Proc. Int. Conf. Knowledge-Based Systems (KBCS 2002), Vikas Publications, pp. 117-126.
Fuzzy Spatial Concepts in Finite Discrete Space Domains Girish Keshav Palshikar Tata Research Development and Design Centre (TRDDC), 54B, Hadapsar Industrial Estate, Pune 411013 Maharashtra, India. Email:
[email protected]
Abstract Many practical applications involving spatial aspects work with finite discrete space domains; e.g., map grids, railways track layouts and road networks. Such space domains are computationally tractable and often include specialized forms of spatial reasoning. Moreover, in such applications the spatial information naturally includes various forms of approximation, uncertainty or inexactness. Fuzzy representations are then appropriate. In this paper, we reformulate the region connection calculus (RCC) framework for finite, discrete space domains in simple set-theoretical terms. We generalize RCC framework and develop several fuzzy spatial concepts like fuzzy regions, fuzzy directions, fuzzy named distances. We propose a fuzzification of standard spatial relations in RCC. For this purpose, we enhance the fuzzy set theory to include fuzzy definitions for membership, subset and set equality crisp binary relations between sets (fuzzy or crisp). We illustrate the approach using a discrete finite 2-D map grid as the space domain.
1.
Introduction
There are many practical applications where spatial concerns arise naturally – telecom, power, construction, civil and municipal services, geographical information systems, CAD, robotics, defense, airtraffic control, railways, battlefield surveillance, resource management (spatial allocation, positioning, deployment, transport, planning, scheduling and monitoring), courier and freight services etc. Spatial aspects in these applications involve many issues including: representation of positional and other spatial data, spatial relationships among regions and objects, querying of spatial databases, analysis of spatial data, automatic deduction of implicit spatial relationships etc. In many applications, the model of underlying space domain and spatial regions is finite and discrete. In such applications, the spatial domain can be represented by means of a graph. In contrast, much work in qualitative spatial reasoning concerns itself with the Euclidean R2 (or R3 ) space, which is infinite, real and continuous and which is a topological space characterized by axioms of connectivity, convexity etc. For those applications where finite, discrete space domains suffice, the Euclidean model of space is over-abstract and over-generalized, leading to unnecessary complications. Throughout the paper, we use a specific and popular model of finite, discrete space, which is a finite 2-dimensional grid of squares. Each square in the grid is a point in space, which is connected to its neighbouring points in the grid; there is no further structure or details within a square. The approach can also deal with other finite discrete models
Proc. Int. Conf. Knowledge-Based Systems (KBCS 2002), Vikas Publications, pp. 117-126.
of space, represented, for instance, by a graph (e.g., for a railway track layout or a road network). An additional requirement in many practical applications is that the representations of spatial regions and associated information are inherently approximate, uncertain or inexact. Fuzzy representations are then likely to be appropriate. For example, a satellite picture may show some green, red and brown regions, where the spread of colours is not uniform and the relationships between the coloured regions (e.g., overlap) are not precise. In this paper, we propose a fuzzy approach to the representation of (finite, discrete) spatial domains and information associated with these domains. We also generalize the relationships in the region connection calculus (RCC) to handle fuzzy aspects. In section 2, we present the finite discrete space domain and an example of a 2-D grid, along with a simple re-formulation of the crisp RCC framework for this space domain. In section 3, we present our approach where we define fuzzy spatial aspects fuzzy regions, fuzzy distances and fuzzy directions and fuzzy positions. In section 4, we extend the RCC framework to work with fuzzy binary relations. The last section presents conclusions and further work.
1.1.
