Adaptive NonStationary Kernel Regression for Terrain Modeling Tobias Lang Christian Plagemann - PDF document

AdaptiveNon-StationaryKernelRegressionforTerrainModelingTobiasLangChristianPlagemannWolframBurgardAlbert-Ludwigs-UniversityofFreiburg,DepartmentforComputerScience,79110Freiburg,Germanyflangt,plagem,burgardg@informatik.uni-freiburg.deAbstract—Three-dimensionaldigitalterrainmodelsareoffundamentalimportanceinmanyareassuchasthegeo-sciencesandoutdoorrobotics.Accuratemodelingrequirestheabilitytodealwithavaryingdatadensityandtobalancesmoothingagainstthepreservationofdiscontinuities.Thelatterisparticularlyimportantforroboticsapplications,asdiscontinuitiesthatarise,forexample,atsteps,stairs,orbuildingwallsareimportantfeaturesforpathplanningorterrainsegmentationtasks.Inthispaper,wepresentanextensionofthewell-establishedGaussianprocessregressiontechnique,thatutilizesnon-stationarycovari-ancefunctionstolocallyadapttothestructureoftheterraindata.Inthisway,weachievestrongsmoothinginatareasandalongedgesandatthesametimepreserveedgesandcorners.Thederivedmodelyieldspredictiveheightdistributionsforarbitrarylocationsoftheterrainandthereforeallowsustollgapsinthedataandtoperformconservativepredictionsinoccludedareas.I.INTRODUCTIONThemodelingofthree-dimensionalterrainhasbeenwidelystudiedacrossdifferentresearchareaslikethegeo-sciencesorrobotics.Importantapplicationsinthelattercaseincludemo-bileroboticsforagriculture,searchandrescue,orsurveillance.Inthesedomains,accurateanddensemodelsofthethree-dimensionalstructureoftheenvironmentenabletherobottoestimatethetraversabilityoflocations,toplanitspathtoagoallocation,ortolocalizeitselfusingrangesensormeasurements.Buildingadigitalterrainmodelmeanstotransformasetofsensoryinputs,typicallya3Dpointcloudortherawrangesensorreadings,toafunctionmapping2-dimensionalposecoordinatestoelevationvalues.Whilegeologicalapplicationsoftenoperateonalargerspatialscale,wherelocalterrainfeaturescanbeneglected,autonomousrobotsgreatlyrelyondistinctstructuralfeatureslikeedgesorcornerstoguidenavigation,localization,orterrainsegmentation.Wethereforehavetwo,attherstglancecontradictingrequirementsforterrainmodels:First,rawsensorydataneedstobesmoothedinordertoremovenoiseandtobeabletoperformelevationpredictionsatalllocationsand,second,discontinuitiesneedtobepreservedastheyareimportantfeaturesforpathplanning,localizationandobjectrecognition.Inthispaper,wepresentanovelterrainmodelingapproachbasedonanextendedGaussianprocessformulation.Ourmodelusesnon-stationarycovariancefunctionsasproposedbyPacioreketal.[7]toallowforlocaladaptationoftheregressionkernelstotheunderlyingstructure.Thisadapta-tionisachievedbyiterativelyttingthelocalkernelstothestructureoftheunderlyingfunctionusinglocalgradientfeaturesandthelocalmarginaldatalikelihood(seeFigure1foranillustration).Indeed,thisideaisakintoadaptiveimagesmoothingstudiedincomputervision,wherethetaskistoachievede-noisingofanimagewithoutreducingthecontrastofedgesandcorners[14,6].Althoughtheseapproachesfromthecomputervisionliteraturearenotspecicallydesignedfordealingwithavaryingdensityofdatapointsorwithpotentialgapstoll,theyneverthelessservedasaninspirationforourkerneladaptationapproach. Fig.1.Ahard,syntheticregressionproblem(left).Thecontinuousregionsshouldbesmoothedwithoutremovingthestrongedgefeature.Ourapproachachievesthisbyadaptinglocalkernelstotheterraindata(right).Thepaperisstructuredasfollows.Werstdiscussrelatedworkinthenextsection.InSectionIII,weformalizetheter-rainmodelingproblemusingGaussianprocessesandintroduceourapproachtonon-stationaryadaptiveregression.SectionIVpresentsourexperimentalresultsonrealandsimulatedterraindatasets.II.RELATEDORKAbroadoverviewovermethodsusedformodelingterraindataisgivenbyHugentorp[5].Elevationmapshavebeenusedasanefcientdatastructureforrepresentingdenseterraindata[1,8]andhavelaterbeenextendedtomulti-levelprobabilisticsurfacemaps[16].Fr¨uhetal.