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2 ISPRS Journl of Photogrmmetry nd Remote Sensing 65 (2010) Contents lists ville t ScienceDirect ISPRS Journl of Photogrmmetry nd Remote Sensing journl homepge: Modeling nd querying pproximte direction reltions Shihong Du,, Luo Guo Institute of Remote Sensing nd GIS, Peking University, Beijing, , Chin School of Life nd Environmentl Science, Minzu University of Chin, Beijing, , Chin r t i c l e i n f o s t r c t Article history: Received 10 June 2008 Received in revised form 14 Mrch 2010 Accepted 18 Mrch 2010 Aville online 18 April 2010 Keywords: Geogrphicl informtion system Direction reltions Uncertin regions Sptil dt query Existing models of direction reltions re minly designed to hndle crisp regions. To ccommodte uncertin sptil dt, it is necessry to investigte the formliztion, uncertin semntics, nd composition opertors for uncertin direction reltions. In this study, direction reltions out uncertin regions, i.e., pproximte direction reltions, re modeled s the comintions of four crisp direction reltions. The pproximte reltions cn e interpreted from two spects: the lower prt (only including crisp reltions out uncertin region) nd the uncertin prt (the uncertin directions out uncertin regions). The uncertin semntics of uncertin directions re formlized, such s possily north, possily south, possily southest, etc. Both crisp nd uncertin prts re used to simplify nd hndle the composition nd the query of uncertin direction reltions. Approximte direction reltions re helpful to model directions concerned with oth crisp nd uncertin regions; they therefore cn ply importnt roles in hndling uncertin dt (in our cse, querying uncertin dt) Interntionl Society for Photogrmmetry nd Remote Sensing, Inc. (ISPRS). Pulished y Elsevier B.V. All rights reserved. 1. Introduction Direction reltions descrie the crdinl directions of trget oject with respect to given reference oject, from which the directions re locted (Goyl, 2000). The formliztion of direction reltions is out uilding model to mp the reltive position of sptil ojects into the symolic directions (for exmple, region A is north of region B, nd region C is southest of region D). The composition of direction reltions is out deriving unknown reltions from existing direction reltions sed on the symolic opertions. For exmple, ssuming two direction reltions: region A is north of region B, nd region B is north of region C, then without geometricl informtion out regions eing required, only through common knowledge resoning it cn e concluded tht region A is lso north of region C. Since direction reltions use symols to express people s cognition to reltions, they re fundmentl to sptil dt query (Goyl nd Egenhofer, 2001), pictoril retrievl systems (Yu nd Zhng, 2004), nd sptil resoning systems (Adelmoty nd Jones, 1997), wyfinding ssistnce systems (Klippel et l., 2004), nd the detection of consistency of direction reltions (Skidopoulos nd Kourkis, 2005). Corresponding uthor. Tel.: ; fx: E-mil ddress: dshgis@hotmil.com (S. Du). Trditionl geogrphicl informtion systems (GISs) only del with geogrphicl phenomen tht cn e clerly defined or hve crisp oundries, like rods nd uildings. Such phenomen re modeled s crisp points, lines, nd regions. While the sptil dt re inherently uncertin due to multiple sources (Clementini nd Di Felice, 1996; Woroys nd Clementini, 2001). For exmple, t the stge of cquiring dt, due to imprecise oservtion nd the inherent vgue, the extent nd oundry of sptil ojects cnnot e defined or otined ccurtely; t the stge of process nd integrtion, multiple representtions nd dt formt trnsformtions cn introduce uncertinty. Uncertin dt from vrious sources (Clementini nd Di Felice, 1996; Woroys nd Clementini, 2001) cn e modeled y the regions with rod oundries (BBRs). As the uncertin phenomen ecome more nd more fmilir, GISs should hve the ility to hndle comintion of these dt. To model direction reltions etween crisp ojects, mny models hve een proposed, such s projection-sed (Frnk, 1996), cone-sed (Hr, 1976; Frnk, 1996), tringulr model (Hr, 1976; Peuquet nd Zhng, 1987), 2D string (Chng et l., 1987), direction reltions etween minimum ounding rectngles (MBRs) (Ppdis nd Theodoridis, 1997) nd direction-reltion mtrix (Goyl, 2000). In generl, these models used the pproximte shpes of sptil ojects to compute direction reltions. The direction-reltion mtrix (Goyl, 2000) cn cpture directions more precisely thn other models s it uses the exct representtion of the trget oject. Furthermore, this model not only considers the sizes nd the shpes of two ojects, ut lso represents /$ see front mtter 2010 Interntionl Society for Photogrmmetry nd Remote Sensing, Inc. (ISPRS). Pulished y Elsevier B.V. All rights reserved. doi: /j.isprsjprs

3 S. Du, L. Guo / ISPRS Journl of Photogrmmetry nd Remote Sensing 65 (2010) the reltions s symols, thus it is dopted comprehensively to descrie direction reltions. To query uncertin dt in the Sptil- Query-y-Sketch context, the two-vlue, 3 3 direction-reltion mtrix for crisp regions ws extended to four-vlue, 5 5 mtrix to model direction reltions etween simple regions with rod oundries (SBBRs) (Cicerone nd Di Felice, 2000). The gol of the composition of direction reltions is out computing ll possile reltions etween ojects A nd C from existing direction reltions etween A nd B, nd B nd C. Currently, mny opertions concerning crisp reltions hve een proposed, such s the compositions for crisp points (Frnk, 1996), for crisp regions (Skidopoulos nd Kourkis, 2004), nd for points orgnized hierrchiclly (Ppdis nd Egenhofer, 1997). Cicerone nd Di Felice (2004) investigted directionl pirwise-consistency out crisp regions sed on the direction-reltion mtrix. Most of models developed thus fr hve focused on hndling direction reltions etween crisp regions. Recently, topologicl reltions out uncertin regions hve een investigted (Clementini nd Di Felice, 1996, 1997, 2001; Cohn nd Gotts, 1996; Liu nd Shi, 2006; Du et l., 2006, 2008,), while there is reltively little work on direction reltions out uncertin regions. Therefore, most pplictions cn only del with crisp direction reltions, due to the lck of such model for uncertin dt. The 4-vlue, 5 5 direction-reltion mtrix cnnot directly represent direction reltions etween uncertin regions s smll set of symols (est, west, north, south, etc.), ut s geometricl configurtion. It lso cnnot mesure the uncertin semntics of direction reltions of uncertin dt. Therefore, the model pplies to the Sptil-Query-y-Sketch (Egenhofer, 1997), ut not other pplictions (like sptil query lnguges (Aufure-Portier, 1995; Egenhofer, 1994; Morris et l., 2004), nd sptil dt mining (Koperski nd Hn, 1995). These pplictions require oth symolic directions nd the compositionl opertions out these symols. Accordingly, model is required to model, query nd compose symolic directions nd their uncertin semntics out uncertin regions. This pper focuses on modeling, composing nd querying direction reltions out multi-source uncertin dt, which is represented s two crisp regions: the inner region nd outer region. Therefore, direction reltions out uncertin dt cn lso e pproximted y four crisp reltions etween inner nd inner, inner nd outer, outer nd inner, nd outer nd outer regions. As these kinds of reltions re pproximte representtions, they re clled pproximte direction reltions compred with the crisp reltions etween