Related Work
Computational techniques for spatial aspects have attracted many researchers from diverse areas: databases, multi-media systems, image processing, multi-media, artificial intelligence and mathematical logic, natural language processing, computer graphics, robotics, computational geometry as also experts from disciplines like geography and civil engineering. Here we focus on qualitative spatial aspects in AI and mathematical logic with issues like qualitative representation, analysis and reasoning with spatial aspects like connectivity, shape, direction, position etc. The idea of treating parts of images as fuzzy regions is not new [Rosenfeld, 1979; Resenfeld and Klette, 1985; Altman, 1994; Matsakis and Wendling, 1999; Clementini and DiFelice, 1996; Erwig and Schneider, 1997; Molenaar, 1996; Schneider, 2000; Zhan, 1998] – see also the proceedings of SDH and GISDATA workshops – although most such work refers to either the discrete 2-D raster or Euclidean space and the fuzzification of spatial relations is not under the RCC framework. For example, in [Krishnapuram et al, 1993] fuzzy membership functions are defined to extract fuzzy regions from images and also 8 primitive spatial relationships LEFT-OF, RIGHT-OF, ABOVE, BELOW, BEHIND, INFRONT-OF, NEAR, FAR, INSIDE, OUTSIDE, SURROUND between objects are fuzzified (differently from our definitions here). [Cohn and Gotts, 1996] develop egg-yolk approach for handling regions with indeterminate boundaries within RCC. Our work is related to region connection calculus (RCC) [Cohn et al, 1997]. RCC proposed a set of binary relations between two spatial regions; these relations were generalizations of Allen’s algebra of 13 binary relations between 2 time intervals [Allen, 1983]. There is more work based on RCC – see QSR group web-site http://www.comp.leeds.ac.uk/spacenet/publications.html. See http://www.cs.albany.edu/~amit/spatsites.html for a bibliography about general spatial reasoning. There is much work about representation and analysis of qualitative spatial aspects. Forbus et al [1991] proposed the MV/PD framework particularly oriented towards qualitative physics; they used it to describe and qualitatively simulate a mechanical clock. Liu [1998] has proposed a qualitative trigonometry for 2-D Euclidean space for use in spatial reasoning; the work involves definitions of concepts like qualitative distance and qualitative angles. Frank [1992] has proposed another formulation of the qualitative distances and qualitative directions; interesting aspects of this work include path algebra and composition table for 8 cardinal directions. Clementini et al [1997] present another framework for qualitative descriptions of 2-D positions, distances and directions; their representation allows order-ofmagnitude reasoning on different scales. El-Geresy and Abdelmoty [1996] present an adjacency-matrix based method to decompose and represent all relations between two arbitrary regions and presents a reasoning formalism (composition tables) that works with this representation. Hernandez and Jungert
Proc. Int. Conf. Knowledge-Based Systems (KBCS 2002), Vikas Publications, pp. 117-126.
[1999] introduce qualitative motion vectors and operations on them to represent and analyze qualitative aspects of the motion of point objects. From a logical point of view, work of Sistla and Yu [2000] provides a sound and complete deduction system for qualitative spatial reasoning and an algorithm to deduce implicit spatial relationships from a given set of relationships. Bennett [1994] describes the propositional logic C0 + in which RCC relations can be described and reasoned with. Ioerger [1997] describes a hybrid system in which a spatial reasoning component is integrated with a resolution theorem-prover. Our work is related to fuzzy spatial aspects, which need to be carefully distinguished from qualitative spatial aspects. Qualitative approach makes only as many distinctions as necessary and works with them; qualitative information need not be vague or uncertain. On the other hand, the fuzzy approach is useful to deal with situations that deal with uncertainty. As an example, a qualitative direction system will assign a single direction (say north-east) to another object with respect to a reference object – and that is the only directional relationship between them. However, in the fuzzy approach, the cardinal directions will be have different degrees of truth; e.g., the object may be northward with a degree of truth 0.8, eastward with a degree of truth 0.3 and north-east with a degree of truth 0.8. Fuzzy concepts are also useful to indicate, for instance, degrees of rainfall (high, medium, low) in different places. This is work is a generalization of our previous work [Palshikar, 2000a; Palshikar and Bahulkar, 2000; Palshikar 2001] where we have applied similar fuzzy generalizations to temporal aspects, including Allen’s interval algebra (which is the temporal analogue of RCC). The ideas in this paper are implemented using our FZLOG tool [Palshikar, 2000b] for fuzzy representations.
2.
