RESEARCH ON EQUIVALNCE OF SPATIAL RELATIONS IN AUTOMATIC PROGRESSIVE CARTOGRAPHIC GENERALIZATION Guo Qngsheng Du Xaochu Wuhan Unversty Wuhan Unversty ABSTRCT: In automatc cartographc generalzaton, the most mportant problem whch must be resolved s to keep the valence of spatal relatons between spatal obects, these spatal relatons are dstance relaton, drectonal relaton, topologcal relatons and so on. Among all these spatal relatons, topologcal relaton s consdered as the relaton that can reflect the best spatal nformaton, so t should be the domnant relaton. Because of ths, t s mportant to keep the valence of topologcal relatons n the progress of automatc progressve cartographc generalzaton. In ths paper, the combnatonal reasonng method s used to represent topologcal relatons, and the valence of topologcal relatons n the process of cartographc generalzaton s dscussed. Besdes these, a method for topologcal relaton abstracton s presented. Combnng wth some rules to deal wth spatal relatons n cartographc generalzaton, valence evaluaton model of topologcal relatons n spatal scene s establshed. Keywords: Progressve cartographc generalzaton, Equvalence, Spatal reasonng, Spatal relaton 1 INTRODUCTION In automatc cartographc generalzaton, the most mportant problem whch must be resolved s to keep the valence of spatal relatons between spatal obects, these spatal relatons are dstance relaton, drectonal relaton, topologcal relatons and so on. Among all these spatal relatons, topologcal relaton s consdered as the relaton that can reflect the best spatal nformaton, so t should be the domnant relaton (Egenhofer and Mark, 1995). Because of ths, t s mportant to keep the valence of topologcal relatons n the progress of automatc progressve cartographc generalzaton. The evaluaton of valence of spatal relatons s very complcated, because the valence of topologcal relatons s nfluenced by many factors, such as the topologcal relaton representaton models, change of dmenson of spatal obects and so on. And some works have been done to dscuss ths problem (Egenhofer and Franzosa 1994, Egenhofer et. al. 1994, Abdelmoty and Jones 1997, Servgen, 2000, Bobzen and Morgenstern 2003). In ths paper, the combnatonal reasonng method s used to represent spatal topologcal relatons (Guo, 2000). Based on ths representaton, the valence of topologcal relatons n the process of cartographc generalzaton s dscussed. 2 COMBINATIONAL REASONING FOR TOPOLOGICAL RELATION Combnatonal reasonng method s manly used to represent topologcal relaton between spatal obects n vector space. The basc dea of ths method s to dvde spatal obects nto elementary cells, for example, n R 2, the elementary cells nclude a pont and a lne segment. By examnng topologcal relatons between these elementary cells and combnatoral reasonng n dfferent levels based on these relatons, topologcal relatons can be ganed (Guo, 2000). In vector space, a regon can be consdered as a close lne, so we need to examne only spatal topologcal relatons between two lnes. Topologcal relatons between a lne and a regon, and between two regons can be represented smlarly. In order to represent topologcal relatons between two lnear obects, one of these s consdered as reference lne, then, we can dstngush dfferent topologcal relatons by checkng nner topologcal relatons and boundary topologcal 1
relatons. There are fve knds of nner topologcal relatons between two lnes,.e., dsont, overlapped meetng, meetng, overlapped crossng and crossng, whch are descrbed n fgure1. overlapped meetng meetng overlapped crossng crossng Fgure1. Four knds of nner topologcal relatons between two lnes At the same tme, there are seven topologcal relatons between nodes of two lnes, see fgure2. Relaton name LLBB1 LLBB2 LLBB3 LLBB4 LLBB5 LLBB6 LLBB7 Fgure representaton Fgure2. Relatons between end ponts of two lnear obects If these basc topologcal relatons appear only one tme at most, there are 56 dfferent topologcal relatons between two lnes by combnatonal reasonng method. 3 EQUIVALENCE OF TOPOLOGICAL RELATION Generally, valency s strong constrant, a hundred percent smlarty s valency (Pava, 1998), however, here we dscuss generalzaton valence or abstracton valence, so the constrant could be loosened. Keepng topologcal relaton valency n spatal abstracton s the basc rrement. The valency s a cogntve valency, namely a smlarty. If the overlapped parts are very short, we can say that overlapped crossng s valent wth crossng n abstracton process, and overlapped meetng s valent wth meetng, see fgure3. Fgure3. Equvalence between nner parts of two lnear obects In order to smplfy topologcal relatons, sometmes we need deletng some nner topologcal relatons f they are nessental. There are three man deleton operatons, ncludng deletng meetng part, deletng one crossng part and deletng two crossng parts, see fgure4. Fgure4. Dfferent types of deleton operatons Accordng to these valent operatons, the valent graph of topologcal relatons could be gotten. Fgure5 s only an example. In the valent graph, real lnes represent valency shown n fgure3, and broken lnes represent valency after deleton operatons. 2
