VISUALIZING SCHEMATIC MAPS THROUGH GENERALIZATION BASED ON ADAPTIVE REGULAR SQUARE GRID MODEL

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1 VISUALIZING SCHEMATIC MAPS THROUGH GENERALIZATION BASED ON ADAPTIVE REGULAR SQUARE GRID MODEL Dong Wehua a,b, *, Lu Jpng b, Guo Qngsheng c a Insttute of Geography and Remote Sensng, Beng rmal Unversty, Beng , Chna; b Chnese Academy of Surveyng and Mappng, 16 Tapng Road, Beng ,Chna; c School of Resource and Envronment Scence, Wuhan Unversty,129 Luoyu Road, Wuhan ,Chna. Commsson II, WG II/3 KEY WORDS: Vsualzaton, Schematc Map, Generalzaton, Smplfcaton, Grd Model. ABSTRACT: Schematzaton s a quck vsualzaton way to convey graphcs nformaton. In cartography, graphc schematzaton s one aspect of map generalzaton. Ths paper analyzes and summarzes the multple crtera for schematc maps and quantfes these constrants functonally. Based on ths, schematc map cartographc generalzaton framework based on the regular square grd s proposed. Furthermore, we research topologcal checkng methods for schematc maps. Based on these constrans, an effectve schematc map s schematzed and vsualzed automatcally n a test, whle keepng the topologcal consstency of the network map between orgnal and schematc map. 1. INTRODUCTION In graphcs and language, schematzaton s an mportant method to emphasze certan aspects and to deemphasze others. Dfferent dscplnes use schematzaton for dfferent reasons. In cartography, graphc schematzaton s one aspect of map generalzaton (Klppel, 2005). The man advantage of schematc maps s that they provde a quck overvew of the layout of the network, wthout showng unnecessary nformaton lke the precse shapes of the connectons (Cabello, 2001).Schematc drawngs of route drectons are one of the most common forms of graphc communcaton. Automated producton of schematc maps has been dscussed n several earler papers. Neyer (1999) and Barkowsky (2000) descrbe a lne smplfcaton algorthm wth the am of generatng schematc maps. But lne smplfcaton s only one characterstc of a schematc network. Equally mportant, schematc maps also nvolve the need of preservng the topologcal structure of the lner network and other aesthetc desgn characterstcs. Elro (1988a, 1988b, 1991) and Cabello et al. (2001) developed more specfc approaches to automatze the schematzaton process. In the approach of Elro, a grd s used to ft the lne network n the schematc drectons, but topologcal conflcts among lne segments can stll occur and no explanaton s gven about how to avod them. In the approach of Cabello, the topologcal equvalence between the networks s preserved, but, f certan condtons cannot be met, no schematzaton s found at all. Lately, the topc has reappeared n Agrawala (2001), Avelar (2000) and Mark Ware (2006) papers. They make use of some local operator, such that after applyng t several tmes, the map converges to a schematc one. Dfferent crtera based on dfferent measures are used to decde n whch order, and to whch parts of the map the operator should be appled. Iteratve approaches lke these have prohbtve costs n tme n nteractve stuatons, and n these papers, no theoretcal tme bounds, nor are actual computaton tmes gven. Moreover, they both adopt the Douglas-Peucker (1973) smplfcaton algorthm for graphc smplfcaton whch can generate self-ntersecton and change the topology. Creatng and vsualzng schematc maps s qute complex work. Mapmakers use a varety of cartographc generalzaton technques ncludng dstorton, smplfcaton, and abstracton to mprove the clarty of the map and to emphasze the most mportant nformaton. To automatcally generate schematc network maps and real-tme vsualze network maps clearly and concsely, we adopt new generalzaton methods ncludng smplfcaton and dsplacement based on adaptve square regular grd. 2. GENERATING SCHEMATIC MAPS 2.1 Graph model based on regular square grd The model we use for the abstract representaton of a schematc map s a graph. In ths case, a graph, G (V, E), s a set of vertexes, V, wth connectons between pars of vertexes represented by a set of edges, E. For drawng schematc maps, we use the vertexes to represent turnng ponts on the network and edges to represent a road segment between two turnng ponts. In some cases, there may be several edges connectng two vertexes. A graph model s used because of ts programmatc flexblty whch makes tasks such as fndng neghborng vertexes or ncdent edges relatvely easy. The graph s embedded on a regular square grd, as shown n Fgure 1. Ths means that vertexes can only be centered on grd ntersectons, but there s no requrement for edges to follow grd lnes. The spacng between adacent grds s denoted by g. The grd allows us to dramatcally reduce the number of * wh_whd@sohu.com; phone / ; fax

