New method of creation data for natural objects in MRDB based on new simplification algorithm

Transcription

1 New method of creaton data for natural obects n MRDB based on new smplfcaton algorthm Krystan Kozoł, Stansław Szombara AGH Unversty of Scence and Technology, Faculty of Mnng Surveyng and Envronmental Engneerng, Department of Geomatcs Abstract: In the paper, authors present a prototype of new smplfcaton algorthm whch frst dvde the orgnal polylne nto surecton parts and then convertng nto ths parts n to curve H(x) and then located ponts dependng on the scale of the generalsed map. Ths method s useful for smplfcaton of the natural obects. Addtonally, the extreme ponts of the curve, whch are n accordance wth the standard of the drawng recognton (recognsablty) 1 (Chrobak 2010) are taken nto consderaton whle locatng. Presented algorthm of smplfcaton use as threshold parameter the standard of the drawng recognton, whch s ndependent of the user. Moreover the new algorthm ncreases the automaton of the polylnes smplfcaton process and ncreases the scale range, whch s partcularly mportant n the Multresoluton/ multrepresentaton Data Base (MRDB) applcatons. Introducton One of the man tasks of cartography n the 21st century s vsualsaton of topographcal obects n Multresoluton/multRepresentaton Data Bases n varous scales. Ths task s closely related to dgtal cartographc generalsaton. Mackaness and Ruas (2007) pont out four reasons for the generalsaton process to be so dffcult to automate. They state, that generalsaton s more than ust a geometrc process based on the analyss of man geometrc types. Accordng to Sarakosk (2007), creatng MRDB means usng generalsaton model nto account the followng crtera: most frequently used representaton of obects requrements consderng actualsaton of obects level of automaton, that can be acheved n the process of creatng the representaton of obects, whle obtanng the representaton of obects at lower level of detal. In the consderatons rased by the above mentoned authors, problems of generalsaton model and automaton n terms of MRDB functonng are contnuously dscussed. Sarakosk (2007) suggests the range of necessary research ncludng: modellng of data, automatc nstruments for creatng MRDB, as well as detectng dscontnutes and conflcts n MRDB. 1 The standard of the drawng recognton (recognsablty) was defned by (Chrobak 2000) on the bass of the least drawng dmensons provded by (Salszczew 1998). The length of shorter sde of the recognsablty trangle s n agreement wth the measure of the cartographcal norm set by Natonal Map Accuracy Standards NMAS(Longley et al. 2006). The standard of the drawng recognton s determned by a black colored elementary trangle wth the followng dmensons: lne wdth 0.1mm, shorter sde a = 0.5 M (mm) and base b ( 0,5 0,7 M [mm]. ]

