iNav

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Title:
iNav A Spatial Model Supporting Route Planning in Indoor Space.
Physical Description:
1 online resource (140 p.)
Language:
english
Creator:
Yuan,Wenjie
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Computer Engineering, Computer and Information Science and Engineering
Committee Chair:
Schneider, Markus
Committee Members:
Dobra, Alin
Ho, Jeffrey
Wilson, Joseph N
Beck, Howard W

Subjects

Subjects / Keywords:
indoor -- modeling -- navigation -- spatial
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre:
Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract:
With increasing requests for indoor navigation, route planning for indoor spaces is highly desirable especially when the indoor structure is large and complicated. This dissertation is to design a spatial model for supporting route planning. In our context, the primary goal of route planning is to provide a desired route in a specific environment from a certain source point. Our research mainly focuses on the following three open problems, which have not been solved in the related literature. First, distance-driven routing aims at finding the shortest path to a given target object, which is an essential problem in an indoor navigation system. Second, range-driven routing is to generate a set of routes to the qualifying objects that are in a specific walking-distance range from the source point. The source location can be static or dynamic, and thus, the determination of qualifying target objects can be classified to two types, stationary and continuous range queries. Third, when the object moving through the route (e.g., the wheelchairs or moving carts) cannot be approximated by a point, size-driven routing is necessary to offer a feasible path for the object with a particular size (i.e., the length, width and height) to reach a given target object. Concentrating on the above three open problems, our research makes the following specific contributions: First, for discovering the shortest routes, we need to generate a distance-based graph from a cell-based indoor map, which is completely different from a network-based outdoor map. In our solution, a distance-based Direct Path Graph (DPG) is introduced, where nodes denote the exits and edges represent the shortest path segments between the exits. The segment computation is based on the geometric shapes of different cells (e.g., rooms). Using Dijkstra's (like) algorithm, the shortest path to a given target object can be effectively determined. Second, with the assistance of DPG, we also can address the stationary and continuous range queries in two separate solutions. For answering stationary range queries, we can obtain the qualifying objects by expanding the edges starting from the source point within a walking-distance range. However, this approach is not practical for continuous range queries (i.e., the source point keeps moving). After each movement, the set of qualifying objects need to be recalculated. We observe that the distances between the exits and the potentially qualifying target objects are relatively static. Based on this observation, we propose an efficient solution to continuous range queries by only considering the distance changes between the source point and the adjacent exits. Third, for resolving the size-driven routing, the challenge is how to decide the maximum allowable size for each route. A cube-based model is designed to represent the 3-dimensional information of the indoor environment, where the entire space is modeled using the same-size cubes. These cubes can be merged to form larger rectangular blocks. Based on this model, a new distance-based graph (called LEGO Graph) is constructed in which the blocks are edges and their intersection areas are nodes. The maximum accessible size of each node and edge can be determined from their sizes.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Wenjie Yuan.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Schneider, Markus.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-08-31

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UFRGP
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Applicable rights reserved.
Classification:
lcc - LD1780 2011
System ID:
UFE0043218:00001


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INAV:ASPATIALMODELSUPPORTINGROUTEPLANNINGININDOORSPACEByWENJIEYUANADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011WenjieYuan 2

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ACKNOWLEDGMENTS First,Iwouldliketothankmysupervisor,Dr.MarkusSchneider,forprovidingtheguidanceandsupportthroughoutmyresearch.Iappreciatehisknowledgeandskillandhisassistanceinwritingreports.Second,Iwouldliketothanktheothermembersofmycommittee(Dr.HowardBeck,Dr.AlinDobra,Dr.JeffreyHo,andDr.JosephWilson),fortheassistancetheyprovided.Iwouldalsoliketothankmyfamilyforthesupporttheyprovidemethroughmyentirelife. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 1.1Overview .................................... 13 1.2Motivation .................................... 16 1.3ContributionsandApproaches ........................ 16 1.4Outline ...................................... 18 2RELATEDWORK .................................. 19 2.1IndoorNavigationModels ........................... 19 2.1.1SymbolicModels ............................ 19 2.1.2SemanticModels ............................ 20 2.1.3GeometricModels ........................... 22 2.1.43D-basedModels ............................ 25 2.2SpaceSyntaxGraph .............................. 26 2.3ApproachestoSupportingRangeQueriesinOutdoorSpace ....... 28 2.3.1StationaryRangeQueries ....................... 28 2.3.2ContinuousRangeQueries ...................... 29 3SUPPORTINGTHESHORTESTPATHROUTEPLANNINGININDOORSPACE 31 3.1ProblemsinExistingIndoorNavigationModels ............... 31 3.2ThePathConstruction ............................. 34 3.2.1SimpleCells ............................... 34 3.2.2ComplexCells ............................. 35 3.2.2.1Flatcomplexcells ...................... 36 3.2.2.2Nestedcomplexcells .................... 36 3.2.2.3Implicitpathsegmentsinatcomplexcells ........ 40 3.2.2.4Implicitpathsegmentsinnestedcomplexcells ...... 43 3.2.2.5Analgorithmforcomputingallshortestpathsegmentsincomplexcells ....................... 46 3.2.3OpenCells ............................... 51 3.2.4Connecters ............................... 54 3.2.5Accessibility ............................... 54 3.3TheiNavModel:ShortestPathRouting ................... 56 4

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3.3.1TheDirectPathGraph ......................... 56 3.3.2NavigationThroughtheDirectPathGraph .............. 58 3.4TheiNavModel:ASimpleNavigationLanguage .............. 61 3.4.1NavigationDescriptions ........................ 61 3.4.2AutomaticGenerationofNavigationDescriptions .......... 65 4SUPPORTINGRANGEQUERIESININDOORSPACE ............. 69 4.1ApproachestoSupportingRangeQueriesinOutdoorSpace ....... 69 4.1.1StationaryRangeQueries ....................... 69 4.1.1.1Euclidean-basedapproaches ................ 69 4.1.1.2Network-basedapproaches ................. 70 4.1.2ContinuousRangeQueries ...................... 71 4.1.2.1Euclidean-basedApproaches ................ 72 4.1.2.2Network-basedApproaches ................ 72 4.2SupportingStationaryRangeQueries .................... 74 4.2.1IncrementalNetworkExpansion .................... 74 4.2.2IndoorRangeNetworkExpansion ................... 75 4.3SupportingContinuousRangeQueries ................... 80 5SUPPORTING3DROUTEPLANNINGININDOORSPACE ........... 85 5.13DRepresentationoftheIndoorSpace ................... 86 5.1.1LEGO-BasedApproximation ...................... 86 5.1.1.1Theapproximationofplanes ................ 87 5.1.1.2Theapproximationofstairs ................. 89 5.1.1.3Theapproximationofobstacles ............... 89 5.2TheAccessibilityofDifferentTypesofUser ................. 90 5.2.1TheMaximumWidths ......................... 90 5.2.1.1Blockswiththemaximumwidths .............. 90 5.2.1.2Theimprovementofthemerging .............. 93 5.2.1.3Thealgorithm ........................ 94 5.2.1.4Connectorsbetweenblocks ................. 94 5.2.2TheMaximumHeights ......................... 98 5.2.3TheMaximumLengths ......................... 98 5.2.3.1Theapproachforgeneralcases .............. 100 5.2.3.2Theapproachforspecialcases ............... 103 5.3TheLEGOGraph ................................ 104 6THEIMPLEMENTATION .............................. 107 6.1TheEvaluation ................................. 107 6.2TheDemonstration ............................... 111 6.2.1SystemArchitecture .......................... 111 6.2.1.1Thedataprocessor ..................... 112 6.2.1.2Thequeryprocessor ..................... 116 6.2.1.3Theresultprocessor ..................... 117 5

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6.2.2Experiments ............................... 117 6.2.2.1Experimentsofrouteplaningin2Dindoorspaces .... 117 6.2.2.2Experimentsofrangequeryin2Dindoorspaces ..... 119 6.2.2.3Experimentsofrouteplanningforarbitrary-shapedusers 120 7CONCLUSIONS ................................... 123 ATHEXSDFILEOFTHEFORMATTEDDATAFORTHE2DSYSTEM ...... 125 BANEXAMPLEOFTHEFORMATTEDDATAFORTHE3DSYSTEM ...... 129 REFERENCES ....................................... 131 BIOGRAPHICALSKETCH ................................ 140 6

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LISTOFTABLES Table page 3-1Thepathchainfromroom106toroom103 ..................... 67 4-1IntervalrepresentationforaccesspointsinFigure 4-8 B .............. 84 7

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LISTOFFIGURES Figure page 2-1Modelsforrepresentingindoorspaces. ...................... 26 2-2Multiplecongurationsofconvexspacesforonespace. ............. 28 2-3Exampleofaxialmaps. ............................... 28 3-1Indoorrepresentationmodels. ........................... 32 3-2Medialaxis. ...................................... 32 3-3AnexampleofLorenz'smodel. ........................... 33 3-4Differenttypesofcells. ................................ 36 3-5Nestedcells. ..................................... 37 3-6Pathsegmentsindifferenttypesofcells. ...................... 40 3-7examplesofconcaveverticesandconcaveboundaries. ............. 41 3-8Theshortestpathsinaconcavepolygon. ..................... 41 3-9ProofofLemma 1 .................................. 43 3-10Layerstructureofnestedcells. ........................... 43 3-11Proofcasesfortheinnerpolygonboundariesofacomplexcellwithahierarchyofdepth2 ....................................... 45 3-12Exampleofafacewithtwoaccesspointsandtwopossiblepathsbetweenthem. ......................................... 45 3-13Constructingpathsegmentsinopencells. ..................... 53 3-14Mapsandpathsegments. .............................. 55 3-15Anexampleofnavigation. .............................. 57 3-16Routeplanningwherethestartingnodeisanintermediatepoint. ........ 59 3-17Thenavigationdescriptions ............................. 62 3-18Thecalculationofanglesanddirections ...................... 64 3-19Thestatustransformationofthelanguagegenerationanddirections ...... 66 4-1R-tree. ......................................... 70 4-2RangeNetworkExpansion. ............................. 70 8

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4-3Continuousqueryusingsplitpoint ......................... 73 4-4Euclidean-basedcontinuousrangequery ..................... 73 4-5Network-basedcontinuousrangequery ...................... 74 4-6TheIncrementalNetworkExpansion ........................ 75 4-7TheIndoorRangeNetworkExpansion ....................... 80 4-8Findingsplitpoints. .................................. 81 5-1Examplesofcubes. ................................. 87 5-2Thecuberepresentation. .............................. 88 5-3Mergingcubestogeneratelargerblocks. ..................... 91 5-4Relationshipsbetweenadjacentblocks. ...................... 96 5-5TheconnectorswiththemaximumwidthbetweenaplaneandastaircannotbefoundbyusingthebasicmergingAlgorithm .................. 97 5-6Computingconnectorsbetweenaplaneandastair. ............... 98 5-7Examplesshowingdifcultiestondtheoptimalroutes. ............. 99 5-8Demonstrationsofcheckingaccessibilityindifferentscenarios ......... 102 5-9Renementtheapproachforevaluatingthemaximumlengthfor90corners. 104 5-10TheLEGOgraph. .................................. 104 6-1Themapusedtoevaluatingtheefciency. ..................... 108 6-2Evaluationoftheperformanceforrouteplanning. ................. 109 6-3Evaluationoftheperformanceforrouteplanning ................. 109 6-4Evaluationoftheperformanceforstationaryrangequeries ........... 110 6-5EvaluationcomparisonbetweenIRNEandIRD. .................. 111 6-6ThearchitectureoftheiNavsystem ........................ 112 6-7Theuserinterface .................................. 115 6-8Thevisualizedroutefromroom502to515 .................... 118 6-9TheresultsofExample 1 .............................. 118 6-10TheresultsofExample 2 .............................. 119 9

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6-11TheresultsofExample 3 .............................. 120 6-12TheresultsofExample 4 .............................. 121 6-13TheresultsofExample 5 .............................. 121 6-14TheresultsofExample 6 .............................. 122 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyINAV:ASPATIALMODELSUPPORTINGROUTEPLANNINGININDOORSPACEByWenjieYuanAugust2011Chair:MarkusSchneiderMajor:ComputerEngineering Withincreasingrequestsforindoornavigation,routeplanningforindoorspacesishighlydesirableespeciallywhentheindoorstructureislargeandcomplicated.Thisdissertationistodesignaspatialmodelforsupportingrouteplanning.Inourcontext,theprimarygoalofrouteplanningistoprovideadesiredrouteinaspecicenvironmentfromacertainsourcepoint.Ourresearchmainlyfocusesonthefollowingthreeopenproblems,whichhavenotbeensolvedintherelatedliterature.First,distance-drivenroutingaimsatndingtheshortestpathtoagiventargetobject,whichisanessentialprobleminanindoornavigationsystem.Second,range-drivenroutingistogenerateasetofroutestothequalifyingobjectsthatareinaspecicwalking-distancerangefromthesourcepoint.Thesourcelocationcanbestaticordynamic,andthus,thedeterminationofqualifyingtargetobjectscanbeclassiedtotwotypes,stationaryandcontinuousrangequeries.Third,whentheobjectmovingthroughtheroute(e.g.,thewheelchairsormovingcarts)cannotbeapproximatedbyapoint,size-drivenroutingisnecessarytoofferafeasiblepathfortheobjectwithaparticularsize(i.e.,thelength,widthandheight)toreachagiventargetobject. Concentratingontheabovethreeopenproblems,ourresearchmakesthefollowingspeciccontributions:First,fordiscoveringtheshortestroutes,weneedtogenerateadistance-basedgraphfromacell-basedindoormap,whichiscompletelydifferentfromanetwork-basedoutdoormap.Inoursolution,adistance-basedDirectPathGraph 11

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(DPG)isintroduced,wherenodesdenotetheexitsandedgesrepresenttheshortestpathsegmentsbetweentheexits.Thesegmentcomputationisbasedonthegeometricshapesofdifferentcells(e.g.,rooms).UsingDijkstra's(like)algorithm,theshortestpathtoagiventargetobjectcanbeeffectivelydetermined.Second,withtheassistanceofDPG,wealsocanaddressthestationaryandcontinuousrangequeriesintwoseparatesolutions.Foransweringstationaryrangequeries,wecanobtainthequalifyingobjectsbyexpandingtheedgesstartingfromthesourcepointwithinawalking-distancerange.However,thisapproachisnotpracticalforcontinuousrangequeries(i.e.,thesourcepointkeepsmoving).Aftereachmovement,thesetofqualifyingobjectsneedtoberecalculated.Weobservethatthedistancesbetweentheexitsandthepotentiallyqualifyingtargetobjectsarerelativelystatic.Basedonthisobservation,weproposeanefcientsolutiontocontinuousrangequeriesbyonlyconsideringthedistancechangesbetweenthesourcepointandtheadjacentexits.Third,forresolvingthesize-drivenrouting,thechallengeishowtodecidethemaximumallowablesizeforeachroute.Acube-basedmodelisdesignedtorepresentthe3-dimensionalinformationoftheindoorenvironment,wheretheentirespaceismodeledusingthesame-sizecubes.Thesecubescanbemergedtoformlargerrectangularblocks.Basedonthismodel,anewdistance-basedgraph(calledLEGOGraph)isconstructedinwhichtheblocksareedgesandtheirintersectionareasarenodes.Themaximumaccessiblesizeofeachnodeandedgecanbedeterminedfromtheirsizes. 12

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CHAPTER1INTRODUCTION ResearchonnavigationsystemsforanoutdoorenvironmentbringsmatureGPSnavigationsystemsintodailyuse.Thissignicantlyaffectsournormallife.WiththeassistanceofGPSsystems,wecaneasilylocateadesiredplace,obtainadetailedroutefromlocationAtolocationB,andlearninformationabouttheinterestinglocationsnearby.However,indoornavigationisalsohighlydemandedinsomeindoorenvironments.Imaginewhenweareinanunfamiliarhugebuildingwhosearchitecturalstructuresarecomplex(e.g.,anairport,amuseum,ahospital,orashoppingmall),itisoftendifcultforustondourdesiredtargetobjectsefciently,especiallyifthedestinationnameisnotclear.Wemighttakeacircuitousway,gotoawrongoor,andevengiveupafterafewattempts.Especially,forexample,whenyouhavetovisitahospitalinanemergency,andthetimeavailableforndingtherightplacemaybecritical.Thus,navigationsystemsforindoorspacesaredesiredinourdailylife. 1.1Overview Thestudyofhowhumansndtheirwaysaroundlargebuildingshasalongtradition.Recentresearchonhumanspatialcognition[ 30 ]pointsoutthatifpeopledonotknowtheexactroutetheydesired,they'dprefertostickasmuchaspossibletothefamiliarpartsofthebuildingandheadtowardsthegoalasdirectlyaspossible.However,thesestrategiesrevealaprobleminthattheroutespeopletakemaynotbethemostoptimalones.Itispossiblethataroutecontainingsomeshortcutsismuchshorterthantheonethatpeoplecanndbythemselves.Thus,anindoornavigationsystemisrequiredtoprovideusersanabilitytondbetterwaystonavigateinsidebuildings. In[ 9 ],navigationisdenedastheprocessofreading,andcontrollingthemovementofacraftorvehiclefromoneplacetoanother.Itinvolvestwomainaspects:localizationandrouteplanning.Inoutdoorspaces,thelocalizationdependsontheGPSsignal;whilewithindoorspaces,sensorsormobilesignalsareoftenusedtodetermine 13

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theuser'scurrentlocationandtracktheirmovements.Basicrouteplanningreferstothedeterminationofappropriateroutesfromoneplacetoanother.Theappropriateroutescanbetheshortestroutes,theroutesfordisabledpersonsinwheelchairsortheroutesfortheblindpersons.Therouteplanningcanalsobeextendedtoanswersomequeriesinrelationtonavigation.Forexample,rangequeries,whichareusedtodeterminetheinterestingobjectswithinagivendistance,canbeconsideredasatypeofextendedrouteplanning.Thisdissertationistodesignaspatialmodelforsupportingrouteplanning. Routeplanningishighlydependentonthestructureoftheenvironment.Inoutdoorspaces,mostuserswhousesnavigationsystemsareinvehicles,andinmostcasestheirmovementsarerestrictedtotheconstructedroads.Thus,theroadnetworkisthemostimportantstructureinanoutdoorspace.Thetraditionalrouteplanningstrategiesthatareusedforoutdoorspacesarebasedontheroadnetworks.Withindoorspaces,theentireenvironmentiscomposedofvariouscells.Thus,thecellsarethemostimportantstructureofindoorspaces.Althoughtherearesomecomparableconceptsforindoorandoutdoorspaces,suchascorridorsandroads,withindoorspaces,therearealsoconceptslikeroomsandlobbiesforwhichwedonotndcounterpartsinoutdoorspace.Thus,thestrategiesdesignedforindoorrouteplanningareverydifferentfromthoseusedinoutdoorrouteplanning.Inaddition,someotherarchitecturalconstraints,suchaswallsanddoors,alsoplayveryimportantroleswithindoorrouteplanning,andthusmakeindoorrouteplanningmoredifcult. Inrecentyears,researchonindoorrouteplanninghasattractedalotofattentionintheacademiccommunity.Asaresult,moreandmoremodelsforindoornavigationhavebeenproposedrecently.Someofthemusesymbolstorepresentcells.Thegeneratedroutesarerepresentedbyasequenceofobjects.Someofthemtrytondtherepresentativepointsforcellsbyanalyzingtheirgeometricshapes,andsomeofthemfocusonbuilding3Dmodelsfortheindoorspacetocapturethetopologicalrelationships 14

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betweendifferentcells.However,comparedtothematuredGPSnavigationsystemsforoutdoorspaces,modelsforindoorspacesarefarfrompracticalintheaspectofrouteplanningstrategies.Inoutdoorspaces,thesequeriesareoftencomputedonthebasisofaroadnetwork,whichdoesnotexistinindoorspaces.Constructinganetworkthatcansupportrouteplanningforindoorspacesismorechallenging. Theprocessofnetworkconstructioninvolvesseveraldifculties.Firstly,therearealotofarchitecturalconstraintssuchaswallsanddoorswithindoorspaces.Theseconstraintsareimportantformodelingtheindoornavigationsystems.Forexample,someareasinsidebuildingsareenclosedbywalls,whilesomeareadjacenttootherswithoutanybarriers.Secondly,onecellcannotsimplyberepresentedbyoneedge.Ifonecellhasmorethantwoexits,usersmaytakedifferentexitstogothroughthecell.Evenifthereareonlytwoexitsinthecell,thenumberofimplicitpathsusersmaytakeisstillinnite.Thirdly,theshortestrouteplanningandrangequerycalculationdependontheshortestwalkingdistancesbetweenobjects.Thus,thenetworkforindoorspacesshouldbeabletoprovidetheshortestroutesbetweenanytwoobjects.Infact,evenifsuchanetworkwiththeminimumdistancesbetweenanytwoobjectsexists,itisstillnoteasytodesignapproachestosupportcontinuousrangequeries,inwhichthelocationofthesourcepointisdynamic.Inoutdoorspaces,themovementsofvehiclesarerestrictedbythedirectionsandthespeedlimitsoftheroads.Unlikeoutdoorspaces,thereisnoparticularruletorestricttheuser'movementsinindoorspaces.Thatmeanspeoplecanroamanywhereandwalkinanydirection.Predictingauser'smovementisverydifcult.Fourthly,thepathsthroughindoorspacesmightnotbepassedbyalltheusers.Sometimesthepathsthatpedestrianscantakearenotsuitableforpersonsinwheelchairs,astheirsizemightpreventthemfromaccessingnarrowexitsorturningaroundtightcornersincorridors. Thisdissertationcontributestotheresearchoftheindoornavigationsystemswithaspecicfocusonapproachestosupportingindoorrouteplanning.Theintended 15

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audienceisbroadandvaried.Thoseinterestedinexploringtheindoorenvironmentwillndatooltomodelindoorspace.Thoseinterestedinroutendingforindoorspacesmightgetnewinsightsintotheapproachesforconvertingcell-basedstructuresintothenetworksforanindoorenvironment.Thoseinterestedinansweringcommonnavigationalqueries,suchasrangequeries,couldlearnafeasiblemethodtocomputethesequeries. 1.2Motivation Inourcontext,theprimarygoalofrouteplanningistoprovideadesiredrouteinaspecicenvironmentfromacertainsourcepoint.Ourresearchmainlyfocusesonthefollowingthreeopenproblems,whichhavenotbeensolvedintherelatedliterature. First,distance-drivenroutingaimsatndingtheshortestpathtoagiventargetobject,whichisanessentialprobleminanindoornavigationsystem. Second,range-drivenroutingistogenerateasetofroutestothequalifyingobjectsthatareinaspecicwalking-distancerangefromthesourcepoint.Thesourcelocationcanbestaticordynamic,andthus,thedeterminationofqualifyingtargetobjetscanbeclassiedtotwotypes,stationaryandcontinuousrangequeries. Third,whentheobjectmovingthroughtheroute(e.g.,thewheelchairsormovingcarts)cannotbeapproximatedbyapoint,size-drivenroutingisnecessarytoofferafeasiblepathfortheobjectwithaparticularsize(i.e.,thelength,widthandheight)toreachagiventargetobject. 1.3ContributionsandApproaches Theoverallgoalofthepresenteddissertationistoanalyzetheindoorenvironmentanddesignamodeltosupportrouteplanningthroughindoorspaces.Thedesignofthemodelhasthreesub-goals: 1. Supportingshortestrouteplanning. 2. Supportingrangequeries. 3. Providingfeasibleroutesfordifferenttypesofusers. 16

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Inordertoacheivetherstsub-goal,weneedtogenerateadistance-basedgraphfromacell-basedindoormap,whichiscompletelydifferentfromanetwork-basedoutdoormap.Inoursolution,adistance-basedDirectPathGraph(DPG)isintroduced,wherenodesdenotetheexitsandedgesrepresenttheshortestpathsegmentsbetweentheexits.Thesegmentcomputationisbasedonthegeometricshapesofdifferentcells(e.g.,rooms).UsingDijkstra's(like)algorithm,theshortestpathtoagiventargetobjectcanbeeffectivelydetermined.Thereareseveralfactsthataffectthenetworkconstruction.First,usershavetoenterroomsthroughexits.Asaresult,doorsandopenboundariesofroomsmustbetakenintoaccountduringtheconstruction.Second,arouteprovidedtouserscannotgoacrossabarrier(e.g.awall).Thus,theboundaryandtheshapeofcellsmustbeconsideredtoo.Inthisdissertation,wewillexploretherelatedfactsanddemonstratehowtoconstructpathsegmentsbasedonthegeometricshapesofcells. WiththeassistanceofDPG,wealsocanaddressthesecondsub-goalintwoseparatesolutions.Foransweringstationaryrangequeries,wecanobtainthequalifyingobjectsbyexpandingtheedgesstartingfromthesourcepointwithinawalking-distancerange.However,thisapproachisnotpracticalforcontinuousrangequeries(i.e.,thesourcepointkeepsmoving).Aftereachmovement,thesetofqualifyingobjectsneedtoberecalculated.Weobservethatthedistancesbetweentheexitsandthepotentiallyqualifyingtargetobjectsarerelativelystatic.Basedonthisobservation,weproposeanefcientsolutiontocontinuousrangequeriesbyonlyconsideringthedistancechangesbetweenthesourcepointandtheadjacentexits. Third,forsolvingthethirdsub-goal,wehavetodeterminethemaximumallowablesizeforeachroute.Acube-basedmodelisdesignedtorepresentthe3-dimensionalinformationoftheindoorenvironment,wheretheentirespaceismodeledusingthesame-sizecubes.Thesecubescanbemergedtoformlargerrectangularblocks.Basedonthismodel,anewdistance-basedgraph(calledLEGOGraph)isconstructedinwhich 17

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theblocksareedgesandtheirintersectionareasarenodes.Themaximumaccessiblesizeofeachnodeandedgecanbedeterminedfromtheirsizes. 1.4Outline Therestofthedissertationisorganizedasfollows:Chapter 2 discussedtheavailablemodelsforindoornavigationsystemsandtheexistingapproachestosupportingrangequeriesinoutdoorspace.Chapter 3 introducedtheiNavmodelthatcansupporttheshortestrouteplanningforindoorspace.Chapter 4 presentsarange-drivenroutingapproachtoefcientlydetermineallthequalifyingobjectswithinaspecicwalkingdistance.aLEGOmodelisintroducedinChapter 5 .Basedonthismodel,thegenerationofafeasibleroutecanbefullyautomatedfromthesourcepointtothetargetobject,whenthesizeofamovingobjectcannotbeapproximatedtoapoint.Chapter 6 evaluatesourmodelandshowsomedemosofourimplementedsystem.Finally,weconcludewithasummaryoffutureworkinChapter 7 18