2 The Internatonal Archves of the Photogrammetry, Remote Sensng and Spatal Informaton Scences. Vol. XXXVII. Part B2. Beng 2008 potental locatons for nodes and also makes producng orthogonal maps easer as nodes are more lkely to be n lne wth each other. mnα α( u, Where { x ( u) u) }, { x(, } { 0 ~ 7}, { u, v} V (1), s coordnates of two adacent vertexes of an edge. 2 Dstance Constrant: When overlappng among network segments, we should separate these overlappng lnes and the separatng dstance g whch accordng to the envronment s obect resoluton. Fgure 1 Regular Square Grd Model We start wth a geographc layout (or a close alternatve usng a sketch of the map). However, we need to make sure that vertexes are centered on grd ntersectons, so a way to dsplace the vertexes to the grd s needed. Calculatng the nearest grd ntersecton to a vertex s smple but care must be taken to ensure that more than one vertex does not share the same grd ntersecton, mstake ntersectons and mssng ntersectons do not be produced by checkng duplcated vertexes and segments durng the generalzaton. In the case of contenton for a partcular ntersecton, the node beng dsplaced should be moved to the nearest grd ntersecton that s vacant. Consderng the representaton of a geographc dataset as an magng system wth a CCD camera, what the observer can see are two-dmensonal pxels, where each pxel corresponds to and represents statstcally a regon of the geographc data. Smlar to the prncple of a CCD camera, a grd s mplemented by dvdng the range of the target dataset to a twodmensonally spread cells. The sze of cells s the same wth or proportonal to the target resoluton. 2.2 Multple Crtera Generatng schematc maps, there are many aspects of the mappng stuaton whch may need specal attenton. Besdes characterstcs of the network, cartographc aspects, such as the amount of smplfcaton of lnes, vertexes dsplacement, symbolzaton and the use of colours, are also mportant. All schematc maps shall have graphc smplcty, whle retanng network nformaton and presentaton legblty. Based on those aspects, we set up crtera for schematc maps by quanttve descrpton, whch ncludng or Orentaton Constrant, Dstance Constrant, Length Constrant, Angular Constrant, Dsplacement Constrant. 1 Orentaton Constrant: In schematc map, based on the standard drectons ncludng horzontal, vertcal or dagonal drecton, all network edges only chose one of these drectons whch most close to ther orgnal drectons. Supposng the standard orentaton RR = { α = 45 } and orgnal { 0~7} 1 u) orentaton s α( u, = tan { u, v} V, x( u) x( thus the Orentaton Constran s quantfed by (1). 3 Length Constrant: The shortest lne segment of network must more length than g and mantan the relatve orderng of the lne segments by length. Length Constran s quantfed by (2) and (3). mn L( e ) g L( e ) L( e ) L( e ) L( e ) {, } and β {, } { 1 ~ n}, { e, e } E (2) { 1 ~ n}, { e, e } E (3) Where L( e ) and L( e ) s the length of orgnal network edge, ) and L ) s the fnal length of these edges, g s L( e ( e the target resoluton and β s the threshold assumed by the user. 4Angular Constrant: When meetng at one ntersecton and angular between each other s small; the drecton of overlapped network segments shall be enttled by more detal standard orentaton n [ α ] [ ], α + 1 or α 1, α. Angular Constran s quantfed by (4). α + 1 α α( = α +, f α M α α 1 α ( = α, f α M OJ OJ { 0 ~ 7}, e E (4) Where M s amount of these overlapped network edges, OJ s the collecton of reorented network edges, α( s the adusted orentaton of network edges. 5Dsplacement Constrant: In all possble dsplacement results n whch shortest dsplacement dstance result wll be selected. Dsplacement Constran s quantfed by (5). mn ( y ( v ) ) 2 + ( x ( v ) x( ) 2 v V, { 0 ~ 7} (5) 380