crisp ojects (Hr, 1976; Peuquet nd Zhng, 1987; Ppdis nd Theodoridis, 1997; Ppdis nd Egenhofer, 1997; Cicerone nd Di Felice, 2004; Skidopoulos nd Kourkis, 2004). The constrints mong the four reltions re presented to determine how mny reltions re relizle for uncertin dt. Furthermore, the four crisp reltions re used to define the uncertin semntics of pproximte reltions nd to query uncertin dt. Becuse the presented model is the pproximtion of the directions out uncertin dt, it cn lso process direction reltions etween uncertin regions nd crisp regions. Furthermore, the method directly models symolic directions, thus it cn e pplied to query uncertin sptil dt, detect the direction inconsistencies, nd evlute the equivlence nd similrity of direction reltions. Relted work on crisp direction reltions is introduced in Section 2. Section 3 presents n pproximte representtion of direction reltions etween uncertin regions. Uncertin semntics of direction reltions re defined in Section 4. The composition nd pirwise-consistency of pproximte direction reltions re descried in Section 5. Section 6 pplies the pproximte direction reltions to support the query of uncertin dt. Future work is suggested in Section 7. Fig. 1. Direction-reltion mtrix. 2. Direction reltions etween regions 2.1. Direction reltions etween crisp regions As projection-sed system, the direction-reltion mtrix (Goyl, 2000) uses the minimum ounding rectngle (MBR) of reference region A to divide the spce round A into eight directions: est, west, south, north, southest, southwest, northest, nd northwest, denoted y E A, W A, S A, N A, SE A, SW A, NE A, nd NW A, respectively (Fig. 1). If two regions re coincident, the direction reltion etween them is sme position, denoted yo A. A trget region B must fll inside t lest one of the nine directions. Definition 1. Let A nd B e two crisp regions, then the nine symolic directions cn e defined s the following (Skidopoulos nd Kourkis, 2004): R(A, B) = {O A } iff inf x (A) inf x (B), sup x (A) sup x (B), inf y (A) inf y (B), nd sup y (A) sup y (B) R(A, B) = {N A } iff inf x (A) inf x (B), sup x (A) sup x (B), nd sup y (A) inf y (B) R(A, B) = {S A } iff inf x (A) inf x (B), sup x (A) sup x (B), nd sup y (B) inf y (A) R(A, B) = {E A } iff inf y (A) inf y (B), sup y (A) sup y (B), nd sup x (A) inf x (B) R(A, B) = {W A } iff inf y (A) inf y (B), sup y (A) sup y (B), nd sup x (B) inf x (A) R(A, B) = {NE A } iff sup y (A) inf y (B), nd sup x (A) inf x (B) R(A, B) = {NW A } iff sup y (A) inf y (B), nd sup x (B) inf x (A) R(A, B) = {SE A } iff sup y (B) inf y (A), nd sup x (A) inf x (B) R(A, B) = {SW A } iff sup y (B) inf y (A), nd sup x (B) inf x (A) In the ove equtions, inf x (A), sup x (A), inf y (A), nd sup y (A) re the minimum nd mximum projections of reference region A on x-xis nd y-xis, respectively, so re inf x (B), sup x (B), inf y (B), nd sup y (B) (Fig. 2). In reltion R(A, B), B is the trget oject, nd A is the reference oject. To incorporte direction reltions into symolic context, the reltions should e represented s set of symols. If region B only flls inside one direction of A, R(A, B) is single-item; if B flls inside multiple directions, R(A, B) is multi-item (Goyl, 2000). For exmple, in Fig. 2, R(A, B) = {E A } is single-item, while in Fig. 2,

4 330 S. Du, L. Guo / ISPRS Journl of Photogrmmetry nd Remote Sensing 65 (2010) Modeling pproximte direction reltions To stisfy the requirement for symolic reltions etween SB- BRs, the forml model should provide ech direction with nme. To hndle uncertin regions nd crisp regions in the sme wy, the model should relte uncertin direction reltions to the crisp reltions. Therefore, direction reltions etween SBBRs should e pproximted y reltions out inner nd outer regions. Fig. 2. Direction reltions nd the MBR of the reference region. () Single-item direction reltion. () Multi-item direction reltion. Fig. 3. A simple region with rod oundries. R(A, B) = {N A, NE A, O A, E A } is multi-item. For pir of regions, there re 218 relizle direction reltions in the physicl world (Goyl, 2000; Skidopoulos nd Kourkis, 2004). If R(A, B) is multi-item composed of directions r 1, r 2,..., r n (2 n 9), the trget region B cn e split into n su-regions 1, 2,..., n (2 n 9), such tht B = n i=1 i, R(A, 1 ) = {r 1 }, R(A, 2 ) = {r 2 },..., nd R(A, n ) = {r n } hold. For exmple, in Fig. 2, B = , R(A, 1 ) = {N A }, R(A, 2 ) = {NE A }, R(A, 3 ) = {O A }, nd R(A, 4 ) = {E A }, thus R(A, B) = {N A, NE A, O A, E A } Simple regions with rod oundries To formlize direction reltions etween regions with rod oundries, the definition of region with rod oundries should e introduced first. Definition 2. A crisp region is simple if it is homeomorphic to closed plne disk (Skidopoulos nd Kourkis, 2004). Definition 3. A region with rod oundries (BBR), A, is composed of pir of simple regions: inner region A 1 nd outer region A 2, such tht A 1 A 2 holds (Clementini nd Di Felice, 1997, 2001). If A 1 is proper suset of A 2, A hs rod oundry; otherwise, it is crisp one. The rod oundry A is defined s the difference etween A 2 nd the interior of A 1, i.e., A = A 2 A 1 (Fig. 3). Definition 4. If oth inner region A 1 nd outer region A 2 re simple regions, A is simple region with rod oundries (SBBR), otherwise it is complex region with rod oundries (CBBR). The holes in ojects hve no effect on direction reltions (Skidopoulos et l., 2005). In Definition 1, the minimum nd mximum projections of oth trget nd reference regions re independent on the holes. Although complex regions re composed of disconnected components, they still hve no effect on the projections of reference region; ut they cn ffect the projections of the trget region. In this sitution, the direction reltions out complex regions cn e represented s the set of the reltions etween components of trget regions nd the reference region. Therefore, this study pys much ttention to the reltions out SBBRs The comintionl representtion of pproximte direction reltions Becuse SBBR A is composed of two crisp regions: inner region A 1 nd outer region A 2 (Clementini nd Di Felice, 1996, 1997), two direction prtitions cn e mde on the plne round A 1 nd A 2. Given two SBBRs A nd B, the reltions etween them cn e represented s the four ones etween A 1 nd B 1, A 1 nd B 2, A 2 nd B 1, nd A 2 nd B 2 (Fig. 4), denoted y R(A 1, B 1 ), R(A 1, B 2 ), R(A 2, B 1 ), nd R(A 2, B 2 ), respectively. Like crisp direction reltions, the four reltions re still formlized y Definition 1 with nine crdinl directions E A, W A, S A, N A, SE A, SW A, NE A, NW A, nd O A. For exmple, in Fig. 4, R(A 1, B 1 ) = {N A }, R(A 1, B 2 ) = {N A, NE A, E A }, R(A 2, B 1 ) = {N A, NE A, O A, E A }, R(A 2, B 2 ) = {N A, NE A, O A, E A }. Therefore, direction reltions out uncertin regions cn e pproximted y the comintion of the four reltions. Among the four reltions, R(A 1, B 1 ) nd R(A 1, B 2 ) re sed on the prtition round A 1, thus they descrie how the regions B 1 nd B 2 re relted to region A 1 ; while the two others re relevnt to A 2. Although the four reltions depend on different reference regions, ll of them re expressed s directions with sme nmes s they re closely relted to the sme region A. From the composition perspective, the four reltions together pproximte the direction reltions, thus there must e some constrints mong the four reltions to remove impossile comintions. From the semntic