Author name used in this publication:  邓敏Author name used in this publication: LI Zhi-linAuthor name used in this publication: 李光强2007-2008 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Deng, M

Li, GQ

Li, ZL

PolyU Institutional Repository

? 37 ?  ? 3 ? ?  ?  ?  ? Vol. 37, No . 3 2008 ? 8 ? ACT A GEODAETICA et CART OGRAPHICA SINICA Aug . , 2008  ????: 1001- 1595( 2008) 03- 0330- 08 ?????: P208    ?????: A????????????????????????邓  敏1 , 李志林2 , 李光强11. ??????????????,???? 410083; 2. ?????? ?????????????,??A Hierarchical Approach to Topological Relations between a Simple Area andan Area with HolesDeng M in1, L i Zh-i lin2, L i Guang-qiang11. Dep ar tment of Su rv ey ing and Geo-inf or mati cs , Centr al S outh Univ er si t y , Changsha 410083, China; 2. Dep artment ofL and Sur v eying and G eo-inf ormat ic s, T he H ong K ong P oly tech nic Univ e rsi ty , H ong K ong , Ch inaAbstract: T opo lo gical relat ions have been a focus o f r esear ch in many disciplines such as computer science, ar t-ificial intellig ence, co gnitive science, linguistics, r obotics and geogr aphic informat ion science. Unfo rtunately,they have so far only been defined fo r and applicable to simple objects like sing le points, cont inuous lines andsimple areas, not invo lv ing the design, definition, and description of topo log ical relat ions operating on thecomplex objects. This ar ticle tries to make an effo rt to this g ap and pays attent ion to the spatial ar eas w ithho les. Based upon the idea of space part itio n and object decomposit ion, t opolo gica l components of an area ob-ject wit h holes are defined by the use o f t he concept of neighbo rhood in the point set topo log y, which is a natu-ral ex tension of the definit ions for topolog ical components o f a simple area. And t hen, a hier archical approachto topo log ical relat ions is pr esented fo r two simple a reas, which is indeed necessary for many practical applica-tions. The hierar chical approach is further ex tended to the topolog ical relat ions between a simple area and anarea w ith ho le( s) . I t can be concluded that the propo sed approaches ar e ver y gener al, suitable fo r topo lo gicalrelat ions of both simple ar eas and complex ar eas.Key words: t opolo gical r elation; area object; hole; hierar chica l representation?  ?:带孔洞的面目标是现实中较为常见的一类复杂目标, 它们之间的拓扑关系要比简单面目标复杂得多?本文基于空间划分和目标分解的思想,利用点集(拓扑学)理论中的邻域概念详细分析和描述带孔洞面目标的点集拓扑分量,这种描述方法实质上是简单面目标点集拓扑分量描述的一种自然延展?进而,对简单面目标间拓扑关系的描述和区分方法进行了扩展, 层次地分析和区分简单面目标与带孔洞面目标间的拓扑关系?相比于 Egenho fer等人提出的代数描述和间接表达方法,本文提出的方法是一种直接描述和层次表达的方法,并且与简单面目标间拓扑关系的表达方法是相统一的?  ????: 2007- 06- 22; ????: 2008- 05- 08????: ???????????? ( 40501053) ;????????? RGC????( Polyu 5228/ 06E) ;???????????( 07JJ6075) ;?????????????????????: ??( 1974-) ,?,??,?????, ??,??,?????,??????? GIS ????????????? GIS ?????????????????????E- mail: l sdmin@ polyu . edu. h k???:拓扑关系; 面目标;孔洞; 层次表达1  引  言???????????????????????,???????????????????????[ 1, 2] ?????????????????(???) 3??????, ????????????????? 20??,????????????????????? GIS ?????????? [ 3, 4]??????, ??????????????,?? 4 ???[ 5 ]?9???[ 6]?????????( RCC) [ 7]?????????????? Voronoi?? 9???? 4 ????[ 8~ 10] ????????????????????????????????????,?????????????????,???????????(???????)?????? 3 ? ?  ??:???????????????????????????????????, ???????????,?????????????????,???????????????????????????, ???????????????????[ 11]?????? GIS ??????????????????????????, ??, ?????????????????????????????????????[ 12, 13]???, ?????????????????????????????????????? 2???:??????????????????;?????????????????????????????, ??????????????,?????????????????????????,???????????????????,?????????????????????????2  现有方法中存在的问题??????????????????,Egenhofer???? 4???????????????[ 12] ???????, ?????????????????????????????????????? A ??????????,HA1 , HA2 , ,, H Am ??? A ??m ?????,????[ 12]???,??A = A*- Gmi= 1HAi ( 1)??, A * ??????? A ????????????? 1, ??? A ??????, ??HA1 ? H A2 , ?????????? A ????? 3??????,? A * , H A1 ? H A2 ???,????????????(?? A ? B)??????,??? A ??????????????? B ????????????????????????????????????????????????????, ??,??????????????????, ????????????(? A * )? A ????????????, ????????????????????????????? L?????[ 1 2] , ?L= ( m+ 1) @ ( n+ 1)= mn+ m+ n+ 1 ( 2)??, m, n?????? A , B???????????,??[ 12]????????????????????????????????, ?????????????????????????????? 2????: ????????????????, ????????????????,??, ?? Euler ???( = 1- ???)?????, ?? 1( a)???? A ? Euler????-1, ????????? 3 ??????(?A * , H A1? H A2 )?Euler????? 1;????,????????????????????,?????????????????????,????????????????????,???????????????????????????????? 1 ?????????????F ig . 1  Ex isting method fo r description of ar ea objectw ith holes3  带孔洞面目标的拓扑区分方法3. 1  ???????????? 2, ???? I R2 ??????? A ,????????? 2???: ???( A )??????(??????,?? A - ) , ??? IR 2 =A G A - ???????????, ????????????????? A , ??????????????????????( 5A )???? ( A o) ?????????????,???5A = p | p I A , and p ?E ¤A ( 3)???????????,???Ao= p | p I A , and p ?E < A ( 4)???,??????????A-= p | p | A , p I IR 2 ( 5)??( 3) ~ ?( 5)?, p ????;?????????; E????????????, ??( 3)????,????????????????????? 2, ? p 2 ???(???????)??331???????,???????????, ??? p 2 ?????? p 1 ??????????,? p 1 ???????????, p 3 ????????????????, ? 3 ????????????????????, ?????????????, ?????????? 3???(?: ??????????????????????)???;???????????????????????, ? 4????9???? 4????????????????? 2  2?????( I R2 )?????????????Fig. 2  Ident if ication of point- set topology for a sim ple ar-ea in I R23. 2  ??????????????????????, ????????,????????? 2???,???????????????????????????(???)????????? 3( a) ,??? A ???????,????????5A 1 ,??????5A 2 , ??, ?????????????(?????????)??????(??S 1 ) ,?????????????????????(??? A ) ,????????????(?????????)??????(?? S2 ) ,?????? I R2 ????????: IR 2= S1GA GS 2 ,????????????????, ????????? A ? 3????????, ?????????????????,???????5A 1= p |p I A, p ? E ¤A, and p ? E HS1 X<( 6)Ao= p | p I A, and p ?E < A ( 7)5A 2= p | p I A, p ?E ¤A, and p ? E HS2 X<( 8)?????????( 3) ~ ?( 5)???,????????????????????? 5?????????, ???? p1 , p 2 , p 3 , p4 ? p5 (??3( a) )????,? 5???????????????? 5?????,? S1, 5A 1 , Ao ,5A 2 ? S2 ?????????????,??????????????????,????????? 5??,? p 1 , p 2 , p 3 , p 4 ? p 5 (?? 3( b) )??????????????????,?????????????,??????????,????????????,??????????????????????,??????????????????5???????(?? 3)???,?????????????,????????????????????????? 3  2 ?????( IR2 )??????????????Fig. 3 Ident ification o f po int- set topolog y fo r an area w ith holes in IR24  带孔洞面目标间拓扑关系的层次表达方法4. 1  2????????????????????? 2?????? A ? B , ??[ 10]????? 4????????????????,???C A , B =Ao H Bo  A - BB - A  5A H 5B( 9)??, A o , Bo ??????A , B ???, 5A , 5B????,?????????