Model of Space
In this section, we re-formulate the RCC formalism for our finite discrete model of space, which entails more concrete definitions for the abstract primitives of the RCC model. The definitions are simpler and lead to straightforward deductive implementations (we have implemented them in Prolog). Our formalization is based on spatial relations between points rather than regions as in RCC. In the next section fuzzification of RCC is introduced. Definition. A finite discrete space domain is a tuple (SPACE,C,R) where SPACE = {s0 , s1 , s2 , ..., sM} is a given finite set of unordered constants called space-points; C = {C1 ,C2 ,…} is a given finite set of crisp local spatial relations between space points; R = {R1 ,R2 ,…} is a given finite set of given global spatial relations between space points. Each Ci or Rj is a fully specified crisp relation on SPACE (we assume they are binary relations). Two space points si and sj are neighbours, denoted N(x,y), if there is some relation Ck in C such that Ck(si,sj) = 1. By NH(x), we denote the set of all neighbours of a space point x including itself i.e., NH(x) = {x} ∪ {y | y ∈ SPACE and ∃ some Ci in C such that Ci(x,y)}. We assume that Ci’s are defined in such a way that N(x,y) is a reflexive and symmetric binary relation, akin to the relation C between regions in RCC. Each Ci indicates a local structural spatial relation between two spatially “neighbouring” space points. Each Rj indicates a global spatial relation between two space points that holds between points not necessarily spatial “neighbours”. Not every such tuple (SPACE,C,R) can be treated as an actual physical space; e.g., the Euclidean R2 (or R3 ) space is a topological space additionally characterised by axioms about connectivity, convexity etc. We assume (SPACE,C,R) is a “proper” model of space and obeys all necessary axioms (finding these axioms is an important task not handled here). For example, typically each Ci or Rj would be irreflexive and even anti-symmetric; moreover each Rj would be transitive. Also, domain of each Ci (and Rj) would be the entire set SPACE. Example 1: In many situations, e.g., Defence and GIS, a raster-style space is represented by a (m+1)×(n +1) grid (Figure 1). Here, SPACE = {s i,j|0≤i ≤ m, 0≤j≤n}; space-point s i,j corresponds to grid element or square (i,j). R =
Proc. Int. Conf. Knowledge-Based Systems (KBCS 2002), Vikas Publications, pp. 117-126.
{north,south,east,west} where each element of R is a crisp binary relation on SPACE. To indicate (x1,y1) is north of (x2,y2), we define north((x1,y1),(x2,y2)) = 1 if y1 > y2 and 0 otherwise; e.g., north((2,6),(5,3)) = 1. Other relations in R are similarly defined. C = {adjacent,meets,ec}. We define adjacent((x1,y1),(x2,y2)) = 1 if (|x1 – x2| = 1 ∧ y1 = y2) ∨ (x1 = x2 ∧ |y1 – y2| = 1) and 0 otherwise. We define meets((x1,y1),(x2,y2)) = 1 if |x1 – x2| = 1 ∧ |y1 – y2| = 1 and 0 otherwise; e.g., adjacent((1,1),(2,1))=1, adjacent((1,1),(2,2))=0, meets((1,1),(2,1))=0, meets((1,1),(2,2))=1. We define a binary relation externally connected as ec(x,y) = adjacent(x,y) ∨ meets(x,y). Points (2,3) and (3,4) are neighbours but (2,3) and (5,6) are not.
7 6 5 4 3 2 1 0 0
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Figure 1. A 7 × 8 grid.
North
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Definition. A non-empty set X ⊆ SPACE of space points is a crisp region if either X is a singleton set or for every space point si in X, there is another space point sj in X such that si, sj are neighbours. For example 1, X = {(0,0), (0,1), (0,2), (1,0), (1,1)} is a region. We can define a crisp predicate region(X) which returns true if the given non-empty set of space points (X ⊆ SPACE and X ≠ ∅) forms a region in some semantically well-defined (e.g., topological) sense. For a finite topological space, the topological closure of every region is the region itself. Stronger assumptions may be incorporated in the region predicate, if necessary; e.g., a region is convex or does not contain any holes. Example 1 (continued): We adopt the following definition for the region(X) predicate, where X ⊆ SPACE and X ≠ ∅; region(X) is true if every point in X is adjacent to or meets at least one other point in X. Formally, region(X) = true if X is a singleton set or ∀x∈X ∃y∈X such that x ≠ y and (adjacent(x,y) ∨ meets(x,y)). Note that a region can be non-convex, can contain holes but it is contiguous.