perspective, only reltion R(A 1, B 1 ) is concerned with the crisp prt, nd the other three re relted to the uncertin prt, thus the uncertin semntic of pproximte reltions should e directly defined ccording to the roles of the four reltions. The constrints will e formlized in Section 3.2, nd the uncertin semntic will e defined in Section 4. In specil cses, only two reltions re sufficient to hndle direction reltions. For exmple, to descrie the direction reltion of crisp region B with respect to SBBR A, two reltions R(A 1, B) nd R(A 2, B) should e comined; while to hndle the reltion of SBBR B with respect to crisp region A, R(A, B 1 ) nd R(A, B 2 ) should e provided. Therefore, the pproximte reltions re fvorle to formlize, compose, nd query directions etween uncertin regions, nd uncertin regions nd crisp regions in the sme wy The constrints mong the four direction reltions The four reltions, R(A 1, B 1 ), R(A 1, B 2 ), R(A 2, B 1 ), nd R(A 2, B 2 ), descrie direction informtion etween different pirs of regions. For two SBBRs A nd B, A 1 A 2 nd B 1 B 2 hold, thus there must hve some constrints mong the four reltions. If one reltion is fixed, the others should e confined to some possile cses. These constrints cn help to determine the relizle pproximte reltions The constrints imposed y n uncertin trget region If the reference region is crisp, while the trget region is uncertin, the constrints cn e modeled y nlyzing the reltions R(A, B 1 ) nd R(A, B 2 ). To nlyze the constrints etween reltions R(A, B 1 ) nd R(A, B 2 ), the reltion R(A, B 2 ) cn e otined y expnding trget region B 1 to B 2, or R(A, B 1 ) otined y reducing B 2 to B 1, with the reference region A eing fixed. Accordingly, the following two theorems re useful: Theorem 1

5 S. Du, L. Guo / ISPRS Journl of Photogrmmetry nd Remote Sensing 65 (2010) c d Fig. 4. Approximte representtion of direction reltions etween SBBRs. () The reltion etween A 1 nd B 1. () The reltion etween A 1 nd B 2. (c) The reltion etween A 2 nd B 1. (d) The reltion etween A 2 nd B 2. interprets the constrint of R(A, B 2 ) on R(A, B 1 ) y deriving the possile R(A, B 1 ) from known R(A, B 2 ); Theorem 2 indictes the constrint of R(A, B 1 ) on R(A, B 2 ) y deriving the possile R(A, B 2 ) from known R(A, B 1 ). Definition 5. Let R e direction reltion consisting of directions r 1,..., r n (2 n 9), function δ(r) = δ(r 1,..., r n ) represents the set of relizle direction reltions generted y the disjunction of directions r 1,..., r n. Tht is, δ(r) = {R 1,..., R m }, with ech R i (1 i m) eing one of the 218 relizle crisp reltions, denoted y symol D (Skidopoulos nd Kourkis, 2004). There re only up to 218 relizle reltions in set D, ut δ(r) cn produce more reltions thn 218. Therefore, those reltions, which re not included in set D, should e removed. For exmple, δ({nw A, N A, NE A }) = {{NW A }, {N A }, {NE A }, {NW A, N A }, {N A, NE A }, {NW A, N A, NE A }}. The reltion {NW A, NE A } is excluded s it is not in D. Theorem 1. Let R(A, B 1 ) nd R(A, B 2 ) e two direction reltions nd B 1 B 2 holds, then R(A, B 1 ) should elong to δ(r(a, B 2 )), i.e., R(A, B 1 ) δ(r(a, B 2 )). Proof. Assuming tht R(A, B 2 ) is composed of directions r 1, r 2,..., r n (1 n 9) (Fig. 5), then the trget region B 2 is split into suregions 21, 22,..., 2n, such tht R(A, 21 ) = {r 1 }, R(A, 22 ) = {r 2 },..., nd R(A, 2n ) = {r n } hold. Furthermore, s B 1 B 2 holds, region B 1 must e inside one or multiple su-regions mong 21, 22,..., 2n (Fig. 5). Therefore, R(A, B 1 ) cn e ny reltion composed of directions in {r 1, r 2,..., r n }, i.e., R(A, B 1 ) δ({r 1, r 2,..., r n }) = δ(r(a, B 2 )). In Fig. 5, R(A, B 2 ) = {N A, NE A, E A }, δ(r(a, B 2 )) = {{N A }, {NE A }, {E A }, {N A, NE A }, {NE A, E A }, {N A, NE A, E A }}, thus R(A, B 1 ) cn e ny one in set δ(r(a, B 2 )). Definition 6. Let M 1 = {R 11, R 12,..., R 1m } (1 m 9) nd M 2 = {R 21, R 22,..., R 2n } (1 n 9) e two sets of reltions, the Crtesin opertion of M 1 nd M 2 (M 1 M 2 ), is set of ll relizle direction reltions y joining ech R 1i M 1 (1 i m) nd ech R 2j M 2 (1 j n), i.e., M 1 M 2 = {R 1i R 2j R 1i M 1 R 2j M 2 R 1i R 2j D}. Fig. 5. The nlysis of reducing the trget region. For exmple, ssuming M 1 = {{E A }, {O A, E A }, {W A, O A }} nd M 2 = {{SW A }, {S A, SE A }}, then M 1 M 2 = {{E A, S A, SE A }, {O A, E A, S A, SE A }, {W A, O A, SW A }, {W A, O A, S A, SE A }}. Direction reltions {E A, SW A }, {E A, O A, SW A }, etc., re removed s they re unrelizle. Theorem 2. Let R(A, B 1 ) nd R(A, B 2 ) e two direction reltions nd B 1 B 2 holds, then R(A, B 2 ) should elong to R(A, B 1 ) D, i.e., R(A, B 2 ) R(A, B 1 ) D. Proof. Let R(A, B 1 ) e composed of direction r 1, r 2,..., r n (1 n 9), then the trget region B 1 cn e split into su-regions 1, 2,..., n, such tht R(A, 1 ) = {r 1 }, R(A, 2 ) = {r 2 },..., nd R(A, n ) = {r n } hold. Furthermore, since B 1 B 2, region B 2 cn e composed of su-regions 1, 2,..., n, n+1 ( n+1 = B 2 B 1 ) (Fig. 6). Furthermore, R(A, n+1 ) cn e ny reltion in D. Therefore, R(A, B 2 ) is ny relizle reltion contining directions r 1, r 2,..., r n, i.e., R(A, B 2 ) R(A, B 1 ) D. As shown in Fig. 6, R(A, B 1 ) = {NW A, N A, NE A, O A, E A, SE A }, R(A, B 1 ) D = {{NW A, N A, NE A, O A, E A, SE A }, {NW A, N A, NE A, O A, E A, S A, SE A }, {NW A, N A, NE A, W A, O A, E A, SE A }, {NW A, N A, NE A, W A, O A, E A, SW A, SE A }, {NW A, N A, NE A, O A, E A, SW A, S A, SE A }, {NW A, N A, NE A, W A, O A, E A, S A, SE A }, {NW A, N A, NE A, W A, O A, E A, SW A, S A, SE A }}. Therefore, R(A, B 2 ) cn e ny reltion mong the seven.

6 332 S. Du, L. Guo / ISPRS Journl of Photogrmmetry nd Remote Sensing 65 (2010) Tle 1 Composition tle for deriving R(A 2, B) from R(A 1, B). R(A 1, B) R(A 2, B) R(A 1, B) R(A 2, B) R(A 1, B) R(A 2, B) {N A } δ({n A, O A }) {W A } δ({w A, O A }) {NW A } δ({nw A, N A, W A, O A }) {S A } δ({o A, S A }) {NE A } δ({n A, NE A, O A, E A }) {SE A } δ({o A, E A, S A, SE A }) {E A } δ({o A, E A }) {SW A } δ({sw A, W A, O A, S A }) {O A } δ({o A }) Fig. 6. The nlysis of expnding the reference region. () Proof. () An exmple. In order to descrie the direction reltion of n uncertin region B with respect of crisp region A, two reltions R(A, B 1 ) nd R(A, B 2 ) re needed. Therefore, Theorems 1 nd 2 re useful to check the relizility of = comintions. Only the comintions stisfying Theorems 1 nd 2 re relizle, thus there re 5444 relizle reltions etween n uncertin region nd crisp region The constrints imposed y n uncertin reference region If the reference region is uncertin, while the trget region is crisp, the constrints cn e modeled y nlyzing the reltions R(A 1, B) nd R(A 2, B). To nlyze the constrints etween reltions R(A 1, B) nd R(A 2, B), the reltion R(A 2, B) cn e derived y expnding the reference region A 1 to A 2, or reducing A 2 to A 1, while the trget region B is fixed. Therefore, we hve the following two theorems: Theorem 3 indictes how R(A 1, B) restricts R(A 2, B) when the reference region A 1 is expnded to A 2 ; while Theorem 4 shows how R(A 2, B) constrins R(A 1, B) when the reference region A 2 is reduced to A 1. Theorem 3. Let R(A 1, B) nd R(A 2, B) e two direction reltions (A 1 A 2 holds). If R(A 1, B) is known, the possile R(A 2, B) derived from R(A 1, B), denoted y CDR 1 (R(A 1, B)), cn e computed y the following situtions: (1) if R(A 1, B) is single-item, CDR 1 (R(A 1, B)) cn e those in Tle 1; (2) if R(A 1, B) = {r 