3 The Internatonal Archves of the Photogrammetry, Remote Sensng and Spatal Informaton Scences. Vol. XXXVII. Part B2. Beng 2008 Where { ( x, } { ( ),y ( )} v v are coordnates of orgnal vertex, x s coordnates of dsplaced vertex. 2.3 Schematzaton Process x( x ( = g g y ( = g g (7) Input Orgnal maps of wth M road segments Where x(, s the vertex coordnates before lattced. Schematzaton For each nner vertex v M () Compute Grd Coordnate of vertex v Checkng Duplcated Vertex or Segment >M For each segment M() x (, y ( s the vertex coordnates after lattced. g s the target resoluton(usually actual dstance represents by 1 px.[ ] s the nteger of [z]. Durng the grd, we should check duplcated vertexes and segments and adopt generalzaton methods accordng to dfferent cases. Duplcated vertexes and segments, whch are those ponts whch are orgnally dfferent but locates at the same grd after beng lattced, and those overlappng segments after beng lattced. If the duplcated nner vertexes and segments are from the same feature and adonng wth each other (see n Fgure3 (a), (b), (c)), the duplcated one should be deleted to keep the unqueness of the smplfed data. If they belong to the same feature but do not adon each other (see n Fgure3 (d), () or belong to dfferent feature (see n Fgure3 (f)), we should separatng overlappng segments and duplcated vertexes. vertex v Dsplacement Checkng Topology >M OutPut Schematc map Fgure2 Generatng Schematc Maps Framework Based on the regular square grd model, we propose a framework (see n Fgure2) wth the goal to produce a schematc network map whch meets topologcal, geometrc and semantc constrants. As the frst step, drven by the regular square grd, all network vertexes wll be moved to the nearest corner of the grd (see n Fgure 1) and the coordnates of these vertexes wll be computed by formulaton (7). Fgure3 Generatng Schematc Maps Framework Then, for the orentaton constrant, vertex q wll be dsplaced accordng to eght drectons and make the schematc map more regular. The new locaton q s chosen among the locatons that ft q n one of the allowed schematc drectons. In each lne segment wth q, e.g. pq, we use the vertex q to compute the dstance to the nearest schematc drecton. In order to meet dstance constrant, the nearest dsplacement dstance determnes the new locaton for q (see n Fgure 4). All network segments orentaton wll be adusted to the eght standard drectons whch s collecton RR={0º,45º,90º,135º,180º,225º,270º,315º}.Where postve X axes representaton 0º and 360 º, and orentaton of network segments s the rotaton angular calculated by counterclockwse based on X axes. 381

4 The Internatonal Archves of the Photogrammetry, Remote Sensng and Spatal Informaton Scences. Vol. XXXVII. Part B2. Beng 2008 Fgure 4 Dsplacement of Vertex If there s no pont nsde the trangle Δ( pqq ) and no lne segment crossng the edge qq of Δ( pqq ) the topology wll be preserved, so the move of q to q s allowed (see Fgure 5(a)). If there s at least one lne segment ntersectng edge qq of Δ( pqq ) (see Fgure 5(b)) or there s at least one pont v nsde the trangle Δ( pqq ) (see Fgure 5(c)), map topology wll also change., map topology wll change, then a new locaton for q has to be recomputed and obtaned. In order to satsfy the orentaton constran, the orentaton and length of every adusted network segment shall be calculated by usng formulaton (8) and (9). α( = α, f mnα α( α( = 0, f α = 360 and L( r L( = cos( α( α( ), f mn( L( e )) < r mn( L( e )) L( = L( cos( α( α( ), f mn( L( e )) r 1 ~ n, e E { 0 ~ 7}, { e} E (8) { } (9) We select ntersecton ponts of network segments, fnd the network segments collecton adacent to these ponts and get the adusted orentaton collecton OJ. Frstly, udge whether there s overlappng network segments whch have been adusted, f have t s to say that there s orentaton conflct at ths ntersecton pont, the method to solve the problem s adoptng Angular Constran. The orentatons of overlappng network segments are reorented durng the collecton of [ RR 1, RR ] or [ RR, RR +1 ] and OR 2 the length of these segments - are recalculated. Next, we grow all network segments that are shorter than a predefned mnmum pxel length; mn(ol) to be r pxels long by usng formulaton (6). (a) (b) (c) Fgure 5 Topologcal Checkng 3. RESULTS AND CONCLUSION Based on the vector database of 1: road networks, we set square grd wth 1000m x 1000m accordng to target resoluton. Includng 334 road segments, 506 ntersectons and 2286 vertexes, orgnal road network map are lattced and schematzed. Experment results are shown n Fgure Topology Consstence Checkng Durng the course of shape smplfcaton and pont dsplacement, topology of network must be checked to and mantaned. Before movng a pont from ts orgnal poston q q to the new schematc locaton we perform a test to detect stuatons that can lead to a change n the map topology. For t we frst create a trangle. The vertces of the trangle are q, q and the other endpont of the lne segment beng analyzed p (Avelar, 1997). We have to fnd out f there s any lne segment of the map crossed by the boundary edge qq of the trangle Δ( pqq ) (Berg et al, 1997). We also have to check f the trangle Δ ( pqq ) contans nsde t any pont. If topology q would change, pont dsplacement must be recomputed. We dstngush the followng three cases to check the topology. (a) Orgnal Road Network Map (b) Schematc Road Network Map Fgure 6 Dsplacement of Vertex 382

