Preserving Spatial Patterns in Point Data: A Generalization Approach Using Agent-Based Modeling

6.92.8 P ese ing Spa ial Pa e ns in Poin Da a: A Gene aliza ion App oach Using Agen -Based Modeling Ma in Knu a and Jochen Schiewe A icle h ps://doi.o g/10.3390/ijgi13120431 Ci a ion: Knu a, M.; Schiewe, J. P ese ing Spa ial Pa e ns in Poin Da a: A Gene aliza ion App oach Using Agen -Based Modeling. ISPRS In . J. Geo-In . 2024,13, 431. h ps:// doi.o g/10.3390/ijgi13120431 Academic Edi o s: Wol gang Kainz and Flo ian H uby Recei ed: 17 Sep embe 2024 Re ised: 19 No embe 2024 Accep ed: 27 No embe 2024 Published: 30 No embe 2024 Copy igh : © 2024 by he au ho s. Published by MDPI on behal o he In e na ional Socie y o Pho og amme y and Remo e Sensing. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). A icle P ese ing Spa ial Pa e ns in Poin Da a: A Gene aliza ion App oach Using Agen -Based Modeling Ma in Knu a * and Jochen Schiewe Lab o Geoin o ma ics and Geo isualiza ion (g2lab), Ha enCi y Uni e si y Hambu g, Henning-Vosche au-Pla z 1, 20457 Hambu g, Ge many; jochen.schiewe@hcu-hambu g.de *Co espondence: ma in.knu a@hcu-hambu g.de Abs ac : Visualiza ion and in e p e a ion o use -gene a ed spa ial con en such as Volun ee ed Geog aphic In o ma ion (VGI) is challenging because i combines eno mous da a olume and he e o- genei y wi h a spa ial bias. When dealing wi h poin da a on a map, hese cha ac e is ics can lead o poin clu e , educing he eadabili y o he map p oduc and misleading use s o alse in e p e a ions o pa e ns in he da a, e.g., ega ding speci ic clus e s o ex eme alues. Wi h his wo k, we p o ide a amewo k ha is able o gene alize poin da a, p ese ing spa ial clus e s and ex eme alues simul aneously. The amewo k consis s o an agen -based gene aliza ion model using p ede ined cons ain s and measu es. We p esen he a chi ec u e o he model and compa e he esul s wi h me hods ocusing on ex eme alue p ese a ion as well as clu e educ ion. As a esul , we can s a e ha ou agen -based model is able o p ese e elemen a y cha ac e is ics o poin da ase s, such as he poin densi y o clus e s, while also e aining he exis ing ex eme alues in he da a. Keywo ds: poin gene aliza ion; agen -based modeling; cons ain s; spa ial pa e n 1. In oduc ion Use -gene a ed geog aphic con en (UGGC) has eme ged as one o he main da a sou ces o esea che s in ecen yea s, o en e e ed o as Volun ee ed Geog aphic In o - ma ion (VGI) [ 1 ]. The isualiza ion and in e p e a ion o VGI da a a e challenging because o i s eno mous olume and he e ogenei y, and when compa ed o adi ional spa ial sam- pling echniques, VGI poin da a samples such as poin s o in e es o loca ions o social media pos s o en ha e a spa ial bias [ 2 ] (see Figu e 1, whe e he numbe o Flick pos s does no e lec he quali y o he iew owa ds he landma k bu mainly he popula i y o he place). I VGI poin da a a e p esen ed on a map, hese da a cha ac e is ics could educe he eadabili y due o o e lapping poin symbols, which could possibly hide speci ic spa ial pa e ns in he da a—such as ex eme alues, clus e s o ho spo s. Acco dingly, a educ ion in he o e all numbe o poin s is needed o imp o e he eadabili y o he map, while a he same ime, he spa ial pa e ns wi hin he da a ha e o be p ese ed. The ca og aphic solu ion o he p oblem o o e lapping poin symbols—i.e., he dis- play o clu e [ 3 , 4 ]—is poin gene aliza ion, using ope a ions such as selec ion, agg ega ion, simpli ica ion, o displacemen . Howe e , i hese gene aliza ions a e applied incau iously, speci ic cha ac e is ics o he da a, such as ex eme alues, may disappea , misleading use s o alse in e p e a ions o he unde lying spa ial phenomena (see Figu e 2). The e o e, i is key o p ese e spa ial pa e ns du ing he gene aliza ion p ocess. Mo e gene ally, p ese ing spa ial pa e ns con ibu es o he p inciple ask o map gene aliza ion, which is o p o ide he bes ep esen a ion o he map con en wi hou neglec ing eadabili y. In ecen decades, di e en app oaches e ol ed o mimic he wo k o human ca og a- phe s wi h he aim o au oma ing he map gene aliza ion p ocess. The ule-based app oaches we e buil on s epwise, local ans o ma ions o map objec s, ollowing unambiguous p e- de ined ules [ 5 ]. While ule-based gene aliza ion was—and s ill is—a e y p omising ool ISPRS In . J. Geo-In . 