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CHAPTER2RELATEDWORK 2.1IndoorNavigationModels Theearlierattemptsofmodelingindoornavigationsystemsfocusondesigningmodelsforrobots.ThemostfamousonesarePolly,whichisamobilerobotactingasaguideforMITATlab[ 32 ],andMinerva,whichisanautonomousguideusedintheNationalMuseumofAmericanHistoryinWashington[ 95 ].Infact,theideasofdesigningnavigationsystemsforrobotsandforhumanaredifferent.Navigationsystemsforrobotsarebasedonlocalviews,whileaglobalviewoftheentireindoorspacesismoreimportantfornavigationsystemsusedbyhumans.Inrecentyears,researchonhuman-orientedindoornavigationsystemsbecomesapopularresearcharea.Consideringtheclassicationoflocation-basedmodelsin[ 6 ],thespatialmodelsforindoorenvironmentcanbemainlyclassiedintothreecategories:symbolic,semanticandgeometricmodels. 2.1.1SymbolicModels Symbolicmodelsdeneobjectsintermsofuniquesymbols.Mostmodelsfocusontherepresentationofanindoorenvironmentandtheexplorationofthetopologicalrelationsbetweenitsobjects.Infact,topologicalrelations,suchasmeet,inside,andcontains,areveryusefulintheprocessofrouteplanning.Byndingadjacentobjectsonthebasisofthetopologicalrelationsfromthestartingobjecttothetargetobject,weareabletoprovideuserswitharoughroutecontainingasequenceofobjects.However,withoutgeometricinformationlikethedistancebetweentwoobjects,topologicalrelationscannotbedeterminedautomatically.Thus,symbolicmodelshavetomanuallyexploreandmaintainthetopologicalrelations.BrumittandShafer[ 10 ]denesymbolsbycreatingasemanticspace,inwhicheachobjecthasitsuniqueidentieranditsrelatedinformation,suchasthetopologicalrelationswithitsneighbors.Sincethetopologicalrelationsofoneobjectarestoredseparatelyandthereisnostructureto 19

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managealltherelations,thedeterminationoftherelationsamongmultipleobjectsisnotsostraightforward.Abetterwaytomaintainandusethesetopologicalrelationsistoorganizesymbolsbyusinghierarchicalstructures,suchastrees([ 79 ])andlattices,whichcanreectthetopologicalrelationsamongalltheobjects.Targetobjectscanbefoundbyreferringtotheirsymbolsthroughthehierarchicalstructure,andtheroutetoitcanbeobtainedbycomputingtheadjacentobjectsfromthestartingobject. Theauthorsin[ 31 70 71 ]aimtodevelopasetofformaldenitionsforthesymbolsrepresentingobjectssuchaswalls,doors,andanglesinanindoorenvironment.Althoughtheirideashaveoriginallybeenusedtoinstructrobotssaction,theseformalizedsymbolsalsocanbeusedforhumanwaynding.Byusingthesesymbols,possiblepathsandchangesbetweenpathscanberecognizedtoachievegoodnavigationdescriptions. Theaforementionedmodelsaregoodforrepresentingindoorspace.However,itneedsalotofworktoderiveasequenceofpathstoperformnavigationbyusingthesemodels.RaubalandWorboys[ 74 75 ]processnavigationqueriesinanindoorenvironmentbyperformingspatialreasoningonaknowledgebase.Intheknowledgebase,knowledgeobservedfromtheenvironmentisdescribedintermsofpropositionsthatareeithertrueorfalse.Thereasoningisconductedbycombiningthesepropositionswithsomelogicoperatorstoderivenewknowledge. Theprimaryadvantageofsymbolicmodelsisthatthesemanticmeaningofeachobjectisunderstandableforusersandthestructureoftherelationshipsamongobjectscanbewelldesigned.However,lackinggeometricinformationpreventsthemtobeappliedtotheapplicationsthatrequirepreciselocations.Inaddition,itrequiresalargeamountofmanualworktoconstructandmaintainthestructureofthesymbols. 2.1.2SemanticModels In[ 44 45 ],KuipersandLevitthaveobservedthecomplexityofhandlinggeographicdatainreal-worldnavigation.Someapplicationssuchaslocation-basedpervasive 20

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applicationsandintelligenceapplicationsrequirerichsemanticinformationsuchasnames,purposes,neighbors,andrelatedfeaturesforrepresentingindoorenvironments.Objectsintheworldallhavegeographicpropertieswithrespecttoshapes,constraints,locationsandneighborhood.However,symbolsusedinsymbolicmodelsmayhavelittleornosemanticmeaning.Thus,researchersareinterestedindevelopingsemanticmodelsforindoorenvironments. In[ 72 ],Pradhanproposedarelativesimplewaytorepresentobjectssemantically.IntheirCoolTownproject,differentplacesarelabeledwithURLs.ThensemanticinformationofdifferentplacesisembeddedintotheirURLs. AverycommonwaytodesignsemanticmodelsistouseOntology.In[ 21 ],Egenhoferdenestheontologyofavehiclenavigationsystemasthespatialobjectsofroads,intersections,landmarks,andplacesthatuserswanttogo.Themodelsproposedin[ 2 11 16 19 58 61 73 98 ]allapplyontologytomanagethesemanticinformationofthenavigationspace.CoronaandWinterdevelopanontologyin[ 16 ]bycollectingconceptsfromdifferentauthorsofwayndingliterature.In[ 2 ],Anagnostopoulos,etc.proposeanontology-basedmodel,calledOntoNav,bycombininggeometricandsemanticinformationtogether.ThereasonwhyweclassifyitintothesemanticcategoryisbecauseoneofitsimportantcontributionsisthattheydevelopanIndoorNaviga-tionOntology(INO),whichcansupportboththerepresentationandrouteplanningofanavigationsystem.BasedontheINO,thismodelisabletoobtainasuitablepathaccordingtousers'requirementsandproles.Theapproachproposedin[ 73 ],simulatepeople'swayndingbehaviorbyusinganagentgroundedintheontology.Thewayndingstrategyusedintheagentisbasedonagraphrepresentingtheenvironment.Inthegraph,nodesrepresentdecisionpointsandedgesdenotelinesofmovements.In[ 58 ],theauthordetailedanalyzesdifferentarchitecturalobjectsinindoorspace,andorganizestherelationshipsbetweentheseobjectsbyusingontology. 21

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Otherthanontology,Stoffel,ect.proposedanovelstrategytobuildahierarchicalgraphin[ 89 ]toorganizetheinformationofdifferentobjectsinindoorspace.Hu,Lee,etc.proposeamodelcontaininglocationsandexits,whicharedecoratedbysemanticinformation([ 33 ]).Theymaintainthesemanticinformationbyorganizethelocationsandexitsinahierarchicalstructureaccordingtotheirreachabilities.Anadvantageofthismodelisitsabilityofgeneratingsemanticallydecorateddescriptionsforpaths.Later,LiandLeeextendHu'smodeltofurtherexplorethetopologicalrelationsbetweendifferentobjectsin[ 49 50 ].In[ 50 ],Liobservesthatinareachabilitygraph,inwhichnodesareexitsandedgesrepresenttheirreachabilityinformation,somepolygonsboundedbymultipleedgesrepresentrealareasinindoorspace,somepolygonsdonot.Accordingtothisobservation,shedevelopsanalgorithmtodetecttherealareasinthereachabilitygraph.In[ 49 ],LideducesalatticefromHu'smodelbasedonthetheoryofformalconceptanalysis.Thegeneratedlatticecanrevealthebasicrelationshipsbetweentwoentities,suchascontainmentandoverlap.Then,theoptimaldistancerepresentedbythenumberofexitsbetweentwoentitiescanbecalculatedaccordingtothenearestneighborrelationshiponthelattice. Comparedtothesymbolicmodels,semanticmodelscanhandlemoreinformationforeachobjectinanindoorenvironment.Thesemanticinformationisusuallymanagedbyformalstructuresandmaintainedbystandardrules.Therefore,itiseasiertounderstandandprocesstheinformationbycomputers.However,semanticinformationaloneisstillnotenoughforapracticalnavigationsystem.Inaddition,theabilitytocalculatetheshortestpathisanimportantcriterion.Withoutgeometricinformation,modelscanneitherrecordtheareasofdifferentobjectsnorcalculateshortestpaths. 2.1.3GeometricModels Geometricmodelsusetraditionalspatialdatatypes[]forpoints,lines,andregionstorepresentgeometricobjectsintheEuclideanspace.informationenablesonetorepresentobjectsbyaccuratepositionssothatlocationinformationcanberetrievedina 22

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moreexibleway.However,itisnotenoughforapracticalnavigationsystemtoonlyusegeometricinformationbecauseusersareusedtorepresentingobjectsbytheirnamesortheirsemanticinformation.Thus,mostgeometricmodelsarehybridmodelscombiningsymbolicorsemanticinformationtogetherwithgeometricinformation. JiangandSteenkiste[ 38 ]representandmanipulatedifferentobjectsbydesigningtheso-calledAuralocationidentieraswellassomecorrespondingoperators.AnAuraidentieriscomposedofbothsymbolicandgeometricinformationpartsdescribinganobject.BypassinganobjectsAuralocationidentierasaparametertodifferentfunctions,themodelisabletoobtainthecorrespondinggeometricdistancesandtopologicalrelations.TheadvantageofthismodelisthatitnicelyrepresentseachobjectbyusingtheAuralocationidentier.However,sincethedistanceitreturnsistheEuclideandistancebetweentwoobjectswithoutconsideringtheobstaclesbetweenthem,itisnotsuitablefornavigation. Atime-dependentoptimalroutingmodelisproposedin[ 66 ]foremergencyevacuation.Thepathnetworkisbuiltonthebasisofthelocationofsensors,andtheoptimalroutesaredeterminedafterconsideringenvironmentalinformationonthepositionsofevacuees.Althoughthismodelcanprovideatime-dependentoptimalrouteforaquickevacuation,therouteitprovidesishighlydependentonthelocationofsensorsandnotonthearchitecturalstructureitself.Thus,apoorlysettledsensornetworkmayleadtoimproperresults. UsinggridrepresentationisacommonwayusedintheRobertresearcharea.Therearetwogeneraltypesofmethodsforthegrid-basedmodels.Oneistrytodecomposetheavailablespaceintodifferentshapesofcells,suchastrianglesandpolygons.Theunionofthegeneratedcellsisexactlytheavailablespace[ 46 100 ].themodelpresentedin[ 46 ]decomposethespacebyusingdifferentshapedcells.Itrstsubdividethespaceintoseveraltrianglesbydetectingthebottlenecksinsiderooms.Thenthesetrianglesaremergedintoseveralconvexcellsindifferentsizes. 23

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However,thisapproachislimitedto2Datplanes.Thesekindofmodelscanpreciselyrepresentthespace.However,theyshareacommonproblemofinefciency.Theotherone[ 3 4 82 83 96 ]istorepresenttheavailablespacebyuniedshapes(e.g.,rectangles).Theunionofthegeneratedcellsmaynotbeexactlytheavailablespace,especiallyforthespaceontheboundaries.However,sincetheyusesimpleanduniedrepresentativeunits,theyareusuallymoreefcientfortherouteplanning.Themodelproposedin[ 4 ]isoneofthemostpopulargrid-basedmodels.Inthismodel,theavailablespaceisdecomposedintocellsmarkedasobstacleornon-obstacle.Basedonthisrepresentation,routescanbecomputedbycheckingtheavailabilityofcells'movementstotheir8neighbors.Thismodelalsosupportnavigationin3Dspacesbyllingouttheindoorspaceswiththeobstacleandnon-obstaclecubes(asshowninFigure 2-1 ).Theobstaclecubesarefurtherclassiedintoinsurmountableandsurmountableonestofacilitatethe3Dnavigation.In[ 3 ],ahierarchicalmodelisproposedbymergingcellstoformtopologicalmaps.Thismodelcanmoreefcientlycomputetherouteswhenthenumberoftherepresentativecellsislarge. Somemodelsrepresentcellsbyusingtherepresentativepoints.Mostofthemusethecenterpointofacellasitsrepresentativepoint[ 23 54 55 ].Someofthemtrytodeterminetherepresentativeaccordingtotheshapeofthecell[ 52 53 ].Someparticularcells(e.g.corridors),arerepresentedbymultiplepoints.modelsin[ 34 76 ],aconnectivitygraph,inwhichnodesdenotecellsandedgesrepresentexits,isbuiltforthenavigationpurpose.Someofthemodelsonlyconsiderthelocationsandtheconnectionsbetweenexits[ 33 42 49 50 ].Themodelsproposedin[ 12 47 48 ]considersboththeshapesandtheexitsofthecells.ThesemodelsalltrytondthemedialaxisofacertaincellbyconstructingVoronoidiagram.Theconstructedpathsarealwaysonthelinewhosedistancestoitstwosidesarealwaysequal. 24

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2.1.43D-basedModels Theabovementionedmodelsassumethattheoorofthecellsare2Dplanes.Actually,thisassumptionisinsufcienttorepresenttheentireindoorspace.Placeswiththesamexandycoordinatevaluemaylocateindifferentoors.Eveninthesameoor,cellsmayhavedifferentheights.Inaddition,itiscommontohavemultiplelayersconnectedbysmallstairsandslopesinoneoor.Thus,somemodelstrytopropose3Dmodelsforindoorrepresentationsandnavigation. In[ 47 48 ],Leeproposea3Dmodeltorepresentthetopologicalrelationsinindoorspace.Inthismodel,Poincaredualitycombinedwithahierarchicalnetworkstructureareusedtoexploretherelationsbetweenobjects.Althoughthisisa3Dmodel,the3Dfeaturesareonlyusedtodistinguishdifferentoors.Theoorsareconsideredtobeatandtheconstructionofthepathsinoneoorstillfocuson2Ddata.Thesameproblemhappensinthemodelsin[ 51 93 ].In[ 56 ],theauthorspresentasemanticmodelofinteriorspacestofacilitatethecalculationoftheevacuationroutes.Thismodeltakesintoaccountdifferentfeaturesoftheinteriors,suchasthetypesofthepassing(e.g.uni-orbi-directional)andthetypesoftheboundaries(e.g.persistentboundarieslikewallsandvirtualboundarieslikeopenings).Byusingthesefeatures,thismodelisabletodistinguishtheaccessiblepartsandnon-accessiblepartsintheindoorspace.However,thismodefocusesonthesurroundings,butignoresthestructureoftheoorplane.Themodelproposedin[ 85 86 ]istheonlymodelwehavefoundtalkingaboutthestructureoftheoorplane.AsshowninFigure 2-1 C,thismodeldenesdifferenttypesofheighttostorethelayeredstructureoftheoor.Thus,thesmallstairsandslopescanbecapturedduringtherouteplanning.However,whentheauthorextendthismodeltosupportnavigation,the3Dstructureoftheoorisonlyusedtorepresentdiscreteandconnectedspaces,nottoconstructdifferentkindsofpaths. 25

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ABC Figure2-1. Modelsforrepresentingindoorspaces.A)cuberepresentation.B)modelsforwheelchairs.C)3Drepresentation. 2.2SpaceSyntaxGraph Inthelate1970sandearly1980s,BillHillier,JulienneHanson[ 28 ]developedasetoftheories,calledspacesyntax,whichisusedtoanalyzethestructureofindoorspacesorsmallpartofurbanspace.Later,Daltonin[ 17 ]andHillierin[ 26 27 ]furtherextendedthistheory.Thegeneralideaofthespacesyntaxistodecomposethecontinuousspaceintoaconnectedsetofdiscretesubspaces.Eachsubspacecanbemeasuredintermsofitsnearbyspaceandaccessibility. Accordingtothegeneralidea,threebasicconcepts,isovists,axialspaceandconvexspace,aredevelopedtodescribetheconnectivityandintegrationofdifferentcomponents.Convexspaceisdenedasaleastnumberofconvexsubspacesthatcancovertheentirespace[ 28 ].ProposedbyBenedikt[ 8 ],isovistsanalysisisusedtoexplorethevolumeofvisiblespacefromagivenpoint.Anisovististheareathatdirectlyvisiblefromagivenpoint.Thisideaprovidesausefulwaytoexplorespatialenvironmentandattractedalotofresearchersworkingonit.Traditionalisovistgeneratethevisiblespacein360fromthegivenpoint.In[ 59 ],Meilingeretal.proposedaconceptofhalfisovist,whichspinsasmallerdegree,ratherthan360tosimulatetheviewofhumans.Axialspace(alsocalledaxialmap)isthedominantapproachofthespacesyntax.Itsideaisbasedontheaxiallines,whicharedenedasthelongestvisibilitylinesbetweentwospaces.Axialspaceiscomposedoftheleastnumberofaxiallinesthataremutually 26

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intersectandcoverthewholespace.Figure 2-3 BisthecorrespondingaxialspaceofFigure 2-3 A. Althoughspacesyntaxprovideatheoreticalwaytoanalyzethespatialcongurations,itisnoteasiertoapplythistheoryintherealworld.First,asdiscussedin[ 67 ],thedecompositionofsubconvexspacesarenotxed.Foragivenspace,theremightbemultiplewaystodecomposethespacewiththesamenumberofsubspaces(asshowninFigure 2-2 ).Therefore,howtoconstructaconvexspaceasdenedin[ 28 ]isnon-determined.Second,fromthecognitiveperspective,onegoalofspacerepresentationapproachesistoretainasmuchinformationaspossible.However,usingaxiallinestorepresentthespacewilldenitelylosesomeimportantspacecongurationinformation. Despitetheproblemsthatwementionedabove,thetheoreticalbasesofspacesyntaxstillholdspromiseasanimportantanalyticaltoolinmanyelds,especiallyintheareaofarchitecturaldesignsandspatiallearning.Inrecentdecades,itshowsitsusageinmanyways,includingroutechoosing,spatialknowledgeacquisition,orientationanddisorientation[ 60 ].Jiangetal.discussedthepossibilityofusingspacesyntaxwithinGISinmethodologicalperspective[ 35 37 ].[ 68 ]hasdemonstratedthebenetofapplyingaxialmaptospatiallearningandwayndingstrategies.In[ 69 ],convexmapandaxialmapareconstructedtoanalyzeandforecasttherelationshipsbetweenthespatialpatternsofthelandandtheowsofthepedestriansinFukuokaCity,Japan.Benediktrstintroducedapproachestodescribespatialspacesbyusingisovistanalysisin[ 8 ].Sincetheisovistofeachpointdependsonthephysicalstructure,itisveryusefultoapplytheisovistanalysisinindoorspace.SomeofthemmaysharethesameIsovist,andsomeofthemhavetheirownisovist.Mostiftheresearchonisovists[ 5 22 101 ]areintheeldofarchitecturetodesignoorplanandarrangearchitecturalcomponents.In[ 99 ],theauthorsproposedanapproachtoconvertasetofisovistsintoagraphof 27

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Figure2-2. Multiplecongurationsofconvexspacesforonespace. AB Figure2-3. Exampleofaxialmaps.A)Asimplemap.B)Theaxialmapof(A). mutualvisiblelocations.Thisgraphcanbeusedinmultipleapplicationslikeway-nding,movementandspaceusage. 2.3ApproachestoSupportingRangeQueriesinOutdoorSpace Atypicalrangequeryisacommondatabasequerythatretrievesalltherecordsbetweenanupperandalowerboundary.InGISarea,arangequeriesiacommon,navigation-relatedquery,whichisusedtoobtainallqualifyingobjectswithinagivenrangeaccordingtotheuser'sinterests.AtypicalrangequeryinGISareacanbeProvidemealltheshoestoreswithin50metersfrommycurrentlocation.Currentstudyonhowtosupportrangequeriesmainlyfocusontheoutdoorspace. 2.3.1StationaryRangeQueries Therangequerythatasksforinterestingobjectswithinagivendistancewithrespecttoastaticquerypointiscalledastationaryrangequery.Theearlierapproachestosupportstationaryrangequeries[ 29 41 62 ]useR-Trees[ 25 ],R+-Trees[ 91 ]andR-Trees[ 7 ]forqueryprocessing.TheseapproachesretrievethequalifyingobjectsbasedontheEuclideandistances.Thereforetheseapproachesarecalledthe 28

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Euclidean-basedapproaches.TheEuclidean-basedapproachesareusefulwhentheobjectsareinaclearingspace.However,theyarenotsuitableforthecasesthattheobjectsareseparatedbyobstacles.Therefore,thenetwork-basedapproaches,whichtakeintoaccounttheactualreachabledistancesoftheobjects,appeartoanswertherangequerybasedontherealreachabledistances.Therstnetwork-basedapproachisproposedin[ 65 ]byPapadiasandzhang.Thisapproach,whichiscalledrangenetworkexpansion(RNE),expandsthenetworkfromthequerypointtocheckallthenodesalongtheexpansion.BecauseofthenicefeaturesoftheVoronoidiagram,usingVoronoidiagramisanotherchoiceofthenetwork-basedapproaches.ThemosttypicalonesaretheVN3modelproposedbyKolahdouzanandShahabiin[ 39 ],thePINEmodelpresentedbySafaetal.in[ 78 ]. 2.3.2ContinuousRangeQueries Approachesforstationaryrangequeriesassumethatthequerypointisalwaysinastaticposition.Infact,usersareusuallyinamovingstatuswhentheyissueanavigationquery.Therangequerythatkeepsupdatingresultsaccordingtotheuser'smovementsiscalledacontinuousrangequery.TheimportanceofcontinuousqueriesismentionedbySistlaetal.in[ 84 ].Inrecentyears,moreandmoreapproachessupportingcontinuousrangequerieshaveappeared.Similartothestationaryrangequeries,therearetwocategoriesofapproachestosupportingcontinuousrangequeriesbasedonwhichdistancetheyuse.AnearlierEuclidean-basedapproachintroducedin[ 87 ]performsstationaryrangequeriesonseveralselectedsamplepointsrespectively.Thentheapproachcalledtime-parameterizedqueriesproposedin[ 92 ]trytoanswertherangequerybyincrementallycalculatingthenextresultandtheobjectsthatmightaffectitbasedonthecurrentresult.Lateron,approachestosupportingcontinuousrangequeriestrytondthesplitpointswhichindicatethechangesofthequaliedobjects.Theideaofthesplitpoints,whichisintroducedbyTaoetal.in[ 103 ],isfutureusedbyotherapproaches[ 40 77 90 102 106 ].Tothebestofourknowledge,theapproach 29

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proposedin[ 102 ]istheonlyoneforsupportingcontinuousrangequeriesbasedonnetworkdistances. 30

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CHAPTER3SUPPORTINGTHESHORTESTPATHROUTEPLANNINGININDOORSPACE 3.1ProblemsinExistingIndoorNavigationModels Thereareacoupleofmodelsthattrytoconvertarchitecturalstructureintopathnetworks.GillironandMerminod[ 23 ]describeastrategytoconvertCADdataintoatopologicalmodel(calledNode-Link),whichsupportstheobjectslikecorridors,rooms,waysandpaths(Figure 3-1 A).Eachlinkisassignedavalueofcostcorrespondingtothetimetotravelfromitsstartnodetoitsendnode.However,thismodelsuffersfromtwomainproblems.First,thecostbetweentwonodesmustbeprovidedinadvance.Second,itlackstheconsiderationofconstraints,suchasdoors,windowsandwalls,inanindoorspace.Thus,thismodelcannotleaduserstotheexactentryofthetargetcell.Moreimportantly,thismodelmaygeneratecircuitouspaths.AsshowninFigure1b,thewallsrepresentedbytheboldedlinespreventadirectreachabilitybetweentherooms.TheroutegeneratedbytheNode-Linkmodelisacircuitousonecomposedofthecenterpointsofcells.TosimplifythenalpathsgeneratedbyNode-Link,Lyardetetal.[ 54 55 ]proposeamodelcalledCoINS,whicheliminatessomeunnecessarynodessothatthesegmentscanbeoptimized.Forexample,Figure 3-1 CshowstheimprovedpathforthesituationshowninFigure 3-1 B.AlthoughtheCoINSmodelcanoptimizethenalpath,itdoesnotconsideraccessibilityconstraintssuchasdoors,windows,andwalls. Werner,Krieg-bruckneretal.[ 43 ]proposearoutegraphmodelwhichbuildsabstractroutesaccordingtoexits,walls,andsomeotherconstraintsinindoorenvironments.However,theydonotdiscusshowtobuildtheroutegraphforanentireindoorspace.Inaddition,theyassumeroutesegmentshavedirections.However,inanindoorspace,thereisnospecieddirectionforplacessincepeoplecanwalkinanydirection.Modelsin[ 12 47 48 ]trytondthemedialaxisofacertaincellbyconstructingVoronoidiagram.Theconstructedpathsarealwaysonthelinewhosedistancestoitstwosidesarealwaysequal.AsshowninFigure 3-2 ,Figure 3-2 AistheVoronoidiagram 31

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ABC Figure3-1. Indoorrepresentationmodels.A)Thenode-linkmodel.B)Circuitouspathgeneratedbynode-linkmodel.C)PathsimplicationbytheCoINSmodel AB Figure3-2. Medialaxis.A)Voronoidiagramofasimplepolygon.B)Medialaxistransformmodel. ofthesimplepolygonandFigure 3-2 BisthecomputedmedialaxistransformedfromFigure 3-2 A.Thegeneratedpathsinthesemodelsarealwaysonthemedialaxis.Thisstrategyguaranteesenoughspacefortheuser.However,itwillcauseproblemwhenthecellisveryspacious. Lorenz,Ohlbach,andStoffel[ 53 ]developanimprovedmodelwhichtakessomearchitecturalconstraintsintoaccountwhenbuildingtheroutegraph.AsshowninFigure 3-3 A,themodelemployssomerepresentativepointstorepresentrooms,corridors,andsomeotherobjects.ThenthecalculationofthepathisperformedamongtheserepresentativenodesaswellassomearchitecturalconstraintslikedoorsinFigure 3-3 A.Theauthorsextendtheirmodelin[ 88 ]byconsideringtheshapesofcells.Theypointoutthatiftheinteriorofthecellinthenalpathisaconcaveregion, 32

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AB Figure3-3. AnexampleofLorenz'smodel.A)ThesolidlineindicatethepathsthataregeneratedbyLorenz'smodel.B)isanexampleofregionpartitioninginLorenz'smodel. severalrouteinstructionsmayberequiredtoshowthewaytogothroughthecell,becauseusersmaynotbeabletoseethenextexitatsomelocationsinthecell.Thus,theydecomposetheconcaveregionsbydetectingtheconcaveverticesasshowninFigure 3-3 B.Thegenerationofnaldescriptionsisintroducedin[ 63 ].Oneadvantageofthismodelisthatitconsidersconstraintsandaccessibilityinanindoorspacesothatthenalpathitgeneratesisafeasiblewayinrealsituations.However,itstillsuffersfromsomeproblems.First,thedeterminationoftherepresentativepointsisnoteasy.Questionsarewhichlocationisthecenterpointforanobject,orhowtodeterminethenumberofpointsandtheirpositionsforcorridors.Second,thenalpathsthemodelgeneratesarenotoptimal.Forexample,thesolidlineinFigure 3-3 Aisthepathfromdoor2toroom106thatthemodelgenerates.Aswecansee,thisrouteisnotanoptimalone.Ifroom105isverylarge,theresultwillbeunacceptabletousers.UserswillpreferthepathspeciedbythedashedlineinFigure 3-3 A.Asimilarproblemexistsregardingthepartitioningofconcaveregions.Sinceuserscannotseed4fromd3inFigure 3-3 B,theregioniscomposed,andpointaistakenasanintermediatelocationinthepathfromd3tod4.Infact,theoptimalwayfromd3tod4isfromd3toc,andthenfromctod4. 33