For finite domains, we need the concept of boundary points. Intuitively, if a notion of distance between space point is available (which we will add later to our space domain), then boundary points are “farthest” from the “centres” of the graph of the space domain; e.g., points like (2,7) and (5,0) are boundary points of the space domain in Example 1. We leave out this formalization and assume an appropriately defined predicate boundary(x) which returns true if x is a boundary point of SPACE. Let X, Y ⊆ SPACE be two crisp regions (region(X) = region(Y) = true). C(X,Y) is the basic relation in RCC where it is used as a building-block for defining further relations between regions. It can be defined in out model (in terms of the basic spatial relations between space points) as: C(X,Y) (X ∩ Y ≠ ∅) ∨ (∃x ∈ X, y ∈ Y such that N(x,y)). Clearly, as in RCC, the relation C between regions is reflexive and symmetric. RCC defines the following 8 binary relations between regions which are jointly exhaustive and pair-wise disjoint (JEPD). RCC defines these relations in terms of the basic C relation as follows. We have specialized RCC definitions of these relations for our finite discrete space model. Note that original RCC definitions for relations like TPP need modification for finite discrete space domains.
Proc. Int. Conf. Knowledge-Based Systems (KBCS 2002), Vikas Publications, pp. 117-126.
disconnected from Y DC(X,Y) X ∩ Y = ∅ ∧ ¬C(X,Y) is externally connected to Y EC(X,Y) X ∩ Y = ∅ ∧ C(X,Y) partially overlaps Y PO(X,Y) X ∩ Y ≠ ∅ ∧ ¬(X ⊆ Y) ∧ ¬(Y ⊆ X) is a tangential proper part of Y TPP(X,Y) X ⊂ Y ∧ ( (∃z [z ∈ SPACE ∧ z ∉ X ∧ z ∉ Y ∧ EC({z},X) ∧ EC({z},Y)]) ∨ (∃b [b ∈ X ∧ b ∈ Y ∧ boundary(b)) ) X is a non-tangential proper part of Y NTPP(X,Y) X ⊂ Y ∧ (1 – TPP(X,Y)) Y is a tangential proper part of X TPPi(X,Y) TPP(Y,X) Y is a non-tangential proper part of X NTPPi(X,Y) NTPP(Y,X) X X X X
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Y DC(X,Y) EC(X,Y) PO(X,Y) TPP(X,Y)
TPPi(X,Y) NTTPi(X,Y)
Figure 2. Graphical illustration of the 8 basic relations in RCC [Cohn et al, 1997].
3.
Fuzzy Spatial Concepts
3.1.
Fuzzy Finite Discrete Space Domain
In many practical situations, the set R of spatial relationships includes some naturally fuzzy relationships. In Example 1, the binary spatial relation north is often actually a fuzzy binary relation which associates a “degree of northward-ness” to each pair of space points in SPACE×SPACE. Many times the given set R of spatial relationships also contains fuzzy named distance relations; e.g., near and far. Each such fuzzy named distance relation can be modeled as a fuzzy binary relation, which associates a fuzzy degree of membership with each tuple in SPACE × SPACE. Definition. A fuzzy finite discrete spatial domain is a tuple (SPACE,C,R) where SPACE is a nonempty finite crisp set of space points, C and R are finite non-empty sets of fuzzy binary relations (each fuzzy binary relation associates a fuzzy degree of membership with each tuple in SPACE × SPACE). Example 1 (continued): We define fuzzy binary relation north on a given grid as: north( (x1,y1), (x2,y2) ) = 1 if x1 = x2 and y1 > y2 = 0 if y1 ≤ y2 = { tan-1(|y1-y2|/|x1 – x2|) } / 90 For example, north((6,7),(0,0)) = 0.55, north((3,7),(0,0)) = 0.74, north((1,7),(0,0)) = 0.91, north((0,7),(0,0)) = 1 and north((6,0), (4,3)) = 0. Fuzzy binary relationships for other cardinal directions south, east, west etc. can be defined similarly. There are of course other ways to define these fuzzy relations. The set R can also include fuzzy named distances like near, far etc. which we model as fuzzy binary relations. The fuzzy named distance far(x,y) is modeled as a fuzzy binary relation over SPACE×SPACE associating a degree of membership = dmh (x,y)/D with each tuple (x,y) of space-points; e.g., far((2,3), (4,6)) = dmh ((2,3), (4,6)) / 13 = 5/13 ≈ 0.38 and far(0,0), (6,7)) = 1 and far((2,2),(2,2)) = 0. The fuzzy named distance near can be similarly modeled as a fuzzy binary relation defined by near(x,y) = 1 – far(x,y), for any space-points x and y in SPACE. Here, the fuzzy relations near and far are symmetric i.e., near(x,y)= near(y,x) and far(x,y) = far(y,x) for all x, y in SPACE. However, the relation near is reflexive (i.e., near(x,x) = 1 for all x in SPACE) and far is irreflexive (i.e.,
Proc. Int. Conf. Knowledge-Based Systems (KBCS 2002), Vikas Publications, pp. 117-126.