1, r 2,..., r n } (2 n 9), nd CDR 1 ({r 1 }), CDR 1 ({r 2 }),..., CDR 1 ({r n }) denote the corresponding sets derived from r 1, r 2,..., r n, respectively, the possile R(A 2, B) is the Crtesin product of these sets, i.e., CDR 1 (R(A 1, B)) = CDR 1 ({r 1 }) CDR 1 ({r 2 }),..., CDR 1 ({r n }). Proof. (1) Becuse A 2 is otined y expnding A 1, inf x (A 2 ) inf x (A 1 ), inf y (A 2 ) inf y (A 1 ), sup x (A 2 ) sup x (A 1 ), nd sup y (A 2 ) sup y (A 1 ) must hold. When R(A 1, B) is single-item, e.g., R(A 1, B) = {N A }, inf x (A 1 ) inf x (B), sup x (A 1 ) sup x (B) nd sup y (A 1 ) inf y (B) hold. When the reference region A 1 is expnded to A 2, inf x (A 2 ) inf x (A 1 ) inf x (A 1 ) inf x (B) inf x (A 2 ) inf x (B), sup x (A 2 ) sup x (A 1 ) sup x (A 1 ) sup x (B) sup x (A 2 ) sup x (B), nd inf y (A 2 ) inf y (A 1 ) inf y (A 1 ) inf y (B) inf y (A 2 ) inf y (B) hold. Fig. 7. Two exmples () nd () of constrints of R(A 1, B) on R(A 2, B). However, ccording to sup y (A 2 ) sup y (A 1 ) nd sup y (A 1 ) inf y (B), the reltion etween sup y (A 2 ) nd inf y (B) cnnot e identified. Tht is, either sup y (A 2 ) inf y (B) or sup y (A 2 ) inf y (B) is possile. According to Definition 1, the direction reltion etween A 2 nd B cn e {N A }, {O A }, or {N A, O A }. Similrly, when R(A 1, B) is {N A }, {W A }, {E A }, {S A }, {SW A }, {SE A }, {NW A }, {NE A }, nd {O A }, the possile R(A 2, B) derived from R(A 1, B) is included in Tle 1. The first column denotes the direction reltion R(A 1, B), nd the second column refers to the possile R(A 2, B). For exmple, when R(A 1, B) is {S A }, the set of possile R(A 2, B) is δ({s A, O A }). Tht is, R(A 2, B) cn e {S A }, {O A }, or {O A, S A }. (2) If R(A 1, B) is composed of multiple directions r 1, r 2,..., r n, B cn e split into n exclusive su-regions, 1, 2,..., n, such tht R(A 1, 1 ) = {r 1 }, R(A 1, 2 ) = {r 2 },..., nd R(A 1, n ) = {r n } hold. For ech single-item R(A 1, i ) (1 i n), the possile R(A 2, i ) cn e otined from Tle 1. B is composed of n exclusive su-regions, nd ech su-region restrict R(A 2, i ) to set CDR 1 ({r i }), thus possile R(A 2, B) is the Crtesin product of CDR 1 ({r 1 }), CDR 1 ({r 2 }),..., CDR 1 ({r n }) (Definition 1). Tht is, CDR 1 (R(A 1, B)) = CDR 1 ({r 1 }) CDR 1 ({r 2 }),..., CDR 1 ({r n }). In Fig. 7, R(A 1, B) = {N A, NE A }, CDR 1 (R(A 1, B)) = CDR 1 ({N A }) CDR 1 ({NE A }) = δ({n A, O A }) δ({n A, NE A, O A, E A }) = {{N A }, {O A }, {N A, NE A }, {N A, O A }, {O A, E A }, {N A, NE A, E A }, {N A, O A, E A }, {N A, NE A, O A }, {NE A, O A, E A }, {N A, NE A, O A, E A }}. Tht is, R(A 2, B) must e one of the 10 reltions. In Fig. 7, R(A 1, B) = {NE A, E A, SE A }, CDR 1 (R(A 1, B)) = CDR 1 ({NE A }) CDR 1 ({E A }) CDR 1 ({SE A }) = δ({n A, NE A, O A, E A }) δ({o A, E A }) δ({o A, E A, S A, SE A }). In this cse, CDR 1 (R(A 1, B)) is composed of 34 reltions. Theorem 4. Let R(A 1, B) nd R(A 2, B) e two direction reltions (A 1 A 2 holds). If R(A 2, B) is known, the possile R(A 1, B) derived from R(A 2, B), denoted y CDR 2 (R(A 2, B)), cn e computed y the following two cses: (1) if R(A 2, B) is single-item, CDR 2 (R(A 2, B)) is listed in Tle 2; (2) if R(A 2, B) = {r 1, r 2,..., r n } (2 n 9), nd CDR 2 ({r 1 }), CDR 2 ({r 2 }),..., CDR 2 ({r n }) re the corresponding sets derived from r 1, r 2,..., r n, respectively, possile R(A 1, B) derived from R(A 2, B) re the Crtesin products, i.e., CDR 2 (R(A 2, B)) = CDR 2 ({r 1 }) CDR 2 ({r 2 }),..., CDR 2 ({r n }). Proof. (1) Becuse A 1 is otined y reducing A 2, inf x (A 2 ) inf x (A 1 ), inf y (A 2 ) inf y (A 1 ), sup x (A 2 ) sup x (A 1 ), nd sup y (A 2 ) sup y (A 1 ) hold.

7 S. Du, L. Guo / ISPRS Journl of Photogrmmetry nd Remote Sensing 65 (2010) Tle 2 Composition tle for deriving R(A 1, B) from R(A 2, B). R(A 2, B) R(A 1, B) R(A 2, B) R(A 1, B) R(A 2, B) R(A 1, B) {N A } δ({nw A, N A, NE A }) {W A } δ({nw A, W A, SW A }) {NW A } δ({nw A }) {S A } δ({sw A, S A, SE A }) {NE A } δ({ne A }) {SE A } δ({se A }) {E A } δ({ne A, E A, SE A }) {SW A } δ({sw A }) {O A } δ({nw A, N A, NE A, W A, O A, E A, SW A, S A, SE A }) c d e f Fig. 8. Possile R(A 1, B) derived from R(A 2, B) when R(A 2, B) = {N A }. () {NE A }. () {N A, NE A }. (c) {N A }. (d) {NW A, N A, NE A }. (e) {NW A }. (f) {NW A, N A }. If R(A 2, B) = {NE A }, sup x (A 2 ) inf x (B) nd sup y (A 2 ) inf y (B) hold. Anywy, sup x (A 2 ) sup x (A 1 ) sup x (A 2 ) inf x (B) sup x (A 1 ) inf x (B) nd sup y (A 2 ) sup y (A 1 ) sup y (A 2 ) inf y (B) sup y (A 1 ) inf y (B) hold. Accordingly, when R(A 2, B) = {NE A }, nd reference region A 2 is reduced to A 1, trget region B is still northest of A 1 ccording to Definition 1. Similrly, when R(A 2, B) = {NW A }, {SW A }, or {SE A }, R(A 1, B) is still {NW A }, {SW A }, nd {SE A }, respectively. If R(A 2, B) = {N A }, inf x (A 2 ) inf x (B), sup x (A 2 ) sup x (B) nd sup y (A 2 ) inf y (B) hold. Anywy, } inf x (A 2 ) inf x (A 1 ) inf x (A 2 ) inf x (B) } sup x (A 2 ) sup x (A 1 ) sup x (A 2 ) sup x (B) nd { infx (A 1 ) inf x (B) inf x (A 1 ) inf x (B), { supx (A 1 ) sup x (B) sup x (A 1 ) sup x (B), sup y (A 1 ) sup y (A 2 ) sup y (A 2 ) inf y (B) sup y (A 1 ) inf y (B) hold. Accordingly, there re four cses: (i) inf x (A 1 ) inf x (B), sup x (A 1 ) sup x (B), nd sup y (A 1 ) inf y (B); (ii) inf x (A 1 ) inf x (B), sup x (A 1 ) sup x (B), nd sup y (A 1 ) inf y (B); (iii) inf x (A 1 ) inf x (B), sup x (A 1 ) sup x (B), nd sup y (A 1 ) inf y (B); (iv) inf x (A 1 ) inf x (B), sup x (A 1 ) sup x (B), nd sup y (A 1 ) inf y (B). In terms of these cses, R(A 1, B) cn e derived s: (i) {NE A } (Fig. 8), {N A, NE A } (Fig. 8); (ii) {N A } (Fig. 8c); (iii) {NW A, N A, NE A } (Fig. 8d); (iv) {NW A } (Fig. 8e), nd {NW A, N A } (Fig. 8f), respectively. Tht is, when R(A 2, B) = {N A }, the possile R(A 1, B) is δ({nw A, N A, NE A }). Similrly, when R(A 2, B) = Fig. 9. Two exmples () nd () of constrints of R(A 2, B) on R(A 1, B). {S A }, {E A }, or {W A }, the possile R(A 1, B) is δ({sw A, S A, SE A }), δ({ne A, E A, SE A }), nd δ({nw A, W A, SW A }), respectively. When R(A 2, B) = {O A }, ecuse the reltion etween the projections of A 1 nd B on x-xis nd y-xis cnnot e identified exctly, ny relizle reltions re possile for R(A 1, B), i.e., R(A 1, B) δ({nw A, N A, NE A, W A, O A, E A, SW A, S A, SE A }). (2) The proof of conclusion 2 is similr to tht of Theorem 3. In Fig. 9, R(A 2, B) = {NW A, N A, NE A }, CDR 2 (R(A 2, B)) = CDR 2 ({NW A }) CDR 2 ({N A }) CDR 2 ({NE A }) = δ({nw A }) δ({nw A, N A, NE A }) δ({ne A }) = {NW A } {{NE A }, {N A }, {NW A }, {NW A, N A }, {N A, NE A }, {NW A, N A, NE A }} {NE A } = {NW A, N A, NE A }. Tht is, R(A 1, B) is {NW A, N A, NE A } when R(A 2, B) is {NW A, N A, NE A }. In Fig. 9, R(A 2, B) = {NE A, E A }, CDR 2 (R(A 2, B)) = CDR 2 ({E A }) CDR 2 ({NE A }) = δ({ne A, E A, SE A }) δ({ne A }) = {{SE A }, {E A }, {NE A }, {E A, SE A }, {NE A, E A }, {NE A, E A, SE A }} {NE A } = {{NE A }, {NE A, E A }, {NE A, E A, SE A }}. Tht is, R(A 1, B) is one of the three reltions when R(A 2, B) = {NE A, E A } holds. To model the direction reltion of crisp region B with respect to n uncertin region A, two reltions R(A 1, B) nd R(A 2, B) re sufficient. The two reltions correspond to = comintions, ut not ll of these comintions re relizle. Theorems 3 nd 4 re helpful to remove unrelizle reltions, nd preserve the relizle ones. There re 9162 relizle reltions of this kind mong the comintions.