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3.2ThePathConstruction Cellssuchasrooms,corridors,andlobbiesareconsideredasthebasicunitsinanindoorspace.However,ifwewanttoprovidedetailedroutinginformation,itisinsufcienttoonlyconsidertheorderofthecells.Otherneededinformationlikethelocationofdoorsandexitsisalsoveryimportant.Afurtherimportantaspectareimplicitpathsinanindoorspaceastheyarecommonlyusedbypeople.Forexample,inFigure 3-3 A,iftheuserisinthecorridorandintendstogotoroom105,shewillgostraighttowardsdoor4.Thestraightlinetodoor4isanimplicitpath;itisalsotheshortestpathtoroom105.Inordertoobtaintheshortestpathsinanindoorspace,wewillexploreinthissectionhowtoconstructtheimplicitpathsforthedifferentcellcategories.Wewillshowthatandwhythegeometricshapesandarchitecturalconstraintsofdifferentkindsofcellsaffecttheconstructionoftheimplicitpathsegments,andthenproposeanapproachtoconstructtheimplicitshortestpathsforthedifferentcellcategories. Thereareanumberofdifferentkindsofarchitecturalcellsinanindoorspace.Someofthemhavesimilarshapesbutservedifferentpurposes,andsomeofthemaretotallydifferentwithrespecttotheirshapesbutplaythesameroleduringtherouting.Forexample,roomswithmultipledoorscanbeapartofapassagetoacertaindestination,whileroomswithonlyonedoorcannot.Accordingtothegeometricandarchitecturalfeaturesofthecells,weclassifythemintofourcategories:simplecells(Section 3.2.1 ),complexcells(Section 3.2.2 ),opencells(Section 3.2.3 ),andconnectors(Section 3.2.4 ).Eachcellhasoneormoreaccesspointsthatarespeciclocationswhereacellcanbeenteredorleft.Inthissection,wewillintroducethesefourcellcategoriesandshowhowonecanndtheimplicitpathsegmentsineachofthem. 3.2.1SimpleCells Asimplecellisacellthatisenclosedbywallsandhasonlyonedoor.Sinceitcannotfunctionasapassage,itdoesnothaveanyinnerimplicitpathsegments.Itcan 34

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onlyplaytheroleofastartobjectoratargetobject,anditsentryandexitareatthelocationofthedoor.Figure 3-4 Ashowsaschematicinstanceofasimplecell.Theblackdotrepresentstheonlydoorofthecell.Anapplicationexampleisanofceroomattheendofaoorthatcanbeenteredandleftbyasingledoor.Denition 1 providesaformalandmoreabstractdenitionofsimplecells. Denition1. Asimplecellscisgivenasatriplesc=(id,r,a).Itrepresentsacellwithageometricstructurethatisenclosedbyaboundary(usuallywalls)andcanbeaccessedbyexactlyoneaccesspoint(usuallyadoor).Thecomponentidistheuniqueidentierofthiscell.Thecomponentrisavalueofthespatialdatatyperegion[ 81 ]thatstoresthearealgeometricstructureandextentofthiscellasasimple,single-componentpolygonwithoutholes.Thecomponentaisavalueofthespatialdatatypepoint[ 81 ],denotes(thelocation,thatis,thexy-coordinates,of)theaccesspoint,andliesontheboundaryofr.Itisthearchitecturalconstraintcontrollingtheaccessibilityofthecell. 3.2.2ComplexCells Acomplexcellisacellthatisenclosedbywallsandcanbeaccessedbymultipledoorsand/orhasanestedstructure.Itcanbeconsideredaseitherastartobject,atargetobject,oranintermediateobjectthatcontainspartsofpathsaspassagestothedestinations.Wedistinguishatcomplexcells(Section 3.2.2.1 )andnestedcomplexcells(Section 3.2.2.2 ).Simplecellsarespecialcasesofatcomplexcellsthatarespecialcasesofnestedcomplexcells.Eachimplicitpathsegmentinacomplexcellisrequiredtobeastraightline(thatis,asegment)connectingtwoaccesspointswithoutintersectinganyboundary.Thus,theconstructionoftheseimplicitpathsegmentshighlydependsonthegeometricshapesandthelocationsoftheaccesspointsofacell.ForatcomplexcellsweconsidertheimplicitpathsegmentconstructioninSection 3.2.2.3 .FornestedcomplexcellsweperformthisconsiderationinSection 3.2.2.4 .Finally,wepresentanalgorithmforcomputingtheimplicitpathsegmentsinatandnestedcomplexcellsinSection 3.2.2.5 35

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ABCD Figure3-4. Differenttypesofcells.A)Simplecell.B)Flatcomplexcell.C)Opencell.D)Connecter 3.2.2.1Flatcomplexcells Thegeometryofaatcomplexcellisgivenbyasimplepolygon,thatis,itconsistsofasinglearealcomponentanddoesnothaveholes.Further,ithasatleasttwoaccesspointsonitsboundary.Figure 3-4 Bshowsaschematicinstanceofaatcomplexcell.Theblackdotsrepresenttheaccesspointsofthecell.Flatcomplexcellscanbeoftenfound.Anapplicationexampleisawaitingroominadoctor'sofcethatcanbeenteredandleftthroughanumberofdoors.Denition 2 providesaformaldenitionofatcomplexcells. Denition2. Aatcomplexcellfccisgivenbyatuplefcc=(id,r,m,a1,...,am,t,s1,...,st)withm2andt1.Itrepresentsacellwithageometricstructurethatisenclosedbyaboundary(usuallywalls)andcanbeaccessedbytwoormoreaccesspoints(usuallydoors).Thecomponentiduniquelyidentiesthiscell.Thecomponentrisavalueofthespatialdatatyperegionthatstoresthearealgeometricstructureandextentofthiscellasasimple,single-componentpolygonwithoutholes.Thecomponentmdenotesthenumberofaccesspointsinthiscell.Thecomponentsa1,...,amarevaluesofthespatialdatatypepoint,represent(thelocations,thatis,thexy-coordinates,of)theseaccesspoints,andlieontheboundaryofr.Thecomponenttkeepsthenumberofimplicitpathsegmentsinthiscell,ands1,...,strepresentthesesegments. 3.2.2.2Nestedcomplexcells Mostroomsinanindoorspacecanberepresentedbysimplecellsoratcomplexcells,thatis,geometricallybysimpleregions.However,itisalsocommontondstructureslikeroomscontainingotherrooms,orcorridorswithacircularshape. 36

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Figure3-5. Nestedcells. Anapplicationexampleisanopenofceenvironmentthatinalargeofcecontainsseparateofcesformanagersorsupervisors.Anotherexampleisalaboratorythatissubdividedintoacollectionofadjacentandconnectedrooms.Theseroomsthemselvescanalsocontainrooms.Further,itcanbeofinteresttomodelobstacleslikedesks,chairs,andcabinetsinaroomsinceroutescannotpassthem.Inthissection,wewillhencediscussmorecomplicatedscenariosinwhichthegeometricshapeofacellismodeledbyahierarchyofsimpleregions.Asimplescenarioisgivenbytwosimpleregionswhereonesimpleregionislocatedinanothersimpleregion(Figure 3-5 A).Thesimplestscenarioisgiveniftheoutersimpleregionwouldhaveexactlyoneaccesspoint.Ifaroomcontainsmultiplerooms,thestructurecanberepresentedbyasimpleregionthatcontainsseveraldisjointormeetingsimpleregions(Figure 3-5 B).Otherstructureslikeroomsthatarerecursivelyinsideotherrooms(Figures 3-5 Cand 3-5 D)leadtoahierarchyofsimpleregionsthatisorganizedwithrespecttothegeometriccontainmentrelationship,thatis,cellscancontainsubcells.Wecallsuchahierarchyanestedcomplexcell. Denition 3 providesitsformaldenition.Itrequiresanumberofconceptsthatweintroducerst.Thedenitionmakesuseofthetopologicalpredicates[ 20 80 ]disjoint,meet,contains,andcoversthatcharacterizetherelativepositionbetweentwosimpleregionsandthathaveapreciseandmutuallyexclusivesemantics.Intuitively,thepredicatedisjointmeansthattwosimpleregionsdonotshareanypartwitheachother.Thepredicatemeetimpliesthattwosimpleregionsshareanitecollectionofboundarypointsorboundarylines.Boundarylinesasone-dimensionalobjectsarerequiredforthe 37

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dimensionaltopologicalpredicate1-meet.Thepredicatescontainsandcoversexpressthatasimpleregionislocatedwithinanothersimpleregion,withoutandbytouchingtheboundaryrespectively.Ifbothregionstouchinaline,thedimensionaltopologicalpredicate1-coversholds.Further,theoperationcommonBorder[ 81 ]computesthesharedboundaryoftworegionobjectsasanobjectofthespatialdatatypeline.ThefunctiongetBorderPointsgetsalineobjectandalistofpointsasoperandsandreturnsthesetofthosepointsthatarelocatedonthelineobject. Denition3. Anestedcomplexcellnccisgivenbythetuple ncc=(id,d,k1=1,k2,...,kd,c1,1,c2,1,...,c2,k2,...,cd,1,...,cd,kd,t,s1,...,st) withd2,ki1forall1id,andt1.Itrepresentsahierarchicalcellstructurewhichconsistsofacollectionofcellsci,jforall1idand1jkjandinwhicheachcellisenclosedbyaboundary(usuallywalls),canbeaccessedbyoneormoreaccesspoints(usuallydoors),canhavebrothercells,andcancontainothercells.Thecomponentiduniquelyidentiesncc.Thecomponentddenotesthedepthofthecellhierarchyoritsnumberoflevels.Thedepthoftherootlevelis1.Thecomponentskidescribethenumberofcellsatthehierarchyleveli.Thecellci,jisthejthcellatthehierarchyleveliandisrepresentedasatuple(idi,j,ri,j,mi,j,(ai,j,1,...,ai,j,mi,j)).Thecomponentidi,juniquelyidentiesci,j.Eachcomponentri,jisavalueofthespatialdatatyperegionandstoresthearealgeometricstructureandextentofthiscellasasimple,single-componentpolygonwithoutholes.Thecomponentmi,jdenotesthenumberofaccesspointsinthiscell,andai,j,1,...,ai,j,mi,jrepresent(thelocations,thatis,thexy-coordinates,of)theseaccesspointslocatedontheboundaryofri,j.Thehierarchylevel1hasonlyonecellwhichistherootofthehierarchy.Allregionobjectsmustsatisfythefollowingfourconditions: 38

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(i)81id81j
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3.2.2.3Implicitpathsegmentsinatcomplexcells Thesimplestcasetoconstructimplicitpathsegmentsinatcomplexcellsisacellwithonlytwoaccesspointsthatcanbeconnectedbyastraightlineinsidethecell.Theshortestpathfromoneaccesspointtotheotheristhenthestraightlinebetweenthem(seeFigure 3-6 A).Ifaatcomplexcellhasmorethantwoaccesspoints,andallofthemcanbedirectlyreachedfromeachotherinsidethecell,weobtainmultipleimplicitpathsegmentsinthecell.Theshortestpathsegmentsinsuchacellareallstraightlinesconnectinganytwoaccesspoints.Forexample,inFigure 3-6 B,wendsiximplicitshortestpathsegmentsinacellwithfouraccesspoints.Thedeeperreasonforthedirectreachabilitybetweenallaccesspointsofthisregionisthatthisregionisconvex,thatis,theinneranglebetweenanypairofconsecutiveboundarysegmentsislessthan180.However,incellswithconcaveshapes,whichhaveatleastonepairofconsecutiveboundarysegmentswithaninnerangleofmorethan180,twoaccesspointsmaynotbedirectlyreachablefromeachother.InFigure 3-8 A,thedashedlinesshowsomecaseswhereastraightlineconnectingtwoaccesspointsisblockedbytheboundary(forexample,thesegmentc)]TJ /F8 11.955 Tf 11.95 0 Td[(d). Wenowdescribeourapproachtoconstructingimplicitpathsegmentsforthecaseofarbitrary,non-selntersecting,simpleregions(polygons).FromComputationalGeometry[ 18 ]weknowthatiftheinteriorofasegmentconnectingtwoboundarypointsofapolygoneitherintersectstheboundaryorisoutsidetheboundary,thispolygonmustbeaconcavepolygon.Thatis,thereisatleastonevertexwhoseinteriorangle ABC Figure3-6. Pathsegmentsindifferenttypesofcells.A)Pathsegmentinaatcomplexcellwithtwoaccesspoints.B)Pathsegmentsinaatcomplexcellwithmultipleaccesspoints.C)Pathsegmentsinaconnecter. 40

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Figure3-7. examplesofconcaveverticesandconcaveboundaries. isareexangle(degree>180)ononepartoftheboundarybetweenthetwoaccesspoints.Wecallthiskindofvertexaconcavevertexandthepartoftheboundarythatcontainsconcaveverticesaconcaveboundary.Forexample,inFigure 3-7 A,xisaconcavevertex,andtheboundarypartbetweenaandbcontainingxisaconcaveboundary.Itispossibletohaveseveralconcaveverticesonaconcaveboundary.AsshowninFigure 3-7 B,bothxandyareconcavevertices.Figure 3-7 Cshowsthatnotallconcaveverticesbetweentwopointsonaconcaveboundarynecessarilycontributetotheshortestpathsegments.AnexampleofasegmentthatislocatedcompletelyoutsideapolygonandisthusnotavalidpathsegmentforthepolygonisgivenbycanddinFigure 3-7 A. Ourapproachtoobtainingtheshortestpathinthiskindofsituationistoselectanappropriateconcavevertexontheconcaveboundaryasanintermediatepointand ABC Figure3-8. Theshortestpathsinaconcavepolygon.A)Twoaccesspointsinaconcaveregion(likebandc)cannotreacheachotheronastraightpathsegment.B)Potentialpathsegments.C)Finalpathsegments. 41

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topartitionthesegmentintotwosegments.Thepartitioningprocesscontinuesuntilallgeneratedsegmentsdonotintersecttheboundary.Forexample,inFigure 3-8 A,thestraightsegmentconnectingtheaccesspointsaandeintersectstheboundary.Figure 3-8 Bshowsthatthissegmentisthenpartitionedintothesegments(a,v3),(v3,v15),(v15,v14),and(v14,e)bytheintermediatepointsv3,v14,andv15betweenthem.Weobservethata)]TJ /F8 11.955 Tf 11.71 0 Td[(v3)]TJ /F8 11.955 Tf 11.7 0 Td[(v15)]TJ /F8 11.955 Tf 11.7 0 Td[(v14)]TJ /F8 11.955 Tf 11.71 0 Td[(eistheshortestpathbetweenaande.Lemma 1 showsthattheintermediatepointsoftheshortestpathbetweentwoaccesspointslocatedontheboundaryofasimpleregion(polygon)arealwaysconcaveverticesofthisboundary. Lemma1. Theshortestpathbetweenanytwoaccesspointsinaatcomplexcellcisapolygonalpath(thatis,aconnectedseriesofstraightlinesegments)whoseintermediatepointsareconcaveverticesofc. Proof. Werstshowthatanysuchshortestpathsispolygonal.Supposethatsisnotpolygonal.Sincesisentirelyinsidec,andcispolygonal,theremustbeapointponsthatisintheinteriorofc,andthepartofscontainingpisnotalinesegment.Sincepisintheinteriorofc,theremustbeacircleofpositiveradiusthatiscenteredatpandcompletelyinsidec(Figure 3-9 A).Sincethepartofscontainingpisnotastraightlinesegment,wecanalwaysshortenthispartbyreplacingitwithalinesegmentconnectingthepointwherethepartentersthecircletothepointwhereitleavesthecircle.Butthiscontradictstheshortestpathpropertysinceanyshortestpathmustbelocallyshortest,thatis,anysubpathconnectingtwopointsqandrmustbetheshortestpathbetweenqandr. Nowweshowthattheintermediatepointsofanysuchshortestpathsareconcaveverticesontheboundaryofc.Anyintermediatepointvofscannotlieintheinteriorofcsincewecouldconstructacircleofpositiveradiusthatiscenteredatvandliescompletelyincsothatwecouldreplacethesubpathofsthatisinsidethecircleandturnsatvbyashorterstraightlinesegment.Similarly,anyintermediatepointvofscannotlieintheinteriorofaboundaryedge.Otherwise,theremustbeacircleof 42

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ABC Figure3-9. ProofofLemma 1 A)Theshortestpathbetweenanytwoaccesspointsmustbepolygonal.B)Anintermediatepointofthepathcannotbeapointintheinteriorofaboundaryedge,C)itcannotbeaconvexvertexontheboundaryofthecell. positiveradiuscenteredatv,andhalfofthecirclewhichcontainsthelinesegmentpartsemanatingfromvisinsidec.Butwecanthenreplacetheselinesegmentpartsinsidethecirclebyusingasingleshorterlinesegment(Figure 3-9 B).Thisisagainaviolationoftheshortestpathproperty.Itisalsoimpossiblethatanintermediatepointvisaconvexvertexofsbecauseifwedrawacircleofpositiveradiuscenteredatv,someportionofthecirclewillbeinsidec,andwecanndashorterstraightlinesegmenttoreplacethelinesegmentpartsinsidethecircle(Figure 3-9 C).Theonlypossibilityleftisthattheintermediatepointvofsisaconcavevertexoftheboundaryofc. 3.2.2.4Implicitpathsegmentsinnestedcomplexcells Thehierarchicalstructureofanestedcomplexcellseemstoindicatethatamethodforthedeterminationofitsimplicitpathsegmentsisrathercomplicated.Butwewillarguenowthatthisdeterminationcanbeperformedlocallyinthehierarchybyconsideringanytwoconsecutivehierarchylevels(exceptfortheleafnodesofthe ABCD Figure3-10. Layerstructureofnestedcells.A)Exampleofanestedcomplexcellwithahierarchyofdepth3.B)Thefaceformedbythehierarchylevels1and2.C)Thefaceformedbythehierarchylevels2and3.D)thefaceformedbythehierarchylevel3. 43

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hierarchy)togetheratatime,determiningtheimplicitpathsegmentslocallyforsuchtwolevels,andcollectingallpathsegments.Figure 3-10 Ashowsascenarioofanestedcomplexcellwithahierarchyofdepth3.Wehavetwoconsecutivepairsofhierarchylevelstogetherwiththespacethattheyenclose.TheyareshowninFigure 3-10 Bforthelevels1and2,andinFigure 3-10 Cforthelevels2and3.Atlevel3(Figure 3-10 D),wecannotndinnercells.Otherwise,thedepthofthehierarchywouldbe4.Thespaceenclosedisshowningraycolorinallthreecases.Geometrically,eacharealledbygraycolorhasaconnectedinteriorandformsaspecialinstanceofthespatialdatatyperegionforcomplexregions.Thisspecialinstancedescribesasingle-componentregionobjectwithholes.Suchaninstanceisusuallycalledaface[ 81 ].Inthissense,Figure 3-10 BhasonegrayfacewhereasFigures 3-10 Cand 3-10 Dhavetwofaceseach.Further,therightfaceofFigure 3-10 CandthetwofacesofFigure 3-10 Ddonothaveholes.Ifafaceshouldhaveanyholesthatshareacommonboundarypart,theseholesaregeometricallymergedtoasingleholetobeinaccordwiththefacedenition. Sincetheinteriorsofallfacesdeterminedasdescribedabovearedisjoint,itissufcienttoonlycomputetheimplicitpathsegmentsforeachfacelocallyandindividually,andthentocollectallpathsegmentsfound.Inafacewithholes,segmentsbetweenaccesspointsmayintersectnotonlytheouterboundarybutalsotheinnerboundaries.Whenasegmentbetweentwoaccesspointsintersectsaninnerboundary,someoftheverticesontheinnerboundarywillbecomeintermediatepointsoftheshortestpathbetweenthetwoaccesspoints.AsLemma 2 shows,onlyconcaveverticesoftheouterboundaryandtheinnerboundariesofafacecanbecomeintermediatepoints. Lemma2. Theshortestpathbetweenanytwoaccesspointsinanestedcomplexcellcofdepth2,whosegeometricshapeisgivenbyaface,thatis,byanouterpolygon(simpleregion)andzero,one,ormoreinnerdisjointormeetingpolygons,isapolygonalpathwhoseintermediatepointsareconcaveverticesofc. 44

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ABC Figure3-11. Proofcasesfortheinnerpolygonboundariesofacomplexcellwithahierarchyofdepth2:A)Anintermediatepointofthepathcannotbeapointintheinteriorofaboundaryedge.B)italsocannotbeaconvexvertexontheboundary.C)Itmustbeaconcavevertexoftheinnerboundary. Proof. Lemma 1 coversthecasethatthedepthofthecellhierarchyis1.Inthiscase,therearenoinnerholepolygons.Wehaveasimplecellifthereisexactlyoneaccesspointonitsboundary.Otherwise,wehaveaatcomplexcellcwithatleasttwoaccesspoints.Anintermediatepointoftheshortestpathsbetweenanytwoaccesspointsofcisthenalwaysaconcavevertexoftheboundaryofc.ThesameargumentationthatwehaveusedinLemma 1 canalsobeappliedtotheinnerpolygon(hole)boundariesofcifthedepthofthecellhierarchyis2.Anintermediatepointofscannotlieintheinteriorofanedgeofanyinnerboundary(Figure 3-11 A)andalsonotbeaconvexvertexofanyinnerboundaryduetotheradiusargument(Figure 3-11 B).Hence,ifanintermediatepointofsislocatedonaninnerboundary,itmustbeaconcavevertexofthisinnerboundary(Figure 3-11 C).Notethattheanglesthatdecidewhetheranendpointofanedgeofaninnerboundaryisconvexorconcavemustbetakenfromtheinteriorsideofc. Figure3-12. Exampleofafacewithtwoaccesspointsandtwopossiblepathsbetweenthem. 45

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Sincetherearetwopossibledirections(clockwiseandcounterclockwise)tobypassaninnerholeboundary,therearepotentiallytworoutesthatcouldleadtoashortestpath.AsshowninFigure 3-12 ,thedashedlinea)]TJ /F8 11.955 Tf 12.36 0 Td[(iv1)]TJ /F8 11.955 Tf 12.35 0 Td[(iv2)]TJ /F8 11.955 Tf 12.35 0 Td[(ov1)]TJ /F8 11.955 Tf 12.36 0 Td[(ov2)]TJ /F8 11.955 Tf 12.36 0 Td[(bandthedottedlinea)]TJ /F8 11.955 Tf 12.52 0 Td[(iv4)]TJ /F8 11.955 Tf 12.52 0 Td[(iv3)]TJ /F8 11.955 Tf 12.51 0 Td[(ov2)]TJ /F8 11.955 Tf 12.52 0 Td[(baretwopotentialpathsfromtheaccesspointatotheaccesspointb.Bothsetsofpointsfiv1,iv2,ov1,ov2gandfiv4,iv3,ov2gcouldbeintermediatepoints.Allintermediatepointsareconcavepoints. 3.2.2.5Analgorithmforcomputingallshortestpathsegmentsincomplexcells Algorithm 1 showshowtocomputeallshortestpathsegmentsinanestedcomplexcellbyleveragingtheinsightsfromSections 3.2.2.3 and 3.2.2.4 .Thestrategyistoiterativelyextractallcellhierarchiesfromanestedcomplexcellthathavethedepth2(or1attheleaflevel)andthusthegeometricstructureofaface(withholesifthedepthis2andwithoutholesifthedepthis1).Foreachsuchfaceanditsrelevantaccesspoints,Algorithm 2 locallydeterminesallpotentialpathsegments.Thesearesegmentsthatcouldbelaterpartofthesolution.Thepotentialshortestpathsegmentsarecollectedforallfaces.Togetherwithallaccesspointsofthenestedcomplexcell,theyformtheinputofawellknownalgorithmtocalculatetheshortestpathsbetweenanytwoaccesspoints. Algorithm 1 traversesallinnerhierarchylevelsofthenestedcomplexcellfromtherootleveldowntothelastbutone,deepestlevel(line2).Allcellsofeachhierarchylevelaretraversed(line3),andforeachsuchcellcwerstcollectitsaccesspoints(line4).Thesymbolshandiencloseasequenceorlistofelements.Thenwecheckforcwhetheritcontainsorcoversanycellsatthenextdeeperlevel(lines5and6).TheBooleanpredicatescontainsandcoversarethewellknowntopologicalpredicatesusedinDenition 3 .Thosecellsthatarecontainedorcoveredbycaregeometricallymergedintoacomplexregionobjectofthespatialdatatyperegion(line7);theyformtheholecellsofc.Theoperationdenotesthegeometricunionoperation[ 80 81 ].Further,theaccesspointsofeachholecellareaddedtotheaccesspointsofcandthealready 46

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determinedaccesspointsofpreviouslyconsideredholecells(line7).Theoperatordenotestheconcatenationoperatoronlists.Next,weaddtheaccesspointsofcanditsholescellstoaglobalvariableaccessPointsthatissupposedtokeepallaccesspointsofthenestedcomplexcell(line8).Itcanhappenthattwoholecellsofc,orcandoneofitsholecells,haveacommonaccesspointonasharedboundarypart.Intherstcase,thetwoholecellsgeometricallymergetoalargerholecellsincetheirsharedboundarypartdisappears,andtheircommonaccesspointbecomesirrelevantatthispoint.Inthesecondcase,theholecellformsabayinc,andtheircommonaccesspointbecomesirrelevanttoo.Inbothcases,bothoccurrencesofthesameaccesspointkeptforcmustberemoved.Thisisperformedbythefunctionrpairs(removepairs)(line9).Foreachcellcanditsholecells,weformafacebyapplyingthegeometricdifferenceoperation[ 80 81 ]tothesimpleregionrepresentingcandthecomplexregionobjectrepresentingitsholecells.TheresultingfaceanditsaccesspointsformtheoperandsoftheoperationDeterminePotentialPathSegs(line10)thatlocallydeterminesallpotentialpathsegments Algorithm1: Computationoftheshortestpathsegmentsinanestedcomplexcell Input: nestedcomplexcellncc=(id,d,k1,k2,...,kd,c1,1,c2,1,...,c2,k2,...,cd,1,...,cd,kd,n,s1,...,sn)whoseconstructionisincompletewithrespecttotheshortestpathsegmentss1,...,standtheirnumbert Output: nestedcomplexcellnccincludingtheshortestpathsegmentinformation accessPoints hi;potShortestPathSegs hi;1 fori 1tod)]TJ /F17 10.909 Tf 10.91 0 Td[(1do2 forj 1tokido3 hi,j hi;ai,j hai,j,1,...,ai,j,mi,ji;4 forl 1toki+1do5 ifcontains(ri,j,ri+1,l)orcovers(ri,j,ri+1,l)then6 hi,j hi,jri+1,l;ai,j ai,jhai+1,l,1,...,ai+1,l,mi+1,li;7 accessPoints accessPointsai,j;8 ai,j rpairs(ai,j);9 potShortestPathSegs 10potShortestPathSegsDeterminePotentialShortestPathSegs(ri,jhi,j,ai,j); hs1,...,sti AllPairsShortestPath(rdup(accessPoints),potShortestPathSegs);11 47