far(x,x) = 0 for all x in SPACE). Note the use of finiteness of the space; since the set SPACE is finite, the distance D is also finite. We now fuzzify the basic crisp binary relations in C: adjacent(x,y), meets(x,y) and ec(x,y) between two spacepoints x, y in SPACE. We define two fuzzy binary relations ≈X and ≈Y, called approximately equal, between two x-coordinates x1 and x2 and also between two y-coordinates y1 and y2, as follows. x1 ≈X x2 = 1 – [ |x1 – x2| ÷ (NX – 1) ] y1 ≈Y y2 = 1 – [ |y1 – y2| ÷ (NY – 1) ] Here, NX and NY are the total no. of columns and rows in the grid respectively. Clearly, when x1 = x2, then x1 ≈X x2 = 1 and when y1 = y2, then y1 ≈Y y2 = 1. For the given example grid with NX = 7 and NY = 8, we have 0 ≈X 6 = 0.0, 3 ≈X 3 = 1.0, 2 ≈X 5 = 0.43, 5 ≈X 6 = 0.71, 0 ≈Y 7 = 0.0, 3 ≈Y 3 = 1.0, 2 ≈Y 5 = 0.5 and 5 ≈Y 6 = 0.75. There are, of course, other fuzzy generalizations of the equality crisp binary relation between 2 rows (and two columns). Given two x-coordinates x1 and x2, we define a fuzzy binary relation 1≈X between x1 and x2 as: x1 1≈X x2 = x1 ≈X (x2+1) if x1 > x2 = (x1+1) ≈X x2 if x1 < x2 = 1 – (1 ÷ (NX – 1) if x1 = x2 Essentially the relation x1 1≈X x2 says how close |x1 – x2| is to 1. Similar fuzzy binary relation y1 1≈Y y2 can be defined between any two y-coordinates y1 and y2. For the given example, 1 1≈X 2 = 1, 2 1≈X 1 = 1, 3 1≈X 3 = 0.83, 2 1≈X 5 = 0.67. Note that x1 1≈X x2 ≠ 0 for any values of x1 and x2. Using these fuzzy equality relations and using the fuzzy ∨ connective, we can generalize the definitions of the basic crisp binary relations adjacent(x,y), meets(x,y) and ec(x,y) between two space-points x, y in SPACE. Clearly, the fuzzy versions are conservative i.e., fz_adjacent((x1,y1),(x2,y2)) = 1 if adjacent((x1,y1,(x2,y2)) = 1 and fz_meets((x1,y1),(x2,y2)) = 1 if meets((x1,y1),(x2,y2)) = 1. For the grid of Figure 1, it is easy to check that fz_adjacent((1,1),(2,3)) = 0.83. fz_adjacent((x1,y1),(x2,y2)) = ((x1 1≈X x2) ∧ y1 ≈Y y2) ∨ ((x1 ≈X x2) ∧ (y1 1≈Y y2)) fz_meets((x1,y1),(x2,y2)) = (x1 1≈X x2) ∧ (y1 1≈Y y2) fz_ec((x1,y1),(x2,y2)) = fz_adjacent((x1,y1),(x2,y2)) ∨ fz_meets((x1,y1,(x2,y2)).
3.2.