2 Accordng to the authors of ths artcle, the degree of process automaton depends on: data orderlness, operators nteroperablty, unambguty of results and also possblty of results verfcaton. For the smplfcaton operator the correct order of data s a superor element, sgnfcant whle treatng the arsng conflcts. Unambguty of smplfcaton results s acheved by determnng the process parameters basng on the standard of the drawng recognton and not knowledge of the user. Verfcaton of the results s possble only f standard whch were used retan the role of comparng a pattern to a case. Number of conflcts occurrng n the smplfcaton process s an addtonal element of verfcaton. Smplfcaton operator The spatal data generalsaton models under elaboraton show how complcated the process s. For nstance, n the Brassel-Webel model (Brassel and Webel 1988), there s a stage of modellng called process and parameter recognton, n whch the proper generalsaton operators are created. The same approach s used n the McMaster and Shea model (Shea and McMaster 1989), whose authors defned the generalsaton operators, and so dd McMaster and Monmoner n ther model (McMaster and Monmoner 1989). The authors underlned the necessty to use dfferent operators for generalsaton of dfferent obects. Clamed to be one of the basc operators, the smplfcaton operator s used at the very begnnng of the modellng process, dealng wth the lnear and surface obects. After long research on smplfcaton operator functons, t s stll mpossble to pont out the algorthm whch would be optmal n ths applcaton. It s clamed that the frst smplfcaton algorthm s the one created by Perkal (1966) and Lang (1969). However most of the algorthms were created n the second half of the 20th century, ncludng one of the most popular, namely the Douglas-Peücker algorthm (Douglas and Peucker 1973). In general, the basc parameter of the Douglas-Peücker algorthm s the length determned by the user. A dfferent approach was proposed by Vsvalngam and Whyatt (1993). Ther algorthm bases on the determned surface of a trangle. As far as the analyss of the orgnal polylne s concerned, one of the most advanced algorthms seems to be the one created by Wang (1996). He has proposed examnng the orgnal polylne and adustng the algorthm to ts shape.l (2007), n hs research on generalsaton process, has dvded the smplfcaton algorthms nto the pont-reducton and scale drven generalsaton algorthms. The frst group conssts n selecton of crtcal ponts of the polylne and subsequent elmnaton resultng from the set target scale. Ths group s further dvded nto: ndependent pont-based procedures, local processng procedures, extended uncondtonal local processng procedures, extended condtonal local processng procedures and global procedures (McMaster 1991). The global algorthms are represented by: Douglas-Peücker algorthm, Vsvalngam-Whyatt algorthm, Wang and Chrobak algorthm Chrobak (2000). Dyken, Dæhlen, and Sevaldrud (2009) presented a method for smultaneous smplfcaton of a collecton of pecewse lnear curves. Ths method based on trangulatons removng lne segments from the pecewse lnear curves wthout changng the topologcal relatons between the curves. Descrbed algorthms of smplfyng have one shared feature they are elmnatng ponts from the orgnal polylne accordng to the procedures determned by the algorthms. Whereas, n

3 fact, the process of smplfcaton s such a choce of vertces from the orgnal polylne, that the smlarty of shape, locaton and measure of the generalsed polylne are retaned (Fg. 1). The value of parameter n the algorthms depends on the knowledge and experence of the user. Fgure 1. Dfferent locaton of the obect vertces n varous scales. The authors present a new prototype of smplfcaton algorthm for polylnes n the paper. The proected smplfyng method s sutable for the obects of natural (smooth) character, such as rvers, lakes and coastlne. The stages of the algorthm are: locatng a characterstc vertex on to polylne whch dvde ths polylne n to parts whch are surectons, transformaton of the polylne nto a curve wth an nfnte number of ponts, usng the Hermte polynomal nterpolaton, determnaton of the extreme ponts of the orgnal polylne and ther herarchy, satsfyng the standard of the drawng recognton, selecton of new ntermedate ponts of the curve between neghbourng extreme ponts and at dstances satsfyng the standard of the drawng recognton. In result of the smplfcaton of a polylne, the shape and locaton of both the orgnal open polylne and the smplfed one are preserved wthn the accuracy lmts set by the recognsablty norm. Furthermore, due to the recognsablty of the new algorthm, the value of the smplfcaton parameter s not determned by the user and thus t s possble to obtan an unambguous result. Both, the unambguty of result and the fact that the user does not have to determne the parameter, ncrease the process automaton. Thus, the new soluton s more a method than ust an algorthm. The new smplfcaton algorthm The flowchart presented below (Fg. 25) was mplemented n MATLAB usng ts nner programmng language. MATLAB was chosen due to the possblty to use ts exstng lbrares, whch nclude the Hermte polynomal nterpolaton. The algorthm s based on: a loop search for global extreme of a functon satsfyng the standard drawng recognton, a loop n whch the ntermedate ponts are nserted n agreement wth the standard drawng recognton.