2024,13, 431. h ps://doi.o g/10.3390/ijgi13120431 h ps://www.mdpi.com/jou nal/ijgi ISPRS In . J. Geo-In . 2024,13, 431 2 o 13 o a a ie y o applica ions, a majo sho coming in he pas was he p oblem ha hese au oma ed sys ems do no ha e he abili y o espond o da a a ia ion o speci ic, ask- ela ed equi emen s in a way ha human ca og aphe s could do by modi ying hese ules. This led o he in oduc ion o he cons ain -based app oach, whe e cons ain s ope a e as equi emen s ha shall be ul illed in he inal—i.e., gene alized—map bu wi hou any p ede ined ac ions bound o hem. Map gene aliza ion using a cons ain -based app oach is, he e o e, an op imiza ion p oblem, whe e he ask is o ind a map s a e ha bes ul ils all p ede ined—and some imes con adic ing—cons ain s, while a se o gene aliza ion ope a ions is used o each his op imal map s a e by manipula ing he map objec s. Figu e 1. Loca ion o Flick pos s agged wi h a ia ions in he names o ei he C is o Reden o (“Tag A”) o he Suga loa Moun ain (“Tag B”) in Rio de Janei o. The spa ial dis ibu ion o poin s in his da ase mainly o igina es om he popula i y o he places and no only in he quali y o hei line o sigh owa d he wo landma ks. The e o e, i is no possible o de i e he bes pho o spo s jus om he numbe o pos s a he espec i e loca ions. Figu e 2. Example o poin gene aliza ion. (a) O iginal da a o 50 poin s wi h wo main clus e s and eigh ex eme alues (in ed). (b) Gene alized da a o 25 poin s using poin selec ion based on alue. All eigh ex eme alues a e p ese ed du ing gene aliza ion, while he wo-poin clus e s disappea ed in he esul ing map. (c) Gene alized da a o 25 poin s using poin simpli ica ion based on loca ion. Bo h clus e s a e p ese ed, while only ou o he eigh ex eme alues a e p ese ed. I bo h he spa ial dis ibu ion o e en s and he occu ence o ex eme alues a e o in e es , bo h gene aliza ion esul s can mislead use s o alse in e p e a ions o he unde lying spa ial phenomena. ISPRS In . J. Geo-In . 2024,13, 431 3 o 13 Fo he op imiza ion p ocess, Ha ie and Weibel [6] a gued ha agen -based modelling (ABM) is he mos powe ul modelling me hod in e ms o applicabili y. In his case, he agen s a e au onomous map objec s a emp ing o minimize a gi en cos unc ion, which consis s o cons ain measu es. Duchêne e al. [7] desc ibed his echnique in de ail. In his pape , we wan o ollow his esea ch, implemen ing an agen -based model using p ede ined cons ain s, which a e deduced by Knu a and Schiewe [8] based on a s udy analyzing use beha io when sol ing in e p e a ion asks on poin da a [9]. The emainde o his pape is s uc u ed as ollows: In he second chap e , we summa ize app oaches om he li e a u e o poin gene aliza ion (Sec ion 2.1), pa e n p ese a ion (Sec ion 2.2) and agen -based modeling in ca og aphy (Sec ion 2.3). In he hi d sec ion, we desc ibe he cons ain s we used du ing he op imiza ion p ocess (Sec ion 3.1 and in oduce ou model (Sec ion 3.2). We hen conduc expe imen s (Sec ion 4) and discuss he esul s (Sec ion 5) be o e concluding ou wo k (Sec ion 6). 