5 The Internatonal Archves of the Photogrammetry, Remote Sensng and Spatal Informaton Scences. Vol. XXXVII. Part B2. Beng 2008 Experment results show that 156 nteror vertexes of orgnal road network (Fgure 6(a)) are elmnated successfully whle keepng the topology. Then, drven by square grd, orgnal road networks are dsplaced and segments are orented to horzontal, vertcal or dagonal drecton. Fnally, a desred schematc road network (Fgure 6(b)) s generated. ACKNOWLEDGEMENTS The work descrbed n ths paper s supported by a grant from the 863 Scence Foundaton of Chna (. 2007AA12Z215). REFERENCES Agrawala, M., Stolte, C., 2001.Renderng effectve route maps: Improvng usablty through generalzaton. In Computer Graphcs - SIGGRAPH 2001 Proceedngs, E.Fume,Ed. ACM Press, pp Avelar, S., The problem of contour n the generaton of dgtal topographc maps. In Auto-Carto13, Seattle. ACSM/ASPRS. pp Avelar, S., Müller, M., 2000.Generatng topologcally correct schematc maps, n Proc.9th Int. Symposum on Spatal Data Handlng, pp. 4a.28 4a.35. Barbowsky, T., Lateck,L. J., Rchter, K.-F., 2000.Schematzng maps: Smplfcaton of geographc shape by dscrete curve evoluton, n Spatal Cognton II, LNAI 1849, pp Berg, M. de, Kreveld, M.van, Overmars, M., Schwarzkopf, O., 1997.Computatonal Geometry: Algorthms and Applcatons. Sprnger-Verlag. Cabello, S., Berg, M.de, van Dk Sm van K M.reveld and Strk, T., Automated Producton of Schematc Networks. Cabello,S., M.de Berg, van Dk Sm van K M.reveld and T.Strk, 2001.Schematzaton of road networks. In Proceedngs of the Seventeenth Annual Symposum on Computatonal Geometry, Medford, Massachusetts: Douglas, D. H., Peucker, T. K., Algorthms for the reducton of the number of ponts requred to represent a dgtzed lne or ts carcature. The Canadan Cartographer, 10(2), pp Elro, D., 1988a. Desgnng a network lne-map schematzaton software-enhancement package, n Proc.8th Ann. ESRI User Conference. Elro, D., 1988b. GIS and schematc maps: A new symbotc relatonshp,nproc.gis/lis 88. catons.html. Elro, D., Schematc vews of networks: Why not have t all, n Proc. of the 1991 GIS for Transportaton Symposum, pp Klppel, A., Rchter.F., Barkowsky, T. and Freksa, C., The Cogntve Realty of Schematc Maps, n Mapbased Moble Servces - Theores, Methods and Implementatons, L. Meng, A. Zpf, and T.Rechenbacher, Eds.: Sprnger; Berln, pp Neyer, G., Lne smplfcaton wth restrcted orentatons. Techncal Report 320, Department of Computer Scence, ETH Zurch, Swtzerland. Ware, J. M., Anand, S. G. Taylor, E., Thomas, N, 2006.Automated producton of schematc maps for moble applcatons. Transactons n GIS. 10(1):

6 The Internatonal Archves of the Photogrammetry, Remote Sensng and Spatal Informaton Scences. Vol. XXXVII. Part B2. Beng

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