Rough Sets in Approximate Spatial Reasoning

Transcription

1 ough Sets in Aroximate Satial easoning Thomas Bittner and John G. Stell Centre de recherche en geomatique, Laval University, Quebec, Canada. Deartment of Comuter Science, Keele University, UK. Abstract. In satial reasoning the qualitative descrition of relations between s- atial regions is of ractical imortance and has been widely studied. Examles of such relations are that two regions may meet only at their boundaries or that one region is a roer art of another. This aer shows how systems of relations between regions can be extended from recisely known regions to aroximate ones. One way of aroximating regions with resect to a artition of the lane is that rovided by rough set theory for aroximating subsets of a set. elations between regions aroximated in this way can be described by an extension of the CC5 system of relations for recise regions. Two techniques for extending C- C5 are resented, and the equivalence between them is roved. A more elaborate aroximation technique for regions (boundary sensitive aroximation) takes account of some of the toological structure of regions. Using this technique, an extension to the CC8 system of satial relations is resented. Keywords: qualitative satial reasoning, aroximate regions. 1 Introduction ough set theory [Paw91] rovides a way of aroximating subsets of a set when the set is equied with a artition or equivalence relation. Given a set with a artition, an arbitrary subset can be aroximated by a function fo o no. The value of is defined to be fo if, it is no if #", and otherwise the value is o. The three values fo, o, and no stand resectively for full overla, artial overla and no overla ; they measure the extent to which overlas the elements of the artition of. In satial reasoning it is often necessary to aroximate not subsets of an arbitrary set, but arts of a set with toological or geometric structure. For examle the set above might be relaced by a regular closed subset of the lane, and we might want to aroximate regular closed subsets of. This aroximation might be with resect to a artition of where the cells (elements of the artition) might overla on their boundaries, but not their interiors. Because of the additional toological structure, it is ossible to make a more detailed classification of overlas between subsets and cells in the artition. An account of how this can be done was given in our earlier aer [BS98]. This is, however, only one of the directions in which the basic rough sets aroach to aroximation can be generalized to satial aroximation.

2 * * $ Our concern in the resent aer is relationshis between satial regions when these regions have been given aroximate descritions. The study of relationshis between satial regions is of ractical imortance in Geograhic Information Systems (GIS), and has resulted in many aers [EF91,CC92,SP92,Ben96]. Examles of relationshis might be that two regions meet only at their boundaries or that one region is a roer art of another. While such relationshis have been widely studied, the toic of relationshis between aroximate regions has received little attention. The structure of the aer is as follows. In section 2 we set out the articular tye of aroximate regions we use in the main art of the aer. In section 3 we discuss one articular scheme of relationshis between regions, known as the CC5, and in section 4 we show how this can be generalized to aroximate regions. In section 5 we briefly consider how our work can be extended to deal with more detailed boundary-sensitive aroximations and the CC8 system of relationshis between regions. Finally in section 6 we resent conclusions and suggest directions for further work. 2 Aroximating regions Satial regions can be described by secifying how they relate to a frame of reference. In the case of two-dimensional regions, the frame of reference could be a artition of the lane into cells which may share boundaries but which do not overla. A region can then be described by giving the relationshi between the region and each cell. 2.1 Boundary insensitive aroximation Aroximation functions Suose a sace $ of detailed or recise regions. By imosing a artition, %, on $ we can aroximate elements of $ by elements of &(' ). That is, we aroximate regions in $ by functions from % to the set & ) fo o no. The function which assigns to each region * $ its aroximation will be denoted ) the boundary insensitive aroximations of regions + ),$ &(' ). The value of + ).- * is fo if * - covers all the of the cell, it is o if * covers - - some but not all of the interior of, and it is no if there is no overla between * and. We call the elements of &(' $ with resect to the underlying regional artition %. Each aroximate region &/' recise regions having the aroximation. This set which will be denoted vides a semantics for aroximate regions: ) *. ) stands for a set of recise regions, i.e. all those ro- & ) Oerations on aroximation functions The domain of regions is equied with a meet oeration interreted as the intersection of regions. In the domain of aroximation functions the meet oeration between regions is aroximated by airs of greatest minimal,, and least maximal, 87:9, meet oerations on aroximation maings [BS98]. Consider the oerations and 87:9 on the set fo o no that are defined as follows.