8 334 S. Du, L. Guo / ISPRS Journl of Photogrmmetry nd Remote Sensing 65 (2010) The constrints imposed y oth uncertin reference nd trget regions There exist mny constrints mong the four reltions. The constrints of R(A 1, B 1 ) on R(A 1, B 2 ), nd R(A 2, B 1 ) on R(A 2, B 2 ) cn e computed y Theorem 2, nd the inverse constrints y Theorem 1. The constrints of R(A 1, B 1 ) on R(A 2, B 1 ), nd R(A 1, B 2 ) on R(A 2, B 2 ) cn e computed y Theorem 3, nd the inverse constrints y Theorem 4. The trnsformtion from R(A 1, B 1 ) to R(A 2, B 2 ) cn e completed y two methods. The first is to expnd reference region A 1 to A 2, nd then expnd trget region B 1 to B 2. In this wy, the restriction of the expnsion of the reference region is equl to tht of R(A 1, B 1 ) to R(A 2, B 1 ), nd the expnsion of the trget region is equl to tht of R(A 2, B 1 ) on R(A 2, B 2 ). The restriction of R(A 1, B 1 ) on R(A 2, B 2 ) is therefore implied in the restriction of R(A 1, B 1 ) on R(A 2, B 1 ) nd R(A 2, B 1 ) on R(A 2, B 2 ). The second wy is to expnd trget region B 1 to B 2, nd then expnd reference region A 1 to A 2. The restriction of R(A 1, B 1 ) on R(A 2, B 2 ) is therefore implied in tht of R(A 1, B 1 ) on R(A 1, B 2 ), nd R(A 1, B 2 ) on R(A 2, B 2 ). Accordingly, the restrictions of R(A 1, B 1 ) on R(A 2, B 2 ), R(A 2, B 2 ) on R(A 1, B 1 ), R(A 1, B 2 ) on R(A 2, B 1 ), nd R(A 2, B 1 ) on R(A 1, B 2 ), re indicted in Theorems 1 4. The four theorems re sufficient to hndle the constrints mong the four reltions R(A 1, B 1 ), R(A 1, B 2 ), R(A 2, B 1 ), nd R(A 2, B 2 ). Definition 7. A symolic direction reltion etween SBBRs, R = R 11, R 12, R 21, R 22, is relizle, if nd only if R 11 D, R 12 D, R 21 D, nd R 22 D hold, nd they stisfy the two conditions: (1) R 11 δ(r 12 ), R 21 δ(r 22 ), R 11 CDR 2 (R 21 ), nd R 12 CDR 2 (R 22 ); (2) R 12 R 11 D, R 22 R 21 D, R 21 CDR 1 (R 11 ), nd R 22 CDR 1 (R 12 ). Whether symolic reltion is relizle depends on two spects: Direction reltions etween SBBRs re the comintions of four crisp reltions. Therefore, the four reltions should e relizle for crisp regions. The four reltions stisfy the constrints (Theorems 1 4. R 11 δ(r 12 ) mens tht R 11 nd R 12 cn e the direction reltions etween regions A 1 nd B 1, s well s A 1 nd B 2, such tht B 1 B 2 hold. Therefore, if the four reltions stisfy Theorems 1 4, they must e reltions etween two SBBRs. Definition 7 indictes tht not ll comintions of the 218 relizle direction reltions etween crisp regions re relizle for SBBRs, nd only those complying with corresponding conditions re relizle. 4. Defining uncertin semntics of pproximte direction reltions Although the four reltions re used to model direction reltions etween SBBRs, R(A 1, B 2 ), R(A 2, B 1 ), nd R(A 2, B 2 ) hve different menings from R(A 1, B 1 ). If B 1 flls inside multiple directions round A 1, B certinly flls inside these directions relevnt to A. Tht is, the direction reltion R(A 1, B 1 ) is the crisp prt of the reltion etween A nd B. However, if B 2 flls inside multiple directions with respect to A 1 or A 2, B my or not fll inside these directions. The reltions, R(A 1, B 2 ), R(A 2, B 1 ), nd R(A 2, B 2 ), re uncertin, s they re closely relted to the uncertin prts of A nd B. Accordingly, lthough the four reltions hve the sme nmes, they ply different roles in defining uncertin semntics of pproximte direction reltions. In Section 3, the constrints mong the four reltions re presented to determine wht comintions re relizle, nd how mny relizle reltions exist in the physicl world for uncertin regions. This section will use the four reltions to define the uncertin semntics of pproximte reltions, which re fundmentl to query uncertin dt. Fig. 10. Exmple of uncertin directions The uncertin semntics of pproximte direction reltions The uncertin semntics of direction reltions is similr with tht of regions with rod oundries. A SBBR A cn e regrded s pir of regions: inner region A 1 nd outer region A 2, such tht A 1 A 2 holds. Region A 1 is the crisp prt, nd A 2 the upper prt, while the rod oundry represents the uncertin prt. A direction reltion R etween SBBRs A nd B cn lso e considered s pir of reltions: lower prt R nd uncertin prt R. Those directions in R men tht trget SBBR B must fll inside them; while for those directions in R, B possily flls inside them. Becuse directions in R re uncertin nd hve different semntics from crisp ones in R, some new nmes re offered to express uncertin semntics, such s possily northest, possily north, possily northwest, possily west, possily southwest, possily south, possily southest, nd possily est, denoted y PNE A, PN A, PNW A, PW A, PSW A, PS A, PSE A, PE A, nd PO A. Definition 8. Let R = R(A 1, B 1 ), R(A 1, B 2 ), R(A 2, B 1 ), R(A 2, B 2 ) e the reltion etween SBBRs A nd B, the uncertinty cn e formlized s: The lower or crisp prt, denoted y R, is defined s R = R(A 1, B 1 ). The uncertin prt, denoted y R, depends on the roles tht uncertin regions ply in direction reltions, thus it needs to e derived from the four reltions. R is crisp reltion, thus it must e one of the 218 relizle reltions, while R does not. R must e non-empty, while R cn e empty. An empty R mens R is crisp reltion; otherwise, n uncertin one. Generlly, if N A R, B is possily north of A. Similrly, uncertin directions cn e defined y the following equtions R(A, B) = {PO A } iff O A R R(A, B) = {PN A } iff N A R R(A, B) = {PS A } iff S A R R(A, B) = {PE A } iff E A R R(A, B) = {PW A } iff W A R R(A, B) = {PNE A } iff NE A R R(A, B) = {PNW A } iff NW A R R(A, B) = {PSE A } iff SE A R R(A, B) = {PSW A } iff SW A R As shown in Fig. 10, R(A, B) = {{NE A, E A }, {NE A, E A, SE A }, {E A }, {NE A, E A, SE A, O A }}, thus R = {NE A, E A }, R = {SE A, O A }. Therefore, the uncertin directions re PSE A nd PO A. The uncertinty of direction reltions is cused y the uncertinty of the trget or reference oject, it therefore hs three sources. The first is relted to direction prtitions nd cused y

9 S. Du, L. Guo / ISPRS Journl of Photogrmmetry nd Remote Sensing 65 (2010) the distortions of reference regions, thus cn e used to mesure the uncertinty of the direction reltions etween crisp region B nd reference SBBR A. The second is relevnt to the reltions nd cused y the distortion of trget region. Therefore, it cn model the uncertinty of direction reltions etween trget SBBR B nd crisp region A. The third is the comintion of the first nd the second. For direction reltions etween two crisp regions, the uncertin prt is empty. This mens tht reltions etween crisp regions re lso crisp. In generl, the numer of elements in R reflects the uncertin degree of pproximte direction reltions. The smller the R is, the more crisp the direction reltion is. An empty R mens tht the reltion is completely crisp, while non-empty R indictes tht the direction reltion is uncertin. The uncertin semntics cn hndle oth crisp nd uncertin regions; therefore, they re helpful to hndle uncertin nd crisp dt in the sme wy. Remrk 1. If the lower nd uncertin prts of n pproximtion reltion R re denoted y R nd R, the comintion R, R cn e regrded s further strction of reltion R. This