4 Fgure 52. The flowchart of the smplfcaton algorthm. At the begnnng the user loads the fle (ESRI Shapefle format) whch contans source data. He also enters the scale denomnator of the source data and the destnaton scale denomnator M0 and Mk, whch determne the values ε01 ε02. Next the polylne s dvded nto the surecton parts. Only then the dscrete data can be (wth use of nterpolaton) expressed as a splne functon whch satsfes the condton 18. The process of dvdng the polylne nto

5 surecton parts s descrbed n the next chapter. The transformaton of a polylne nto a curve takes place for the surecton parts and allows us to determne the extremum of ths functon. Ths enables us to check the recognzablty norm (Equaton 23). If the trangle begnnng pont-extremum-end pont satsfes the condton of the norm, then the other vertces of the curve n the gven nterval may be recognzable. These vertces are searched for as the further determnaton of extremes s beng carred out and the standard of the drawng recognton s checked. As a result a new polylne s created. Ths polylne undergoes verfcaton whch s descrbed n detals n the further chapters of ths paper. Locatng the characterstc vertex on to polylne (creaton of surecton parts) Characterstc pont s an mportant concept n many dscplnes such as computer vson, mage processng, pattern recognton, computer graphcs, and geospatal scence. For example, n computer vson and pattern recognton, the concept of crtcal ponts s used n algorthms for feature extracton, shape recognton, pont- based moton estmaton, and codng. In geospatal scence t s used for data compresson, for lne carcature, and msleadngly for mult-scale representaton of lnes (L 1993). Unambguous nterpolaton performed wth use of the Hermet method requres vertces and edges to satsfy the condton of surecton and therefore any polylne has to be dvded nto parts. Those parts have to satsfy the followng condton: n local rectangular Cartesan coordnates system wth orgn n the frst node of a R part and x-axs drected towards the last node, all the vertces and edges of a part must be a surecton and satsfy the condton: y[ymn[y,r ],ymax[y,r ]] x [0,xmax[x,R ]]:f(x)= y, (1) where: x maxx, R - maxmum x-coordnate (abscssa) of the vertces of R, y mny, R y, - mnmum y-coordnate (ordnate) of the vertces of R, max y R - maxmum y-coordnate (ordnate) of the vertces of R, and the number of such parts should tend to mnmum. Such parts wll be called unambguous and the vertces whch determnate them as characterstc ponts (L 2007). In the Fgure 32 a polylne wth ts unambguous parts s presented.

6 a b Fgure 32. Orgnal polylne a) and polylne dvded nto unambguous parts b). a b c d e f g h Fgure 34. Determnaton of unambguous parts (descrbed n text). The algorthm of unambguous parts determnaton was descrbed wth use of notaton based on the (Molenaar 1998) notaton. For the polylne P, the set of ts vertces s gven as:

7 v V P 1,..., v,..., v and the set of ts edges as: P n, (2) e e,..., e v, v,.., v, v,..., v v E, 1,..., n 1 2 k k1 n1 The Drecton of P1 (Fgure 2a) s specfed by: P v 1 v n n. (3) Dr,. (4) Let R(P) be an unambguous part and VR ts set of vertces: R 1, v2,..., v k,..., v n 1 n V v, v, (5) whose coordnates can be treated as attrbutes: x Xv oraz y Yv, (6) The fgure 43 depcts the algorthm of unambguous parts determnaton. Ths algorthm begns wth creatng R1 and assgnng of vertces v1, v2 to t. Next, the followng vertces are examned one after another and added to R1 as long as the condton (1) s satsfed. If the condton (1) s not satsfed a new part s created. The algorthm examnes all the vertces of the polylne. Fgure 43a shows addng of v3 to R1. The x-axs of local cartesan coordnate system (orgnatng n v1) s drected towards ths vertex. For R1(P) (at Fgure 34a) contanng v3 the followng can be stated: the set of ts vertces accordng to (2) V R v, v, (7) 1 2, v 1 3 the set of ts edges accordng to (3) e, e v, v, v, (8) E R , v 1 3 ts drecton accordng to (4) DrR v 1,v 3, (9) the range of ts coordnates (n local system): x x - mnmum x-coordnate (abscssa) of the vertces of R1, mn,r 1 max x,r 1 mn y,r 1 y x - maxmum x-coordnate (abscssa) of the vertces of R1, y - mnmum y-coordnate (ordnate) of the vertces of R1, ymax,r 1 - maxmum y-coordnate (ordnate) of the vertces of R1, relatonshps of the begnnng and end of the part: BegnRR 1 v 1, (10) EndRR 1 v 3. (11)