Fuzzy Regions
A fuzzy region is a fuzzy subset of SPACE that associates a degree of membership (in the fuzzy region) with each space-point; we require that the support set of a fuzzy region be a crisp region. Fuzzy regions are useful to represent concepts like area of influence or effectiveness (say, of an anti-tank gun placed at a particular position). The concept of fuzzy region is also useful to describe non-object spatial phenomena. For example, to describe rainfall in the given space, each location may have a different degree of truth for the fuzzy proposition, indicating the strength of the rainfall there (e.g., light, medium, heavy, none etc.). Many times, there is some uncertainty about the location (or position) of an object. Instantaneous position of such an object can be elegantly represented by a fuzzy region, say P, over SPACE, that associates a degree of truth with each space-point, which indicates the possibility that the object is at that space-point. We say that P is a fuzzy position set. Some fuzzy regions in Example 1 are: X1 = {(2,3) : 0.4, (2,4) : 0.7, (2,5) : 0.8, (3,3) : 0.4, (3,4) : 0.7, (3,5) : 0.8, (4,3) : 0.4, (4,4) : 0.7}, X2 = {(1,4) : 0.5, (1,5) : 0.5, (1,6) : 0.5, (2,4) : 0.7, (2,5) : 0.7, (2,6) : 0.7} etc. Unlike RCC, we allow application-specific spatial relations (as given in set R). Each relation in R holds between a pair of space points. We provide a natural extension so that they hold between regions. Example 1 (continued): we define the center of gravity of k space-points in a given region X = {(x1, y1):d1, …, (xk, yk):dk} is defined to be the point Xcg = ( (d1 * x1 + … + dk * xk) / (d1 + … + dk), (d1 * y1 + … + dk * yk) / (d1 + … + dk) ). The coordinates of the center of mass (and center of gravity) can be converted, if necessary, to integers. Now we can extend the fuzzy binary relations in R (which are over space-points) to fuzzy regions as follows. Let X, Y be two fuzzy regions and let Xcg and Ycg respectively be the space-points corresponding to the centers of gravity of X and Y respectively. Then the fuzzy direction relation north can be extended to fuzzy regions as north(X,Y) = north(Xcg,Ycg).
Proc. Int. Conf. Knowledge-Based Systems (KBCS 2002), Vikas Publications, pp. 117-126.
4.
Fuzzy Region Connection Calculus (FRCC)
In some applications, one may still treat fuzzy regions as crisp regions by considering only their support sets. In that case all the RCC relations remain crisp binary relations between 2 fuzzy regions. However, there is often a need to consider the “extent” to which the RCC relations hold between two fuzzy regions. For example, if support(X) ∩ support(X) = ∅ then one may want to know “how far” disconnected the 2 fuzzy regions X and Y are. Or if support(X) ∩ support(X) ≠ ∅ then one may want to know “how much” disconnected the 2 fuzzy regions X and Y are. That is there is a need to treat the relations in RCC as fuzzy relations rather than crisp relations. For this purpose, we now present a fuzzification of the RCC framework for finite discrete space domains.
4.1.
Extended Fuzzy Set Theory
Membership, subset and set equality are crisp binary relations between sets (fuzzy or crisp); e.g., X ⊆ Y = 1 if all members of X are also members of Y and 0 otherwise. We now define these relations as fuzzy binary relations, written in bold with a subscript fz, to distinguish from their crisp versions. Each of the statements x ∈fz X, X ⊆ fz Y and X =fz Y now has a fuzzy degree of truth, where X and Y are two finite fuzzy sets over the crisp finite universe U and x ∈ U. Example 1 statement x x ∈fz X = 1 =d
(continued): Suppose X is a finite fuzzy region in SPACE. The degree of truth associated with the ∈fz X, where x is a space-point in SPACE, is computed as follows: if degree of membership of x in X is d and d ≠ 0 if degree of membership of x in X is 0 and d = 1 – [dmh (x,Xcg) / D] where Xcg is the center of gravity of the region X and D is the largest distance in SPACE For the fuzzy region X1 = {(2,3):0.4,(2,4):0.7,(2,5):0.8,(3,3):0.4,(3,4):0.7,(3,5):0.8,(4,3):0.4,(4,4):0.7} and the space-point (3,4), the truth-value of the statement (3,4) ∈fz X1 = 1 but the truth-value of the statement (3,2) ∈fz X1 is 0.85. This is because cog(X1) = (3,4) and 1 – [dmh ((3,4),(3,2))/D] = 