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Algorithm2: (DeterminePotentialShortestPathSegs)Computationofthepotentialshortestpathsegmentsinanestedcomplexcellrepresentedbyafaceanditsaccesspoints Input: faceF(withzero,one,ormoreholes),listarrayaccessPointsofaccesspointsonF'sboundary Output: listarraypotentialShortestPathSegsofallpotentialShortestpathsegmentsinF intersectionFound false;m size(accessPoints);potentialShortestPathSegs hi;1 fori 1tom)]TJ /F17 10.909 Tf 10.91 0 Td[(1do2 forj i+1tomdo3 p accessPoints[i];q accessPoints[j];4 ifnotintersects(segment(p,q),boundary(F))andin(center(segment(p,q)),F)5then potentialShortestPathSegs potentialShortestPathSegsh(p,q)i;6 else7 intersectionFound true;8 ifintersectionFoundthen9 concaveVertices ndConcaveVertices(F);//listarrayofconcavevertexpoints10 n size(concaveVertices);//numberofconcavevertices11 fori 1ton)]TJ /F17 10.909 Tf 10.91 0 Td[(1do12 forj i+1tondo13 p concaveVertices[i];q concaveVertices[j];14 ifnotintersects(segment(p,q),boundary(F))andin(center(segment(p,q)),F)15then potentialShortestPathSegs potentialShortestPathSegsh(p,q)i;16 fori 1tomdo17 forj 1tondo18 p accessPoints[i];q concaveVertices[j];19 ifnotintersects(segment(p,q),boundary(F))andin(center(segment(p,q)),F)20then potentialShortestPathSegs potentialShortestPathSegsh(p,q)i;21 returnpotentialShortestPathSegs;22 ofthisfaceandthatispresentedinAlgorithm 2 .Aftertheexecutionoflines1to10,wehaveascertainedallpotentialpathsegmentsofthenestedcomplexcell.Inalaststep(line11),werstremoveallduplicatesfromthelistofaccesspointsbythefunctionrdup(removeduplicates)sinceinthenetworktobeformedeachaccesspoint(vertex)isonlyallowedtoappearonce.Wethentakeallremainingaccesspointsandallpotentialshortestpathsegmentsasinput,andapplythealgorithmAllPairsShortestPathtothem 48

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tocomputetheshortestpathsandthustheirsegmentsbetweenanytwoaccesspointsinthenetworkformedbythepotentialshortestpathsegments.WellknownalgorithmstoperformthistaskaretheFloyd-Warshallalgorithm[ 13 ]andJohnson'salgorithm[ 14 ].TheresultofapplyingsuchanalgorithmtothescenarioinFigure 3-8 BisshowninFigure 3-8 C.Becauseonlyconcaveverticescanbeintermediatepoints,allshortestpathsbeginandendwithanaccesspointandhave0ormoreconcaveverticesasintermediatepoints. ItremainstodescribetheoperationDeterminePotentialPathSegswhosetaskitistodetermineallpotentialpathsegmentsforagivenfaceanditsaccesspointsasinput.WegivethisdescriptioninAlgorithm 2 .Thealgorithmisgeneralinthesensethatafacecancontainzero,one,ormoreholes.Theoverallstrategyconsistsofthreesteps. Inarststep(lines1to8),weconstructallsegmentsbetweenanytwoaccesspointsandcheckwhethertheyarelocatedinsidetheface(lines4and5).Ifthisisthecase,theyarepotentialshortestpathsegments(line6).Otherwise,theyeitherintersectthefaceboundary(thatis,theouterboundaryoraholeboundary)orareoutsidetheface(line8).Ifnointersectionhasbeenfound,thealgorithmterminatessincewehavefoundallpotentialshortestpathsegments(line22). Acheckwhetherasegmentislocatedinsideafaceisneededatthreeplacesinthealgorithm,namelyinlines5,15,and20.Totestwhetherasegmentintersectsthefaceboundary,weuseageometricprimitiveintersectswhichtraversesallboundarysegmentsofthefaceuntilitndsonethatintersectsthesegmentbuiltfortwoconnec-tionpoints(accesspointsorconcavepoints)bytheconstructorsegment.Ifwendoutthatafacedoesnotintersectthesegmentbuiltfortwoconnectionpoints,itcanstillbethatthesegmentisoutsidetheface(seeFigure 3-7 A).Wetakeavariationofthewellknownplumblinealgorithmforpoint-in-polygontesting(calledinhere)tocheckwhetherthecenterofthissegment(determinedbythefunctioncenter)islocatedinsidetheface.Insummary,ifasegmentbetweentwoconnectionpointsdoesnotintersectthe 49

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faceboundaryandthecenterofthesegmentisinsidetheface,thenthesegmentisapotentialshortestpathsegment. Fromthesecondstep(lines9to16)on,wehavetodealwiththefactthattherearestraightsegmentsthatintersectthefaceboundaryorlieoutsidetheface.Thismeansthatconcaveverticesmustbepartofshortestpathsbetweenaccesspointsintheface.Sincewedonotknowwhichconcaveverticeswilllaterplayaroleintheshortestpathndingprocess,wedetermineallofthembyemployingthefunctionndConcaveVertices(line10).Thisfunctiontraversesallpairsofconsecutiveface(outerandhole)boundarysegmentsandcheckswhethertheirinnerangleisreex.Weconstructallsegmentsbetweenanytwoconcaveverticesandcheckwhethertheyarelocatedinsidethepolygon(lines14and15)sinceonlythosearepotentialshortestpathsegments(line16). Inathirdstep(lines17to21),wecreatetheconnectionsbetweenallaccesspointsandallconcavevertices.Againwebuildallsegmentsbetweenthemandcheckwhethertheyarelocatedinsidetheface.Asanalresult,weobtainandreturnallpotentialshortestpathsegmentsintheface(line22). Finally,weexploretheruntimecomplexityofthecompletealgorithm.Notethatthealgorithmisonlyappliedonetimeforeachnestedcomplexcellsincethenitsall-pairsshortestpathgraphisstoredinitsrepresentationinthesegmentss1,...,st.SinceAlgorithm 2 isappliedtoeachfaceconstructedinAlgorithm 1 ,weassumethatinAlgorithm 2 ,misthemaximumnumberofaccesspointsofanyface,andnisthemaximumnumberofconcaveverticesofanyface.Further,letbbethemaximumnumberofboundarysegmentsofanyface.ThepredicateintersectsrequiresO(b)time,andthefunctioncentercanbeperformedinconstanttime.Hence,thecomplexityofndingallpotentialshortestpathsegmentsforanyfaceisO(b(m2+n2+mn)).InAlgorithm 1 ,weformafaceforeachcellofthehierarchylevels1tod)]TJ /F3 11.955 Tf 13 0 Td[(1.ThismeansweformPd)]TJ /F7 7.97 Tf 6.58 0 Td[(1i=1kifaces.Inordertobuildaparticularface,foreachcellofa 50

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hierarchyleveli,wetraverseallcellsofthehierarchyleveli+1inordertondoutacontainmentrelationshipbetweenbothcells.Thatis,wehavePd)]TJ /F7 7.97 Tf 6.59 0 Td[(1i=1kiki+1containmenttests.EachcontainmenttestrequiresO((b+b)log(b+b))=O(blogb)timeforaneededplanesweepalgorithm[ 18 ].Intotal,thetimeneededforallcontainmenttestsisO(blogbPd)]TJ /F7 7.97 Tf 6.59 0 Td[(1i=1kiki+1).Fromthecollectedlistofaccesspointsforeachface,wehavetoremoveallpairsofequalaccesspoints(functionrpairs)sincetheyarenotrelevantforthedeterminationofpotentialshortestpathsegments.Thisrequiressortingthelistandtraversingthesortedlisttodeleteequalpairs.ThisimpliesO(mlogm)timeforeachfaceandO(mlogmPd)]TJ /F7 7.97 Tf 6.59 0 Td[(1i=1ki)timeforallfaces.Finally,wehavetodeterminethecostoftheall-pairsshortestpathalgorithm(liketheFloyd-Warshall)algorithmtocomputetheshortestpathsbetweenallpairsofnodes.Inagraphwithvvertices,theworstcaseperformanceofthisalgorithmisO(v3).Foreachface,thenumberofendpointsofallpotentialpathsegmentsarerestrictedbythemaccesspointsandthenconcavevertices.SincewehavePd)]TJ /F7 7.97 Tf 6.59 0 Td[(1i=1kifaces,theworsttimecomplexityoftheall-pairsshortestpathalgorithmisO((m+n)3Pd)]TJ /F7 7.97 Tf 6.58 0 Td[(1i=1ki).Astheoverallworstcasetimecomplexityweobtainnally: O(blogbd)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xi=1kiki+1+b(m2+n2+mn)d)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xi=1ki+mlogmd)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xi=1ki+(m+n)3d)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xi=1ki) Withk=maxfkij1idgandfurthersimplications,weobtainastheoverallworstcasetimecomplexity: O(blogbdk2(m+n)3) Sincethedeterminationoftheshortestpathsegmentsisonlyperformedonetimeforeachnestedcomplexcell,theefciencyofthealgorithmisnotthemajorrequirement.Ontheotherhand,thealgorithmicstepsareneeded,quitecomplex,anddependontheveparametersb,d,k,m,andn. 3.2.3OpenCells Anopencellisacellforwhichatleastapartofitsboundaryisnotclosedbyexplicitwallsorotherspatialconstraints.Examplesofopencellsareconcoursesinairports 51

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aswellashallsandlobbiesinbuildings.Differentopencellscanhavevariousopenboundarypartswithdifferentwidths.Eveninthesamecell,multipleopenboundarypartsofdifferentwidthsmayexist.Thesevariablewidthsmakeitdifculttodeterminetheaccesspointsintheopenboundarypartssincethelatterhavemorethecharacterofaccesslines.Solidboundarypartsmayhaveaccesspointslikedoors.Figure 3-4 Cshowsanexampleofanopencell;adashedlinerepresentsanopenpartoftheboundary,andasolidlineindicatesawall.TheformaldenitionofanopencellisgiveninDenition 4 Denition4. Anopencellocisgivenbyatupleoc:=(id,r,l,m,a1,...,am,t,s1,...,st)withm0andt1.Itrepresentsageometricstructurethatispartiallyenclosedbyasolidboundary(usuallywalls)andpartiallyenclosedbyanopenboundaryenablingfreeaccessandtransit.Itmusthaveatleastoneopenboundarypart.Thecomponentidu-niquelyidentiesthiscell.Thecomponentrisavalueofthespatialdatatyperegionthatstoresthearealstructureandextentofthiscellasasimple,single-componentpolygonwithoutholes.Thecomponentlisavalueofthespatialdatatypelineandrepresentstheopenboundarypartsoftheopencell.Thatis,thepredicateinside(l,boundary(r))mustholdwherethefunctionboundarydeterminestheborderofaregionobjectasalineobjectandthepredicateinsidetestsforspatialcontainment.Thecomponentmdenotesthenumberofaccesspointsonthesolidboundarypartsofthiscell.Thecom-ponentsa1,...,amarevaluesofthespatialdatatypepoint,represent(thelocations,thatis,thexy-coordinates,of)theseaccesspoints,andlieonthesolidboundarypartsofr.Thatis,on(ai,boundary(r)l)mustholdforall1imwherethepredicateoncheckswhetherapointislocatedonaline.Thecomponenttkeepsthenumberofimplicitpathsegmentsinthiscell,ands1,...,strepresentthesesegments. Anopencellisdifferentfromacomplexcellwithrespecttotheopenboundaryparts.Further,itisdifculttodeterminetheaccesspointsonitsopenboundaryparts.Whenanopencellisthetargetobjectinaquery,itisreasonabletoconsiderthecenter 52

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AB Figure3-13. Constructingpathsegmentsinopencells.A)Finalpathsegments.B)potentialpathsegments pointoftheopenboundarypartasoneoftheaccesspointsoftheopencell.However,whenanopencellplaystheroleofapassage,wecannotsimplyconsiderthecenterpointbecausetheshortestpathcangothroughanypointoftheopenboundary.Tosolvethisproblem,ourapproachtoobtainingthepathsegmentsinanopencellconsistsofthreesteps.Intherststep,wecombineallopencellssharingopenboundarypartsuntilthecombinedcellisacomplexcellclosedbywalls.Inthesecondstep,weapplythestrategyofndingpathsegmentsincomplexcells,thatis,Algorithm 1 ,tothiscombinedcelltoobtainthepathsegmentsbetweentheaccesspointsontheouterboundary.Inthethirdstep,forallopenboundaryparts,weselectthecenterpositionofeachopenboundarypartasitsaccesspoint.Thenweconstructpathsegmentsbetweenthesenewaccesspointsandallotherexistingaccesspoints. AsshowninFigure 3-13 A,acistheshortestpathwhenauserwantstogothroughtheseopencells.Whenanopencellisthetargetobjectinaquery,thebestpositiontoleadusersisthecenteroftheopenboundary.Thus,inthethirdstep,thecenterpointofeachopenboundaryisselectedasanaccesspoint.Asanexample,inFigure 3-13 B,dandearetwoaccesspointsoftheregionsAandBrespectively.WhenauserstandinginthedoorbofthecombinedcellwantstogotoregionB,thebestwayforheristhesegmentbe. 53

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3.2.4Connecters Aconnecterisacellthatconnectsdifferentlevelsoroorsinabuilding.Aconnectercanbeastair,anelevator,apaternosterelevator,oranotherobjectthatcanbeusedtoreachdifferentoors.Thelocationwhereaconnecterandaoormeetformsanaccesspointinthisconnecter.Figure 3-4 Dshowsanexampleofaconnecter.Thevedotsrepresentaccesspointsandindicatethatthisconnecterconnectsveoors.Thereareonlytwodirectionsinaconnecter:upstairsanddownstairs.Onceauserknowsthenumberofhiscurrentoorandthenumberofthedestinationoor,heimmediatelyknowsthedirectiontothedestinationoor.Thus,weignoretheshapeofconnectersandassumethateverytwooorsarestraightlyreachable.Thenthepathsegmentsinaconnecterarethepredenedsegmentsconnectingeachpairofaccesspoints(seeFigure 3-6 C).Denition 5 givestheformaldenitionofaconnecter. Denition5. Aconnectercoisgivenasatupleco:=(id,m,a1,...,am,t,s1,...,st)withm1andt=(m)]TJ /F7 7.97 Tf 6.59 .01 Td[(1)m 2.Itrepresentsageometricstructurethatconnectsalloorsinabuilding.Thecomponentiduniquelyidentiesthiscell.Thecomponentmdenotesthenumberofaccesspointsofthiscell.Thecomponentsa1,...,amarevaluesofthespatialdatatypepoint,represent(thelocations,thatis,thexy-coordinates,of)theseaccesspoints,andformtheconnectinglocationsbetweentheconnecterandtheoors.Thecomponenttkeepsthenumberofimplicitpathsegmentsinthiscell,ands1,...,strepresentthesesegments.Eachsegmentdescribesthetransitionfromoneoortoanotheroor. 3.2.5Accessibility Animportantissueofthewayndingprocessistheaspectofaccessibilityofarchitecturalcellsintheindoorspace.Forexample,whileanemployeeinabuildingmayhaveaccesstocertainofcerooms,theseroomsmightbeinaccessibletoacustomer.Anotherexampleisaconstructionsitethatpreventspeoplefromwalkingthrougha 54

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name accessibility d1 8am-2pmd2 8am-5pmd3 8am-5pmd4 8am-5pmd5 8am-4pmd6 8am-5pmd7 8am-1pmd8 8am-5pmd9 8am-5pm name accessibility length d1d2 8am-2pm 3d2d3 8am-5pm 5d2d6 8am-5pm 12d2d7 8am-1pm 4d2d8 8am-5pm 15d3d4 8am-5pm 9d3d5 8am-4pm 4d3d6 8am-5pm 7d3d7 8am-1pm 2d3d8 8am-5pm 9d4d5 8am-4pm 9d5d6 8am-4pm 3d6d7 8am-1pm 9d6d8 8am-5pm 2d7d8 8am-1pm 10d7d9 8am-1pm 6d8d9 8am-5pm 8ABC Figure3-14. Mapsandpathsegments.A)themapofasimpleshoppingarea.B)theavailabletimeslots.C)thegeneratedpathsegments. corridorandforcesthemtobypassit.Inourmodelwecontroltheaccessibilityofcellsbyassigningaccessibilityattributestobothaccesspointsandshortestpathsegments. Theaccessibilityinanaccesspointorinashortestpathsegmentiscontrolledbyatimestampindicatingwhenanaccesspointorashortestpathsegmentisaccessible.Figure 3-14 Ashowsthearchitecturalmapofasmallmalltogetherwithitsaccesspoints(doors)anditsshortestpathsegments.TheaccessibilitytimesareshowninFigure 3-14 BforeachdoorandinFigure 3-14 Cforeachshortestpathsegment.Weseethattheaccessibilityofthedoorsd3andd4aswellasthepathd3)]TJ /F8 11.955 Tf 12.86 0 Td[(d4isgivenbetween8amand5pm.Onlyduringthisperiodoftime,userscanenterthemusicstorethroughthedoord3andleavethemallthroughthedoord4byusingthepathd3)]TJ /F8 11.955 Tf 12.57 0 Td[(d4.Thereasonwhywealsoassignanaccessibilityattributetoeachpathsegmentisthattheaccessibilityoftheinteriorofapathsegmentcannotbecontrolledbyitstwoendpoints.Forexample,apathsegmentmaynotbeaccessiblebecauseofaconstructionsitewhileitstwoendpointsareaccessible.Thismeansthatanaccessibilityattributeonlyforaccesspointsoronlyforpathsegmentswouldbeinsufcient. Therelationshipbetweentheaccessibilityofaccesspointsandtheaccessibilityofpathsegmentscanbesummarizedbythreeobservations.First,ifanaccesspointis 55

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inaccessible,allitsemanatingpathsegmentsareinaccessible,andviceversa.Itdoesnotmakesensethatanyofitsincidentpathsegmentsisaccessiblesincetheaccesspointcanneverbereached.Second,ifanaccesspointisaccessible,thereisatleastoneemanatingpathsegmentthatisaccessible,andviceversa.Inotherwords,anaccessibleaccesspointmusthaveatleastonepathsegmentthatleadstoit.Third,eveniftwoendpointsofapathsegmentareaccessible,thepathsegmentcanbeinaccessible.Thisindicatesthatapathsegmentcanbeblockedalthoughitstwoendpointsareaccessibleduetootherpathstraversingthem. 3.3TheiNavModel:ShortestPathRouting Navigationisaprocessthatsuccessfullyandefcientlyleadsusersfromasourcetoadestinationthatuserswanttoreach.Usually,itincludestwomainrequirements.First,itshouldbeabletondthemostappropriateroutestodestinations.Themostappropriateroutesmaybetheshortestrouteswithrespecttotime,theshortestrouteswithrespecttodistance,therouteswithoutpayingfees,oranyotherroutesaccordingtosomeuserrequirements.Second,navigationshouldbeabletoprovideclearandaccuratedescriptionsforpathssothatuserscangettodestinationsbyfollowingthedescriptions.Inthissection,wewillexplainhowwecanobtaintheshortestroutebasedontheimplicitpathsegmentsconstructedinSection 3.2 .InSection 3.4 wewillintroduceasimplenavigationlanguagetogenerateclearandusefuldescriptionsforroutes. 3.3.1TheDirectPathGraph Themostefcientmethodtocalculatepathsistheapplicationoftheshortestpathalgorithmtographs.Inourcase,weobtainanappropriategraphfornavigationbyassemblingtheaccesspoints,intermediatepoints,andshortestpathsegmentsobtainedfromAlgorithm 1 intoaso-calleddirectpathgraph(DPG).Denition 6 speciessuchagraphthatwewilluseforrunningashortestpathalgorithm. Denition6. AdirectpathgraphG=(V,E)isagraphwhichreectsallpossibleshortestpathconstructionsinagivenindoorspacescenario.Visasetofaccesspoints 56

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AB Figure3-15. Anexampleofnavigation:A)TheindoorspacescenariofromFigure 3-14 Aincludingthecurrentposition(cp)ofauser.B)ThecorrespondingDPG andintermediatepoints,andEVVisthesetofshortestpathsegmentsobtainedfromAlgorithm 1 InSection 3.2 ,wehavedividedtheindoorspaceintoseveralnon-overlappingcellsandhaveshownhowtoconstructtheimplicitpathsegmentsaccordingtotheshapesofthecellsandthelocationsoftheaccesspoints.ADPGforanindoorspaceiscomposedoftheaccesspoints,intermediatepoints,andpathsegmentsderivedfromallcellsembeddedinthisspace.Thereasonwhywecallitdirectpathgraphisthateveryedgeinthegraphrepresentsastraightpassagethatisfullyinsideitscorrespondingcell.Figure 3-15 BshowsthedirectpathgraphfortheindoorspaceintheFigures 3-14 Aand 3-15 Arespectively.Thenodesandedgesinsideeachdashedovalbelongtothesamecellintheindoorspace. ADPGhasseveralnicepropertiesinheritedfromtheshortestpathsegments.First,aDPGrepresentsthewholeinfrastructureoftheshortestpathsegmentsofanindoorspacescenario.Therefore,apathfromthecurrentlocationtoatargetlocationcanbeobtainedbyapplyingtheshortestpathalgorithmtoaDPG.Second,edgesinaDPGrepresentpathsegmentsinanindoorspaceandarestraightlinesbetween 57

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accesspoints,orbetweenaccesspointsandintermediatepoints.Therefore,everyedgerepresentstheshortestpathbetweenanytwoconnectednodes.Third,anytwolocationsbetweentwoconnectednodesinthegrapharevisiblefromeachotherintheindoorspaceandcanreacheachotherwithoutencounteringanobstaclelikeawall.Hence,thenexttargetonarouteduringnavigationcanalwaysbeseenbyauser.Fourth,thelengthofeachedgeinaDPGisstoredintheattributelengthforeachpathsegmentandcalculatedbytheEuclideandistancebetweenthesegmentendpoints.ThelengthinformationfortheedgelabelsoftheDFGinFigure 3-15 BstemsfromthetableinFigure 3-14 C. 3.3.2NavigationThroughtheDirectPathGraph Usually,usersareinterestedinndingapathtoacertaindestinationlikeanofceorastore.Forexample,inFigure 3-15 A,theymightaskHowcanIgettothemusicstore?.TheproblemnowisthatthenodesinaDPGaretheaccesspointsandintermediatepointsintheindoorspace.Thisimpliesthatthereisnoexplicitobjectinformationaboutrooms,corridors,stairs,andsoonrepresentedinthisgraph.Thus,aDPGcannotbedirectlyusedforansweringnavigationqueries.Butinthissubsection,wewillshowhowwecanneverthelessleverageaDPGtondthedesiredshortestpaths. ADPGcontainsalltheaccesspoints,intermediatepoints,andpathsegmentsforacertainindoorspace.SincethecurrentpositionandthetargetofausermightnotbeidenticaltoanyofthenodesintheDPG,ourrststepistodetermineasuitablestartingnodeandasuitabletargetnodeinaDPG.Forrepresentationpurposes,wehavedividedtheentireindoorspaceintoseveralnon-overlappingcells,andforeachcellwekeepitsaccesspointsexplicitlyanditsintermediatepointsimplicitlyastheendpointsoftheshortestpathsegments.Thus,welocatethesurroundingcellregardingthecurrentpositionoftheuserandobtainthepertainingaccesspointsofthiscell.Wechoosetheaccesspointorintermediatepointthatisnearesttothecurrentpositionandsetit 58

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Figure3-16. Routeplanningwherethestartingnodeisanintermediatepoint. asthestartingnodefortheshortestpathalgorithm.Forexample,inFigure 3-15 Athecurrentpositioncpoftheuserismarkedbyatriangle.Theclothingstoreisdeterminedasthesurroundingcellofthecurrentposition.Itsaccesspointsared8andd9;therearenointermediatepoints.Thenearestaccesspointtothecurrentpositionisd9,whichishencethestartingnode.Astartingnodecanalsobeanintermediatepoint.AsshowninFigure 3-16 ,theintermediatepointsinthiscellarev3,v6,v9,v14,andv15.Forthecurrentpositionmarkedbyatriangle,thestartingnodechosenistheintermediatepointv6. Choosingthenearestaccesspointorintermediatepointasthestartingpointdoesnotmeanthatauserneedstogotothisstartingpointatthebeginning.Itisonlyusedtorepresenttheuser'scurrentpositionanddeterminetheaccesspointwheretheuserhastoleavethesurroundingcell.ForourexampleinFigure 3-15 A,weassumethatauserwantstogotototheshoeBstorefromhiscurrentlocationmarkedbythetriangleintheclothingstore.Weknowthatd9isthenearestaccesspoint,andwendoutthatd9)]TJ /F8 11.955 Tf 12.55 0 Td[(d8)]TJ /F8 11.955 Tf 12.55 0 Td[(d6istheshortestpathfromd9totheshoeBstore.However,thedirectpathfromthecurrentpositiontod8isshorter,andhencetheusercanavoidthepathsegmentd9)]TJ /F8 11.955 Tf 12.01 0 Td[(d8insidethecell.IncasethattheuserwouldliketogototheshoeAstore,welearnthattheshortestpathfromd9totheshoeAstoreisd9)]TJ /F8 11.955 Tf 12.55 0 Td[(d7)]TJ /F8 11.955 Tf 12.55 0 Td[(d2.Theaccesspointd9isthesuitablepointtoenabletheusertoleavethecellimmediately.Inamoregeneralsituation,iftheaccesspointwheretheuserhastoleavethecellisnotvisiblefromtheuser'scurrentpositionandhencecannotbedirectlyapproachedonastraightwayduetoobstacleslikewalls,weapplythestrategyfromtheSections 3.2.2.3 and 3.2.2.4 59

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andcomputetheshortestpathintheinteriorofthecellfromthecurrentpositiontothisaccesspoint.Againtheintermediatepointsofthisshortestpathwillbeconcaveverticesofthecell.Therstpathsegmentfromthecurrentpositiontotherstintermediatepointwillnotbeoneofthedeterminedshortestpathsegments.However,theshortestpathfromtherstintermediatepointtotheaccesspointwheretheuserleavesthecellwillconsistofdeterminedshortestpathsegments.Fromtheaccesspointwheretheuserleavesthecell,theshortestpathwillcorrespondtotheonecomputedfromtheoriginalstartingnode. Thewaywedeterminethetargetnodeisdifferentfromthedecisiononthestartnode.Usually,ifwewanttoknowthewaytoatargetcell,wejustneedtondtheshortestwaytoanyofitsaccesspoints(e.g.,doorsandsomeopenings)onitsboundary.Thus,allaccesspointsontheboundaryofthetargetcellarepotentialtargetnodes.Forexample,inFigure 3-15 ,ifourtargetcellisthemusicstore,thend3,d4andd5arethepotentialtargetnodes.Sincetheshortestpathalgorithmisabletoreturntheshortestpathsfromthestartingnodetoanyothernodeinagraph[ 15 ],werunthisalgorithmforallpotentialtargetnodesanddeterminethatnode(accesspoint)astargetnodewiththeshortestdistancefromthestartingnode. Duringtheshortestpathalgorithm,weneedtochecktheaccessibilityofpathsegmentsbecausesomeoftheedgesmightnotbeaccessible.Forexample,inFigure 3-14 C,theaccessibilityofallpathsegmentsiscontrolledbytheaccessibilityofitstwoendpoints.Ifweassumethatthecurrenttimeis3pm,thesegmentd7)]TJ /F8 11.955 Tf 12.51 0 Td[(d9willnotbetakenintoaccountforanyshortestpathcomputationinthiscasesincetheaccessibilityofd7islimitedtotheperiodfrom8amto1pmonly. Somemodelsrepresenteachobjectbyasinglenodeonly,asshowninFigure 3-3 A.Thispreventsthemfromprovidingshortestpaths.Assaidbefore,inFigure 3-3 A,thepathfromdoor2toroom106thatthesemodelswillprovideisshownbythesolidline.Thisisacircuitousandunrealisticpath,andtheproblembecomesseriouswhen 60