8 Upon addng v3 and change of the orentaton of the local coordnaton system the condton (1) s examned, whch can be noted as: [y [y,r ],y [y,r ]] x [0,x [x,r ]]:f(x)=y. (12) y mn 1 max 1 max 1 If the condton (12) s satsfed, R1 wll be enlarged by v3 (as depcted n fg. 43a) and algorthm wll move on to the next vertex. Fgures 43b, 34c, 3e show the stuaton n whch the condton (1) s satsfed and the algorthm moves on to the next vertex, whereas the fgure 43g presents the case where the condton (1) s satsfed but algorthm fnshes ts work. If the condton (1) s not satsfed (fg. 43d, 43f) then: R1 s saved wthout the examned vertex, a new orgn of a local coordnate system d determned n R a new part R+1 s created wth begnnng n R BegnRR After the algorthm has been performed (fg 42h) we obtan: EndR, EndR. 1 the fnal set of unambguous parts of the polylne: P R 1, R 2, R 3, (13) R the set of vertces whch determne parts (characterstc ponts (CP) ) s descrbed as: CP P v 1, v7, v11, v16 BegnRR 1, BegnRR 2, BegnRR 3, EndR 3. (14) 1 R Fgure 54. Characterstc ponts of the polylne determned wth use of algorthm performed reverse to ts drecton. It can be easly observed that, accordng to the drecton n whch we perform the algorthm, whether from the begnnng to the end of a polylne (Fg. 43h) or reverse (Fg. 54), the number of the obtaned parts (as well as the number of characterstc ponts) may dffer. For the polylne we choose ths drecton whch has the smaller number of R parts R(P). If both drectons gve the same number of parts we take the parts (and of course the drecton) on whch the characterstc ponts are placed more regularly. Ths regularty s measured by coeffcent of varaton of parts lengths. Creatng curve usng 3rd degree Hermte Interpolaton

9 The frst step of the algorthm s convertng polylne nto a curve, contnuous wthn the gven nterval, wth use of a chosen nterpolaton method (Fg. 52). The chosen nterpolaton method should satsfy the followng condtons for a curve: the curve should pass through all the ponts of a polylne f(x) = H(x), the local extreme of a polylne should be preserved f (x) = H (x). These condtons are satsfed by the Hermte method. We know the number of ponts and the values of ther coordnates, for whch exsts the functon of class C 1 [a, b]: f 1 xc a b[ x, x, x,..., x,..., x ] a b, n, where: x x 1 for = 1,2,3,,n-1., We use the Hermte polynomal, compatble wth f(x) and wth f (x) at ponts x for = 0,1,2,n, furthermore the polynomal s of no greater degree than 2n+1. The form of the polynomal (Boor 1978) s as follows: H 2 n 1 ', 0 0 where: H n n x f x H n, x f x H n x, (16) ' 2 x 1 x x L x L x H x x L x, n, 2 n, n, n, 2 n, Ln, s a th coeffcent of the Lagrange polynomal of n th degree. If f(x)c 2n+2 [a, b], then there s the followng relaton between the functon f(x) and the Hermte polynomal: f x 2 x x 0... x x n 2n 2! 2 2n 2 H 2n 1x f, (17) where: a < < b In order to be able to clam that Hermte nterpolaton satsfes the condtons of an unknown nterpolaton functon (preserves ts shape), t s enough to prove that: H 2n1 x fx where: = 0, 1, 2,, n. H' 2n1 x f ' x (15), (18) H n Thus, we should prove that expressons, and H n, determned by equatons (16) satsfy smultaneously the followng four condtons:

10 H n x for 1, 0 for (19) d dx H Hn, n, x 0, x 0, (20) (21) d dx H n x for, 1 0, for,, (22) what has been proven by de Boor n hs book (De Boor 1978). After convertng the polylne nto a curve (wth use of an nterpolaton method) we obtan the most probable outlne of shape of the orgnal obect. The next stage of the algorthm s to arrange the ntermedate ponts on the orgnal curve, dependng on the scale. Arrangement of ponts on the orgnal curve accordng to the standard drawng recognton All the smplfcaton algorthms are based on the geometrcal condton, whch should be satsfed for the vertces remanng after the smplfcaton process. By use of the proposed algorthm, after the nterpolaton process, we obtan a curve wth unlmted number of vertces defned by functon H(x). It s therefore necessary to choose only the vertces, whch wll be characterstc of the outlne and locaton of a shape n the target scale. Fgure 6. Choosng the vertces of a curve. Checkng of the elementary trangle condton for the trangle (Nb, Vmax, Ne) enables decdng on the recognsablty (the standard of the drawng recognton) of the vertex, whch s the global extreme Vmax (Fgure 6). In such a way t s determned f the trangle wll be represented n the analysed scale. Then, n the same way, the algorthm checks all the local extremes on the bass of the adequate trangles (Fgure 6). After determnng the recognsablty of the local extremes, the algorthm arranges the remanng ntermedate vertces. In order to dstngush ponts of the orgnal curve (16) for preparaton of a map n the scale 1:Mk we have used:

11 the recognsablty based on the elementary trangle (Chrobak 2010): ε01 = 0,5[mm]*Mk b є [0,5mm 0,7mm)*Mk], ε02 > 0,5[mm]*Mk b є [0,4mm 0,5mm)*Mk] (23) where: ε01 ε02 - lengths of the shorter sdes of the trangle (Fgure 63), b lengths of the base of the trangle, M map scale denomnator, ƒthe extreme sm (m=1,2,...,l ) of the functon (15) and the polynomal (16), n agreement wth recognsablty, The presented condtons for arrangement of the new ntermedate ponts on the curve (16), enable us to: verfy the nterpolaton, by comparson of the extreme ponts of the polylne (15) wth the extreme of the orgnal curve (16). The result of the comparson s dentty, whch proves that the Hermte polynomal nterpolaton s correct, determne unambguously wth no operator nterventon needed extreme ponts of the polylne under smplfcaton nto the map scale 1:Mk (k = 0,1,2,,n), provded both standard of the drawng recognsng and the orgnal herarchy of crtcal ponts are preserved, create, n the map scale 1:Mk (k = 0,1,2,,n), ntervals based on the determned extreme ponts of the orgnal curve, wthn whch new ntermedate ponts wll be chosen at dstances compatble wth the standard of the drawng recognton sutable for the scale 1:Mk. The created polylne, after beng smplfed, undergoes verfcaton. Verfcaton of results Pątkowsk (1969) defned the norm of the accuracy of acqurng obects wth character of the curve, callng her the prmtve generalzaton. The orgnal generalsaton dea conssts n use of lne segments (chords) n place of correspondng curvlnear sectons of a lnear obect. Pątkowsk determned, emprcally, the length of the ordnate - e - between the chord and the arc of the curve, called the rse. Length of the rse depends on the scale of the map as follows: e 0,3 M0 [mm], (24) where: M0 s the source map scale denomnator (the map, from whch the data s drawn). The scale denomnator M0, n every smplfcaton process, satsfes the condton: ek 0,3 Mk,[mm], (25) where: Mk the target scale denomnator.