3 D * D $ no o fo no o fo no no no no no o no o fo These oerations extend to elements of &/' 3 4 =.- 87:9 no o fo no o fo no no no no o o no o fo ) (i.e. the set of functions from % to & ) ) by > -? -@ 3 4 and similarly for 87:9. This definition of the oerations on &/' ) is equivalent to the construction for oerations given by Bittner and Stell [BS98, age 108]. 2.2 Boundary sensitive aroximation We can further refine the aroximation of regions $ with resect to the artition % by taking boundary segments shared by neighboring artition cells into account. That is, we aroximate regions in $ by functions from %BAC% to the set &8DC fo fbo bo nbo no. The function which assigns to each region * $ its boundary sensitive aroximation will be denoted + DEF$ & 'HG?'. The value of + -J -LK DI* is- fo- Kif * -J covers all of the - cell -, K it is fbo if * covers all of the boundary - segment,, shared by the cell and and some but not all of the interior of, it is bo if * covers - some - - K but not all of the boundary segment and some- but - not K all of the interior of, it is nbo if * does not intersect with boundary segment and some but not all of the - Q- - interior K of, and it is no if there is no overla between * - and. Let NMOP - - K be the boundary segment shared by the cell and. We define boundary sensitive aroximation in terms of airs of aroximation functions, + ), as follows [BS98]: SUT@V W X YZ[X:\]WU^ S`_V W[a:bc^ fo S`_VLW[adbe^ o S`_V W[a:bc^ no S`_VLWfX Y,^ fo fo - - S _ VLWfX Y ^ o fbo bo nbo S _ VLWfX Y ^ no - - no Each aroximate region & 'HG@' recise regions having the aroximation : D */ We define the oeration 87:9 on the set & D ) stands for a set of recise regions, i.e. all those. fo fbo bo nbo no as: 3 487:9 no nbo bo fbo fo no no no no no no nbo no nbo nbo nbo nbo bo no nbo bo bo bo fbo no nbo bo fbo fbo fo no nbo bo fbo fo This oeration extends - to - K elements - of -& K 'G?' j to & D ) by 87:9 i 87:9Q oeration [BS98]. (i.e. the- set - of K k functions from %gah%. The definition of the is similar but silightly more comlicated the details can be found in

4 z { 3 CC5 relations Qualitative satial reasoning (QS) is a well-established subfield of AI. It is concerned with the reresentation and rocessing of knowledge of sace and activities which deend on sace. However, the reresentations used for this are qualitative, rather than the quantitative ones of conventional coordinate geometry. One of the most widely studied formal aroaches to QS is the egion-connection Calculus (CC) [CBGG97]. This system rovides an axiomatization of sace in which regions themselves are rimitives, rather than being constructed from more rimitive sets of oints. An imortant asect of the body of work on CC is the treatment of relations between regions. For examle two regions could be overlaing, or erhas only touch at their boundaries. There are two rincial schemes of relations between CC regions: five boundary insensitive relations known as CC5, and eight boundary sensitive relations known as CC8. In this aer we roose a secific style of defining CC relations. This style allows to define CC relations exclusively based on constraints regarding the outcome of the meet oeration between (one and two dimensional) regions. Furthermore this style of definitions allows us to obtain a artial ordering with minimal and maximal element on the relations defined. Both asects are critical for the generalization of these relations to the aroximation case. Given two regions l and m the CC5 relation between them can be determined by considering the trile of boolean values: Ny nl3omh #qrlq3smtulvwl3smexm The corresondence between such triles and the CC5 classification is given in the following table. Possible geometric interretations are given in figure 1. l3smh #q l=3omtl l3omtum CC5 F F F D T F F PO T T F PP T F T PPi T T T EQ { set of triles is artially ordered by setting ) 8} for ~?~ƒ, where the Boolean values are ordered by F f ~ ~ ) iff } T. This is the same ordering induced by the CC5 concetual grah [GC94]. But note that the concetual grah has PO and EQ as neighbours which is not the case in the Hasse diagram for the artially ordered set (ight diagram in figure 1). We refer to this as the CC5 lattice to distinguish it from the concetual neighborhood grah. 4 Semantic and Syntactic Generalizations of CC5 The original formulation of CC dealt with ideal regions which did not suffer from imerfections such as vagueness, indeterminacy or limited resolution. However, these are factors which affect satial data in ractical examles, and which are significant in