1 – (2/13) = 0.85. Note that there are other ways to define the degree of truth for the statement x ∈fz X; e.g., we may alter the above definition to use the distance between x and the nearest space-point of X (rather than the center of gravity of X). The degree of truth of the statement x ∉fz X is defined as 1 – degree of truth of the statement x ∈fz X. Suppose X and Y are two finite fuzzy regions in SPACE. Then the degree of truth associated with the statement X ⊆ fz Y is computed as follows: X ⊆ fz Y = 1 – [ | support(X) – support(Y)| ÷ |support(X)| ] |S| indicates the number of elements in a crisp set S. The numerator in the second term denotes the number of elements that have a non-zero degree of membership in X but have a zero degree of membership in Y i.e., which “fall outside” Y. The definition is conservative; when support(X) ⊆ support(Y) then the numerator in second term is zero and X ⊆ fz Y = 1. The degree of truth for the statement X = fz ∅ (where X is a non-empty fuzzy region, with the support set support(X)) is defined as 1 if support(X) is also empty; otherwise, it is 1 – (|support(X)| ÷ |SPACE|). We have assumed that SPACE is a finite non-empty set. The degree of truth of the statement X = fz Y between two non-empty fuzzy regions X and Y can now be defined as X ⊆ fz Y ∧ Y ⊆ fz X. The degrees of truth of the statements X ≠ fz Y and X ⊂ fz Y are respectively defined as those of the statements ¬(X = fz Y) and X ⊆ fz Y ∧ X ≠ fz Y. As an example, consider the 2 fuzzy regions in Figure 1: X1 = {(2,3) : 0.4, (2,4) : 0.7, (2,5) : 0.8, (3,3) : 0.4, (3,4) : 0.7, (3,5) : 0.8, (4,3) : 0.4, (4,4) : 0.7}, X2 = {(1,4) : 0.5, (1,5) : 0.5, (1,6) : 0.5, (2,4) : 0.7, (2,5) : 0.7, (2,6) : 0.7} Here, support(X1) = {(2,3), (2,4), (2,5), (3,3), (3,4), (3,5), (4,3), (4,4)} and support(X2) = {(1,4), (1,5), (1,6), (2,4), (2,5), (2,6)}. Then support(X1) ∩ support(X2) = {(2,4), (2,5)}. Then the truth-value of the statement X2 ⊆ fz X1 is 1 – [ | support(X2) – support(X1)| ÷ |support(X2)| ] = 0.5.
4.2.
Fuzzy Spatial Relations
Recall that two space points x and y are neighbours, denoted N(x,y), if there is some relation Ci in C such that Ci(x,y) = 1. We need to fuzzify the binary relation N(x,y). Since each Ci in C is a fuzzy binary relation, we define FZ_N(x,y) C1 (x,y) ∨ C2 (x,y) ∨ … over all (finite number of) relations in C.
Proc. Int. Conf. Knowledge-Based Systems (KBCS 2002), Vikas Publications, pp. 117-126.
Recall that the basic spatial relation C between two crisp regions was defined as C(X,Y) (X ∩ Y ≠ ∅) ∨ (∃x ∈ X, y ∈ Y such that N(x,y)). We fuzzify the relation C keeping in mind that X, Y are fuzzy regions: FZ_C(X,Y) (support(X) ∩ support(Y) ≠ ∅) ∨ max{FZ_N(x,y) | x ∈ X, y∈Y}. We have already presented definitions of the 8 basic binary spatial relations between crisp regions in RCC, after adapting them for discrete finite space domains. We now extend these 8 spatial relations by adapting them for fuzzy spatial domains (i.e., fuzzy spatial regions).
how far X is disconnected from Y FZ_DC1(X,Y) ¬FZ_C(X,Y) if support(X) ∩ support(Y) = ∅ 0 otherwise how closely X is externally connected to Y FZ_EC1(X,Y) FZ_C(X,Y) if support(X) ∩ support(Y) = ∅ 1 otherwise
There is actually another possible way for defining these two fuzzy RCC relations. In this approach, the degree of truth of FZ_DC2(X,Y) indicates “how much” of X is disconnected from Y (rather than “how far” X is disconnected from Y). Note the use of fuzzy set operations like =fz, defined earlier. DC(support(X),support(Y)) if support(X) ∩ support(Y) = ∅ FZ_DC2(X,Y) X ∩ Y =fz ∅ otherwise EC(support(X),support(Y)) if support(X) ∩ support(Y) = ∅ FZ_EC2(X,Y) X ∩ Y ≠ fz ∅ otherwise “extent” to which X partially overlaps Y FZ_PO(X,Y) X ∩ Y ≠ fz ∅ ∧ ¬(X ⊆ fz Y) ∧ ¬(Y ⊆ fz X) “extent” to which X is a tangential proper part of Y FZ_TPP(X,Y) X ⊂ fz Y ∧ ∃z [z ∈ SPACE ∧ z ∉ support(X) ∧ z ∉ support(Y) ∧ FZ_EC({z:1},X) ∧ FZ_EC({z:1},Y) Definitions of the fuzzy versions of the remaining RCC relations are simple: X is fuzzily a non-tangential proper part of Y FZ_NTPP(X,Y) 1 – FZ_TPP(X,Y) Y is fuzzily a tangential proper part of X FZ_TPPi(X,Y) FZ_TPP(Y,X) Y is fuzzily a non-tangential proper part of X FZ_NTPPi(X,Y) FZ_NTPP(Y,X) X is fuzzily identical to Y FZ_EQ(X,Y) X = fz Y
5.