2. Rela ed Wo k 2.1. Poin Gene aliza ion Focusing on he p oblem o poin gene aliza ion, ope a ions such as agg ega ion, simpli ica ion, selec ion, and displacemen a e o majo in e es . When using agg ega ion (i.e., poin clus e ing), poin clus e s a e eplaced wi h agg ega o ma ke s. Di e en clus e ini ializa ion me hods can he eby igge qui e di e en esul s o he same da a (e.g., see [ 10 ]). Fu he mo e, Meie [11] e alua es and compa es ma ke clus e echniques and simila app oaches, including hea maps and iled hea maps. Poin simpli ica ion desc ibes a poin educ ion based on geome ic c i e ia, such as minimum dis ances be ween poin s [ 12 ]. When seman ic c i e ia a e used o educe he o e all numbe o poin s, a poin selec ion akes place, e.g., based on scale [ 13 ]. In con as , poin displacemen eloca es poin s o educe poin clu e , using an i e a i e wo k low o o e lap de ec ion, eloca ion, and e- e alua ion [14]. While hese ope a ions all ep esen he adi ional me hod o poin gene aliza ion based on ca og aphic scene judgemen , he e a e also da a-d i en me hods such as deep lea ning [ 15 ], which can be u ilized o he applica ion o poin gene aliza ion, as pe o med by Xiao e al. [16] . Based on aining da a c ea ed h ough manual labeling, hei model p edic s he p obabili y o each poin in he da ase o be e ained a e gene aliza ion. Depending on he numbe o poin s eques ed o he inal map, he espec i e poin s wi h he highes e aining p edic ion a e hen selec ed. 2.2. P ese ing Spa ial Pa e ns The a o emen ioned ope a ions ocus on he i s ask o map gene aliza ion—inc easing map legibili y—mainly by emo ing a ce ain amoun o poin s. By con as , p ese ing he p esen in o ma ion o he poin da a as much as possible is he second ask o map gene aliza ion—and possibly con adic ing he i s . The eby, he ques ion a ises as o which spa ial poin pa e ns a e o in e es . A s udy conduc ed by Knu a and Schiewe [9] analyzed use beha io when in e p e ing spa ial poin pa e ns and he eby e ealed wo main aspec s. Fi s , he p opo ion o poin s be ween di e en pa e ns, as well as be ween dense and spa se a eas, was c ucial o he ask-sol ing p ocess o he pa icipan s. Second, he p opo ion be- ween di e en classes wi hin an a ea—o hei espec i e absence—was equen ly desc ibed du ing decision-making. As a esul o his s udy, poin pa e n p ese a ion can be desc ibed as a mul i-c i e ia decision, a echnique ha has al eady been p oposed in ca og aphy o speci ic asks wi hin he wo k low o au oma ed map gene aliza ion [ 17 ] and o class in e al selec ion [ 18 ]. Based on he la e wo k on cho ople h maps, Chang and Schiewe [19] we e able o p ese e spa ial pa e ns such as local ex eme alues and ho o cold spo s. As an example o p ese ing a spa ial poin pa e n o isualiza ion pu poses, Qiang e al. [20] used a py amid modeling amewo k and poin densi y me ics in hei wo k. Fo some applica ions, i could also be use ul o isualize poin pa e ns wi h espec o o he geome y ISPRS In . J. Geo-In . 2024,13, 431 4 o 13 objec s, such as s ee ne wo ks [ 21 ]. To he bes o ou knowledge, app oaches o poin pa e n p ese a ion wi h espec o poin densi y and local ex eme alues a e s ill missing. 2.3. Agen -Based Modeling in Map Gene aliza ion In addi ion o he nume ous ad ances in he op imiza ion o indi idual gene aliza ion ope a ions, he e is also conside able wo k ega ding he o ches a ion o hese ope a ions. E en be o e Ha ie and Weibel [6] iden i ied agen -based modeling as he mos powe ul applicable modelling me hod, he e we e mul iple esea ch s udies o au oma ed gene - aliza ion sys ems ha elied on his app oach. Fo a de ailed o e iew, we e e o he wo k o Duchêne e al. [7] , who explain he basic p inciples o his app oach, desc ibe di e en implemen a ions o agen -based models a he F ench Na ional Mapping Agency (IGN), and discuss he ad an ages and d awbacks o mul i-agen sys ems o ca og aphic gene aliza ion. Acco dingly, he implemen a ions in oduced in he ollowing sec ions o his pape a e based on his wo k. These mul i-agen sys ems ely on he de ini ion o a se o cons ain s and hei espec i e measu es, which de ine he le el o