5 l Š Œ Š T T T EQ ˆ ˆ ˆr T T F PP T F T PPi ˆ ˆ ˆr D(x,y) PO(x,y) PP(x,y) PPI(x,y) EQ(x,y) T F F PO F F F D Fig. 1. CC5 relations and CC5 lattice alications such as geograhic information systems (GIS)[BF95]. The issue of vagueness and indeterminacy has been tackled in the work of [CG96]. The toic of the resent aer is not vagueness or indeterminacy in the widest sense, but rather the secial case where satial data is aroximated by being given a limited resolution descrition. 4.1 Syntactic and semantic generalizations There are two aroaches we can take to generalizing the CC5 classification from recise regions to aroximate ones. These two may be called the semantic and the syntactic. The syntactic has many variants. Semantic We can define the CC5 relationshi between aroximate regions and to be the set of relationshis which occur between any air of recise regions reresenting and. That is, we can define Syntactic FŽ $ F l jm 8 00 h11 and m y We can take a formal definition of CC5 in the recise case which uses oerations on $ and generalize this to work with aroximate regions by relacing the oerations on $ by analogous ones for &(' Syntactic generalization The above formulation of the CC5 relations can be extended to aroximate regions. One way to do this is to relace the oeration 3 with an aroriate oeration for 1 This technique has many variants since there are many different ways in which the CC5 can be formally defined in the recise case, and some of these can be generalized in different ways to the aroximate case. The fact that several different generalizations can arise from the same formula is because some of the oerations in (such as and ) have themselves more than one generalization to oerations on Õ.

6 In the context of aroximate regions, the bottom element, q, is the function from % to & ) which takes the value no for every element of %. Each of the above triles rovides an CC5 relation, so the relation between and can be measured by a air of CC5 relations. These relations will be denoted by $ 4 and $ 487:9. Theorem œ} Qk 1 The airs n$(4 j$(487:9f which can occur are all airs d where with the excetion of PP EQ and PPi EQ. =H} = Proof z { First we show that $(4 $(47N9F. Suose that $O4 ) = and that $O487:9 f ~ ~ ) }. We have to show that for -. Taking the first comonent, if Ÿq - then for each such that - Q no, we also have, by examining the tables for 3 4 and 3 487:9, that 3 487:9 no. Hence 3 487:9 q. Taking the second comonent, if then 3 47N9 - because from it follows that 3-487:9. This can be seen from the tables for 3 4 and 3 487:9 by considering each of the three ossible - values for. The case of the third comonent z { aroximate regions. If and are aroximate regions (i.e. functions from % to & ) ) we can consider the two triles of Boolean values: / / / 3 4 šq Ny (1) 7N9 q 87:9 87:9 follows from the second since and 87:9 are commutative. Finally= we have to show that the airs PP EQ and PPi EQ cannot occur. If $O487:9 EQ, then so must take the same value as. Thus the only triles which are ossible for $(4 are those where the= second and third comonents are equal. This rules out the ossibility that is PP or PPi. $O4 4.3 Corresondence of semantic and syntactic generalization Let the syntactic generalization of CC5 defined by, = f$o4 j$(487:9f where $(4 and $O487:9 are as defined above. Theorem 2 For any aroximate regions and, the two ways of measuring the relationshi of to are equivalent in the sense that FŽ I t H} œ} Qd CC5 $O4 where CC5 is the set EQ PP PPi PO } D, and lattice. =j $(487:9@ is the ordering in the CC5 The roof of this theorem deends on assumtions about the set of recise regions $. We assume that $ is a model of the CC axioms so that we are aroximating continuous sace, and not aroximating a sace of already aroximated regions.