kind of strction cn not only reduce the numer of relizle reltions out uncertin regions, ut keep the crisp nd uncertin semntics of pproximte directions. This is useful to simplify the results of composing pproximte reltions nd to process the query out uncertin reltions The uncertin semntics cused y n uncertin reference region Since the reference region determines the direction prtition, its distortion cn result in some uncertin directions. The direction titles of sme directions round A 1 nd A 2 re different. Since the titles of NE A, NW A, SW A, nd SE A of A 2 re susets of the ones of A 1, if the trget region flls inside the four directions of A 2, the region must fll inside the sme ones of A 1. According to Theorem 4 nd Tle 2, when reltion R(A 2, B) contins directions NE A, NW A, SW A, nd SE A, reltion R(A 1, B) lwys contins the sme ones. As shown in Fig. 11, R(A 2, B) = {NE A }, thus R(A 1, B) = {NE A } must hold. Therefore, in this cse the uncertin directions PNE A, PNW A, PSW A nd PSE A cn never hppen. Remrk 2. Regrding directions N A, S A, W A, E A, nd O A, the titles round region A 1 re susets of the ones of A 2. If trget region flls inside one of the five titles round A 2, it my or not fll the sme titles of A 1. This difference leds to five new directions, PN A, PS A, PW A, PE A, nd PO A (Fig. 11). Therefore, in this cse only the five uncertin directions re possile. There re four cses relting reltion R(A 2, B) to R(A 1, B), from which the uncertin semntics cn e interpreted. If R(A 1, B) = R(A 2, B) holds, the reltion is crisp s the two reltions hve sme menings. In Fig. 11, R(A 1, B) = R(A 2, B) = {NE A, E A }, R(A 1, C) = R(A 2, C) = {N A }, nd R(A 1, D) = R(A 2, D) = {NW A }. As trget region B flls inside the sme directions with respect to regions A 1 nd A 2, reltion R(A, B) is crisp, thus region B is prtly northest nd prtly est of region A. In this cse, R =. If R(A 1, B) R(A 2, B) = holds, the reltion is uncertin. Both crisp nd uncertin directions exist. In Fig. 11, R(A 1, B) = {NE A }, R(A 2, B) = {E A }, R(A 1, C) = {NW A }, nd R(A 2, C) = {W A, O A }. Therefore, region B is northest of region A, nd possily est of A. The trget region cn fll inside the titles of uncertin directions. In this cse, R = R(A 2, B). c d Fig. 11. Uncertin semntics induced y the distortions of reference regions. () R(A 1, B) = R(A 2, B). () R(A 1, B) R(A 2, B) =. (c) R(A 1, B) R(A 2, B). (d) R(A 1, B) R(A 2, B).

10 336 S. Du, L. Guo / ISPRS Journl of Photogrmmetry nd Remote Sensing 65 (2010) Fig. 12. Uncertin direction nlysis for the distortions of trget regions. () Existence of uncertin directions. () Non-existence of uncertin directions. If R(A 1, B) R(A 2, B) holds, the reltion is uncertin. In this cse, the directions in R(A 2, B) re still kept s uncertin s R(A 2, B) hs different mening from R(A 1, B). The trget region flls inside fewer uncertin directions thn crisp directions. In Fig. 11c, R(A 1, B) = {NE A, E A }, R(A 2, B) = {E A }. This cn e interpreted s region B eing prtly northest nd prtly est of region A, nd possily est of A. In this cse, R = R(A 2, B). If R(A 1, B) R(A 2, B) holds, the reltion is uncertin. In this cse, only those directions in the set R(A 2, B) R(A 1, B) re regrded s uncertin. The trget region flls inside more uncertin directions thn crisp directions. In Fig. 11d, R(A 1, B) = {NE A }, R(A 2, B) = {N A, NE A, E A }. Thus, region B is northest of A, ut lso possily nd prtly north, northest nd est of A. In this cse, R = R(A 2, B) R(A 1, B). The comintion of two reltions R(A 1, B) nd R(A 2, B) cn descrie the uncertin semntics of direction reltions of crisp regions with respect to uncertin regions The uncertin semntics cused y n uncertin trget region To descrie the uncertinty of direction reltions of n uncertin region with respect to crisp region, two reltions R(A, B 2 ) nd R(A, B 1 ) re sufficient. The trget region cn ffect the reltions etween the prtition nd the trget region. If the trget region is expnded from B 1 to B 2, or reduced from B 2 to B 1, lthough reference region A is fixed, the reltions still chnge. The uncertinty of this kind cn e defined s the difference set R = R(A, B 2 ) R(A, B 1 ). According to Theorem 2, R(A, B 2 ) R(A, B 1 ) D. Therefore, R is composed of directions in R(A, B 2 ) with the exception of those in R(A, B 1 ). Remrk 3. As the uncertin directions re defined y the difference etween R(A, B 2 ) nd R(A, B 1 ), the nine directions PNE A, PN A, PNW A, PW A, PSW A, PS A, PSE A, PE A, nd PO A re possile in this cse. The differences of uncertin semntics etween the first nd the second lie in: (1) in the first cse, the sptil titles of uncertin directions clerly exist (those shdow regions in Fig. 11), thus if region (either crisp or uncertin) flls inside the title, it cn led to uncertin directions; nd (2) in the second cse, uncertin directions hve no corresponding titles, ut re decided y the difference etween reltions R(A, B 2 ) nd R(A, B 1 ) (Fig. 12). In Fig. 12, inner region B 1 flls inside direction NE A, nd outer region B 2 directions NE A, N A, nd E A, the uncertin directions re therefore N A nd E A, denoted y PN A nd PE A. While in Fig. 12, oth regions B 1 nd B 2 fll inside the sme direction NE A, thus no uncertin directions exist The uncertin semntics cused y oth uncertin reference nd trget regions To hndle the uncertinty etween two uncertin regions, the four reltions, R(A 1, B 1 ), R(A 1, B 2 ), R(A 2, B 1 ) nd R(A 2, B 2 ), re required. The uncertinty of the third kind cn e nlyzed y comining the first nd the second. Becuse oth the trget nd reference regions re uncertin, the direction prtition nd the ssocition etween the prtition nd the trget region re uncertin. The uncertin prtition is cused y expnding reference region A 1 to A 2, or reducing A 2 to A 1, while the uncertin ssocition is led y expnding trget region B 1 to B 2, or reducing B 2 to B 1 (Fig. 13). This kind of uncertinty cn e derived y comining the first nd second kinds. R cn e set union derived from three pirs of reltions: R(A 1, B 1 ) nd R(A 1, B 2 ), R(A 1, B 1 ) nd R(A 2, B 1 ), s well s R(A 1, B 1 ) nd R(A 2, B 2 ). The uncertinty of the first pir is cused y the distortion of trget region, the uncertinties of the ltter two pirs cn e computed y the method in Section 4.2. Tht is, R includes the first nd second kinds of uncertinties. In Fig. 13, R(A 2, B 1 ) = R(A 2, B 2 ) = {NE A, N A, E A, O A }, nd R(A 1, B 2 ) = {NE A, N A, E A }, R(A 1, B 1 ) = {NE A }, thus R = {N A, E A, O A }. Directions N A, E A nd O A re uncertin, denoted y PN A, PE A nd PO A. The uncertin directions PN A nd PE A re led y the distortions of oth the trget nd reference regions, while PO A is only induced y the distortion of the reference region. In Fig. 13, R(A 2, B 2 ) = {NE A, N A, E A, O A }, R(A 2, B 1 ) = R(A 1, B 2 ) = R(A 1, B 1 ) = {NE A }, thus R = {N A, E A, O A }. There re still three uncertin directions PN A, PE A nd PO A, while ll three directions re led y the distortion of the trget region. 