12 For practcal reasons of the smplfcaton process, the rght hand sde of the equaton (17) can be replaced by the condton of equalty n relaton (24), whch allows: verfcaton of the newly arranged ponts of the curve wth use of the Pątkowsk condton (25). unambguous test of the condton (25) for the rses from the deleted ponts of the orgnal polylne to the sdes of the smplfed polylne, the evaluaton of precson of the data obtaned n 1:Mk scale. Fgure 7. Example of applyng the descrbed algorthm on the sngle part: a) orgnal, b) 1 : 5000, c) 1 : , d) 1 : , e) 1: (for b, c, d and e below vsualsaton n the scale)

13 Fgure 8. Example of the symplfcaton on large dataset (mor then vertex): b) 1 : 5000, c) 1 : , d) 1 : , e) 1: (the data becomes from remote sensng for VHR)

14 Fgure 9. Example of symplfcaton few parts from fgure 8: b) 1 : 5000, c) 1 : , d) 1 : , e) 1: (the data becomes from remote sensng for VHR)

15 Fgure 10. An example of the smplfcaton of a coastlne wth use of the other algorthms for chosen scales (value of threshold parameter for other algorthms: Douglas and Wang 15, 30, 60 and Vsvalgam 50, 250, 1250)

16 The results of the polylnes smplfed wth Wang algorthm do not exceed the acceptable length of rse up to the 1:10,000 scale and are smlar to these obtaned n the smplfcaton process usng the Chrobak algorthm and the new method. In the 1:25,000 and 1:50,000 scales, the dfferences n the rse lengths ncrease sgnfcantly, whch results from the nfluence of the arrangement of the ponts on the orgnal polylne on the sze of the (ncreasng) scale dfference, whch has been neglected n the smplfcaton algorthms used before. Table 1. Lengths L of the polylnes smplfed from fgure 7 wth the compared algorthms and the dfferences between them and the orgnal polylne L0 Algorthm Lengths and ther dfference s Scales 1:1000 1:2000 1:5000 1: : :50000 Douglas - Peückera Vsvalngham Whyatt Wang Chrobak The new algorthm L ΔL L ΔL L ΔL L ΔL L ΔL The dfferences -ΔL of the length between the orgnal polylnes and these smplfed usng the Doulas-Peücker and Vsvalngham-Whatt algorthms are smlar (Table 1). In case of the three other algorthms, the dfferences -ΔL are sgnfcantly smaller. The lengths of the polylnes smplfed wth Wang algorthm show no mportant dfference to the lengths of the polylnes smplfed wth Chrobak algorthm and the new method. The dfferences are more sgnfcant n smaller scales (1:25,000 and 1:50,000), what proves that t s crucal to arrange the curve vertces accordng to the scale. The great advantage of the new method s the extended range of scales, what s partcularly useful n MRDB. Conclusons The new algorthm uses the followng norms: Patkowsk norm of the orgnal generalsaton and the standard of the drawng recognton. It enables verfcaton of the htherto appled algorthms n a way not dependent on the decson of the user.

17 The converson of a polylne nto a curve (lnes of at least the class C2) allows arrangement of the ponts on the orgnal curve dependng on the target map scale and not source map scale, as practsed before. The presented method can be mplemented to any database of MRDB type under the assumpton that the hgh-detal data wll consttute the source data, whereas the results wll be data for the purpose of vsualzaton at low level of detal. The Brassel-Webel (Brassel and Webel 1988) model for cartographc generalsaton dstngushes fve stages of processng: a) structure recognton, b) process recognton, c) process modellng (by proper selecton of parameters), d) process executon, e) verfcaton and vsualsaton of results. The lack of ndependent parameters n the dgtal cartographc generalsaton made t mpossble to carry out the last stage of the Brassel-Webel model: verfcaton. However, the new algorthm enables us to mplement t. The standard of the drawng recognton and determnstc method mplemented n the smplfcaton process wth ths partcular algorthm gves the unambguous result as well as the opportunty for a measurable verfcaton. The smplfcaton process wth use of the new algorthm fulfls condtons of a method, because the value of ts parameter does not depend on the user, and the result s congruent wth accuracy of the appled norms. References De Boor, C., A Practcal Gude to Splnes. Berln: Sprnger-Verlag. Brassel, K.E. and Webel, R., A Revew and Conceptual Framework of Automated Map Generalzaton. Internatonal Journal of Geographcal Informaton Scence, 2 (3), Chrobak, T., A Numercal Method for Generalzng the Lnear Elements of Large-Scale Maps, Based on the Example of Rvers. Cartographca: The Internatonal Journal for Geographc Informaton and Geovsualzaton, 37 (1), Chrobak, T., The role of least mage dmensons n generalzed of obect n spatal databases. Geodesy and Cartography, 59 (2), Douglas, D.H. and Peucker, T.K., Algorthms for the Reducton of the Number of Ponts Requred to Represent a Dgtsed Lne or ts Carcature. Cartographca: The