7 m Proof There are three things to demonstrate. Firstly that for all l 00 h11, and u}, that (} $O4 = jm. Secondly, for all l and m as before, that jm =} œ} $O487:9, and thirdly that if is any CC5 relation such that $ 4 $ 487:9 then there exist articular l and m which stand in the relation to each other. To Q rove the first of these it is necessary to consider each of the three comonents 3 4 q =, 3 4 Q v and 3 4 in turn. If 3 4 #q is true, we have to show for all l and m that l3 m šq is also true. From Pq - it follows that there is at least one cell where one of and fully overlas -, and the other at least artially overlas. Hence there are interretations of and having - non-emty intersection. If - is true then for all cells we have - no or fo. In each case every interretation must satisfy le3omœl. Note that this deends on the fact - /- that the combination o cannot occur. The case of the final comonent is similar. Thus we have demonstrated for all l and m } o} that Q $O4 $ F l jm. The task of showing that jm $(487:9F is accomlished by a similar } s} analysis. Finally, we have to show that for each CC5 relation,, where $O4 $(487:9@, there are l and m such that the relation = of l to m is Q. This is done by considering the various ossibilities for $ 4 and $ 487:9. We will only consider one = of the cases here, but the others are similar. If $ 4 PO and $ 487: EQ, then for each cell, the values of and are equal and there must be some cells where this value is o and some cells where the value is fo. Precise regions l and m can be constructed by selecting sub-regions of each cell say lfª and m«ª, and defining l and m to be the unions of these sets of - sub-regions. /- In this articular case, there is sufficient freedom with those cells where o to be able to select lfª and m«ª so that the relation of l to m } œ} can be any where PO EQ. 5 Generalizing CC8 relations 5.1 CC8 relations CC8 relations take the toological distinction betwen interior and boundary into account. In order to describe CC8 relations we define the relationshi between l and m by using a trile, but where the three entries may take one of three truth values rather than the two Boolean ones. The scheme has the form nl3omh qjl3sm l klq3sm m where l3smh qš T if dy ± y the interiors of l and m overla jl=3sm² šq M if dy ± y only the boundaries l and m overla jl=3smtšq and ³LlQ3o³Lm² šq F if dy ± y there is no overla between l and m, jl=3smtšq and ³LlQ3o³Lmtšq

8 and where l,3rm ĺ T if dy ± y l is contained in m and the boundaries are either disjoint or k identical kl=3omtl and n³?nl 3 ³?nm #q or ³?nl 3 ³?nm š³? l M if dy ± y l is contained in m and the boundaries are not disjoint and not identical kl=3omtl and ³Ll3 ³Lmh #q dy ± yand ³?nl 3 ³? m š³? l F if l is not contained within mµ kl=3om² ul and similarly for lq3sm m. The corresondence between such triles and the CC8 classification is given in the following table. Possible geometric interretations can be found in figure 2. l3smh q l3om l l3om m CC8 F F F DC M F F EC T F F PO T M F TPP T T F NTPP T F M TPPi T F T NTPPi T T T EQ The CC5 relation D refines to DC and EC and the CC5 relation PP refines to TPP and NTPP. We define F M T and call the corresonding Hasse diagram CC8 lattice (figure 2 to distinguish it from the concetual neighborhood grah. DC(x,y) EC(x,y) PO(x,y) TPP(x,y) NTPP(x,y) EQ(x,y) Fig. 2. CC8 lattice 5.2 Syntactic generalization of CC8 Let scheme has the form and be boundary sensitive aroximations of regions l and m. The generalized j ( :9 q q / :9 / 3 4 = 3 487:9 Qk