5. Composing pproximte direction reltions The directionl opertions, like composition, consistency nd pirwise- consistency, re fundmentl to the ppliction of reltions. These opertions hve een developed well for crisp reltions, ut not for pproximte reltions. Therefore, new opertions should e redeveloped. The presented model uses the four crisp reltions to pproximte direction reltions etween SBBRs. In ddition, the representtion cn hndle crisp nd uncertin regions in the sme wy. These chrcteristics help to implement the opertions for SBBRs. In this study, existing composition nd pirwiseconsistency out direction reltions etween crisp regions re extended to hndle the counterprts etween SBBRs Composition of pproximte direction reltions Since direction reltions etween SBBRs re formlized y the four reltions, composing pproximte direction reltions cn e implemented y computing the possile four reltions from nother four pirs of reltions. Definition 9. Let the pproximte direction reltion etween SBBRs A nd B e R(A, B) = R(A 1, B 1 ), R(A 1, B 2 ), R(A 2, B 1 ), R(A 2, B 2 ), nd the one etween B nd C e R(B, C) = R(B 1, C 1 ), R(B 1, C 2 ), R(B 2, C 1 ), R(B 2, C 2 ). Then the possile four reltions ssocited with A nd C re s follows: R 11 = R(A 1, B 1 ) R(B 1, C 1 ) R(A 1, B 2 ) R(B 2, C 1 ), R 12 = R(A 1, B 1 ) R(B 1, C 2 ) R(A 1, B 2 ) R(B 2, C 2 ), R 21 = R(A 2, B 1 ) R(B 1, C 1 ) R(A 2, B 2 ) R(B 2, C 1 ), nd R 22 = R(A 2, B 1 ) R(B 1, C 2 ) R(A 2, B 2 ) R(B 2, C 2 ). In Definition 9, R(A 1, B 1 ) R(B 1, C 1 ) is to derive possile direction reltions etween crisp regions A 1 nd C 1 from crisp reltions R(A 1, B 1 ) nd R(B 1, C 1 ), thus it cn e computed y existing pproch for crisp regions (Skidopoulos nd Kourkis, 2004). Similrly, seven other compositions cn e resolved. Possile reltions of R(A 1, C 1 ) cn e derived from the constrints etween R(A 1, B 1 ) nd R(B 1, C 1 ), nd etween R(A 1, B 2 ) nd R(B 2, C 1 ), so re R(A 1, C 2 ), R(A 2, C 1 ) nd R(A 2, C 2 ).

11 S. Du, L. Guo / ISPRS Journl of Photogrmmetry nd Remote Sensing 65 (2010) Fig. 13. Two exmples () nd () of uncertin semntics for the distortion of oth regions. Fig. 14. Exmple for composing direction reltions. Definition 10. Let R 11, R 12, R 21, nd R 22 e the four sets computed y Definition 9. Possile reltions etween A nd C re ll relizle comintions R(A 1, C 1 ), R(A 1, C 2 ), R(A 2, C 1 ), R(A 2, C 2 ), where R(A 1, C 1 ) R 11, R(A 1, C 2 ) R 12, R(A 2, C 1 ) R 21, R(A 2, C 2 ) R 22, nd they should stisfy Definition 7. All relizle comintions, computed y Definition 10, re possile reltions etween SBBRs A nd C, the cuses re: R 11, R 12, R 21 nd R 22 re the minimum sets of relizle crisp reltions s the composition uses ll constrints mong the known reltions. The comintions defined on sets R 11, R 12, R 21 nd R 22 re minimum s the results contin ll possile cses. The four reltions in ech comintion oey the constrints tht reltion etween SBBRs should comply (Definition 7). The ove nlysis indictes tht Definitions 9 nd 10 generte minimum numer comintions, nd certify ech comintion is relizle for SBBRs. Therefore, they cn correctly compute the composition of direction reltions etween SBBRs. In Fig. 14, R(A, B) = {NE A }, {NE A, E A }, {NE A, E A }, {N A, NE A, E A, O A }, R(B, C) = {SE B, S B }, {SE B, S B }, {SE B, S B }, {SE B, S B }. Firstly, R(A 1, B 1 ) R(B 1, C 1 ) = δ({ne A, E A, SE A }); secondly, R(A 1, B 2 ) R(B 2, C 1 ) = δ({e A, SE A }); thus, R 11 = δ({e A, SE A }) = {{E A }, {SE A }, {E A, SE A }}. Similrly, R 12 = R 21 = R 22 = {{E A }, {SE A }, {E A, SE A }}. Among the 9 possile comintions of R(A 1, C 1 ) nd R(A 1, C 2 ), only five ones stisfy the constrints (Theorems 1 nd 2). The relizle comintions of R(A 1, C 1 ), R(A 1, C 2 ) re: {E A }, {E A }, {E A }, {E A, SE A }, {SE A }, {SE A }, {SE A }, {E A, SE A }, nd {E A, SE A }, {E A, SE A }. Similrly, the relizle comintions of R(A 2, C 1 ), R(A 2, C 2 ) re: {E A }, {E A }, {E A }, {E A, SE A }, {SE A }, {SE A }, {SE A }, {E A, SE A }, nd {E A, SE A }, {E A, SE A }. There re 25 comintions for A nd C. When R(A 1, C 1 ), R(A 1, C 2 ) = {E A }, {E A } holds, R(A 2, C 1 ) nd R(A 2, C 2 ) should e ny reltion in δ({o A, E A }) ccording to Theorem 3. Only {E A }, {E A } of the five relizle comintions of R(A 2, C 1 ) nd R(A 2, C 2 ) oeys Theorem 3, thus only the reltion {E A }, {E A }, {E A }, {E A } is relizle in this sitution. When R(A 1, C 1 ), R(A 1, C 2 ) = {E A }, {E A, SE A } holds, ccording to Theorem 3, R(A 2, C 1 ) should e in set δ({o A, E A }), nd R(A 2, C 2 ) in set δ({s A, E A, SE A, O A }). Only {E A }, {E A } nd {E A }, {E A, SE A } stisfy Theorem 3 mong the five relizle comintions of R(A 2, C 1 ) nd R(A 2, C 2 ), thus two reltions, {E A }, {E A, SE A }, {E A }, {E A } nd {E A }, {E A, SE A }, {E A }, {E A, SE A }, re relizle. Similrly, when R(A 1, C 1 ), R(A 1, C 2 ) is {SE A }, {SE A }, {SE A }, {E A, SE A } nd {E A, SE A }, {E A, SE A }, the relizle reltions cn e derived. Totlly, 15 relizle reltions etween A nd C cn e derived from the reltions etween A nd B, nd B nd C: (1) {E A }, {E A }, {E A }, {E A } ; (2) {E A }, {E A, SE A }, {E A }, {E A } ; (3) {E A }, {E A, SE A }, {E A }, {E A, SE A } ; (4) {SE A }, {SE A }, {E A }, {E A } ; (5) {SE A }, {SE A }, {E A }, {E A, SE A } ; (6) {SE A }, {SE A }, {SE A }, {SE A } ; (7) {SE A }, {SE A }, {SE A }, {E A, SE A } ; (8) {SE A }, {SE A }, {E A, SE A }, {E A, SE A } ; (9) {SE A }, {E A, SE A }, {E A }, {E A } ; (10) {SE A }, {E A, SE A }, {E A }, {E A, SE A } ; (11) {SE A }, {E A, SE A }, {SE A }, {E A, SE A } ; (12) {SE A }, {E A, SE A }, {E A, SE A }, {E A, SE A } ; (13) {E A, SE A }, {E A, SE A }, {E A }, {E A } ; (14) {E A, SE A }, {E A, SE A }, {E A }, {E A, SE A } ; nd (15) {E A, SE A }, {E A, SE A }, {E A, SE A }, {E A, SE A } Clssifiction of derived direction reltions In generl, there re so mny derived reltions tht it is insignificnt to list ll possile ones. Therefore, the derived reltions should e clustered or clssified into severl ctegories. The conceptul neighorhood of topologicl reltions is helpful for users to understnd nd use topologicl reltions (Egenhofer nd Mrk, 1995; Clementini nd Di Felice, 1997). However, it is difficult or insignificnt to provide similr conceptul neighorhoods for pproximte direction reltions, s there re thousnds of relizle direction reltions, while only tens for topologicl reltions. Therefore, the sic ide is to clssify the derived reltions into severl smll sets y using the uncertin semntics (Definition 8). The clssifiction helps not only to simplify the derived results, ut lso to process the query of pproximte direction reltions in terms of the crisp nd uncertin prts (Section 6). Let M = {R 1, R 2,..., R n }(n 1) e the derived set of pproximte direction reltions, then M cn e grouped into severl smll sets ccording to the lower nd uncertin prts of pproximte reltions (Definition 8). Crisp-set sed prtition clssifies those pproximte reltions with sme crisp prts into one ctegory. The crisp set of M is defined s R(M) = {R(R i ) R i M}, where R(R i ) denotes the lower prt of reltion R i. Assuming R(M) contin r elements R 1, R 2,..., R r (1 r n), then set M cn e prtitioned into r susets M 1, M 2,..., M r, such tht M j M nd M j = {R i R(Ri ) = R j R i M} (1 j r).