Fgure5. Equvalent graphs of topologcal relatons 4 GENERALIZAITON METHOD OF TOPOLOGICAL RELATION Though topologcal relaton s consdered as a qualtatve relaton, t s necessary to represent them wth some metrcs. In order to abstract spatal topologcal relatons, the metrc features of topologcal relaton must be represented n detals. When topologcal relatons between two spatal obects are complcated, t can be represented wth component sence. A component s an nner topologcal relaton between two lnes, and component sence s a set of all components, whch arrange n terms of certan order. Fgure6 s a component sence of two lnes. Fgure6. Component sence of two lnes Fgure7. Component correlaton regons Each component sence can be expressed wth a component character table. Table1 s the component table of fgure6, where, overlapped meetng, meetng, overlapped crossng and crossng s denoted by T 1, T 0, I 1 and I 0 respectvely. Table1. Component character table of fgure6 Component C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 Type I 0 T 0 I 1 T 1 I 0 I 1 I 0 T 1 I 0 Start and end pont P 1 P 2 (P 31, P 32 ) (P 41, P 42 ) P 5 (P 61, P 62 ) P 7 (P 81, P 82 ) P 9 3
Besdes component sence, we must represent some metrc characters of these components, for example, the length of one-dmenson components, the ntervals between two adacent components, the area of component regon (the regon composed of two lne segments between two components, see fgure7). In topologcal relaton generalzaton, the abstracton operatons nclude deleton, amalgamaton and collapse operaton. The deleton operaton s shown n fgure4. And there are three dfferent amalgamaton operatons, see fgure8. The collapse operaton s the reducton of dmenson of component. Fgure8. Amalgamaton operatons of component In ths paper, a progressve generalzaton method s used to abstract topologcal relatons. The man steps are as follow: frstly, calculate the pertnent metrcs of each component, secondly, calculate the mportance of each component n terms of these metrcs, and determne the most nessental component(s), at last, accordng to mportance of components and shape feature, spatal topologcal relatons are abstracted by deleton, amalgamaton and collapse operatons. 5 EQUIVALENCY EVALUATION MODEL OF TOPOLOGICAL RELATIONS Generally, the progressve cartographc generalzaton methods could keep the generalzaton valency, however, an evaluaton model s needed to check ths valence. Assumng a spatal scene (SB) denotes a set of spatal obects whch has not been generalzed, and SA denotes a set of spatal obects after generalzaton. In SA, the number of spatal obects OA 1, OA 2,, OA m s m, and the topologcal relatons between these obects are denoted by r(oa, OA ) (1 < m= respectvely. Accordng to generalzaton method and valent rules, we can dentfy topologcal relatons generalzed, they are denoted by r (OA, OA ) (1 < m) respectvely. These topologcal relatons may be valent wth r(oa, OA ), or may be not. We can calculate the valency between them by followng formula: EQU ( r( OA ),r ( OA T 1 0,fr( OA )),fr( OA ) r ) r ( OA ( OA Then, the valency of topologcal relatons between SB and SA can be calculated: EQU ( SB,SA ) ACKNOWLEDGEMENT T m m 1 1 1 EQU T ( r( OA m( m 1) 2 ),r ( OA Guo Qngsheng and Du Xaochu are supported by Foundaton for key laboratory of Geo-Informatcs of State Bureau of Surveyng and Mappng. )) ) ) REFERENCE 1. Egenhofer Max J. and Mark D. Nave Geography, In: Spatal Informaton Theory-A Theoretcal Bass for GIS, Internatonal Conference, COSIT 95, Semmerng, Austra, Sprng- Verlag, Berln, pp. 1-15, 1995 2. Egenhofer Max J. and Robert D. Franzosa. On the valence of topologcal relatons, Internatonal Journal of Geographcal Informaton Systems, 1994, 8(6): 133-152 3. Egenhofer Max J., Clementn Elseo and Paolno D Felce. Evaluatng nconsstences among multple 4
representatons, Sxth Internatonal Symposum on Spatal Data Handng, Ednburgh, Scotland, September 1994, 901-920 4. Abdelmoty Ala I. and Jones Chrs B. Towards mantanng consstency of spatal databases, Proceedngs of the sxth nternatonal conference on nformaton and knowledge management, Las Vegas Nevada, Unted States, 1997, page. 293-300 5. Bobzen M. and Morgenstern D. Abstractng and formalzng model generalzaton, ICA ffth Workshop on Progress n Automated Map Generalzaton, Pars, Aprl 28-30, 2003 6. Sylve Servgen, Therry Ubeda, Alan Purcell and Robert Laurn. A methodology for spatal consstency mprovement of geographc databases, GeoInformatca, 2000, 4:1, 7-34 7. Joao Argemro de Carvalho Pava. Topologcal valence and smlarty n mult-representaton geographc database, Doctor Thess, Unversty of Mane, December 1998 8. Guo Qngsheng. Combnatoral representaton of spatal relatonshps on 2D vector map, Acta Geodaetca et Cartographca Snca, 2000, 29(2): 155-160 (n Chnese) 5