18 Internatonal Journal for Geographc Informaton and Geovsualzaton, 10 (2), Dyken, C., Dæhlen, M., and Sevaldrud, T., Smultaneous curve smplfcaton. Journal of Geographcal Systems, 11 (3), Lang, T., Rules for robot draughtsmen. The Geographcal Magazne, 42, L, Z., Some observatons on the ssue of lne generalsaton. Cartographc Journal, The, 30 (1), L, Z., Algorthmc Foundaton of Mult-Scale Spatal Representaton. London: CRC Press. Longley, P.A. et al., GIS, Teora praktyka. Warszawa, Polska: Wydawnctwo Naukowe PWN. Mackaness, W.A. and Ruas, A., Evaluaton n the Map Generalsaton Process. In: W.A. Mackaness, A. Ruas, and L.T. Sarakosk, eds. Generalsaton of Geographc Informaton: Cartographc Modellng and Applcatons. Amsterdam, McMaster, R.B., Conceptual frameworks for geographcal knowledge. In: B.P. Buttenfeld and R.B. Mcmaster, eds. Map Generalzaton: Makng Rules for Knowledge Representaton. London: Longman Scentfc & Techncal, McMaster, R.B. and Monmoner, M., A conceptual framework for quanttatve and qualtatve raster-mode generalzaton. In: Proceedngs of GIS/LIS 89, Orlando, November. Bethesda, MD: Amercan Congress on Surveyng and Mappng, Molenaar, M., An Introducton to the Theory of Spatal Obect Modellng for GIS. 1st ed. London: Taylor & Francs. Perkal, J., An attempt at obectve generalzaton. In: J.D. Nystuen, ed. Dscusson Papers of the Mchgan Interunversty Communty of Mathematcal Geographers. Dscusson Paper 10. MI: Ann Arbor: Department of Geography, Unversty of Mchgan. Ptkowsk, F., Kartografa: redakca map reprodukca kartografczna. Warszawa: Wydawnctwo Naukowe PWN. Salszczew, K.A., Cartography. 2nd ed. Warszawa: Wydawnctwo Naukowe PWN. Sarakosk, L.T., Conceptual Models of Generalsaton and Multple Representaton. In: W.A. Mackaness, A. Ruas, and L.T. Sarakosk, eds. Generalsaton of Geographc Informaton: Cartographc Modellng and Applcatons. Amsterdam: Elsever, Shea, K.S. and McMaster, R.B., Cartographc generalzaton n a dgtal envronment: When and how to generalze. In: Proceedngs Auto-Carto 9, Nnth Internatonal Symposum on Computer-Asssted Cartography, Baltmore, 2-7 Aprl. Baltmore,

19 Vsvalngam, M. and Whyatt, J.D., Lne generalsaton by repeated elmnaton of ponts. The Cartographc Journal, 30 (1), Wang, Z., Manual versus Automated Lne Generalzaton. In: Proceedngs of GIS/LIS 96, Denver, November. Bethesda, MD: Amercan Socety for Photogrammetry and Remote Sensng,