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room105isverylarge.Ourmodelovercomesthisproblembyviewingeachobject(cell)ingeneral,bytakingintoaccountitsgeometricstructure,andbyrecordingallpossiblepathsegmentsinit.Thus,ourmodelcanprovideamuchmoreappropriatepathfromdoor2toroom106,asshownbythedashedlineinFigure 3-3 A. 3.4TheiNavModel:ASimpleNavigationLanguage Aclearanddetailedroutedescriptionisanimportantcriterionforthequalityofanavigationsystem.RoutedescriptionsprovidedbyGPSbasedoutdoornavigationsystemsareusuallymadeupofthenames,lengths,andturnsofroads.Whilewecanalsoderivelengthandturninformationinanindoorspacefromtheaccesspointsandintermediatepoints,wedonothaveexplicitroadsinanindoorspacebutonlyimplicitandinvisiblepathsegments.Inparticular,theimplicitpathsegmentswecreatedonothavespecicnamesthatwecoulduseinroutedescriptions.Thus,itisimpossibletoprovideroutedescriptionsthataresimilartothoseGPSsystemsprovide.Inthissection,wewillintroduceasimplenavigationlanguagethatisdesignedfortheiNavsystem.Wedonotclaimthatournavigationlanguageisanappropriateenduserlanguageandsatisesallthelinguisticandcognitiverequirementsneededforsuchalanguage.Butitcouldbethetargetlanguagetowhichanenduserlanguageistranslated.Section 3.4.1 discussesnavigationdescriptionsintheiNavsystemandthelinguisticmeansweneedforthem.Section 3.4.2 explainsourmethodtoautomaticallygeneratenavigationdescriptionsfromadeterminedshortestpath. 3.4.1NavigationDescriptions IntheiNavmodel,weuseimplicitandinvisiblepathsegmentstodescriberoutes.Duetotheirinvisibilityandlackofname,wecannotusethesepathsegmentsdirectlyforthedescriptionofaroute.Instead,weusethesequenceofconnectionpoints(i.e.,accesspointsandintermediatepoints)oftheidentiedshortestpathinordertodescribetheroutetheuserhastowalk.Thisispossiblesincetheconnectionpointsphysically 61

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Figure3-17. Thenavigationdescriptions existandarevisibletotheusers.Forexample,inFigure 3-14 A,theinstructiontoindicatethepathfromd2tod7is: walkxmeterstod7incorridor; wherethevaluexisthelengthofthesegmentd2)]TJ /F8 11.955 Tf 12.9 0 Td[(d7,andcorridoristhecellthatcontainsthispathsegment. Inourmodel,connectionpointsandpathsegmentshavetheveryimportantpropertythatanytwoconsecutiveconnectionpointsthatarelinkedbyapathsegmentarevisiblefromeachotherandnotimpededbyobstacleslikewalls.Thus,userscaneasilyndthenextconnectionpointintheinstruction.However,ifanintermediatepointisinvolved,wecannotusethesameinstructionasabovetodescribethepath.Afeasiblewaytodescribesuchapathistouseanglesanddirections.Forexample,assumingtheusercomesfromroom2,theinstructionforthepathfromatobinFigure 3-17 is: turn25totheright,andwalk20metersincorridor; where25isthedegreeoftheangle2,and20metersisthelengthofthesegmenta)]TJ /F8 11.955 Tf 11.8 0 Td[(b. Fromtheabovetwoexamples,wecanlearnthatdifferentwordsplaydifferentrolesinnavigationdescriptions.Forexample,thetermsrightandleftareusedtoindicateachangeofdirection,andcorridordenotesalandmark.Accordingtotherolesofdifferent 62

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words,weclassifythemintofourtermcategories.Topologicaltermslikeinspecifythetopologicalrelationshipsbetweenusersandobjects.Directionaltermsspecifythedirectionsusershavetotake.Wedistinguishabsolutedirections,knownascardinaldirections,likeeastandwest,aswellasrelativedirectionslikerightandleft.Cardinaldirectionsareseldomusedinanindoorspacebutmuchmoreimportantinoutdoorspaces.Weapplyrelativedirectionsinournavigationdescriptions.Therelativedirectiondependsontheorientationofthesurroundingspaceandtheuser'scurrentpositionandwalkingdirection.Landmarktermspointtodistinguishablesignsorobjectsthatcanhelpusersndinglocations.Forexample,thesignsofroadnamesarethemostimportantlandmarksinanoutdoorspace.Inanindoorspace,thedoorofaroomoraroomitselfcanbeconsideredasadistinguishablelandmark.Sincethelandmarksareusedtoshowusersthewaytotheirdestination,theyhavetobedistinguishableandtobeeasilyfound.Inourmodel,thelandmarkscorrespondtothenamesofcells,andwemakesurethateverylandmarkweprovideisvisibletousers.Metrictermsaddressquantitativeaspectsincontrasttotheaforementionedtermcategoriesthatfocusonqualitativeaspectsandonlyprovideageneraldirection.Weusemetrictermstoprovideprecisequantitativeinformationsothatusersareabletoobtainmoredetailedinformationoftheroutes.Inthepreviousexample,thelengthofthecurrentpath(20meters)andthedegreeofanangle(25)arethemetricterms. WeusethescenarioshowninFigure 3-18 toexplainhowweobtainthedirectionalandmetrictermsofanavigationdescription.Assumeu)]TJ /F8 11.955 Tf 12.12 0 Td[(visthecurrentpathsegmentandv)]TJ /F8 11.955 Tf 12.01 0 Td[(wisthenextpathsegmentinthesequenceofapath.Theangleiscalculatedbythelawofcosinesas2abcos(180)]TJ /F13 11.955 Tf 12.55 0 Td[()=a2+b2)]TJ /F8 11.955 Tf 12.55 0 Td[(c2,wherea,b,andcarethelengthsofu)]TJ /F8 11.955 Tf 12.48 0 Td[(v,v)]TJ /F8 11.955 Tf 12.48 0 Td[(w,andu)]TJ /F8 11.955 Tf 12.48 0 Td[(wrespectivelyinFigure 3-18 .ThecalculationofthedirectionisbasedonLemma 3 (withoutproof). Lemma3. Assumeu,v,andwarepoints,u)]TJ /F8 11.955 Tf 12.19 0 Td[(visasegment,andwelookfromutov(vectorviewu!v).Therelativedirectionbetweenu)]TJ /F8 11.955 Tf 11.95 0 Td[(vandwisdenedasfollows: 63

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Figure3-18. Thecalculationofanglesanddirections (i)wistotherightofu)]TJ /F8 11.955 Tf 11.96 0 Td[(vif,andonlyif,(v.x)]TJ /F8 11.955 Tf 11.96 0 Td[(u.x)(w.y)]TJ /F8 11.955 Tf 11.96 0 Td[(u.y)+(w.x)]TJ /F8 11.955 Tf 11.95 0 Td[(u.x)(u.y)]TJ /F8 11.955 Tf 11.96 0 Td[(v.y)<0(ii)wistotheleftofu)]TJ /F8 11.955 Tf 11.95 0 Td[(vif,andonlyif,(v.x)]TJ /F8 11.955 Tf 11.96 0 Td[(u.x)(w.y)]TJ /F8 11.955 Tf 11.96 0 Td[(u.y)+(w.x)]TJ /F8 11.955 Tf 11.95 0 Td[(u.x)(u.y)]TJ /F8 11.955 Tf 11.96 0 Td[(v.y)>0(iii)wisonu)]TJ /F8 11.955 Tf 11.96 0 Td[(vif,andonlyif,(v.x)]TJ /F8 11.955 Tf 11.96 0 Td[(u.x)(w.y)]TJ /F8 11.955 Tf 11.96 0 Td[(u.y)+(w.x)]TJ /F8 11.955 Tf 11.95 0 Td[(u.x)(u.y)]TJ /F8 11.955 Tf 11.96 0 Td[(v.y)=0 Forexample,inFigure 3-18 ,letusassumethatu)]TJ /F8 11.955 Tf 12.29 0 Td[(visapathsegmentinroom1,v)]TJ /F8 11.955 Tf 12.59 0 Td[(wisapathsegmentinroom2,visanexplicitaccesspoint,wisanintermediatepoint,andthecoordinatesofu,v,andware(2,1),(1,3),and(4,2)respectively.Thenthelengthofu)]TJ /F8 11.955 Tf 12.8 0 Td[(visa=p (2)]TJ /F3 11.955 Tf 11.96 0 Td[(1)2+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(3)2=p 5,thelengthofw)]TJ /F8 11.955 Tf 12.8 0 Td[(visb=p (4)]TJ /F3 11.955 Tf 11.96 0 Td[(1)2+(2)]TJ /F3 11.955 Tf 11.96 0 Td[(3)2=p 10,andthelengthofu)]TJ /F8 11.955 Tf 12.07 0 Td[(wisc=p (2)]TJ /F3 11.955 Tf 11.95 0 Td[(4)2+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)2=p 5.Accordingtothesevalues,theanglecanbeobtainedbyapplyingtheformulacos(180)]TJ /F13 11.955 Tf 11.95 0 Td[()=a2+b2)]TJ /F9 7.97 Tf 6.59 0 Td[(c2 2ab=5+10)]TJ /F7 7.97 Tf 6.59 0 Td[(5 2p 50=p 2 2.Finally,weobtainthatequalsto135. Next,weneedtodeterminetherelativedirectionbetweenthesegmentu)]TJ /F8 11.955 Tf 12.39 0 Td[(vandthepointw.AccordingtoLemma 3 ,wedeterminetherelativedirectionbycomparingtheresultof(v.x)]TJ /F8 11.955 Tf 12.22 0 Td[(u.x)(w.y)]TJ /F8 11.955 Tf 12.22 0 Td[(u.y)+(w.x)]TJ /F8 11.955 Tf 12.23 0 Td[(u.x)(u.y)]TJ /F8 11.955 Tf 12.22 0 Td[(v.y)with0.Since(1)]TJ /F3 11.955 Tf 12.22 0 Td[(2)(2)]TJ /F3 11.955 Tf -449.26 -23.91 Td[(1)+(4)]TJ /F3 11.955 Tf 12.1 0 Td[(2)(1)]TJ /F3 11.955 Tf 12.1 0 Td[(3)=)]TJ /F3 11.955 Tf 9.3 0 Td[(5<0,weseethatwisontherightsideofthesegment(vector)u)]TJ /F8 11.955 Tf 11.04 0 Td[(v.Therefore,ifauserwalksfromutovandwantstogotow,sheneedstoturnrightby135.Thenalnavigationdescriptionalongthepathsegmentsu)]TJ /F8 11.955 Tf 11.95 0 Td[(vandv)]TJ /F8 11.955 Tf 11.95 0 Td[(wis: (1)walk2meterstovinroom1; (2)turn135totheright,andwalk3metersinroom2. 64

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3.4.2AutomaticGenerationofNavigationDescriptions Inthissection,weshowhownavigationdescriptionscanbeautomaticallygeneratedbyleveragingthefourtermcategoriesintroducedinSection 3.4.1 .InSection 3.2 ,wehavedistinguishedtheexplicitlyknownaccesspointsandtheimplicitlygeneratedintermediatepointsasthesegmentendpointsfoundinshortestpaths. Therstnodeistheuser'scurrentlocation.Consideringthattheusercanbeanywhereintheindoorspace,thecurrentlocationmaynotbeanaccesspointoranintermediatepoint.Ifthereareonlytwonodesinvolved,thesecondnodemustbeanaccesspoint.Iftherearemorethantwonodes,thesecondnodecanbeanintermediatepoint.Thismeansthatifwereadnodesonebyone,thentherstnodeisneitheraccesspointnorintermediatepoint.Themodiedgureisattached.Inthisgure,PfromstatusSisthesecondnodeinthegeneratedpath.Ashortestpathisalwaysboundedbytwoaccesspointsindicatingthestartnodeandthetargetnode.Intermediatepointscannotbeendpointsofashortestpath. Wehaveintroducedhowtocalculatethedirectionaldescriptionsandvaluedescriptions.Nowwewillshowhowtogeneratetheformalnavigationlanguagebyusingthefourdescriptions.InSection 3.2 ,welearnthattherearetwokindsofnodes,theaccesspointswhicharetheexplicitexitsofeachroom,andtheintermediatepointswhicharegeneratedduringtheprocessofthepathconstruction.Thegenerationofthenavigationdescriptiondependsonthetypesofthenodes.Ifthenextnodeisanaccesspoint,thenitcanbeconsideredasavisiblelandmark,otherwise,moredetailedinformationshouldbeprovidedinordertogettothenextnode.Figure 3-19 showsthestatustransformationdiagramofthenavigationlanguage: inputdata:asequenceofnodesindicatingthegeneratedshortestroute.wisthecurrentnodethatisreadfromthesequence. 65

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Figure3-19. Thestatustransformationofthelanguagegenerationanddirections S:Sisthestartstatusofthelanguagegeneration.Inthisstatus,thesystemreadtherstthreenodesfromtheinputdata,andcheckthetypeofthethirdnodew.Ifwisanaccesspoint,gotothestatusP,elsegotothestatusU. P:Inthisstatus,thecurrentnodeisanaccesspoint.Thecorrespondinginstructiongeneratedinthisstatusisbasedonthecurrentnode.Thenreadthenextnode,checkitstype,andgotothecorrespondingstatus. U:Inthisstatus,thecurrentnodeisanintermediatepoint.Thecorrespondinginstructiongeneratedinthisstatusisbasedonthecurrentnode.Thenreadthenextnode,checkitstype,andgotothecorrespondingstatus. E:Theendoftheprocess. InthestatusP,thecurrentnodeisanaccesspoint,whichcanbeusedasavisiblelandmark.Assumingthepreviousnodeispvandthecurrentnodeispw,thentheinstructiongeneratedinthisstatusis: Walklengthtopwinroomx lengthisavaluedescriptionindicatingthedistancebetweenpvandpw.Itiscalculatedbyapplyingtheformulap (pvx)]TJ /F8 11.955 Tf 11.96 0 Td[(pwx)2+(pvy)]TJ /F8 11.955 Tf 11.96 0 Td[(pwy)2.pwisalandmarkandinisatopologicaldescriptionindicatingtherelationbetweenthispathsegmentandroomx. 66

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Table3-1. Thepathchainfromroom106toroom103 startnode nextnode length object currentlocation d9 10 room106 d9 d7 14 room104 d7 d3 13 corridor InthestatusU,thecurrentnodeisanintermediatepoint,whichisnotadistinguishablelandmarkinindoorspace.Thus,thenavigationinstructionshouldcontainmoredetailedinformationtohelpuserstoreachthatposition.Assumingtheprevioustwonodesarepu,pvandthecurrentnodeispw,theinstructiongeneratedinthisstatusis: Rotatedegreetothedirection,andwalklengthinroomx Degreeandlengtharevaluedescriptions.Thedegreeindicatesthedegreeoftheanglethattheusershouldtake,andthedistanceprovidethedistancesbetweenpvandpw.directionisadirectionaldescriptionwhichtellsuserstoturnrightorleft,andinisatopologicaldescriptionindicatingtherelationbetweenthispathsegmentandroomx. Forexample,Table 3-1 showstheshortestpathfromthelocationmarkedbythetriangleintheclothingstoretothemusicstoreinFigure 3-14 A.Thenaldescriptiongeneratedforthispathis (1)walk10meterstod9inroom106; (2)walk14meterstod7inroom104; (3)walk13meterstod3incorridor. Theinformationofpathsegmentsisimplicitlyindicatedineachinstruction.Forexample,thepathsegmentindicatedbytherstinstructionisfromthecurrentlocationtod9,andthepathsegmentshownbythesecondinstructionisfromd9tod7. Figure 3-17 givesanexamplethatsomeintermediatepointsareinvolved.Assumingthedegreeofangle1,2and3are50,25and30respectively,thedescriptionofthepathfromroom2toroom9is: 67

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(1)rotate50totheright,walk30metersinthelobby; (2)rotate25totheright,walk20metersinthecorridor; (3)walk10meterstod14inthecorridor. 68

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CHAPTER4SUPPORTINGRANGEQUERIESININDOORSPACE 4.1ApproachestoSupportingRangeQueriesinOutdoorSpace Tothebestofourknowledge,currentstudiesmainlyfocusonrangequeriesinoutdoorspacesbasedontheEuclideanornetworkdistances,butrarelyanyapproachisproposedforindoorspaces. 4.1.1StationaryRangeQueries Astationaryrangequeryasksforinterestingobjectswithinagivendistancewithrespecttoastaticquerypoint.Approachessupportingstationaryrangequeriescanbesubdividedintotwocategories:Euclidean-basedapproachesandnetwork-basedapproaches.Euclidean-basedapproachescheckthedistancesbetweenobjectsbytheirrelativepositionsinthespacewhilenetwork-basedapproachesconsidertheiractualreachabledistances. 4.1.1.1Euclidean-basedapproaches ThemosttypicalEuclidean-basedapproachesproposedin[ 29 41 62 ]useR-Trees[ 25 ],R+-Trees[ 91 ]andR-Trees[ 7 ]forqueryprocessing.Figure 4-1 showsanexampleofR-treewiththecapacityof3entriespernode.Neighboringpoints(e.g.a,binFigure 4-1 )aregatheredintoonenodeandrepresentedbyaminimumboundingrectangle(E1).Thennodesarerecursivelyclustereduntiltherootnodeisformed.Theprocessofarangequerystartsfromcheckingtheroot,toeachnodeoverlappedwiththerangecirclewhosecenteristhequerypointqandtheradiusistherequireddistance,sothatallqualifyingobjectswithinthiscirclecanbeobtained. Duetoitssimplicityandefciency,R-treebecomesapopularindexstructureforspatialdatabases.However,themajordisadvantageofR-treeisthatobjectsareevaluatedbasedontheirEuclideandistances,nottheactualreachabledistances.However,inindoorspace,objectsareseparatedbywalls,whichmadeitimpossible 69

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Figure4-1. R-tree. Figure4-2. RangeNetworkExpansion. toevaluatethedistancesbetweenobjectsbytheirEuclideandistances.Thus,theEuclidean-basedapproachesarenotsuitableforindoorspace. 4.1.1.2Network-basedapproaches Morepracticalsolutionstosupportrangequeriesarebasedonspatialnetworks.In[ 65 ],Papadiasandzhangproposeanetwork-basedapproach,calledrangenetworkexpansion(RNE),tosupportstationaryrangequeries.Itrstselectsallsegmentswhosedistancestothequerypointarelessthanthegivenrangedistance.Thenforeachselectedsegmentsegment,checkthecorrespondingnodesontheR-tree.Forexample,inFigure 4-2 ,assumingsegments(n1,n2),(n1,n3),(n3,n4),and(n4,n5)arethesegmentsinsidethegivenrange.ThenthegraynodesintheR-treearethevisitednodes,andthenalobjectsoftherangequeryareb,dandf. UsingVoronoidiagramisanotherchoiceofthenetwork-basedapproaches.In[ 64 ],Okabeetal.rstintroducetheapproachtoconvertthenetworkintoaVoronoi 70

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diagram.ThisideaisthenusedbyKolahdouzanandShahabiintheirVoronoi-basedNetworkNearestNeighbor(VN3)modelproposedin[ 39 ].InVN3,theentirenetworkisdecomposedintoseveralsmallerparts,eachofwhichisrepresentedbyarst-orderVoronoidiagram,calledNVP.ThentheintraandinterdistancesbetweenthecellsoftheVoronoidiagramarepre-computedforthedistanceevaluationpurpose.ThePINEmodelpresentedbySafaetal.in[ 78 ]usesarstorderVoronoidiagramtopartitionthenetworkintoseveralsub-networksandcomputestheirinterdistances.ThenamethodsimilartotheRNEapproachisappliedtocomputetheintradistancesofthesesub-networks.In[ 78 ],theauthorsmentionthatPINErequireslessdiskaccesstimeandlessCPUtimethanVN3. Thenetwork-basedapproachesprovidetheabilityofndingtheexactnetworkdistancesbetweenobjects.Thus,theyaresuperiortotheEuclidean-basedapproacheswhenconsideringtheirpracticalutilization.Moreover,Theseapproachescansupportnotonlytherangequeries,butalsosomeothercommonqueriesliketheKNearestNeighbor(KNN)queriesandClosestPairqueries.Generally,theRNEapproachhasbetterperformancethantheVoronoi-basedapproachesforrangequeries.However,iftheobjectdistributionisdense,theVoronoi-basedapproachesareusuallybetter,becauseRNEhavetoretrievemoredatafromthedatabasetoobtainthenalobjects. 4.1.2ContinuousRangeQueries Approachesforstationaryrangequeriesassumethatthequerypointisalwaysinastaticposition.Infact,usersareofteninamovingstatuswhentheyissueanavigationquery.TheimportanceofcontinuousqueriesismentionedbySistlaetal.in[ 84 ].TheyproposeadatamodelcalledMovingObjectsSpatio-Temporal(MOST)forpresentingmovingobjectsindatabasesystems,whichfocusesonthesyntaxandsemanticsofthequeryprocessing.Inrecentyears,moreandmoreapproachestosupportingcontinuousrangequerieshaveappearedwhichalsocanbeclassiedintoEuclidean-basedapproachesandnetwork-basedapproaches. 71

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4.1.2.1Euclidean-basedApproaches AnearlierEuclidean-basedapproachintroducedin[ 87 ]rstselectsseveralsamplepointsonthepath,andthenperformsstationaryrangequeriesoneachselectedsamplepoint.Thisapproachsuffersfromaperformanceandaccuracyproblem:themoresamplepointsweselect,themoreaccuratebutpoorertheperformancewillbe.Thelesssamplepointswetake,thebetterbutlessaccuratetheperformancewillbe. Inordertoovercometheperformanceandaccuracyproblem,laterapproachesextendtheconceptoftime-parameterizedqueriesproposedin[ 92 ]tobettersupportcontinuousqueries.Thetime-parameterizedapproachincrementallycalculatesthenextresultandtheobjectsthatmightaffectitbasedonthecurrentresult.TheresultofTime-parameterizedqueryisintheformof,whereRisthecurrentresult,TistheperiodtimewhenRisvalid,andCisthesetofobjectsthatmightaffectRafterthetimeofT.Thetime-parameterizedapproachcansupportthecontinuousrangequerybutthecostofitsincrementalcalculationislarge.In[ 103 ]Taoetal.proposeanideaofndingsplitpointswhichindicatethechangesofthequaliedobjects,tosupportcontinuousnearestneighborqueries.AsshowninFigure 4-3 ,aandbaretwointerestingobjectsonthewayfromthestartStotheendE.Thesplitpoint,whichisdeterminedbythepositionofaandb,indicatesthatthenearestobjectexchangesfromatob.Theapproachproposedin[ 102 ](showninFigure 4-4 )rstdeterminesallpossiblyqualiedobjectsbydrawingaregionaccordingtothegivenrange(indicatedbysolidlines).Then,thesplitpoints(s1,s2,...,s7)aredeterminedbytheintersectionsbetweenthepathandthedashedcircles.Thedistancesfromthecenterpointofeachcircletoitsrelatedsplitpointsarethegivenrange. 4.1.2.2Network-basedApproaches Tothebestofourknowledge,theapproachproposedin[ 102 ]istheonlyoneforsupportingcontinuousrangequeriesbasedonnetworkdistances.ThisapproachismotivatedbytheideaofcontinuousK-nearestneighborqueriesproposedin[ 40 ] 72

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Figure4-3. Continuousqueryusingsplitpoint Figure4-4. Euclidean-basedcontinuousrangequery and[ 77 ].AsshowninFigure 4-5 ,thisapproachrstselectsasegmentthatcontainsnointersectionnode(e.g.AB).Thenitincrementallychecksthenodesinnearbysegmentswithinagivenrange.ThesplitpointsaredeterminedbycheckingtheconditionSp[A,x]>dis[A,B],wherexisaninterestingobject,dis[A,B]isthedistancebetweenAandB,andSp[A,x]iscomputedbytheformulagivenrange)]TJ /F8 11.955 Tf 9.3 0 Td[(dis[A,x].Forexample,assumingthegivenrangedistanceis10meters,wecanlearnthatSp[A,o1]=10)]TJ /F3 11.955 Tf 12.21 0 Td[(3=7.Therefore,thesplitpoints o1onthesegmentABindicatesthevalidityoftheobjecto1. InEuclidean-basedapproaches,theobjectsareevaluatedbasedontheirEuclideandistancesandnotontheactuallyreachabledistances.Consideringthewallsseparatingtheindoorspace,itisimpossibletoapplyEuclidean-basedapproachestoindoorspace.Thenetwork-basedapproachesprovidetheabilityofndingtheexactnetworkdistances.Thus,theyaresuperiortotheEuclidean-basedapproacheswhenconsideringtheirpracticalutilization.However,noneofthemcanbedirectlyappliedtoindoorspacesforaccuratecomputationofrangequeries.Inthispaper,wewillproposetwoapproachestosupportingcontinuousrangequeriesinindoorspacesrespectively. 73

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Figure4-5. Network-basedcontinuousrangequery 4.2SupportingStationaryRangeQueries InChapter 3 ,wehavedemonstratedhowtoconstructthenetworkstructureforindoorspace.Theconstructionofthenetworkhastwoimportantfeaturesthatcannicelysupportrangequeries: AllcellsinindoorspacesarerepresentedbyatleastonenodeinaDPG. Foranynodeinthenetwork,wecanndtheshortestdistancesbetweenthisnodeandallothernodes. Therstfeaturemakessurethatnocellwillbemissinginthegraph,andthesecondfeatureensuresthatitispossibletondallthecellsthatarereachablewithinagivendistance.Inthissection,weproposetheIndoorRangeNetworkExpansion(IRNE)algorithmtosupportstationaryrangequeries,andtheIndoorRangeRegionDivision(IRRDalgorithmtosupportcontinuousrangequeriesinindoorspace.ThesetwoalgorithmsutilizetheideaoftheIncrementalNetworkExpansionproposedin[ 65 ],andextendsittottheindoorenvironment. 4.2.1IncrementalNetworkExpansion TheIncrementalNetworkExpansion(INE)approachpresentedin[ 65 ]isanetwork-basedapproachusedfortheNearestNeighborSearch.Itsbasicideaistoexpandthenetworksegmentsfromthequerypointtoobtainthequalifyingobjects.AsshowninFigure 4-6 ,assumingauserwantstondthenearestobjectofinterest(denotedbyblackrectangles)fromq,INEwillrstchecktheobjectsinthesegment 74