sa is ac ion o each cons ain . The eby, Mackaness and Ruas [22] dis inguish be ween h ee le els o measu es: mic o-measu es ocus on indi idual ea u es o map objec s, meso-measu es desc ibe p ope ies o g oups o objec s, and mac o measu es deal wi h cha ac e is ics o he whole map da a. Fu he mo e, he au ho s dis in- guish be ween in e nal measu es ha desc ibe single da ase s, and ex e nal measu es ha desc ibe ela ions be ween di e en da ase s o map s a es, o example be ween he o iginal and he gene alized map. Wi h ega d o con en , Bea d [5] classi ied cons ain s in o six hema ic ca ego ies: posi ion, opology, shape, s uc u e, unc ion, and legibili y. Consequen ly, ou wo k also elies on a subse o hese cons ain s. 3. Me hod 3.1. De ini ion o Cons ain s We de ine h ee di e en ypes o cons ain s and espec i e measu es ha can guide he gene aliza ion p ocess wi hin ou agen -based model. The i s ype is cons ain s ha co espond wi h he speci ic kind o asks we aim o , such as he iden i ica ion o ex eme alues o dense clus e s, and he e o e ensu e ha spa ial pa e ns a e main ained. The second ype o cons ain s suppo s he ask-sol ing p ocess o he use s and esul om he a o emen ioned use s udy [ 9 ]. The las ype o cons ain e lec s he undamen al equi emen s o success ul poin gene aliza ion, such as a educ ion in clu e o he p ese a ion o Ges al Law ules wi hin he map. Table 1lis s hese cons ain s. Fo mo e de ails on he de ini ion o hese measu es, we e e o he wo k o Knu a and Schiewe [8] , in which he au ho s iden i ied a se o cons ain s by ansla ing he ou comes o hei s udy [9] in o measu able alues. Table 1. Rele an cons ain s and measu es based on [ 8 ]. Cons ain s a e de i ed om ei he ask equi emen s ( ype 1), esul s o a use s udy [ 9 ] ( ype 2) o undamen als o poin gene aliza ion ( ype 3). Cons ain (Type) Measu e Re ain p opo ion o poin s be ween a eas (2) spa ial dis ibu ion o poin s [23] P ese e anking o densi ies be ween a eas (2) clus e densi y anking [23] P ese e local ex eme alues (1) local ex eme alue p ese a ion Main ain a leas one poin pe class (2) poin ca ego y p ese a ion P ese e clus e densi y (1) mean dis ance o clus e membe s P ese e spa ial co ec ness (3) dis ance o o igin loca ion Reduce numbe o poin s (3) numbe o poin s ia Radical Law [24] P ese e Ges al law ules o clus e shape (3) con ex hull o alpha shape [25] P ese e Ges al law ules o clus e o ien a ion (3) minimum bounding ec angle ISPRS In . J. Geo-In . 2024,13, 431 5 o 13 3.2. A chi ec u e o he Agen -Based Model 3.2.1. Gene al A chi ec u e The agen -based model is implemen ed using he open-sou ce amewo k Mesa [ 26 ] and i s spa ial ex ension Mesa-Geo [ 27 ]. Mesa is w i en in Py hon and o e s he basic ABM unc ionali ies by p o iding ou co e componen s (Model, Agen , Schedule and Space) alongside wo addi ional componen s o analysis and isualiza ion. The Model class is he main class o he amewo k and con ols he majo componen s o he sys em. In his class, he ini ial s a e o he model is de ined, as well as he ac ions ha happen while he model is unning. The Model class also c ea es he agen s ha implemen he Agen class and he Schedule class, which con ols he ime and he ac i a ions du ing un ime. Mesa o e s ou di e en schedule ac i a ions, namely he BaseSchedule , which ac i a es agen s one a a ime in he s a ing o de , RandomAc i a ion, which ac i a es he agen s in andom o de , Simul aneousAc i a ion, which ac i a es all agen s a he same ime, and S agedAc i a ion, whe e he ac ion wi hin one model s ep is di ided in o se e al s ages, and all agen s execu e one s age be o e mo ing o he nex s age. Fo models ha equi e he concep o space, he espec i e Space class in Mesa has i e gene al de ini ions o space: Con inuousSpace whe e agen s ha e (x,y) posi ions, Ne wo kG id, which implemen s g aphs wi h nodes