Conclusions and Further Work
Finite discrete space domains are important many practical applications. In such applications, there is also a need to represent information over this space domain that is inherently approximate, uncertain or inexact. We presented a fuzzification of the well-known RCC framework adapted for the finite discrete space domain. We illustrated the approach using a discrete finite 2-D map grid as the space domain. The simplifications resulting due to finite discrete nature of the space domain lead to computationally tractable implementation of the fuzzy spatial representation system. We have omitted a discussion composition of our fuzzy spatial relations vis-à-vis that in RCC. We have omitted compositions of fuzzy directions. One interesting extension is to work with a finite but dense space domain; e.g., a finite (bounded) 2-D square in R2 . There is also a need to further explore forms of spatial reasoning especially suitable for fuzzy aspects. We are working on extending the framework for fuzzy spatiotemporal aspects. Another interesting work is to see how spatial query languages can make use of fuzzy concepts. We are interested in applying the ideas in this paper in practical applications like remote sensing and GIS.
Proc. Int. Conf. Knowledge-Based Systems (KBCS 2002), Vikas Publications, pp. 117-126.
Acknowledgements. I thank Prof. Mathai Joseph for his support. I am sincerely grateful to Dr. Manasee Palshikar for her love, patience and understanding. 6.
References
[Clementini et al, 1997] E. Clementini, P. Di Felice, D. Hernandez. Qualitative Representation of Positional Information. Artificial Intelligence, 95, pp. 317-356, 1997. [Cohn et al, 1997] G. Cohn, B. Bennet, J. Gooday, N.M. Gotts. Qualitative Spatial Representation and Reasoning with the Region Connection Calculus. Geoinformatica, 1, pp. 1 – 44, 1997. [El-Geresy and Abdelmoty, 1996] A. El-Geresy, A.I. Abdelmoty. Order in Space: A General Formalism for Spatial Reasoning. Int. Journal of Artificial Intelligence Tools, vol. 6, no. 4, pp. 423-450, 1996. [Frank, 1992] U. Frank. Qualitative Spatial Reasoning about Distances and Directions in Geographic Space. Journal of Visual Languages and Computing. 3, pp. 343-371, 1992. [Hernandez, 1999] E. J. Hernandez. Qualitative Motion of Point-like Objects. Journal of Visial Languages and Computing, 10, pp. 269-289, 1999. [Krishnapuram et al, 1993] R. Krishnapuram, J. M. Keller, Y. Ma. Quantitative Analysis of Properties and Spatial Relations of Fuzzy Image Regions. IEEE Tran. Fuzzy Syst., 1(3), pp. 222-233, 1993. [Liu, 1998] J. Liu. A Method of Qualitative Spatial Reasoning based on Qualitative Trigonometry. Artificial Intelligence, 98, pp. 137-168, 1998. [Matsakis and Wendling, 1999] P. Matsakis, L. Wendling. A New Way to Represent the Relative Position of Areal Objects. IEEE Pattern Analysis and Machine Intelligence, 21(7), pp. 634-643, 1999. [Palshikar, 2001] G. K. Palshikar. A Fuzzy Temporal Notation and Its Applications to Specify Fault Patterns for Diagnosis. Pattern Recognition Letters, vol. 22, no. 3-4, pp. 381-394, March 2001. [Sistla and Yu, 2000] A. Prasad Sistla, C. Yu. Reasoning about Qualitative Spatial Relationships. Journal of Automated Reasoning, vol. 25, no. 4, pp. 291-328, Nov. 2000.