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Figure4-6. TheIncrementalNetworkExpansion n1n2.Then,thesegmentn1n7emanatingfromn1willbechecked,followedbythesegmentsn2n3andn2n4emanatingfromn2.Theexpansionprocesswillcontinueuntilthenearestqualifyingobjectisfound.Finally,p5isreturnedasthenalresult. 4.2.2IndoorRangeNetworkExpansion OncewehavebuilttheDPGfortheentireindoorspace,wecanprocessstationaryspatialgraphrangequeriesbyapplyinganextendedversionoftheINEalgorithmtothegraph.Theunderlyinggraphproblemisavariationofthesingle-sourceshortestpathproblembyDijkstrawithanumberofimprovementssummarized,forexample,in[ 94 ],inwhichwehavetondtheshortestpathsfromasourcevertextoallotherverticesinthegraph.However,inourcase,weareonlyinterestedinthoseshortestpathsthatsatisfyaparticulardistanceconditionandfurtherthematicconditions.Inotherwords,ourgoalistoprunethesearchgraphaspartoftheDFGasmuchaspossible.Anexampleisthestationaryrangequeryndallshoestoreswithin10meters.Heremetersisthedistanceorrangecondition,andshoestoreisathematiccondition.Sinceauserscurrentpositioncp,whichhastheroleofaquerypointandsourcevertex,isnotrestrictedtothelocationsonthesegmentsoftheDPG,itisdifculttodeterminethestartingsegment.Inordertoovercometheproblem,theIndoorRangeNetworkExpansion(IRNE)algorithm(showninAlgorithm 3 )takestwostepstocomputestationaryspatialgraphrangequeries. 75

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Intherststep,ifthequerypointcpisnotonanysegmentoftheDFGG,wetemporarilycomputetheshortestpathsfromcptoallaccesspointsthatbelongtothecellccontainingcp.Inthisway,weconnectcptoGandobtainatemporarilynewgraphG0.WeleveragetheoperationExtendDFG(line1)toaccomplishthis.ItdeploysconceptsthataresimilartothoseinAlgorithm1.Thatis,ifdirectsegmentsfromcptotheaccesspointsofcarenotpossiblesincetheywouldintersecttheboundaryofc,wehavetoidentifytheconcavepointsthatareneededasintermediatepointsontheshortestpaths. Inthesecondstep,weperformtheactualspatialrangesearchbyleveragingamodicationofDijkstrasalgorithm.OurextendedDFGG0isaweighted,undirected Algorithm3: IndoorRangeNetworkExpansion Input: (i)DPGG;(ii)pointcpdenotingthecurrentpositionoftheuser;(iii)cellccontainingcp;(iv)predicatepthatisappliedtoanodeofGandchecksathematiccondition;(v)networkdistancerrepresentingthesearchrange Output: setOofobjectsofinterest;arraypredthatenablesthedeterminationoftheshortestpathsfromcptoanyobjectofinterestinreverseorder G0=ExtendDPG(G,c,cp);//G0=(V0,E0);1 Q fcpg;ndist[cp] 0;pred[cp] ;;2 foreachv2V0)-222(fcpgdo3 ndist[v] 1;pred[v] ?;4 foreache2E0do5 visited[v] false;6 whileQ6=;do7 Findavertexv2Qsuchthatndist[v]=minfndist[u]ku2Qg;8 Q Q)-221(fvg;9 foreachu2Adj[v]do10 ifnotvisited[(v,u)]then11 visited[(v,u)] true;12 ifndist[u]>ndist[v]+weight[(v,u)]then13 ndist[u]landist[v]+weight[(v,u)];14 pred[u]=v;15 ifndist[u]rthen16 Q Q[fug;17 ifap(u)^p(u)then18 O O[fug;19 76

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graphwithnonnegativeedgeweights.Inline2weinitializeasetQ,whichhastheroleofapriorityqueue,withthecurrentpositioncp.Further,wemakeuseoftwoarraysndistandpred.Thearrayndiststoresthenetworkdistancebetweencpandallqualifyingobjectsofinterest.Wesetthedistancebetweencpanditselfto0.Thearraypredallowsustoconstructtheshortestpathbetweencpandallobjectsofinterestinreverseorder.WesetthepredecessorofcptoundenedandinitializethesetOofobjectsofinterestwiththeemptyset.Inlines3and4weinitializethenetworkdistancesfromcptoallothernodesinG0withthevalue1andsetthepredecessorofallthesenodestoundened.ThenwemarkeachedgeofG0asunvisited(line5). ThepriorityqueueQkeepsallthosenodesthatalreadybelongtothesetOofobjectsofinterestandfromwhichwecanhopetondfurtherobjectsofinterest.Thegoalistondtheshortestpathtotheseresultnodestoincreasethechancetondotherobjectsofinterest.Hence,thesamenodecanappearseveraltimesinQ.Thereasonisthatifwendashorterpathfromcptoanodev,thiscanhavepositiveimpactonadjacentnodesofvthatcouldnowqualify.ThealgorithmterminatesassoonasQisempty(line6).FromQwealwaystakethenodevwiththesmallestnetworkdistancefromcp(line7)andremoveitfromQ(line8).Foreachadjacentnodeuofvwecheckwhethertheedge(v,u)isunvisited(line10).If(v,u)hasalreadybeenvisitedbefore,weignoreusincegoingfromutovandbacktoucannotcontributetondingashortestpath.Iftheedge(v,u)hasnotbeenvisitedbefore,wemarkitasvisited(line11).Thenwecheckwhetherweareabletolowerthecurrentnetworkdistancefromcptoutoanewminimumwhenwetraverse(v,u)(line12).Aslongasthenetworkdistancefromcptouisgreaterthanthegivenranger,uisnotpartofthesolution.Butifthenetworkdistanceiswithinthesearchrange(forexample,within10meters)(line13),u,whichcanbeanaccesspointorintermediatepoint,isinsertedintoQ(line14)tolatercontinuethesearchfromhereforpotentiallyotherqualifyingnodes,andthepredecessorofuissettov(line15).WhetheruisaddedtothesetOofobjectsofinterestdependsontwo 77

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furtherconditions(line16).First,umustbeanaccesspointandnotanintermediatepoint.Thisisdeterminedbythepredicateap.Second,additionalthematicconditions(forexample,thetargetmustbeashoestore)musthold.Wehaverepresentedthesethematicconditionsbyagivenpredicatep. Fortherststep,thetimecomplexityofconstructingpathsegmentsfromthecurrentpositiontotheaccesspointsofthecellcisO((m+n)3)wherenisthenumberofaccesspointsandmisthenumberofconcaveverticesinthecurrentcell.Infact,thevaluesofmandncanbeassumedtoberelativelysmallinpractiseandwillespeciallybemuchsmallerthanV0.Forthesecondstep,inpractise,wecanexpectthatonlyasmallpartofG0willbetraversed.Butdependingonthesizeoftheranger,intheworstcase,thecompletegraphG0hastobeexplored.Thisbringsusthenbacktothesingle-sourceshortestpathproblem.Asisknownfromtheliterature,thetotalrunningtimeofDijkstrasalgorithmisO(jE0j+jV0jlogjV0j)ifwestoretheverticesofthepriorityqueueQinaFibonacciheap[28],andO(jE0jlogjV0j)ifwestoretheseverticesinaregularbinaryheap[29].Inbothcases,ndist[v]isthekeyofanodev. AsanexampleoftheIRNEalgorithm,welookbacktoFigure 4-7 AthatshowsanindooroorplanwiththepathsegmentsofitsDFGandacurrentpositioncprepresentedbythetriangleintheclothingstore.Figure18bshowstheexplicitandextendedDFGG0aftertherststepofAlgorithm 3 withallnetworkdistancesbetweenadjacentnodesasedgeweights.Thetemporarypathsegmentsfromcptoallaccesspointsintheclothingstoreareshownbydashedlines.Forreasonsofsimplicity,thisDFGdoesnotcontainintermediatepoints.Wefurtherassumethatalltheaccesspointsareaccessible.Forthequeryndalltheshoestoreswithin10meters,inthesecondstepoftheIRNEalgorithm,Qrstcontainsthenodecp,whichisthenremovedfromQ.Theadjacentnodesofcpared8andd9withndist[d8]=5andndist[d9]=3,andtheedgesfromcptothetwonodesaremarkedasvisited.BothnodesareinsertedintoQsincetheirnetworkdistancefromcpislessthanorequaltor=10.However,both 78

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nodesdonotbelongtothesetOofobjectsofinterestsincetheyarenotdoorsofshoestores.ThenodewiththesmallestnetworkdistanceinQisd9,whichweremovefromQ.Wecheckalladjacentneighborsofd9whicharecp,d8,andd7.Theedgebetweend9andcphasalreadybeenvisitedandisthusignored.Theedgebetweend9andd8ismarkedasvisitedbutleadstoatotalcostof11fromcptod8whichislargerthanrandlargerthanthecurrentndist[d8].Theedgefromd9tod7ismarkedasvisited.Weobtainndist[d7]=ndist[d9]+weight[(d9,d7)]=3+6=910.Hence,d7isinsertedintoQ.Butd7isnotinsertedintoOsinceitisnotthedoorofashoestore.ThenextnodeinQwiththesmallestnetworkdistancefromcpisd8whichisremovedfromQ.Itsadjacentnodesarecp,d9,d7,d2,d3,andd6.Theedges(d8,cp)and(d8,d9)havealreadybeenvisited.Theotherthreeedgesaremarkedasvisited.Forthenoded7wemaintaintheoldvaluendist[d7]=9andinsertd7intoQwhereitalreadyis.Forthenodesd2andd3weobtainndist[d2]=20andndist[d3]=17whichallexceedthethresholdvalue10.Forthenoded6weobtainndist[d6]=710.Therefore,d6isinsertedintoQ.Sincethisnodeisanaccesspoint(ap(d6)=true)ofashoestore(p(d6)=true)namedshoeB,d6isinsertedintoO.ThenextnodeinQwiththesmallestnetworkdistancefromcpisd6whichisremovedfromQ.Itsadjacentnodesared8,d7,d2,d3,andd5.Theedge(d6,d8)hasalreadybeenvisited.Theotheredgesaremarkedasvisited.Forthenoded7wemaintaintheoldvaluendist[d7]=9andinsertd7intoQwhereitalreadyis.Forthenodesd2andd3wegetndist[d2]=19andndist[d3]=14whichbothexceedthethresholdvalue10.Forthenoded5wegetndist[d5]=10;hence,weinsertd5intoQ.Thenoded5isalsoanaccesspointtoashoestorenamedshoeBsothatitisinsertedintoO.Next,d7istakenfromQ.Asonecanseeeasily,alledgesemanatingfromd7haveeitheralreadybeenvisited,ortheshortestpathfromcptothemexceedsthethreshold10.Thesameholdsforthenoded5thatisremovedfromQ.Finally,Qisempty,andthesolutionisO=fd5,d6gleadingtotheshoestoreshoeB. 79

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Figure4-7. TheIndoorRangeNetworkExpansion 4.3SupportingContinuousRangeQueries Traditionally,approachesofsupportingcontinuousrangequeriesforoutdoorspacestrytondthesplitpointsontheroadatwhichthequalifyingobjectswillchange.Whenthequerypointmoveswithintheintervalbetweentwosplitpoints,theresultoftherangequeryremainsthesame,andwhenthequerypointpassesoveronesplitpoint,thesetofthequalifyingobjetschangesaccordingly.Sincewedon'thaveexplicitpathsinindoorspaces,theapproachofndingsplitpointscannotbeappliedtocell-basedindoorstructures.TheworstwaytoanswercontinuousrangequeriesinindoorspacesistoapplytheIRNEalgorithmforeachpositionthattheusermoves.However,thiswaywillconsumesalotofenergyduetoCPUcomputation,andnotfeasibleforarealworldproduct.Inthissection,weintroduceourapproach,calledIndoorRangeDivision(IRD)forcontinuousrangequeriesthatcandramaticallyreducethecomputationtimeandquicklyanswercontinuousrangequeries. Inindoorspaces,whentheusergoesfromoneroomtoanother,sheisactuallypassingthroughmultipleexits.Weobservedthat,althoughthepositionsofthequerypointsarechangingdynamically,thepositionsoftheexitsandtheobjectsarestatic,andthedistancesbetweentheexitsandtheobjectsarexed.Onceweobtainthequalifyingobjectsthatarereachablefromoneexit,wecanmonitorthedistancebetweenthequerypointandtheexittodeterminethequalifyingobjectsthatcanbereachedfromthequerypoint.Forexample,assumingcellchasoneexita,andthequalifyingobjectthatcan 80

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AB Figure4-8. Examplesofndingsplitpoints.A)Acellwithoneaccesspoint.B)Acellwithmultipleaccesspoints. bereachedfromaisOb.Weobservedthatfromanypointpinc,ifthedist(a,p)islessthanthevalueofrange)]TJ /F8 11.955 Tf 12.14 0 Td[(dist(a,Ob),wecangettoObthroughawithinthegivenrange.Here,thevalueofrangeanddist(a,Ob)arexed.Tondthequalifyingobjectsforanypointincwhosepositionischangingdynamically,weonlyneedtomonitorthedistancesfromptoeachexitinc. Inordertoobtainqualifyingobjectsbasedonthedistancesbetweenthequerypointandtheexits,oneofourtasksistondtherelationshipsbetweenthedistance(i.e.thedistancebetweenthequerypointandtheexits)andthequalifyingobjectsthatarereachablefromtheexits.AsshowninFigure 4-8 A,therectanglerepresentsacellcwithoneaccesspointa.Assumingthegivenrangeis10,wecanlearnthatO1andO2aretwoqualifyingobjectsthatcanbereachedfroma.Itisclearthatfromanypositionpinsidec,whosedist(a,p)islessthan4andgreaterthan2,theusercangettoO1within10,andifthedist(a,p)islessthan2,bothO1andO2arereachablefromp.Here<0,2>and<2,4>aretwointervalsindicatingtherelationshipsbetweendist(a,p)andthequalifyingobjects.Ifdist(a,p)isbetween0and2,thequalifyingobjectsforpareO1andO2,andifdist(a,p)isbetween2and4,onlyO1isthequalifyingobjectfromp.WecanrepresenttherelationshipsbetweenthedistancesandthequalifyingobjectsbyusingtheintervalrepresentationthatisdenedinDenition 7 .Theresultfor 81

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Figure 4-8 Acanberepresentedas<0,2>fO1,O2g;<2,4>fO1g.Ifthecellhasmorethanoneaccesspoint,eachaccesspointwillhaveitsownintervalrepresentation.InFigure 4-8 B,therearetwoaccesspointsaandbinthecell.TheintervalrepresentationsforaandbareshowninTabel 4.3 Denition7. Assumingristhegivenrange,andthelistofthequalifyingobjectsthatcanbereachedfromaccesspointaisrepresentedas: ......,rifO1,...,Oi,...,On>g,...,fO1,O2,...,Oig,...,fO1g Ourapproachtosupportcontinuousrangequeriescontainstwophases.Intherstphase,whentheuserentersacell,wegeneratetheintervalrepresentationsforallaccesspointsinthecell.Thestphaseisdoneonceforallwhentheuserentersthecell.Inthesecondphase,wekeeptrackingthedistancesbetweenthequerypointandtheaccesspoints.Thenalqualifyingobjectsareobtainedbylookinguptheintervalrepresentationsgeneratedintherststep. Algorithm 4 showsourapproachtocomputetheintervalrepresentationsforallaccesspointsinonecell.TherearefourstepsinAlgorithm 4 .TherststepistoconstructtheextendedDPGbyaddingtheshortestpathsfromthequerypointtoalltheexitsinthecell(line1).Inthesecondstep,wecomputethequalifyingobjectsthatcanbereachedfromeachexitbyusingthealgorithmforansweringstationaryrangequeries(line3-4).ThecomplexityofthisstepisdiscussedinSection4.2.Inthethirdstep,thelistoftheobjectsaresortedaccordingtotheirdistancestotheexit(line5).Thenearestobjectfromtheexitwillbetherstelementinthesortedlist.Finally,theintervalrepresentationforeachaccesspointisgeneratedatthefourthstep(line7-14).Weomitthedetailedexplanationheresincethisstepissimplyacomputationbasedon 82

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Algorithm4: IndoorRangeDivision Input: (i)DPGG;(ii)pointcpdenotingthecurrentpositionoftheuser;(iii)cellccontainingcp;(iv)predicatepthatisappliedtoanodeofGandchecksathematiccondition;(v)networkdistancerrepresentingtheseachrange Output: setOofobjectsofinterest; G0=ExtendDPG(G,c,cp);//G0=(V0,E0);1 resultSet[][]=;;2 nalList[][]=;;3 foreachaccesspointapincdo4 result[ap][]=IndoorRangeNetworkExpansion(G0,cp,c,p,r);5 sort(resultSet[ap]);6 foreachaccesspointapincdo7 intdist prev=r;8 intk=resultset[ap].length;9 fori=0tokdo10 nalList[ap][i].min=r-dist prev;11 nalList[ap][i].max=r-resultSet[ap][i].dist;12 forj=0toido13 nalList[ap][j].obj.add(resultSet[ap][i].O);14 dist prev=resultSet[ap][i].dist;15 Denition 7 .ThecomplexitytocomputetheintervalrepresentationisO(nm2),wherenisthenumberofaccesspointsandmistheaveragenumberofqualifyingobjectsthatcanbereachedfromeachaccesspoint.Infact,thevaluesofmandncanbeassumedtoberelativelysmallinpractiseandwillespeciallybemuchsmallerthanthenumberofnodesinthegraph. Iftheusermovesfromonepositiontoanotherinsidethesamecell,weonlyneedtocomputethedistancefromthequerypointtoeachexitandndthecorrespondingobjectsbylookinguptheintervalrepresentations.Ifthecellhasmorethanoneaccesspoint,thenalqualifyingobjectswillbegeneratedbycombiningallthequalifyingobjectsobtainedfromeachintervalrepresentation.Forexample,assumetheuseriscurrentlyincellcshowninFigure 4-8 B.Table 4.3 showstheintervalrepresentationofFigure 4-8 byusingAlgorithm 4 .Whentheusermovestopositionpinc,wecancomputethedistancefromptoaandb(i.e.dist(a,p)anddist(b,p)).Assumingthegivenrangeis10,anddist(a,p)is3anddist(b,p)is2,bylookinguptheinterval 83

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accesspoints intervalrepresentation a <0,2>fO1,O2g;<2,4>fO1g b <0,1>fO3,O4g;<1,4>fO3g Table4-1. IntervalrepresentationforaccesspointsinFigure 4-8 B representationsinTable 4.3 ,wecanlearnthatthequalifyingobjectscanbecomputedbyfO1g[fO3g,whichisfO1,O3g. Becauseweonlycomputethedistancesbetweenquerypointsandexits,theCPUtimeforansweringthecontinuousrangequeriesaredramaticallyreduced.InlaterChapters,wewillshowtheperformanceoftheIRDalgorithmbycomparingtheCPUtimewiththeIRNEalgorithm. 84

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CHAPTER5SUPPORTING3DROUTEPLANNINGININDOORSPACE Forindoorrouteplanningmodelsthemostimportantusersarepedestrians.However,besidesthepedestrians,thereareotherpotentialusers,suchaspeopleinwheelchairs,smallindoorautos,robots,andcartscarryingproducts.Inpreviousliterature,severalindoornavigationmodelshavebeenproposedtosupportrouteplanninginindoorspaces,butusuallyonlyforpedestrians.Therehavebeenafewmodelswhicharealsodesignedforpeopleinwheelchairs.However,tothebestofourknowledge,noneofthemareabletoprovideauniversalmodelthatcansupportrouteplanningforallkindsofusers. Themodelsdesignedforpedestriansapproximatethepositionofusersbypoints.Thisapproximationisnotsuitableforotherkindsofusers,asfortheseusers,otherinformationneedstobestudiedtocheckaccessibility,andthisiscomplicatedthanpoint-basedmodels.Anotherproblemisthatmostoftheexistingmodelsignoretheobstaclesthatcanbefoundwithinindoorspaces.Therearemanypotentialobstaclesforindoorspaces,suchastables,chairs,decorativeplants,andstepsthatareusedtoconnectlayersinthesameoor.Thoseobstacleswillaffecttheamountofindoorspacesthatisaccessible,especiallywhentheuserhasalargesize.Inaddition,mostoftheexistingmodelsare2D-basedmodels.Withoutthedataofthethirddimension,thesemodelsarenotabletohandlethestructureoftheentirespace.Asaconsequence,theycannotchecktheaccessibilityfromdifferentdimensions. WehaveproposedaLEGOmodel[ 104 105 ]thatcanprovidefeasibleroutesfordifferenttypesofusersforindoorspaces.Oursolutionconsistsofatwo-phasemethodthatincludesarepresentationphasefollowedbyanaccessibilitycheckingphase.Intherstphase,aLEGO-basedrepresentationmodelisproposedtorepresentthe3Dstructureoftheindoorspace.TheentirespaceisapproximatedbyseveralLEGOcubeswiththreemaincategories:plane cubesrepresentingplanes,stair cubesrepresenting 85

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stairsandobstacle cubesrepresentingobstacles.Byusingthesecubestheentireindoorspacecanberepresentedindetail.Inthesecondphase,theLEGOcubesaremergedintoblocksbyusinganovelstrategytoobtainthemaximumaccessiblewidths,heightsandlengthsindifferentlocations.Thesevaluesarethenusedinordertoevaluatetheaccessibilityofdifferentusers.Finally,aLEGOgraphrecordingalltheaccessibilityinformationisbuilttoapplyefcientroutesearchalgorithms.Thegeneratedrouteconsistsofasequenceofblocksthatareaccessiblebytheuser. 5.13DRepresentationoftheIndoorSpace Anindoorspaceiscomposedofdifferentkindsofcells.Eachcellunitisrestrictedbyalotofarchitecturalconstraintssuchasoors,wallsandexits.Thestructureofacellcouldbeverycomplicatedifthecomponentsofthecellhavecomplexstructures.Forexample,apyramidceilingmayaffecttheaccessibilityofcertainplacesinsidethecell,andsmallstairsontheoorareonlysuitableforpedestrians.Inthissection,wewilldiscussourapproachtorepresentingthe3Dstructureoftheindoorspace. 5.1.1LEGO-BasedApproximation OurrepresentationmodelisinspiredbyLEGO'stoybricks.Byusingdifferentshapesofthetoybricks,weareabletobuildvariousmodelsofcellspaces.Thus,ourapproachtomodel3Dspacesincellsistoapproximatetheentirespacebycubes.Forsimplicity,weonlyusecubes,calledLEGO cubes,withthesamebasicareaasourrepresentativeunits.EachLEGO cubehasitsownheightandtypeaccordingtotheobjectitrepresents.ALEGO cubeisnon-dividable,anditcanonlyrepresentoneobjectorapartofanobject.The2Dprojectionofallthecubesinonecellisagridwithequalsizedsquares.Figure 5-1 showsanexampleofacubeinacube-shapedcellandtwocubeswithdifferentheightsinapyramid-shapedcell.AsshowninFigure 5-1 B,theheightofeachcubedependsonthedistancebetweentheoorandtheceilinginthecorrespondingarea,whichthecubelocates.Thesizeofthesquarecontrolsthegranularityoftherepresentation.Thebasicareaistheminimumunitforrepresentingthe 86

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ABCD Figure5-1. Examplesofcubes.A).Acubeinacellwithregularshape.B)Cubesinapyramidshapedcell.C)The2Dprojectionofaoorwithrectangularshape.D)The2Dprojectionofaoorwithtriangularshape. 2Dstructureofthecells.Thesmallerthesquareis,themorethedetailsoftheindoorspacecanberepresented. Figure 5-1 AisthesimplestexampleofrepresentingtheindoorspacebyusingLEGO cubes.However,inmostcases,thestructuresincellsarenotsosimple.Irregularlyshapedoorsandceilings,obstacles,andthecomplicatedshapesofcellswillincreasethedifcultyoftherepresentation.Inourmodel,weclassifydifferentobjectsintothreemaincategories:planes,stairsandobstacles. TheformaldenitionoftheLEGO cubeisgiveninDenition 8 Denition8. ALEGO cubeLC:=(id,type,slope,w,h)isabasic3Dunitofrepresent-ingobjectsinindoorspace.idistheuniqueidentieroftheLEGO cube.typerecordsthetypeoftheLEGO cube.Thevalueofthetypecanbeobstacle cube,stair cubeorplane cube.slopeistheslopeofaplane cube.IfthetypeoftheLEGO cubeisobsta-cle cubeorstair cube,theslopevalueisalways1.wisthewidthofthebasalareaoftheLEGO cube.histheheightoftheLEGO cube. 5.1.1.1Theapproximationofplanes Whenaoorandaceilingareat,andthereisnoobstaclebetweenthem,itiseasytollouttheavailable3DspacebetweenthembyusingLEGO cubes.AsshowninFigures 5-1 CandD,the2Dprojectionoftherepresentationisagridstructure.Figure 5-1 Cisanexampleofacellwitharectanglularshape.Whentheboundaryof 87

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ABCDE Figure5-2. Thecuberepresentation.A)Thelongitudinalsectionoftherepresentationofaslopingoor.B)HowtodeterminethebasalareaofaLEGO cubeforaslopingoor.C)Aslopingceiling.D)Thelongitudinalsectionoftherepresentationofastair.E)3Drepresentationofhowtomodelastairwithcubes. thecellisnothorizontalorverticaltheboundaryisrepresentedbymultipleLEGO cubesconnectedbythecornersofthecubes'basicareas(asshowninFigure 5-1 D).Theresolutionoftheapproximationdependsonthesizeofthebasicarea.Ifwereducethesizeofthebasicarea,thespacecanberepresentedmoreprecisely. Whenaoororaceilingisnotat,theavailablespacecanbeapproximatedbyusingmultipleLEGO cubeswithdifferentheights.Theverticallocationofacell'sbasic/topareadependsontheaverageheightofthecorrespondingareaintheoor/ceiling.Slopingoorsorceilingsarerepresentedbyasetofgraduallyascending/descendingLEGO cubes.Figure 5-2 Ashowsthelongitudinalsectionoftherepresentationofaslopingoor.Figure 5-2 BshowsawaytodeterminetheverticallocationofthebasicareaofsuchaLEGO cube.Assumingtheverticallocationsoftheendpointsoftheslopingoorare0mand0.2m,thentheverticallocationofthebasicareaoftherepresentativeLEGO cubeis0.1m,whichistheaverageverticallocationofthetwoendpoints.Theapproximationforaslopingceilingisthesameasaslopingoor.AsshowninFigure 5-2 C,thetopareasoftheLEGO cubesaregraduallydescendingaccordingtotheshapeoftheceiling. Sometimes,aplanemayslopetoomuchforsomeuserstouse.Thisplaneisactuallyanobstaclethatisunavailableforthoseusers.Inourmodel,weuseathresholdfortheslopestocontroltheavailabilityoftheplanes.Iftheslopingofaplane 88

