieval. References [1] S.K. Chang, Q.Y. Shi and C.W. Yan, \Iconic Indexing by -D Strings", IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-9, No. 6, pp. 1-8, [] M.J. Egenhofer and R. Franzosa, \Point-Set Topological Spatial Relations", International Journal of Geographic Information Systems, Vol., No., pp , Boundary Interior of region a Interior of region b Common boundary Figure One-unit-wide common boundary [] M.J. Egenhofer and J. Sharma, \Topological relations Between Regions in R and Z ", Proc. rd International Symposium on Large Spatial Databases, LNCS Vol. 69, pp. 16-6, 199. [] M.S. Kankanhalli and X. Jiang, \A Spatial Algebra for Flexible Content-based Retrieval", RWC Technical Report (under preparation), 199. [] M.G. Luby, \Grid Geometries Which Preserve Properties of Euclidean Geometry A Study of Graphics Line Drawing Algorithms", Theoretical Foundations of Computer Graphics and CAD, NATO ASI Series, Vol. F0, Springer-Verlag, pp. 97-, [6] J.K. Wu, \The Dual Raster-Vector Data Structure", Unpublished Technical Report, 199. region a region b visual boundary Figure The visual boundary between two raster regions