and edges, and h ee ypes o g ids (SingleG id,Mul iG id and HexaG id). Howe e , Mesa does no di ec ly suppo he in eg a ion o geog aphical da a in o he model, and so he spa ial ex ension Mesa-Geo was de eloped by Wang e al. [27] , allowing use s o impo , manipula e, isualize and expo geog aphical da a. The e o e, he new class GeoSpace was added, which can consis o mul iple laye s o ec o and as e da a. Fu he mo e, Mesa- Geo dis inguishes be ween Agen Laye s, which con ain GeoAgen s ha ca y ou ac i i ies du ing he simula ion, and Vec o Laye s, which emain s a ic (e.g., oad ne wo ks). Figu e 3shows he a chi ec u e o ou ABM applica ion o poin gene aliza ion, which consis s o h ee modules. The Co e Module con ains he main Model class, a class o di e en ypes o MapAgen s, he Schedule class and se e al ins ances o he GeoSpace class. The Da aCollec o class o he U ili y Module collec s and p o ides he in o ma ion du ing un ime o he Visualiza ion class o he Use In e ace Module, which also con ains wo classes o he speci ica ion o he map and he pa ame iza ion o he cons ain s and measu es o he model. In he ollowing, he modules and hei in e connec ions a e explained in de ail. Figu e 3. A chi ec u e o he agen -based poin gene aliza ion model. ISPRS In . J. Geo-In . 2024,13, 431 6 o 13 3.2.2. Use In e ace Module The i s s ep o un he model is o de ine global map speci ica ions, such as he scale o he o iginal and he a ge map, he bo de s o he map ame, and whe he he o iginal da a se is al eady ul illing legibili y cons ain s. I his is he case, he a ge numbe o poin s a e gene aliza ion can be calcula ed, e.g., using he Radical Law [ 24 ]. Fu he mo e, he desi ed beha io o he map agen s ega ding hei ul illmen o he selec ed cons ain s has o be p ede ined. The common wo k low in agen -based map gene aliza ion o ansla e a lis o cons ain s in o a sa is ac ion alue ep esen ing he s a us o a map agen consis s o wo s eps [ 28 ]: Fi s , he espec i e measu es o he cons ain s a e ansla ed in o a Like -like sa is ac ion scale acco ding o p ede ined h eshold alues, anging om 1 (“unaccep able”) o 8 (“pe ec ”). Second, global sa is ac ion is calcula ed based on he indi idual sa is ac ion alues, o example by calcula ing he mean alue o by u ilizing p inciples om Social Wel a e O de ings (SWO) [28]. Figu e 4lis s he di e en ypes o pa ame e s ha ha e o be de ined be o e un- ning he agen -based model. In addi ion o he a o emen ioned global map speci ica ions, he basic model pa ame e s, and a lis o cons ain s, as p esen ed in Table 1, he de ini- ion o , easonable h eshold alues a e essen ial o he success o an agen -based map gene aliza ion model, bu i is also a complex ask. To help wi h his p ocess o model pa ame iza ion, we adap ed an app oach o Taillandie and Ga u i [29] using a human– machine dialogue. The e o e, we o e a guided use in e ace o adjus he h esholds o each measu e sa is ac ion unc ion and isualize he impac o selec ed h esholds ia samples on a map o ob ain a be e unde s anding o he impac s o he h esholds on he cons ain sa is ac ion and on he expec ed esul s o he gene aliza ion p ocess in gene al. Fo example, dec easing he h eshold alues o he minimum dis ance be ween agen s ensu es ha he agen s a e mo e likely o decide o pe o m gene aliza ion ope a ions ha imp o e hei sa is ac ion ega ding his measu e, such as inc easing he dis ance o hei neighbo s using displacemen , o dele ing hemsel es as a esul o a selec ion ope a ion. As a esul , he dis ance be ween indi idual agen s will likely inc ease on he inal map, while he numbe o poin s will dec ease. Figu e 4. Lis o pa ame e s and examples o manual pa ame e de ini ions using a human–machine dialogue as sugges ed by Taillandie and Ga u i [29] . As an example, adjus ing he o e lay ac- cep ance h esholds o he espec i e measu e-sa