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exceedsthethreshold,thisplaneisconsideredasanobstacleandisrepresentedbyobstacle cubes. 5.1.1.2Theapproximationofstairs Similartotheapproximationoftheslopingplanes,astairisrepresentedbyasetofascendingordescendingLEGO cubes.ThewidthoftheLEGO cubecannotexceedthewidthofthecorrespondingsteps.Thus,eachstepisrepresentedbyoneormoreLEGO cubes.Theverticallocationsofthebasicareasaredeterminedbythelocationsofthecorrespondingsteps.Figures 5-2 Dand 5-2 Eshowsanexampleofthelongitudinalsectionandthe3Ddiagramoftheapproximation. 5.1.1.3Theapproximationofobstacles Obstaclesrefertotheobjectswhoseoccupiedareasarenotavailableforusers.Theycanbewalls,tables,chairs,andotherobjects.Whenanobstaclelaysonaoor,nomatterhowhighitis,thisareaisconsideredasinaccessibleforusersinwheelchairs.However,itmaybeaccessibleforpedestrians.Usually,differentobstacleshavedifferentshapes.However,theirrepresentationscanbeclassiedintothreetypes. Thersttypeistheobstaclesthataretoohightobepassedover.Typicalexamplesarethewalls,furnituresuchastables,andsomedecorationssuchaspottedowers.Amongtheseobstacles,someofthemreachtheceilingandsomeofthemstillhavefreespaceabovethem.Sincethefreespaceabovetheseobstaclescannotbeusedaspassages,itisactuallyunavailableforusers.Thus,theobstaclesofthistypearerepresentedbyLEGO cubeswhosetopareasreachtheceilings.Thismeansthespacefromtheoortotheceilinginthislocationisunavailable. Thesecondtypeistheobstaclesthatpedestrianscanpassover.Inourmodel,theseobstaclesareconsideredascurbs.Theyareaccessibleforpedestrians,butunavailableforusersinwheelchairstopass.Thesecurbsareconsideredassmallstairsandrepresentedbystair cubes. 89

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Thethirdtypeisobstaclesintheair.Thespacesbelowthemareavailableforusers.Thistypeofobstaclewilllimittheheightoftheavailablespacebelowthem.ThespacefromtheoortotheceilinginthisareawillberepresentedbytwodifferentLEGO cubes.Thebottomonerepresentstheavailablespaceandtheupperonerepresentstheobstacle.Thetopareaofthebottomcubewillbethebasicareaoftheuppercube. Sometimes,differentpartsofanobstaclemaybelongtodifferenttypes.Fortheobstacleswithcombinedshapes,werstdividethemintomultiplepartsaccordingtotheclassicationoftheshapes,andthenapproximatethembyusingdifferentstrategies. 5.2TheAccessibilityofDifferentTypesofUser Thegoalinourpaperistoprovidefeasibleroutesforalldifferentusers.Theaccessibilityisaffectedbymultiplethingssuchasthewalls,theexitsandtheobstaclesinsiderooms.Wewillintroducehowwechecktheaccessibilityoftheuser'swidth,height,andlength. 5.2.1TheMaximumWidths Inordertochecktheaccessibilityofthewidth,wehavetondthemaximumavailablewidthinallplaces.Themaximumwidthindifferentplacescanberestrictedbydifferentobstacles.Forexample,Figure 5-3 Aisthe2DprojectionofacellrepresentedbyLEGO cubes.Thewhite,blackandgreycubesrepresenttheavailablespace,obstaclesandstairsrespectively.Fromthegure,wecanlearnthatthemaximumaccessiblehorizontalwidthinthelocationofthecubea,bandcisthesame.Thismaximumwidthcanbeobtainedbymergingtheplanecubesinahorizontaldirectionuntilwemeetanobstaclecube. 5.2.1.1Blockswiththemaximumwidths Thecube-basedrepresentationintroducedinSection 5.1 dividestheentireindoorspaceintomultipleLEGO cubes.Eachcubehasitsownuniquetypeindicatingwhetheritispartofaplane,astair,oranobstacle.Usually,anobjectisrepresentedbyseveralconnectedLEGO cubes.Thus,ifaLEGO cubeisavailablefortheuser(e.g.apart 90

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ABC Figure5-3. Mergingcubestogeneratelargerblocks.A)Themaximumsizeisdeterminedbythelocationsoftheobstacles.B)Theapproachtogeneratelargerblocks.C)Theimprovementofthemergingprocess ofaplane),itsneighborcubesmayalsobeavailableforthisuser.BymergingsimilarLEGO cubes,weareabletondthemaximumavailablewidthforaparticulararea. AsshowninFigure 5-3 A,bymergingnearbycubes,themaximumrectangle-shapedblockcontainingthecubeaistherectanglewiththecornercubesof1,2,3and4.Inreality,thewidthofablockdependsonthedirectionofthemovements.Ifyoumoveinaverticaldirection,thewidthofthisblockcontains8cubes,andifyoumovealongthehorizontaldirectionthewidthoftheblockbecomes3cubes.ThemaximumblocksextendedfromoneLEGO cubemaybedifferentwhenusedtomergedirections.Forexample,themaximumblockcontainingcubecmergedalongthehorizontaldirectionistherectangle(1,2,3,4);whilealongtheverticaldirection,themaximumblockistherectangle(1,5,6,7).Therefore,inourmodel,thereareusuallytwoblocksextendedfromthesamecube. Beforeweintroduceourmergingstrategy,wewillrstdiscussseveralconditionsinvolvedinthemergingprocess.Firstly,onlyLEGO cubesofthesametypecanbemergedtogether.DifferenttypesofLEGO cubesrepresentdifferentkindsofobjectsthroughouttheindoorspace.Itisimpossibleforustowalkfromaplaneintoanobstacle,andforsomeusers,staircubesarenotaccessibletothem.Secondly,althoughtheheightsoftheLEGO cubesreecttheheightoftheavailablespaceinsidethecells,theaccessibleheightofacellisactuallycontrolledbytheheightoftheexits.Assumingthedefaultheightofonecellisthemaximumheightamongalltheexitsofthiscell,then 91

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theLEGO cubeshigherthanthedefaultheightcanbeaccessedwithoutrestriction.Therefore,theLEGO cubeshigherthanthedefaultheightcanbemergedtogether.FortheLEGO cubeslowerthanthedefaultheight,onlythecubeswiththesameheightcanbemergedtogether.Thirdly,thenumberofLEGO cubesusedtorepresentthecellswillaffecttheefciencyofthemergingprocess.IftherearealotofLEGO cubes,theprocessmaytakealongtime.Wealsoobservedthatblocksmergedfromdifferentcubesmaybethesame.Forexample,inFigure 5-3 A,theblockmergedfromthecubeaandfromthecubebisthesameblock.Thus,inordertoavoidduplicatesandimprovethemergingprocess,wecanreducethenumberofstartingcubes.AsshowninFigure 5-3 A,fromanyrandomlyselectedcube,themaximumwidthwecanobtainalwaysdependsontheboundarycubes,whicharenexttoothertypesofcubes.Thus,weonlychoosetheboundaryLEGO cubestobethestartingcubesinordertoperformthemerging. Ourmergingprocessinthehorizontaldirectionstartsfromaboundarycube.Thecubesthatarehorizontallyextendedfromthisstartingcubearemergedonebyonetoobtainthemaximumwidth.AsshowninFigure 5-3 B,thecubeswhicharehorizontallynexttocube1aremergedonebyoneuntilwereachtheboundary.Thisstepproducesatemporaryblockwiththemaximumwidthobtainedfromthestartingcube(e.g.,therectangle(1,2,3)).Then,allthecubesdirectlyabovethetemporaryblockarecheckedtoseeiftheyhavethesametypeandsatisfytheheightcondition.Ifso,theywillbemergedintothetemporaryblock.Forexample,thecubes4,5,6thataredirectlyabovetherectangle(1,2,3)canbemergedintotherectangle(1,2,3).Ifoneofthecubesdoesnotsatisfythiscondition,thecubeextensioninthisdirectionwillstop.Therefore,thecubesabove(4,5,6)cannotbemerged.Thecubesunder(1,2,3)aremergedbyusingthesamestrategy.Thenalblockgeneratedfromcube1istherectangle(4,6,18,16).Ifweapplythealgorithmalongtheverticaldirection,therectangle(10,16)will 92

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bethecorrespondingblockextendedfromcube1thatcontainsthemaximumverticalwidth. Theresultofthemergingprocessisasetofblocks.Therearethreerelationshipsbetweenanytwoblocks.Therstrelationshipisdisjointed.Forexample,inFigure 5-3 B,theblock(10,16)verticallyextendedfromcube1andtheblock(12,18)verticallyextendedfromcube3aredisjointed.Thesecondrelationisadjacent.Twoadjacentblocksshareapartoftheboundary(e.g.theblock(19,20)horizontallyextendedfromcube19,andtheblock(21,22,23,24)horizontallyextendedfromcube22areadjacent).Twoblockscanalsobeoverlapped.InFigure 5-3 B,theblock(10,16)andtheblock(4,6,18,16)areoverlapping.However,oneblockcannotbefullyinsideanotherblock.Thisisbecauseifoneblockisfullyinsideanotherblock,itcanalsobeextendedtoformtheouterblockbymergingwithnearbycubes.ItisalsoimpossibletohavetwoblockswithrelationshipssimilartoFigure 5-4 D,becausetheboundaryaoftheblockBinFigure 5-4 DcanbeextendedtomeettheboundaryboftheblockA. 5.2.1.2Theimprovementofthemerging Althoughthismergingstrategycangenerateblockswiththemaximumwidthsinthehorizontalandverticaldirectionsincells,itmayproduceunnecessaryblocksinsomesituations.Forexample,theirregularobstacleseparatingtwoavailablespacesinFigure 5-3 Cleadstoirregularboundarycubes.Accordingtothemergingstrategy,theblockextendedfromtheboundarycubeswillbeasetofadjacentblocksshownbytheboldlines. However,comparedtotheslashedarea,thesmallcubesneartheboundarydonotusuallycontributetotheavailablespace.Therefore,forsimplicityandefciency,whentheboundarybetweenanobstacleandanavailablespacehasazigzagshape,thisboundarycanbesimpliedintoastraightlinebeforeperformingthemerging. 93

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5.2.1.3Thealgorithm Algorithm 5 showshowwemergethecubestogenerateblockswiththemaximumwidthinahorizontaldirectioninoneroom.Theinputisamatrix,cubes[n][m],thatrecordsallthecubesinoneroom.Fromthebottomleftcube,cubes[0][0],wecheckthecubesonitsrightsideandndtherstcube,c1,withaPLANEtype(seeline4-5).Fromc1,werepeatedlyaddthecubeswithPLANEtypeontherightsideuntilwehitacubewithnon-PLANEtype(seeline6-10). Atthispoint,themaximumhorizontalwidthofc1isfound.Inline11-20,wecheckthetypesofthecubesmovingupwards.FromthelineofthecubesthathavebeenfoundinAlgorithm 5 ,line6-10,wecheckthewholelineofthecubesaboveit.IftheyarePLANEcubes,weaddthewholelineintotheblock.Similarly,line21-30isusedtocheckthecubesgoingdownwards. 5.2.1.4Connectorsbetweenblocks Oncewegeneratetheblocks,theroutesfromthestartingplacetotheusers'destinationcanbeconsideredastheblock-by-blockpaths.Theconnectionsbetweentwoblocks(calledconnectors)controlthemaximumaccessiblewidthandheightbetweenoneblockandthenext.Therearetwotypesofconnector,oneisusedtoconnecttwoblockswiththesametype,andtheotherisusedtoconnecttwoblockswhichareofdifferenttypes(e.g.,theconnectorsbetweenplanesandstairs). ConnectorsBetweenTwoBlockswiththeSameType AccordingtoSection 5.2.1.1 ,therearethreerelationshipsbetweentwoblocks,disjointed,adjacentandoverlapping.Whenthetwoblocksareadjacentoroverlap,userscanmovefromoneblocktotheotheriftheirconnectoriswideenough. Whentwoblocksareadjacent,theconnectorbetweenthemistheintersectedboundary.Forexample,thelineaistheconnectorbetweenblockAandBinFigure 5-4 A. 94

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Algorithm5: CubeMerge Input: Matrixcubes[n][m] Output: listarrayblocks ki down 0;ki up 0;kj down 0;kj up 0;1 fori 1tondo2 forj 1tomdo3 whilecubes[j][i].getType!=PLANEdo4 i++;5 ifi!=mthen6 ki down=i;7 while(i
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ABCD Figure5-4. Relationshipsbetweenadjacentblocks.A)Meet.B)overlap1.C)overlap2.D)impossiblerelationship. Whentwoblocksareoverlapping,theirboundarieswillalwaysbecrisscrossedasshowninFigure 5-4 BandC.Theconnectorbetweentwooverlappingblocksisthediagonaloftheintersectedrectangle.AsshowninFigure 5-4 BandC,thelineaisthemaximumconnectorbetweenblockAandB. ConnectorsBetweenAvailableSpaceswithDifferentTypes Theconnectorsbetweentwoblockswiththesametypecanreecttheaccessiblewidthsbetweentwoareas.However,whentryingtondthemaximumaccessiblewidthsbetweentwodifferenttypesofavailableareas(e.g.,planesandstairs),theblocksgeneratedbyapplyingthemergingstrategymayproduceincorrectmaximumconnectors,especiallywhentheboundariesarenothorizontalorvertical. Figure 5-5 showssomeexamplesofboundariesbetweenaplaneandastair.Thewhitepartistheavailableplaneandthegreypartrepresentstheareaofthestair.Theirregularlinesandtheblackcubesdenotetheaccessibleandinaccessibleboundariesbetweentheplaneandthestairrespectively.Therectangularblocksboundedbyboldlinesintheplanearetheblocksgeneratedbyapplyingthemergingstrategyontheboundarycubes.ThemaximumaccessiblewidthsbetweentheplaneandthestairforFigures 5-5 A,bandcaredenotedbythedashedlines.Fromthesegureswecanseethatnoneoftheblockscancapturethemaximumaccessiblewidthbetweentheplaneandthestair.Therefore,weareunabletoobtainthemaximumaccessiblewidthbetweenthesetwoareas. 96

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ABC Figure5-5. TheconnectorswiththemaximumwidthbetweenaplaneandastaircannotbefoundbyusingthebasicmergingAlgorithm Theseincorrectresultsarecausedbythemergingprocess.Themergingprocesswillstopwhenacubewithanothertypeismet.Iftheboundarybetweenaplaneandastairisnothorizontalorvertical,thisprocesswillnotbeabletogenerateablockcontainingtheentireboundarybetweenthetwoavailablespaces.Inordertocapturethemaximumaccessiblewidthbetweentwodifferentavailableareas,wehavetogenerateblockscontainingasmuchoftheboundaryaspossible. Ourapproachtosolvingthisproblemistogenerateblocksbymergingplane cubesandstair cubestogether.Thismeansthatifastair cube(plane cube)ismetwithmergingplane cubes(stair cubes),thisstair cube(plane cube)isalsomergedintotheblock.AsshowninFigure 5-6 A,Bistheblockcreatedbymergingboththeplane cubesandthestair cubes.TheblockAdenotedbythedashedlinesistheminimumboundingboxoftheaccessibleboundary.TheconnectorbetweenAandBisthediagonalc,whichreectsthemaximumaccessiblewidthbetweentheplaneandthestair. Thisapproachalsoworkswhentheboundaryisaffectedbyobstacles.AsshowninFigure 5-6 B,therectangle(1,2,3,4),(1,8,6,5)and(8,7)demonstratestheblocksproducedbyapplyingthenewapproach.Theconnectorsa,b,andccorrectlyreectthemaximumaccessiblewidthsindifferentdirectionsinthisscenario. ThedenitionofaconnectorisgiveninDenition 9 Denition9. AconnectorC=(loc1,loc2,height)isalistofconnectedcubesrepresent-ingthemaximumaccessiblewidthandheightwhenmovingfromoneblocktotheother. 97

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AB Figure5-6. Computingconnectorsbetweenaplaneandastair.A)Withouttheeffectofobstacles.B)Withtheeffectofanobstacle loc1andloc2arethecenterpointsofthetwoendcubesinthisconnector.heightistheminimumheightofthetwoblockssharingthisconnector. 5.2.2TheMaximumHeights Theaccessibleheightisthesecondconditionwehavetocheck.AlthoughtheheightsofLEGOcubescanreecttheheightsoftheavailablespaceinsidecells,theaccessibleheightofacellisactuallycontrolledbytheheightsoftheexits.Assumingthatthedefaultheightofonecellisthemaximumheightamongallitsexits,anyareahigherthanthedefaultheightcanbeaccessedwithoutrestriction.Therefore,inourmergingprocess,theLEGOcubeshigherthanthedefaultheightwillbemergedtogether.FortheLEGOcubeslowerthanthedefaultheight,onlythecubesofthesameheightcanbemergedtogether. Thisensuresthatallthecubesinoneblockeitherhavethesameheightorarehigherthanthedefaultheight.Inthecaseoftheformer,theheightofthecubesistheheightoftheblock,andinthelattercase,thedefaultheightbecomestheheightoftheblock.Foreachpairofadjacentoroverlappingblocks,themaximumaccessibleheightistheminimumheightofthetwoblocks. 5.2.3TheMaximumLengths Theprocessofcheckingaccessiblelengthsisverycomplicated.Turns,obstacles,anduser'ssizesallhaveanimpactonthemaximumaccessiblelength(examplesareshowninFigures 5-7 A,BandFigure 5-8 A).Inourmodel,weproposeanapproachto 98

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ABCD Figure5-7. Examplesshowingdifcultiestondtheoptimalroutes. provideuserswithafeasiblewaytotheirdestinations.Thegeneratedroutemaynotbeoptimal,butitisguaranteedtobefeasible. Thereareseveralreasonswhywecannotprovidetheoptimalroutes.Firstly,theshapesoftheusersmaybedifferent.Itishardtochecktheavailabilityforeverypartoftheobjectinallplaces.Forexample,inFigure 5-7 C,therectanglecannotpassthoughtheseobstacles.However,inFigure 5-7 D,althoughobjectAsharesthesamerectangularshape,itisabletogothroughthepath.Evenifwecanndawaytochecktheaccessibilityforeverypart,itisinefcientandunpracticaltodoso.Secondly,inarealworldscenario,usersmayprefertousemorecomfortablewaysratherthantheoptimalone.Forexample,itwillnotbearealisticrouteifusershavetomakeseveralattemptstondtherightangletomakeaturn.Thus,itisbettertoprovidearoutewithenoughspace. Asweknow,themovementsofdifferentusers'canbevary.However,ifwedeconstructthemovementsintosmallsteps,theycanbeclassiedintotwomaincategories:goingstraightandmakingturns.Tochecktheaccessibilityofthelengthforastraightpath,weonlyneedtocomparethelengthoftheobjectwiththelengthofthestraightpath.However,checkingtheaccessibilityforturnsisdifcult. Inourpaper,wewillrstintroduceanapproachthatcanchecktheaccessiblelengthsforallgeneralcasesforanindoorspace.Thenwewillrenetheapproachforspecialcases. 99

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5.2.3.1Theapproachforgeneralcases Inreality,nomatterhowsophisticatedthenalpathis,itisalwayscomposedofsegmentsandturns.Therefore,insteadofconsideringthewholeenvironment,wewillfocusoncheckingtheaccessibilityofoneturnfortheobject. Let'sstartfromasimplecase.Figure 5-8 Ashowsatypicalcorridorturn.Oneimportantobservationisthatiftheminimumboundingcircleoftheobjectcanbecontainedintheturningcorner(thegreyarea),thisobjectisabletomaketheturn.ThiscornerconceptcanbeappliedtoourLEGOrepresentationmodel.InSection 5.2.1 ,wediscussedwaysinwhichtondthemaximumaccessiblewidthsbymergingcubesintolargerblocks.Anytwoblocksmaybedisjointed,adjacentoroverlapping(asshowninFigure 5-4 ).Usersmayonlyneedtomaketurnswhentheyaregoingfromoneblocktoanotherthroughthecorrespondingconnector. Lemma4. GivenanobjectOandacornerareathatcanberepresentedbyarectan-gleA,theobjectcanaccessthiscorneriftheminimumboundingcircleofOcanbecontainedinA. Proof. AsshowninFigure 5-8 A,thepolygonArepresentsacornerareaandOrepresentstheobject.disthediameteroftheminimumboundingcircleofO.ItisclearthatiftheminimumboundingcirclecanbecontainedinA,thediameterofthecircledmustbelessthanthewidthandlengthofthecornerareaA.Therefore,theobjectOcanaccessthecornerA. Thescenarioforoverlappingblocks(asshowninFigure 5-4 C)issimilartothetypicalcorridorturns.Iftheminimumboundingcircleoftheobjectcanbecontainedintheoverlaparea,thisobjectisabletomaketheturnintothisconnector.However,forthecornerinthetypicalcorridorturn,theshapeandthesizeisrestrictedbythewalls,whilefortheoverlappedareaoftwoblocks,itsboundarymaynotberestrictedbyobstacles.Onepossiblesolutionistoextenditsboundarytondthemaximumcornerareas. 100

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AsshowninFigure 5-8 C,blocksAandBaregeneratedaccordingtothelayoutoftheobstacles.Inthisscenario,wecanextendtheboundaryoftheoverlappingareatoformalargeraccessiblespace,asindicatedbythedashedlinesinFigure 5-8 C.Infact,thisextensionprocessisunnecessary;becausethemergingprocessoftheLEGOmodelguaranteesthatanyblockwiththemaximumaccessiblespacewillbegenerated.Therefore,anypossibleoverlapareascanbecapturedbyourmodel.Forexample,theareaindicatedbythedashedboundaryinFigure 5-8 CistheoverlapareaoftheblockCandD.WecanndthemaximumalternativecornerareaofAandBbylookingforthelargestcornerareathatcontainsthecurrentone. Figure 5-4 Bshowsanotherkindofoverlaprelationshipbetweentwoblocks.Forthiskindofscenario,accordingtothelocationsofotherconnectors,usersmayormaynothavetomaketurns.TakingFigure 5-8 Easanexample,AandBaretwoadjacentblocks,andcistheconnectorbetweenAandB.AssumingaandbaretwoconnectorsconnectingAandotherblocks,ifoneusergoesfromatoc,shecangostraighttoB.However,ifshegoesfrombtoc,thenshewillprobablyneedtotakeaturn.Ourapproachtohandlethisscenarioistondthemaximumoverlapareaforthetwoblocks. Iftheminimumboundingcircleoftheobjectcanbecontainedinthemaximumoverlaparea,theobjectcangothroughthetwoblockswithoutanyproblem.Forexample,inFigure 5-8 EthegreypartistheoverlapareabetweenAandB.Accordingtothelocationoftheobstacles,thisareacanbeextendedtotheareaindicatedbythedashedlines.Ingure 5-8 F,sincetheminimumboundingcircleoftheobjectOcanbecontainedinthisextendedoverlaparea,OcansuccessfullygofromAtoB. ThesituationofthetwoadjacentblocksshowninFigure 5-4 AissimilartotheoverlappingblocksinFigure 5-4 B.AsshowninFigure 5-8 G,AandBaretwoadjacentblocks,andcistheconnectorbetweenAandB.AssumingaandbaretwoconnectorsconnectingAwithotherblocks,ifoneusergoesfromatoc,shecangostraighttoB.However,ifshegoesfrombtoc,thenshewillprobablyneedtotakeaturn.Tosimplify 101

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ABCD EFGH Figure5-8. Demonstrationsofcheckingaccessibilityindifferentscenarios thetwocases,wehaveobservedthatifweextendtheconnectorctoformalargerblock(asshowninFigure 5-8 H),thescenariobecomesthesameastheoverlappingblockswediscussedbefore.Therefore,wecanapplythesamestrategytochecktheaccessibilityoftheadjacentblocks. Asanextensionoftheminimumboundingcircleapproach,themaximuminnercircleoftheobjectcanalsobeusedtocheckthenon-accessibilityoftheturns.Ifthemaximuminnercircleofoneobjectcannotbecontainedinthecornerarea,thisobjectdenitelycannotaccessthiscorner. Lemma5. GivenanobjectOandacornerareathatcanberepresentedbyarectangleA,theobjectcannotaccessthiscornerifthemaximuminnercircleofOcannotbecontainedinA. Proof. AsshowninFigure 5-8 B,thepolygonArepresentsacornerarea,OrepresentstheobjectandCisthemaximuminnercircle.ItisclearthatifthesmallestobjectwhosemaximuminnercircleisC,cannotaccessthecornerA,alltheotherobjectsthathavethismaximuminnercirclecannotaccessA. ThesmallestobjectwhosemaximuminnercircleisCisactuallythecircleC.SinceCcannotbecontainedinthecornerareaA,itdenitelycannotaccessA.Therefore,any 102

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objectswhosemaximuminnercirclearethesameasorgreaterthanCcannotaccessA. 5.2.3.2Theapproachforspecialcases Thisminimumboundingcircleapproachmakessurethattheprovidedroutesarefeasibleforusers.However,thisapproachisnotpreciseenough.Forsomeparticularscenarios,weareabletorenetheaccessibilitycheck. Forindoorspaces,corridorcornersareoneofthemostcommonareasinwhichusersmayhaveproblemsinsuccessfullypassingthrough.Forthetraditional90corners,wehavedevelopedanapproachtorenetheaccessibilitychecking.AsshowninFigure 5-9 A,therectangle(a,b,c,d)istheminimumboundingboxofanobject(user).Thesegment(o,a)isparalleltoonesideofthecorner,and(o,a)hasthesamewidthofw.Wehavenoticedthatifthelengthof(o,b)equalsorislessthanv,thenthisrectanglecanmaketheturn.Pointbhasthesamesituation.InFigure 5-9 B,thelengthof(o,b)isequaltov.Ifthelengthof(o,a)isequalorislessthanwthenthisrectanglecanmaketheturn. Therefore,ourapproachcontainstwostepstochecktheaccessibilityofa90corner.Firstly,theminimumboundingrectangle(MBR)oftheuserisconstructed(e.g.(a,b,c,d)inFigure 5-9 A).Secondly,assumingtheboundaries(a,b)and(c,d)arelongerthan(a,d)and(b,c),wendapointoontheboundary(c,d)sothatthelengthof(a,o)isequaltothewidthofonesideofthecorner(e.g.,winFigure 5-9 A).Ifthelengthof(b,o)equalsorlessthanthewidthoftheothersideofthecorner(e.g.,vinFigure 5-9 A),thentheusercansuccessfullymaketheturn. Wecanperformthesecondstepinanotherway,inwhichwetrytondapointQsothatthelengthof(b,Q)isequaltothewidthofonesideofthecorner.Ifthelengthof(a,Q)equalsorlessthantheothersideofthecorner,theusercanmaketheturn.Otherwise,thiscornerisnotfeasiblefortheuser. 103

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AB Figure5-9. Renementtheapproachforevaluatingthemaximumlengthfor90corners. A BC Figure5-10. TheLEGOgraph.A)Aoorplanewithobstaclesandstairs.B)Thegraphreectingtheconnectivityoftheblocks.C)ThecorrespondingLEGOgraph 5.3TheLEGOGraph Mostoftheexistingpathsearchingalgorithms(e.g.,theshortestpathsearchandtheA*algorithm)aregraph-basedalgorithms.Inthissection,wewilldiscusshowtobuildagraphtosupportroutesearchingalgorithms. Asdiscussedinprevioussections,theindoorspaceisapproximatedbyLEGOcubes,whicharefurthermergedtoformlargerblocks.Userscanwalkblockbyblock 104