12 338 S. Du, L. Guo / ISPRS Journl of Photogrmmetry nd Remote Sensing 65 (2010) Fig. 15. Querying pproximte direction reltions. () Conflict regions in multi-scle dt. () Uncertin region in remotely sensed imges. Uncertin-set sed prtition strcts the reltions hving sme uncertin prts into one group. Similrly, the uncertin set of M is defined s R(M) = { R(R i ) R i M}, where R(R i ) refers to the uncertin prt of reltion R i. Assuming R(M) consist of s elements R 1, R 2,..., R s (1 s n). Therefore, set M cn e prtitioned into s susets M 1, M 2,..., M s, such tht M k M nd M k = {R i R(R i ) = R k R i M} (1 k s). Composition prtition clusters the reltions with sme crisp nd uncertin prts into one group. Let the comintion of crisp nd uncertin prts e n strction of n pproximte reltion, then set M cn e prtitioned into t susets M 1, M 2,..., M t (1 t n), such tht M l M nd M l = {R i R(Ri ) = R j R(R i ) = R k R i M} (1 j r, 1 k s, 1 l t). All reltions in suset M l cn e strcted s the comintion R j, R k. According to the three methods ove, set M of the derived reltions is clssified s severl ctegories M 1, M 2,..., M r (M s or M t ). Ech ctegory of pproximte reltions cn e further considered s the sme reltion R j ( R k or R j, R k ). The crisp or lower prt is equl to the first reltion of ech comintion representtion (Definition 8). In Fig. 14, the 15 derived reltions correspond to three crisp prts, i.e., R(M) = {{E A }, {SE A }, {E A, SE A }}. There re three ctegories {1, 2, 3}, {4, 5, 6, 7, 8, 9, 10, 11, 12}, nd {13, 14, 15}, where numers 1 15 denote the 15 derived reltions in Fig. 14. Those reltions in ech ctegory correspond to the sme strction R 1 = {E A }, R 2 = {SE A }, R 3 = {E A, SE A }, respectively. The uncertin prt needs to e clculted for ech derived reltion. In Fig. 14, the 15 derived reltions correspond to three uncertin sets, i.e., R(M) = {, {PE A }, {PSE A }}. The three uncertin sets clssify the 15 reltions into three ctegories {1, 6, 15}, {4, 5, 8, 9, 10, 11, 12, 13, 14}, {2, 3, 7}, respectively. The crisp nd uncertin prts of the 15 derived reltions cn generte seven comintions {E A },, {SE A },, {E A, SE A },, {E A }, {PSE A }, {SE A }, {PE A }, {SE A }, {PSE A } nd {E A, SE A }, {PE A }. These comintions clssify the 15 derived reltions into seven ctegories {1}, {6}, {15}, {2, 3}, {4, 5, 8, 9, 10, 11, 12}, {7} nd {13, 14}. The composition prtition cn generte more ctegories thn the crisp- or uncertin-set sed prtitions. The composition prtition uses oth the crisp nd uncertin informtion to clssify the derived reltions, while the other methods only consider prt informtion. Furthermore, the otined comintions, i.e., strcted reltions, cn e used to nlyze the reltions from oth crisp nd uncertin spects. Therefore, the composition prtition is the preferred method Pirwise-consistency of pproximte direction reltions Becuse the direction reltions is not symmetricl, i.e., the direction reltion of region B with respect to A differs to tht of A with respect to B (Cicerone nd Di Felice, 2004). To express completely sptil scene with symolic direction reltions, oth the direction reltions of B with respect to A nd the inverse reltions should e recorded. The pirwise-consistency is out checking for two reltions M 1 nd M 2 whether there exist two ojects A nd B in the physicl world, such tht R(A, B) = M 1 nd R(B, A) = M 2 hold. The existing pproch for pirwise-consistency is intended to hndle crisp regions (Cicerone nd Di Felice, 2004). A similr pproch needs to e developed to resolve the pirwiseconsistency of pproximte direction reltions. Definition 11. For two relizle direction reltions: R 1 = R 11, R 12, R 21, R 22, nd R 2 = R, 11 R, 12 R, 21 R 22, if there exist two SBBRs A nd B, such tht R 11 = R(A 1, B 1 ), R 12 = R(A 1, B 2 ), R 21 = R(A 2, B 1 ), R 22 = R(A 2, B 2 ), R 11 = R(B 1, A 1 ), R 12 = R(B 1, A 2 ), R = 21 R(B 2, A 1 ), nd R = 22 R(B 2, A 2 ), R 1 nd R 2 re pirwiseconsistent. Pirwise-consistency cn e resolved y checking whether R 2 Inverse (R 1 ), where Inverse (R 1 ) represents the set of relizle direction reltions from B to A derived from tht from A to B. Becuse direction reltions etween SBBRs re represented s the comintion of four reltions etween crisp regions, pirwise-consistency for uncertin regions cn e completed y checking whether R 11 Inverse (R 11), R 12 Inverse (R 12), R 21 Inverse(R 21 ), nd R 22 Inverse (R 22 ). These conditions re the pirwise-consistency prolems of direction reltions etween crisp regions, they therefore cn e computed ccording to the pproch for crisp regions (Cicerone nd Di Felice, 2004). 6. Querying pproximte direction reltions The regions with rod oundries cn model the uncertinty of vrious sources, thus the pproximte representtion of direction reltions cn hndle the uncertinty of reltions out uncertin dt. This section will demonstrte how to implement the process of querying uncertin dt sed on pproximte direction reltions nd their uncertin semntics. As the direction reltions out uncertin region include crisp nd uncertin prts, it is criticl to process the direction query out uncertin regions. Fig. 15 provides two exmples out uncertin regions. In Fig. 15, the sme entities hve two different representtions t different resolutions. As the methods for generlizing sptil dt re different t two resolutions, the representtions re inconsistent. In this cse, the intersection set of two