9 l 4 ³ ³ ³ ³ ³ where q# T M F q q and ³ q and ³ q q and where ( 3 4 T 3 4 and f³ 3 4 q ( or 3 4 M 3 4 and ³ 3 4 q and 3 4 F 3 4 and similarly for, 87:9 q, 87:9,and 87:9. In this context the bottom element, q, is the function from %ua % to & D which takes the value no for every element of % A%. The formula ³ #q is true if we can derive from boundary sensitive aroximations and that for all l 00 h11 and m the least relation that can hold between l and m involves boundary intersection 2. Corresondingly, ³ 3 487:9 ³ #q is true if the largest relation that can hold between and m involves boundary intersection. Each of the above triles defines a CC8 relation, so the relation between and can be measured by a air of CC8 relations. These relations will be denoted by $( = = and $/ 47N9. [BS00] show the corresondence between this syntactic generalization and the semantin generalization corresonding to the CC5 case. 6 Conclusions In this aer we discussed aroximations of satial regions with resect to an underlying regional artitions. We used aroximations based on aroximation functions and discussed the close relationshi to ough sets. We defined airs of greatest minimal and least maximal meet oerations on aroximation functions that constrain the ossible outcome of the meet oeration between the aroximated regions themselves. The meet oerations on aroximation maings rovide the basis for aroximate qualitative s- atial reasoning that was roosed in this aer. Aroximate qualitative satial reasoning is based on: 1. Jointly exhaustive and air-wise disjoint sets of qualitative relations between exact regions, which are defined in terms of the meet oeration of the underlying Boolean algebra structure of the domain of regions. As a set these relations must form a lattice with bottom and to element. 2. Aroximations of regions with resect to a regional artition of the underlying sace. Semantically, an aroximation corresonds to the set of regions it aroximates. 3. Pairs of meet oerations on those aroximations, which aroximate the meet oeration on exact regions. 2 For details see [BS00].

10 Based on those ingredients syntactic and semantic generalizations of jointly exhaustive and air-wise disjoint relations between exact regions were defined. Generalized relations hold between aroximations of regions rather than between (exact) regions themselves. Syntactic generalization is based on relacing the meet oeration defining relations between exact regions by its minimal and maximal counterarts on aroximations. Semantically, syntactic generalizations yield uer and lower bounds (within the underlying lattice structure) on relations that can hold between the corresonding aroximated exact regions. There is considerable scoe for further work building on the results in this aer. We have assumed so far that the regions being aroximated are recisely known regions in a continuous sace. However, there are ractical examles where aroximate regions are themselves aroximated. This can occur when satial data is required at several levels of detail, and the less detailed reresentations are aroximations of the more detailed ones. Thus one direction for future investigation is to extend the techniques in this aer to the case where the regions being aroximated are discrete, rather than continuous. This could make use of the algebraic aroach to qualitative discrete sace resented in [?]. Subject of ongoing research is to aly techniques resented in this aer to the temoral domain [Bit00]. eferences [Ben96] B. Bennett. An alication of qualitative satial reasoning to gis. In 1st. International Conference on GeoComutation, Leeds, [BF95] Peter Burrough and Andrew U. Frank, editors. Geograhic Objects with Indeterminate Boundaries. GISDATA Series II. Taylor and Francis, London, [Bit00] T. Bittner. Aroximate temoral reasoning. In Worksho roceedings of the Seventeenth National Conference on Artificial Intelligence, AAAI 2000, [BS98] T. Bittner and J. G. Stell. A boundary-sensitive aroach to qualitative location. Annals of Mathematics and Artificial Intelligence, 24:93 114, [BS00] T. Bittner and J. Stell. Aroximate qualitative satial reasoning. Technical reort, Deartment of Comuting and Information Science, Queen s University, [CBGG97] A.G. Cohn, B. Bennett, J. Goodday, and N. Gotts. Qualitative satial reresentation and reasoning with the region connection calculus. geoinformatica, 1(3):1 44, [CG96] A.G. Cohn and N.M. Gotts. The egg-yolk reresentation of regions with indeterminate boundaries. In P. Burrough and A.U. Frank, editors, Geograhic Objects with Indeterminate Boundaries, GISDATA Series II. Taylor and Francis, London, [EF91] Max J. Egenhofer and obert D. Franzosa. Point-set toological satial relations. International Journal of Geograhical Information Systems, 5(2): , [GC94] J.M. Goodday and A.G. Cohn. Concetual neighborhoods in temoral and satial reasoning. In ECAI-94 Satial and Temoral easoning Worksho, [Paw91] Zdzis aw Pawlak. ough sets : theoretical asects of reasoning about data. Theory and decision library. Series D, System theory, knowledge engineering, and roblem solving ; v. 9. Kluwer Academic Publishers, Dordrecht ; Boston, [CC92] D. A. andell, Z. Cui, and A. G. Cohn. A satial logic based on regions and connection. In 3rd Int. Conference on Knowledge eresentation and easoning. Boston, [SP92] T.. Smith and K. K. Park. Algebraic aroach to satial reasoning. Int. J. Geograhical Information Systems, 6(3): , 1992.