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inordertoreachtheirtarget.Theaccessiblewidths,heightsandlengthsarerestrictedbytheseblocksandtheconnectorsbetweenthem.Inordertosupporttheaccessibilitychecks,thisinformationmustbestoredinthegraph.Onesolutionistobuildagraphinwhichnodesdenoteblocksandedgesrepresentconnectors. Figure 5-10 BissuchagraphandconsistsofalltheblocksandconnectorsforthescenarioshowninFigure 5-10 A.Onebigproblemofthisgraphisthatthedistanceofeachpathisstoredinnodesinsteadofedges.Therefore,itisdifculttoapplytheshortestpathalgorithms. Inreality,theprocessofwalkingblockbyblockisthesameastheprocessofwalkingconnectorbyconnector.Abettersolutionistobuildagraphinwhichthenodesdenotealltheconnectors,andedgesrepresenttheirdistances.Inourmodel,thiskindofgraphiscalledaLEGOgraph.Denition 10 istheformaldenitionoftheLEGOgraph. Denition10. ALEGOgraphLG=(V,E)isagraphwhichreectsallpossiblepathswithdifferentaccessiblewidths,heightsandlengthsinagivenindoorspacescenario.Visasetofconnectorswiththeinformationofthesupportablelengths.Eisasetofimplicitpathsintheformatof,whereNisthenameoftheedge,Disthedistancebetweentwoconnectednodes,W,andHarethemaximumaccessiblewidthandheights.Tisthetypeoftheedges,whichcanbeplane,obstacleorstair. Thevaluesattachedtoeachedgearedeterminedasfollows: D:ThelengthofanedgeinaLEGOgraphisthedistancebetweenthecenterpointsofthetwoendnodes. W:Theaccessiblewidthofanedgedependsonthemaximumwidthsofthetwoendnodes.Itwillbesettobetheminimumwidthofthetwonodes. H:Asdiscussedinprevioussections,ourgeneratedblocksarealwaysrectangles,andtheconnectorsareeitherontheboundaryorinsidetheblock.Thus,thepathbetweentwoconnectorsisalwaysinsidethecorrespondingblock.Theaccessibleheightofanedgeistheheightoftheblock. T:Sinceeachedgeisinsideoneblock,thereisonlyonetypeforeachedge.Forexample,ifthecubesinoneblockareallplane cubes,thepathisplane. 105

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L:Theaccessiblelengthismaintainedinnodes,whichisthediameterofthemaximumcircleintroducedinSection 5.2.3 .Thereasonwedon'tcheckthelengthinedgesisbecauseiftheminimumboundingcircleoftheusercanbecontainedintheextendedconnectingarea,theremustbeenoughspacefortheuser'slengthtot. 106

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CHAPTER6THEIMPLEMENTATION Thissectioncontainstwoparts:theevaluationoftheefciencyforthe2DmodelwhichsupportstheshortestpathsrouteplanningandthedemonstrationofourimplementediNavsystem.Theevaluationpartshowstheelapsedtimesthatarespentforconstructingallpathsegmentsindifferentscenarios,andtheaverageefciencyforcomputingaroutebetweentworandomlyselectedlocations.ThedemonstrationpartpresentsourimplementationoftheiNavsystem,includingthe2Dandthe3Dsystem.The2Dsystemsupportstheshortestpathsearchandrangequeries,andthe3Dsystemprovidesthefeasibleroutesfordifferenttypesofusers. Inreality,thenavigationsystemincludesthreeparts:thedeterminationoftheuser'scurrentlocation,theprocessofthenavigationqueries,andtheprovisionofunderstandableresults.Basedontheassumptionthattheuserscurrentlocationcanbeobtainedwiththeuseofotherequipment,oursystemfocusesonprocessingthequeriesandvisualizingtheresults.Thecapturingoftheuserscurrentlocationsaresimulatedbyclickingthemousetorecordtheposition. 6.1TheEvaluation Inprevioussections,wehaveintroducedoursolutionforsupportingtheshortestrouteplanninginanindoorspace.Thissolutioniscomposedoftwoseparatephases:1)dataloadingandpath-segmentconstruction,2)routeplanning.Thegoalofourexperimentsistoevaluatetheperformanceofthewholesolution.Especially,theexperimentsaredesignedtoanswertwoquestions:1)Isitpracticaltoconstructallthepathsegmentsinabuildingwithacomplicatedstructure?2)Isitefcienttodiscovertheroutesinalarge-scalegraphthatisgeneratedintherstphase? ExperimentSetup:OurexperimentsareconductedonalaptopwithGenuineIntel(R)CPU,1.3GHZand4.0GBRAM.TheoperatingsystemisMicrosoftWindows7,andtheprogramrunsonJava2StandardEditionSDKv1.6.0.Toevaluatethe 107

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A B C Figure6-1. Themapusedtoevaluatingtheefciency.A)Themapoftherstoor.B)Themapoftheoorforoorn(n2).C)Statisticsinformationofthedatasets. performanceofoursolution,wemeasurethetimecostduringthreeprocedures:1)loadingthedataintodatabases,2)constructingpathsegments,3)andplanningtheroutes. DataSet:WecollecttheoormapsfromShandsatUF[ 1 ],whichisanationalrecognizedhospital.Thishospitalresidesina7-oorbuildingwithcomplextopology.Toevaluatetheefciencyofouralgorithm,wechoosethemosttwocomplicatedoormapsasourdataset.Sincethedoorinformationismissing,werandomlychoosedoorlocationineachroom.Figure 6-7 Aillustratesthemapofoor1.Inordertosimulatetheenvironmentscontainingmultipleoors,allthemapsoftheoorn(100n2)arethesame,asshowninFigure 6-7 B.AllthesemapsaremanuallyextractedandrepresentedinXMLasthedatasets.Figure 6-7 Cshowsthestatisticsinformationofthegenerateddatasets.Whenthenumberofoorsis100,thenumberofcells(i.e.,rooms)andaccesspointsreach2695and7972,respectively. Experiment1evaluatestheperformanceduringtherstphase(dataloadingandpath-segmentconstruction).Inoursolution,thisphaseisapre-processingprocedureforgeneratingaDirectPathGraph,whosenodesrepresentaccesspointsandedges 108

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AB Figure6-2. Evaluationoftheperformanceforrouteplanning.A)Thetimespentforloadingdatasets.B)Thetimespentforpathconstruction Figure6-3. Evaluationoftheperformanceforrouteplanning denotepathsegments.Therelatedtimecostisonce-for-allforeachbuildingorconstruction.Figure 6-2 AandBillustratetwosetsofexperimentstomeasurethetimecost,respectively,inloadingthemapdatafromXMLlestodatabasesandconstructingthepathsegments.Ineachset,therearethirteenvaluesofnumberofoorsrangingfrom2to100.Thetimecostiscalculatedbytheaverageofthreeruns.Bothguresindicatethetimeincreaseislinearwithrespecttothenumberofoors.Whenthenumberofooris100,thetotaltimeforthepre-processingis150s(25s+125s). Experiment2examinestheefciencyofrouteplanningbetweenasourcepointandatargetobjectbasedonthegeneratedDirectPathgraph.Onegoalofthesecond 109

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Figure6-4. Evaluationoftheperformanceforstationaryrangequeries phaseistondashortestpathfromasourcepointS(i.e.,aparticularpointonamap)toatargetobjectT(e.g.,aroom).ThehistogramshowninFigure 6-3 demonstratesthetimespentinrouteplanningfor10000samples.EachsamplecorrespondstoanodepairhS,Ti,whicharerandomlyselectedfroma100-oormap.AfterpruningthenodesandedgeswhosecorrespondingoorisdifferentfromtheoneofeitherSorT,oursolutioncangeneratetheexpectedpathin65.36millisecondsinaverage.Themaximumexecutiontimeis167millisecondsand94.55%routeplanningtakeslessthan70milliseconds.Theexperimentsinareal-worldscenarioshowthepotentialofoursolution. Experiment3evaluatestheefciencyofourproposedalgorithmforstationaryrangequeries,whichaimsatndingtheshortestpathsforallthequalifyingobjectswithinaspecicdistancerangefromthesourcepoint.Figure 6-4 illustratesthetimespentontherangequerieswithdifferentrangevalues.Foreachvalue,thetimecostisanaverageof100range-queryexecutiontime,whereeachsourcepointisrandomlyselected.Weobservethatthetimecostlinearlyincreasesfrom55.92millisecondsfortherangeof100metersto237.27millisecondsfortherangeof1000meters.Theseresultsclearlyshowoursolutionispracticalandefcientinareal-worldapplication. Experiment4examinestheperformanceofthealgorithmforcomputingcontinuousrangequeries.Inourexperiment,100querypointsarerandomlyselectedwithinacell. 110

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AB Figure6-5. EvaluationcomparisonbetweenIRNEandIRD.A)Numberofdynamicquerypointsis100.B)Numberofdynamicquerypointsis1000 Then,fortherangevaluesfrom100metersto1000meters,wecomputethequalifyingobjectsforallthequerypointsbyapplyingthealgorithmforcomputingstationaryrangequeries(IRNE)andcontinuousrangequeries(IRD).TheresultsshowninFigure 6-5 AindicatesthattheIRDalgorithmismoreefcientthanIRNE.TheadvantageofthealgorithmIRDwillbemoreobviouswhentheuserispassingthroughalargercell.Inthenextexperiment,werepeatthesametestsbutusealargernumberofquerypointstoindicatethattheuserispassingthroughalargecell.Figure 6-5 Billustratesthat,withthelargernumberofquerypoints,thetimespentforIRDisalmostthesameasthepreviousexperiment(showninFigure 6-5 A),whilethetimeforIRNEisdramaticallyincreased. 6.2TheDemonstration Therearetwosystemsimplemented.Oneisusedtosupporttheshortestrouteplanningandrangequeriesina2Dspace,andtheotherisusedtoproviderouteplanningfordifferenttypesofuserina3Dspace.ThetwosystemssharethesamearchitectureasshowninFigure 6-6 6.2.1SystemArchitecture Thearchitectureconsistsofvecomponents:UserInterface,DataProcessor,QueryProcessor,ResultProcessor,andStorage/RetrievalManager.UserInterfaceprovidesatoolforuserstouploadtheformatteddataandcheckthevisualizedpaths 111

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Figure6-6. ThearchitectureoftheiNavsystem forthecorrectdirections.DataProcessoranalyzesthedataandcreatesimplicitpathsegments.ThestorageandretrievalofthedataaremanagedbytheStorage/RetrievalManagercomponent.Oncethedatabaseisready,QueryProcessorwillprocesstheuser'squery,andResultProcessorwillgeneratethedescriptionsandthevisualizedresultsfortheuser. 6.2.1.1Thedataprocessor DataProcessorisusedtoanalyzetheinputdata,createtheimplicitpathsegments,andstorealltheinformationintothedatabasethroughtheStorage/RetrievalManager.TheinputdatashouldbeanXMLlewhichspeciesthestructureofthebuilding. TheXSDlewhichspeciestheformatoftheXMLdataforthe2DsystemisshowninAppendix A Therearefourtypesofobjectsusedtodescribetheindoorenvironment:room,corridor,stairandelevator.Roomsandcorridorscanbesimplecells,complexcells, 112

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oropencells,dependingontheirstructures.Stairsandelevatorsaretwotypesofconnectors.Anexampleofasimplecellisshownasfollows: 50200office550225298:00-17:00250400400403040 113

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302530
2530250
Thetagdoorspeciesthelocationandtheavailabletimeofanaccesspointinthecell,andthetagshaperecordstheboundaryofthecell.Ifthecellisacomplexone,therewillbemultipledoorsassociatedwiththatcellinthele.Iftheshapeofthecellisaregionwithholes(notanothercellhavingaccesspoints),theinnerboundarytagshouldbeusedtodescribetheshapesoftheholes.Thetagparentandchildareusedtoindicatethecontainmentrelationshipsbetweendifferentcells.Aftertheleisuploadedtothesystem,DataAnalyzerwillanalyzethele,andPathCreatorwillcreatetheimplicitpathsegments. Oncethedataisready,Storage/RetrievalManagerwillstorethestructureandthecreatedpathsegmentsintothedatabase.Inourimplementation,thePostGreSQLdatabaseisusedtostoreallthedata,andthepostGIS,whichaddssupportforgeographicobjectstothePostgreSQL,isusedtohandlethespatialfeaturesofthedata.Thus,accesspoints,pathsegmentsandshapesofcellsarestoredasthetypesofPOINT,LINEandPOLYGONinthedatabase.AftertheoperationofthedataprocessorusersareabletoviewthemapofthebuildingthroughtheUserInterface.Figure 6-7 istheuserinterfaceoftheiNavsystem.Thecenteroftheinterfaceshowsthemapof 114

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Figure6-7. Theuserinterface thebuilding.Ifthebuildinghasmultipleoors,eachoorwillbeshowninaseparatetabpage.Usersareabletoviewthemapsofdifferentoorsbyswitchingthetabs.AsshowninFigure 6-7 ,the5thooriscomposedofonecomplexcell(room502),severalsimplecells,andacorridorwithahole(thecenterregionwithoutanameinit). Theinputdataforthe3Dsystemismorecomplexthanthatusedforthe2Dsystem.Oursystemisbasedoncuberepresentation.However,itisimpracticalforuserstoprovidethecube-basedstructureofdatatothesystem.Thereareseveralapproacheswhichcanbeusedtostore3Dstructures.Onesuchapproachistorepresent3Dobjectsbyrecordingtheirboundaries(e.g.Doubly-ConnectedEdgeList(DCEL)[ 18 ],Quad-Edge[ 24 ],andTriangularIrregularNetwork(TIN)[ 97 ]).Theotheroptionistodeconstruct3Dobjectsintosmallpartsandorganizethesmallpartsusinghierarchicalstructures,suchastheOctree[ 57 ],andtheConstructiveSolidGeometry(CSG)tree.Inourmodel,theboundary-basedapproachisusedtorecordthestructureoftheindoorspace. 115

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Accordingtothemostrepresentativestructureoftheindoorspace,eachcellisdeconstructedintovemaincomponents:oorareas,ceilingareas,walls,doorsandobstacles.Obstaclesarefurtherbrokendownintotopareas,bottomareasandsurroundingfaces.Alltheareasandfacescanberepresentedbyasetofpolygons.Sinceallobjectsfollowastandardstructure,theycanberepresentedbyaformalizedXMLformat.Appendix B showsanexampleoftheinputXMLdatathatrepresentsthestructureofthecellinFigure 5-1 A. 6.2.1.2Thequeryprocessor The2DiNavsystemsupportstwovarietiesofquery:routingqueriesandrangequeries.UserscaninitiateroutingqueriesthroughtheUserInterfacebyindicatingtheirnearestaccesspointsandtheirtargets(asshownintherightsideofFigure 6-7 ).Theycanalsopostrangequeriesbyindicatingtheirnearestaccesspoints,alongwiththeirpreferredrangesandinterests(asshowninthedialogformofFigure 6-7 ).TheiNavsystemwillretrievealltheimplicitpathsegmentsfromthedatabaseandconstructthedirectpathgraph. ThepostedquerieswillbeprocessedbyapplyingtheDijkstra'salgorithmortheIRNEalgorithmproposedinChapter 4 .Inthe2DiNavsystem,ifuserswanttopostcontinuousqueriestheycanshowtheirmovementsbyclickingthemouseonthemap.TheresultproducedbytheQueryProcessorisasequenceofnodesindicatingtheroutes. The3Dsystemsupportsrouteplanningforarbitrary-shapedusers.Similartothe2Dsystem,userscaninitiatetheirqueriesbyindicatingtheirnearestexits,theirtargets,widths,heightsandlengths.The3DsystemwillretrievealltheconnectorsandbuildtheLEGOgraph.Dijkstra'salgorithmwillbeappliedtogeneratethenalpathwhichiscomposedofasequenceofconnectorsbetweenblocks. Inordertomakeresultsmoreunderstandable,thequeryresultswillbefurtherprocessedbytheResultProcessor. 116

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6.2.1.3Theresultprocessor TheResultProcessorinthe2DiNavsystemisusedtovisualizetheresultsproducedbytheQueryProcessorandproducethedescriptionsoftheroutesasdiscussedinSection 3.4 .ItrstanalyzestheresultsobtainedfromtheQueryProcessorandretrievesallthenecessaryinformationrelatedtotheresultsfromthedatabase.Thenitdeterminesthelocationsofthenodesanddrawsthepathaccordingtothesequenceofthenodes.Figure 6-8 showsthevisualizedroutefromroom502toroom515,inwhichtherednodesaretheinvolvedaccesspointsandintermediatepoints.TheResultProcessoralsogeneratesthedescriptionofthisrouteanddisplaysthedescriptionatthebottomoftheinterface.Byusingthevisualizedmapandthedescriptionoftheroute,theuserisabletogettothedestinationintheshortestdistance. Inour3Dmodelthepathsarecomposedofblocksandconnectorsthataretfortheusers'size.Inordertoshowthepathsmoreclearly,theresultvisualizationinour3Dsystemcontainstwoparts:showingthelinesconnectingthedifferentconnectorsinsequence,andhighlightingtheblocksinvolvedinthepath.AsshowninFigure 6-13 ,thepathfromd22toroom6isindicatedbythelinesfromd22tod3,andfromd3tod6.Besidesthelines,therearesomeblueareasindicatingtheblocksinvolvedinthegeneratedpath. 6.2.2Experiments 6.2.2.1Experimentsofrouteplaningin2Dindoorspaces Example1. Findtheshortestpathfromroom502toroom615. Figure 6-9 isthevisualizedresultoftheroute.Figure 6-9 Ashowsthepathonthe5thoorandFigure 6-9 Bshowsthepathonthe6thoor.Figure 6-9 Cdisplaysthechangestotherouteaccordingtotheuser'scurrentlocation,asindicatedbyclickingthemouse(thegreennodeinFigure 6-9 C). Weobtainthedescriptionoftherouteasfollows: Startfromdoor502_1todoor615 117

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Figure6-8. Thevisualizedroutefromroom502to515 ABC Figure6-9. TheresultsofExample 1 .A)Thegeneratedroutefromroom502to615inthe5thoor.B)Thegeneratedroutefromroom502to615inthe6thoor.C)Theroutechangesaccordingtothepositionofthemouseclicking next:walk16meterstodoorelevator1_5incorridornext:walktodoorelevator1_6Next:rotate40degreetotheleft,walk17metersincorridornext:walk48meterstodoor615incorridor Example2. Findpathsfromoutsidetonestedcells. 118

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ABC Figure6-10. TheresultsofExample 2 .A)Thegeneratedroutefromroom1toroom6.B)Thegeneratedroutefromroom1toroom4.C)Thegeneratedroutefromroom1toroom7. InFigure 6-10 ,room2isanestedcellcontainingroom3,room5androom6.Insideroom3,thereisanothercell,room4.Figures 6-10 A,bandcarethreeexamplesofrouteplaninginbuildingswithnestedcells.Therstoneistotestthepathstotherstlevelnestedcells(thecellswhoseparentcellisnotnestedinothercells).TheresultofthiscaseisshowninFigure 6-10 A.Thesecondoneistotestthepathstodeeperlevelnestedcells(thecellswhoseparentisalsonestedinothercells).TheresultofthetestisshowninFigure 6-10 B.Thethirdtest,whichisillustratedbyFigure 6-10 C,istotestpathsthatarefromthecellsinonesideofthenestedcelltothecellsintheothersideofthesamecell,goingthroughtheentirenestedcell. Example3. Findpathsfromnestedcellstooutsidecells. SimilartoExample 2 ,weusethreecasestotestthepathsfromnestedcellstocellsoutside.Therstoneistotestpathsfromdeeperlevelnestedcells(cellsinnestedcells)totherstlevelnestedcells(showninFigure 6-11 A).Thesecondexample,illustratedinFigure 6-11 B,istotestthepathsfromdeeperlevelnestedcellstooutercells.Thethirdexampleteststhepathsfromtherstlevelnestedcellstotheoutercells.Figure 6-11 Cshowstheresultofthiscase. 6.2.2.2Experimentsofrangequeryin2Dindoorspaces Example4. Findalltheofceswithin30metersofthedoorofroom517. 119

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ABC Figure6-11. TheresultsofExample 3 .A)Thegeneratedroutefromroom4toroom2.B)Thegeneratedroutefromroom4toroom7.C)Thegeneratedroutefromroom6toroom7. Figure 6-12 isusedtotesttheresultsofrangequeriesandcontinuousrangequeries.Assumingalltheroomsaremarkedasofces,Figure 6-12 Ashowsthepathstoallofceswithin30metersofthedooroftheroom504.Thebluenodesarethequalifyingofces,andtheredlinesarethepathsfromthestartnodetothecorrespondingtargets.FromFigure 6-12 Bwecanlearnthatthereare7qualifyingofcesinthe5thoorand3moreinthe6thoor.Therednodebetween517and518istheconnectorofthe5thoorandthe6thoor.Theresultchangeswiththemovementsoftheuser.AsshowninFigure 6-12 C,whentheusergoestothelocationindicatedbytherednode,theofceswithin30metersoftherednodechangeaccordingly. 6.2.2.3Experimentsofrouteplanningforarbitrary-shapedusers Example5. Checktheaccessibilityofdoors. ThegoalofExperiment 5 istochecktheaccessibilityofdoors.InFigure 6-13 ,d6is50inches,andallotherdoorsare100inches.Oneuserisinthepositionofd22andwantstogointoroom6.Iftheuser'swidthislessthan50inches,shecangointoroom6byenteringdoord6(asshowninFigure 6-13 A).However,iftheuser'swidthisgreaterthan50inches,shehastogointoroom5rst(asshowninFigure 6-13 B). Example6. Checktheaccessibilitybetweenblocks. 120

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ABC Figure6-12. TheresultsofExample 4 .A)Thetheofceswithin30metersfromtheroom517onthe5thoor.B)Thetheofceswithin30metersfromtheroom517onthe6thoor.C)Thequaliedofceschangesaccordingtothepositionofthemouseclicking. AB Figure6-13. TheresultsofExample 5 .A)Thewidthoftheuseris30inches.B)Thewidthoftheuseris80inches. ThegoalofExperiment 6 istocheckthecorrectnessofthegeneratedpathswhenthereareobstaclesblockingtheway.AsshowninFigure 6-14 ,therearetwoobstaclesinroom3.Thelongeroneis50inchesfromthethreesidesofthewalls.Figure 6-14 aandFigure 6-14 bshowstwodifferentpathsfromd22toroom5.ThepathinFigure 6-14 Aisfortheuserswhosewidthsorlengthsarelessthan50inches,andthepathinFigure 6-14 Bisfortheuserswhosewidthsorlengthsaregreaterthan50inches.Theredcirclesareeitherdoorsorthecenterpointsoftheinvolvedconnectors. 121

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AB CD Figure6-14. TheresultofExample 6 .A)Thelengthoftheuseris30inches.B)Thelengthoftheuseris80inches Figure 6-14 candFigure 6-14 dillustratethepathswhenmoreblocksareinvolved.InFigure 6-14 AandB,thereareonlytwoblocksinvolvedinthegeneratedpathinroom3;whileinFigure 6-14 C,thenumberofinvolvedblocksbecomes4.Thegeneratedpathfromd22toroom4areshowninFigure 6-14 C,whentheuser'swidthandlengtharelessthan50inches,andthepathforuserswiderorlongerthan50inchesisshowninFigure 6-14 D. 122

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CHAPTER7CONCLUSIONS Thisdissertationpresentsaspatialmodelforsupportingrouteplanninginindoorspaces.Mymajorresearchcontributionsincludethefollowingthreeaspects.First,theiNavmodel(describedinChapter 3 )providesaneffectivesolutiontothediscoveryoftheshortestpathbetweenthesourcepointandthetargetobject.Second,myproposedrange-drivenroutingapproach(discussedinChapter 4 )isabletoefcientlydetermineallthequalifyingobjectswithinaspecicwalkingdistance.Third,basedontheLEGOmodel(presentedinChapter 5 ),thegenerationofafeasibleroutecanbefullyautomatedfromthesourcepointtothetargetobject,whenthesizeofamovingobjectcannotbeapproximatedtoapoint. Thepresentedsolutionmakesasignicantcontributiontotheresearchofdesignofindoornavigationsystems,especiallyforaddressingthethreefundamentalrouteplanningissues(i.e.,distance-driven,range-drivenandsize-drivenrouteplanning).Althoughindoornavigationisourmajorapplicationscenario,ourproposedsolutioncanbeappliedintheotherrelatedsystems,suchas,shoppingguidesystems,recommendationsystemsfordesigningarchitecturalstructures,andemergencyevacuationsystems.Intheindustrialcommunity,muchefforthasbeendevotedtodesigningandimplementingoutdoorrouteplanningsystems(e.g.,inGPSnavigationsystems).Withincreasingrequestsforindoornavigation,webelievetheindoorrouteplanningisindispensableinthenextgenerationofnavigationsystems. Thefollowinginterestingdirectionsaresuggestedforfutureresearchtopics: Semantics-basedrouteplanningistointegratesemanticinformationintothegeometric-basedrouteplanning.Inareal-worldscenario,weobservedthereexistvarioussemanticresources,suchasuserproles,roadmarks,andiconicobjects.Througheffectivereasoningontheseresources,indoorroute-planningsystemscanfurtherimprovetheaccuracyofrouterecommendation. 123

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Automationofmapinterpretationisacriticalstepintheproductizationofourproposedindoorrouteplanningprototype.Asmajorsourcesofindoorstructure,buildingmapsarenormallydrawninsomewidely-usedindustrialsoftware,suchasAutoCADandProE.Thus,itishighlydesirabletoautomatemapinterpretationforresolvingsuchaninformation-transformationissue. 124

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APPENDIXATHEXSDFILEOFTHEFORMATTEDDATAFORTHE2DSYSTEM 125

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APPENDIXBANEXAMPLEOFTHEFORMATTEDDATAFORTHE3DSYSTEM room1office1d11(500,200,0)(500,300,0)(500,200,200)(500,300,200)8:00-17:00(0,0,0)(500,0,0)(500,500,0)(0,500,0)(0,0,200)(500,0,200)(500,500,200)(0,500,200)(0,0,0)(0,0,200)(0,500,200)(0,500,0)(0,0,0)(0,0,200)(500,0,200)(500,0,0)(500,0,0)(500,0,200)(500,500,200)(500,500,0)(500,500,0)(500,500,200)(0,500,200)(0,500,200)Table(650,50,0)(750,50,0)(750,450,0)(650,450,0) 129

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(650,50,200)(750,50,200)(750,450,200)(650,450,200)(650,50,0)(650,50,200)(650,450,200)(650,450,0)(650,50,0)(750,50,0)(750,50,200)(650,50,200)(750,50,0)(750,50,200)(750,450,200)(750,450,0)(750,450,0)(750,450,200)(650,450,200)(650,450,0)
Table(250,250,0)(300,250,0)(300,300,0)(250,300,0)(250,250,200)(300,250,200)(300,300,200)(250,300,200)(250,250,0)(250,250,200)(250,300,200)(250,300,0)(250,250,0)(300,250,0)(300,250,200)(300,250,200)(300,250,0)(300,250,200)(300,300,200)(300,300,0)(300,300,0)(300,300,200)(250,300,200)(250,300,0)
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BIOGRAPHICALSKETCH WenjieYuanhasreceivedherM.S.andB.S.fromtheNanjingUniversityofScience&EngineeringinChinain2006and2004.ShestartedherstudyintheUniversityofFloridain2006.Herresearchfocusesonspatialmodelingofindoornavigationsystems,especiallyonthedesignofdatastructuresfordatabasestorageandnavigationstrategiesforwaynding. 140