is ac ion unc ion by allowing a smalle dis ance be ween Poin Agen s will p obably esul in se e al o e lapping poin s s ill exis ing in he inal map. 3.2.3. Co e Module A simpli ied model low is shown in Figu e 5. When ini ializing he model, a Ma- pAgen is c ea ed o e e y poin wi hin he da a se , which should be gene alized. Fu - he mo e, MapAgen s a e c ea ed o e e y clus e wi hin he o iginal poin da a se . MapAgen s a e he cen al pa o ou model and hei decision-making p ocess ollows he wo k o Duchêne e al. [7] , which decomposes he “b ain” o agen s in map gene aliza ion sys ems in o h ee main componen s: capaci ies, men al ep esen a ion and p ocedu al knowledge. The capaci ies o MapAgen s include he abili y o pe cei e hei su ounding space—i.e., poin s in hei neighbo hood— o e alua e hei own s a e, o communica e wi h o he agen s, and o pe o m gene aliza ion ope a ions on hemsel es. Spa ial ope a ions o sel -e alua ion and sel -gene aliza ion a e he eby p o ided and con olled by he ISPRS In . J. Geo-In . 2024,13, 431 7 o 13 GeoSpace class. The pa o he b ain ha ep esen s he men al s a e o he agen s compa es i s cu en s a us wi h he goals he agen s a e aiming o —i.e., he men al ep esen a ion o MapAgen s desc ibes hei ul illmen owa ds he p ede ined map cons ain s, using he measu e sa is ac ion unc ions e ie ed as desc ibed abo e. Fu he mo e, his pa o he MapAgen s’ b ain memo izes all p e ious decisions and hei espec i e ou comes. The ac ual decision-making is unde aken in he p ocedu al knowledge pa o he MapA- gen s. Using he in o ma ion abou i s cu en s a e o cons ain sa is ac ion and i s o me decisions and hei ou comes, each MapAgen decides which gene aliza ion ope a ion i wan s o pe o m nex . Figu e 5. Simpli ied model low cha wi hou de ails o he decomposed “b ain” o he MapAgen s. A e ini ializing he MapAgen s o poin s and clus e s, each model s ep begins by calcula ing he measu es o analyze he ac ual s a e o cons ain ul illmen . Using he measu e-sa is ac ion unc ions o each cons ain , all MapAgen s ecei e a lis o sa is ac ion alues anging om 1 (wo s ) o 8 (bes ) ep esen ing hei own s a e. Based on his, MapAgen s can calcula e hei own o e all sa is ac ion, and a gene al model s a e can be de e mined. I he o e all model s a e does no each an equilib ium model s a e—o ma ch he model e mina ion condi ions—MapAgen s can pe o m gene aliza ion ope a ions on hemsel es, and he nex s ep is ini ialized. 4. Expe imen s We wan o es ou model by gene alizing a es da a se om a scale o 1:15,000 o a scale o 1:30,000. The aim is o suppo he p ese a ion o spa ial pa e ns du ing poin gene aliza ion. In addi ion o speci ic pa e ns such as local ex eme alues, i is also o in e es o main ain poin densi ies wi hin clus e s as well as in spa se a eas. Because app oaches o poin pa e n p ese a ion conside ing bo h poin densi y and local ex eme alues a e s ill missing, we wan o compa e he esul s o ou model wi h di e en me hods ocusing on spa ial dis ibu ion and ex eme alue p ese a ion. Fo he gene aliza ion o spa ial dis ibu ion cha ac e is ics, we u ilized a quad ee s uc u e. We ill his quad ee s uc u e (capaci y = 1) sequen ially wi h a poin da a se and hen dele e all poin s ha a e assigned o lea es wi h an a ea below he poin signa u e size h eshold o he a ge scale. In o he wo ds, we only e ain poin s ha a e assigned o lea es o he quad ee o a p ede ined h eshold le el o highe . Fo he p ese a ion o local ex eme alues, we used he disc e e isola ion algo i hm o G öbe and Bu gha d [ 13 ], which calcula es a poin ’s dis ance o he closes poin wi h a highe alue, which in he nex s ep can be used as a selec ion pa ame e o poin gene aliza ion. 4.1. Da a We u ilized a da a se ha shows he loca ions o social media images om he pla o m Flick ha con ain a leas one bicycle [ 30 ]. Ou es da a se con ains 800 o hese da a poin s, which a e all loca ed in he a ea o D esden, Ge many. We chose his da ase because i ul ils wo equi emen s we wan o es ou model wi h: Fi s , he poin dis ibu ion is clea ly biased in a way ha is cha ac e is ic o VGI da a, as he majo i y o poin s a e loca ed in a ela i ely small a ea a ound he ci y cen e , while he e a e only ew ou lie s sp ead o e he subu bs. Second, he da a se p o ides he numbe o bicycles pe social media ISPRS In . J. Geo-In . 