( 3)??( 4)??332 ?  ?  ?  ?                   ? 37 ?? 3 ? ?  ??:??????????????????????????( 9) , ?????/??????? 8?????,???( Disjoint)???( M eet )???( Over-lap)???( Contain)???? ( Contained by )???( Cover)????( Covered by) ???( Equal)?????,??[ 10] ?[ 16]?????4???????? 4?????????????????????????(?? 4)?????????? 4 ???????????????Fig. 4 Conceptual neighborhood g raph of the topolo g-ical relations between two reg ions???????????????????,????????????????????????????,???????????????????????????,???????????????,??????? 2????(????)????????,????????????????????( Disjoint ) ?????( Connected) ,??????? 2 ????????,??????2????????????????? 2??????????,??? Basic relat ionsA H B = < ] DisjointA H B X < ] Connected( 10)? A H B ???( ­ < )?,????????????????????( M eet )?????( Overlap)???????, A H B ???(??dim( A HB) )? 0? 1;???????, A H B????? 2,???Connecteddim A H B = 0 or 1 ] Meetdim A H B = 2 ] Overlap( 11)??,????????????? ( Part iallyOver lap)?????( Wholly Over lap) 2????????????????????, ?OverlapA- B X <andB- A X < ] Part ially OverlapA- B= <orB- A = < ] Wholly Overlap( 12)??,???????????????? Over-lap,??????????????????????????? 5????????????,? 5??????? 2?????????????,?Wholly OverlapA- B= < andB- A = < ] EqualA- B X<, B - A = <, and 5A H5B= < ] ContainA- B X <, B - A = <, and 5A H5B X < ] CoverA- B= <, B - A X <, and5 A H5B X < ] Containd byA- B= <, B -A X <, and5A H5B X < ] Covered by( 13)??( 10) ~ ?( 13)????, 2 ???????????????????????????????( A H B )??( A - B ?B - A )????????( 5A H 5B )??, ????[ 10]??? 4????(?? 9)??????????? 2??????? 8?????????, ??????????????????????????????????????4. 2  ?????????????????????????????????????????,??????????? 23?????????,?? 5???,?????????????? B ???? A ??????????????????(???)?????????????????, B ?A ??????????????? Disjoint, ? A ??????????????? Disjoint ,??? B ? A ?????? Disjoint-Disjoint?????? A ??????????????, ?????????????,????? 5? 23????????4. 3  ????????? 4 ??? 2??????? 8 ?????????, ????????????????????????? 23 ?????????????, ?? 6????????????????????????, ??,Contain-M eet??(? 14 ???) ?????????????????, ???????Cover-M eet (? 11 ???)?Contain-Disjo int (?13 ???) ? Contain-Overlap( ? 15 ??? )???????, ?????????????????????????? [ 1 4] ?????? ? ?, ? ? ? ? ? ( ? 1-2-3-12-13 )?*-Disjoint??;??????, ??????333? 5  ?????????????? 23 ?????????Fig . 5  Tw ent y- three basic relations betw een a simple area and an area w ith hole? 6  23 ?????????????F ig. 6 Conceptual neighbo rhood g raph of 23 r elations?? Overlap- * ?? ( 3 y 7) , Cover- * ??( 8y 12) , Contain- * ?? ( 13 y 17) ; ?? 13 y 20?? Overlap- * ??( 3 y7) , Cover- * ??( 8y12) , Contain- * ??( 13 y 17) ; ?? 13 y 20?????? Contain- * ?????????(?? 4????) ,??????? 5???( 23, 22, 21,17, 8? 7)?*-Contained by ???????,??????????* ?????????????????4. 4  ??????????????????????????????????????????, ??????????( A )????????( B) ,??????????????????????( Disjoint )??????( Connected) , ??????( 10)??????????, ???????,??????( Internal Disjoint)????( Ex-ternal Disjoint ) ,????? 5?( 20)?( 1)????,??????? B ??? S1 ?????????????,?????????????????( Meet)?????( Overlap)????Basic relat ionsA H B = < ] DisjointB H S1 = < ] Ex ternal Disjoint  Fig. 5(20)B H S1 X < ] InternalDisjoint  Fig . 5( 1)A H B X < ] Connected dim A H B = 0 or 1 ] Meetdim A H B = 2 ] Over lap( 14)334 ?  ?  ?  ?                   ? 37 ?? 3 ? ?  ??:??????????????????????????,????( M eet )????????( Inter-nal M eet )????( Ex ter nal Meet)???, ???????????? B ???? A ??????????????( Wholly Inter nal M eet )??????( Part ially Internal Meet ) , ????? 5?( 18) , ( 19)?( 2) 3???????????????????, ?Meet5A 1 H 5B X < ] Internal Meet5A 1 - 5B = < ] Wholly Internal M eet  Fig. 5( 18)5A 1 - 5B X < ] Pat ially Internal M eet  Fig. 5(19)5A 2 H 5B X < ] Ex ternal Meet  Fig . 5(2)( 15)?????( Overlap)?????????( Par-t ially Overlap)?????( Wholly Overlap)????????????????????,?:OverlappedA - B X < and B- A X < ] Part ially OverlapA - B = <or B - A = < ] Wholly Overlap( 16)????????,? B- A = <?,?/ A ???? B0,??5?( 11) ~ ( 14) 4???,????????????????,??????A Whol ly Overlap B5A 1 H5B X < and 5A 2 H5B X < Fig. 5(11)5A 1 H5B= < and 5A 2 H5B X < Fig. 5(12)5A 1 H5B= < and 5A 2 H5B= < Fig. 5(13)5A 1 H5B X < and 5A 2 H5B= < Fig. 5(14)( 17)? A - B = <?, ?/ B ???? A 0, ?? 5?( 21) ~ ( 23) 3???, ?????????????????,??????B Wholly Overlap A5A 2 H5BX< and 5A 2 - 5B= < Fig. 5( 21)5A 2 H5BX< and 5A 2 - 5BX< Fig. 5( 22)5A 2 H5B= < Fig. 5( 23)( 18)????????, ????????????????? Holes-No Overlapped, Holes-Part ially Over lapped ? Holes-Wholly Over-lapped, ??Part ially OverlappedB HS1 = < ] Holes-No OverlappedB HS1 X< ] Holes-OverlappedS1- B X < ] Holes-Partially OverlappedS1- B= < ] Holes-Wholly Overlapped( 19)??, Hole-No Overlapped ???? 5? ( 3)?( 4) 2???,?????????? B ???A?????????,?Holes-No Overlapped5A 1 H5B = < Fig. 5(3)5A 1H5B X< Fig. 5(4)( 20)Holes-Part ial ly Over lapped ??? 3???, ?? 5 ?( 5) , ( 10)?( 15) 3 ???, ???? BH S2 ?5A 2 H 5B ???, ???Holes- Par tially OverlappedB H S2 X <  Fig. 5(5)B H S2 = <5A 2 H 5B X <  Fig. 5(10)5A 2 H 5B = <  Fig. 5(15)( 21)?Hole- Wholly Overlapped???? 5??( 6) ~( 9) , ( 16)?( 17)???????, ????????B H S2?5A 1 H 5B?5A 2 H 5B ???,???Holes-Wholly OverlappedB H S2 X <5A 1 H 5B X <  Fig. 5(6)5A 1 H 5B = <  Fig. 5(7)B H S2 = <5A 1 H 5B X <5A 2 H 5B X <  Fig. 5(9)5A 2 H 5B = <  Fig. 5(16)5A 1 H 5B = <5A 2 H 5B X <  Fig. 5(8)5A 2 H 5B = <  Fig. 5(17)( 22)  ??????????,??????????????????????????? 2?????????????????????????????????????????????? 2???????????????????, ???????,??? 2 ?????????????335??,??????????????????? (S1?S2)??????????????? 7???? 7  ????????????? 23 ????????????Fig . 7  H ierar chica l representation o f the tw enty- thr ee basic topolog ical relations5  结论与展望?????????(?: ?????)????????????????????????????????????(?????????????)??????????????????????, ?????????????????????????????????????????: ¹???????????????????? 23?????????; º?????????????????????????????????????????????????????????????????????, ??????????? (?5A H Bo ?A o H 5B )????????????????[ 15, 16]??????/???/????????????????????????????? 2??????????????????????????????,?????? GIS??????????????????,?????????????参考文献:[ 1]  CLEMENT INI E, SHARMA J, EGENHOFER M. M od-eling T opological Spat ial Relat ions : St rategies for QueryPr oces sing[ J] . Computer & Graphics, 1994, 18( 6) : 815-822.[ 2]  LIU Ya- bin, LIU Da- you . Reason ing of T opological Rela-t ions b etw een Spat ial Object s in GIS [ J ] . J ou rnal of Sof t-w are, 2001, 12( 12) : 1 859-1 863. (???, ???. ??????????????????? 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简单面目标与带孔洞面目标间拓扑关系的层次表达方法

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