2024,13, 431 8 o 13 image, which can be used o de e mine local ex eme alues, i.e., pho os ha show a high numbe o bicycles. 4.2. Pe o mance Me ics To e alua e he pe o mance o ou model in compa ison o he quad ee gene al- iza ion and he disc e e isola ion algo i hm, we de ine me ics o e alua e poin densi y p ese a ion, as well as local ex eme alue p ese a ion. Fo poin densi y p ese a ion, we i s de ine clus e s wi hin he o iginal da ase using he DBScan algo i hm. Based on hese eigh clus e s, we measu e he poin densi y be o e and a e gene aliza ion o each clus e , and calcula e he mean p ese ed densi y (MPD) and i s s anda d de ia ion (SDPD) o e all clus e s. Fu he mo e, we wan o ake a close look a wo speci ic clus e s ha a e cha ac e is ic o VGI da a. The i s clus e has 44 poin s and is loca ed a ound he Thea e pla z in D esden, whe e se e al his o ic si es such as Sempe ope ,Ho ki che,Zwinge and G ünes Gewölbe a e in p oximi y, bu wi hou a speci ied cen e . The second clus e has 68 poin s and is loca ed a he F auenki che, whe e a high amoun o pho os is loca ed in a small a ea o in e es . We expec ha ou p oposed model and he quad ee gene aliza ion pe o m be e han he disc e e isola ion algo i hm wi h hese clus e s, as he objec i e o he la e is poin selec ion based on high alues, bu we also wan o analyze i hese clus e s a e gene alized di e en ly. Fo he p ese a ion o local ex eme alues, we iden i y all ex eme alues and coun he numbe o p ese ed poin s in he gene alized da ase s. Fo his es , we expec ha he disc e e isola ion algo i hm and ou model pe o m be e han he quad ee gene aliza ion, as he la e akes only he poin loca ion in o accoun du ing gene aliza ion, bu no i s a ibu es. 4.3. Resul s Figu e 6shows he gene aliza ion esul s o he h ee me hods. I can be seen ha he esul o he disc e e isola ion algo i hm is mo e sp ead ou compa ed o he o he me hods, which p ese e he o e all shape o he cen al clus e s be e . In he quad ee gene aliza ion esul (Figu e 6b), all clu e s a e esol ed, bu due o he applied me hod, some o hese clu e s a e comple ely emo ed. The same can be s a ed o he disc e e isola ion algo i hm (Figu e 6c), which is e en mo e sp ead ou in he o me clus e a eas. Looking a he less dense a eas o he map, bo h he quad ee and he disc e e isola ion me hod p ese ed he majo i y o poin s, which is in ac he in ended beha io o bo h algo i hms. In he esul ing map o ou agen -based model (Figu e 6d), he o iginal shape o he poin dis ibu ion, as well as spo s wi h poin s in close p oximi y wi hin he clus e s, a e p ese ed, bu wi h he downside ha he e a e s ill a ew o e lapping poin s exis ing a e he gene aliza ion. Table 2p esen s he esul s o ou expe imen s using he pe o mance me ics, which con i m he indings s a ed abo e. While he poin densi y wi hin he eigh clus e s is educed o 26% o he quad ee and 35% o ou model compa ed o he o iginal da ase , he disc e e isola ion algo i hm main ains only 12% o he o iginal densi y. In e u n, mos o he ex eme alues (25/27) a e p ese ed wi h his me hod, while he quad ee (20/27) and ou model (23/27) p ese e less ex eme alues. Wi h espec o he numbe o poin s e ained in he wo ocus clus e s, all gene aliza ion me hods e ained mo e o he same numbe o poin s in he clus e a ound he Thea e pla z, al hough i has ewe o al poin s han he one a he F auenki che.