[4]presentanapproachtollinglocalgapsin3Dmodelsbasedonlocallinearinterpolation.Astheirapproachhasyieldedpromisingresultsincitymappingapplications,wecompareitsmodelingaccuracytoourapproachinSectionIV.Gaussianprocesses(GPs)havealongtraditioninthegeo-sciencesandstatisticsliterature[11].Classicalapproachesfordealingwithnon-stationarityincludeinput-spacewarping[12,13]andhierarchicalmodelingusinglocalkernels[7].Thelatterapproachprovidesthegeneralframeworkforthiswork. Recently,GPshavebecomepopularinrobotics,e.g.,forlearningmeasurementmodels[2]ormodel-basedfailurede-tection[9].Todealwithvaryingtargetfunctionpropertiesinthecontextofperceptionproblems,Williams[17]usesmixturesofGPsforsegmentingforegroundandbackgroundinimagesinordertoextractdisparityinformationfrombinocularstereoimages.RasmussenandGhahramani[10]extendideasofTresp[15]andpresentaninnitemixtureofexpertsmodelwheretheindividualexpertsaremadeupfromdifferentGPmodels.Agatingnetworkassignsprobabilitiestothedifferentexpertmodelsbasedcompletelyontheinput.DiscontinuitiesinwindeldshavebeendealtwithbyCornfordetal.[3].TheyplaceauxiliaryGPsalongtheedgeonbothsidesofthediscontinuity.ThesearethenusedtolearnGPsrepresentingtheprocessoneithersideofthediscontinuity.Incontrasttoourwork,theyassumeaparameterizedsegmentationoftheinputspace,whichappearstobedisadvantageousinsituationssuchasdepictedinFigure1andonreal-worldterraindatasets.Theproblemofadaptingtolocalstructurehasalsobeenstudiedinthecomputervisioncommunity.Taketaetal.[14]performnon-parametrickernelregressiononimages.Theyadaptkernelsaccordingtoobservedimageintensities.Theiradaptationruleisthusbasedonanonlinearcombinationofbothspatialandintensitydistanceofalldatapointsinthelocalneighborhood.Basedonsingularvaluedecompositionsofintensitygradientmatrices,theydeterminekernelmodi-cations.MiddendorfandNagel[6]proposeanalternativekerneladaptationalgorithm.Theyuseestimatesofgrayvaluestructuretensorstoadaptsmoothingkernelstograyvalueimages.III.DIGITALTERRAINMODELINGDataforbuilding3-dimensionalmodelsofanenvironmentcanbeacquiredfromvarioussources.Inrobotics,laserrangendersarepopularsensorsastheyprovideprecise,high-frequencymeasurementsatahighspatialresolution.Othersensorsincludeon-boardcameras,whicharechosenbecauseoftheirlowweightandcosts,orsatelliteimagery,whichcoverslargerareas,e.g.,forguidingunmannedarealvehi-cles(UAVs)orautonomouscars.Aftervariouspreprocessingsteps,therawmeasurementsaretypicallyrepresentedas3Dpointcloudsoraretransformedintoa3Doccupancygridorelevationmap[1].Inthiswork,weintroduceatechniqueforconstructingcontinuous,probabilisticelevationmapmodelsfromdatapoints,thatyieldpredictivedistributionsforterrainelevationsatarbitraryinputlocations.Theterrainmodelingproblemcanbeformalizedasfollows.GivenasetDf(xi;yi)gni=1ofnlocationsamplesxiR2andthecorrespondingterrainelevationsyiR,thetaskistobuildamodelforp(yjx;D),i.e.,thepredictivedistributionofelevationsyatnewinputlocationsx.Thismodelingtaskisahardoneforseveralreasons.First,sensormeasurementsareinherentlyaffectedbynoise,whichanintelligentmodelshouldbeabletoreduce.Second,thedistributionofavailabledatapointsistypicallyfarfromuniform.Forexample,theproximityofthesensorlocationisusuallymoredenselysampledthanareasfartheraway.Third,smallgapsinthedatashouldbelledwithhighcondencewhilemoresparselysam-pledlocationsshouldresultinhigherpredictiveuncertainties.Toillustratethelastpoint,consideranautonomousvehiclenavigatinginoffroadterrain.Withoutllingsmallgaps,evensinglemissingmeasurementsmayleadtotheperceptionofanun-traversableobstacleandconsequentlytheplannedpathmightdiffersignicantlyfromtheoptimalone.Ontheotherhand,thesystemshouldbeawareoftheincreaseduncertaintywhenllinglargergapstoavoidovercondenceattheselocations.Asalastnon-trivialrequirement,themodelshouldpreservestructuralelementslikeedgesandcornersastheyareimportantfeaturesforvariousapplicationsincludingpathplanningorobjectrecognition.Inthispaper,weproposeamodeltoaccommodateforalloftheabove-mentionedrequirements.Webuildonthewell-establishedframeworkofGaussianprocesses,whichisanon-parametricBayesianapproachtotheregressionproblem.Todealwiththepreservationofstructuralfeatureslikeedgesandcorners,weemploynon-stationarycovariancefunctionsasintroducedbyPaciorekandSchervish[7]andpresentanovelapproachtolocalkerneladaptationbasedongradientfeaturesandthelocalmarginaldatalikelihood.Inthefollowing,werestatethestandardGaussianprocessapproachtonon-parametricregressionbeforeweintroduceourextensionstolocalkerneladaptation.A.GaussianProcessRegressionAsstatedintheprevioussection,theterrainmodelingtaskistoderiveamodelforp(yjx;D),whichisthepredictivedistributionofterrainelevationsy,calledtargets,atinputlocationsx,givenatrainingsetDf(xi;yi)gni=1ofelevationsamples.TheideaofGaussianprocesses(GPs)istoviewanynitesetofsamplesyifromthesoughtafterdistributionasbeingjointlynormallydistributed,p(y1;:::;ynjx1;:::;xn)N(;K);(1)withmeanRnandcovariancematrixK.istypicallyassumed0andKisspeciedintermsofaparametriccovariancefunctionkandaglobalnoisevarianceparametern,Kij:=k(xi;xj)+2nij.ThecovariancefunctionkrepresentsthepriorknowledgeaboutthetargetdistributionanddoesnotdependonthetargetvaluesyiofD.Acommonchoiceisthesquaredexponentialcovariancefunctionk(xi;xj)=2fexp 1 22Xk=1(xi;kxj;k)2 `k!;(2)wherefdenotestheamplitude(orsignalvariance)and`karethecharacteristiclength-scalesoftheindividualdimensions(see[11]).Theseparametersplustheglobalnoisevariancenarecalledhyperparametersoftheprocess.Theyaretypicallydenotedas=(f;`;n).SinceanysetofsamplesfromtheprocessisjointlyGaussiandistributed,thepredictionofanewtargetvalueyatagivenlocationxcanbeperformedbyconditioningthen+1-dimensionaljointGaussianonthe knowntargetvaluesofthetrainingsetD.ThisyieldsapredictivenormaldistributionyN(;v)denedbyE(y)=kTK2nI
1y;(3)vV(y)=k2nkTK2nI
1k;(4)withKRnn,Kijk(xi;xj),kRn,kjk(x;xj),kk(x;x)R,andthetrainingtargetsyRn.LearningintheGaussianprocessframeworkmeansndingtheparametersofthecovariancefunctionk.Throughoutthisworkweuseaconjugategradientbasedalgorithm[11]thatxestheparametersbyoptimizingthemarginaldatalikelihoodofthegiventrainingdataset.Alternatively,theparameterscouldbeintegratedoverusingparameter-specicpriordistri-butions,whichresultsinafullyBayesianmodelbutwhichisalsocomputationallymoredemandingasonehastoemployMarkov-ChainMonteCarlosamplingforapproximatingtheintractableintegral.Thestandardmodelintroducedsofaralreadyaccountsforthreeoftherequirementsdiscussedintheprevioussection,namelyde-noising,dealingwithnon-uniformdatadensities,andprovidingpredictiveuncertainties.Asamajordrawback,however,byusingthestationarycovariancefunctionofEqua-tion(2),whichdependsonlyonthedifferencesbetweeninputlocations,onebasicallyassumesthesamecovariancestruc-tureonthewholeinputspace.Inpractice,thissignicantlyweakensimportantfeatureslikeedgesorcorners.TheleftdiagramofFigure1depictsasyntheticdata-setwhichcontainshomogenousregionswhichshouldbesmoothed,butalsoasharpedgethathastobepreserved.Ourmodel,whichisdetailedinthenextsection,addressesthisproblembyadaptinganon-stationarycovariancefunctiontothelocalterrainproperties.B.Non-StationaryCovarianceFunctionsMostGaussianprocessbasedapproachesfoundintheliteratureusestationarycovariancefunctionsthatdependonthedifferencebetweeninputlocationsxx0ratherthanontheabsolutevaluesxandx0.Apowerfulmodelforbuildingnon-stationarycovariancefunctionsfromarbitrarystationaryoneshasbeenproposedbyPaciorekandSchervish[7].FortheGaussiankernel,theirnon-stationarycovariancefunctiontakesthesimpleformk(xi;xi)=jij1 4jjj1 4i+j 21 2
(5)exp"(xixj)Ti+j 21(xixj)#;whereeachinputlocationx0isassignedanindividualGaus-siankernelmatrix0andthecovariancebetweentwotargetsyiandyjiscalculatedbyaveragingbetweenthetwoindividualkernelsattheinputlocationsxiandxj.Inthisway,thelocalcharacteristicsatbothlocationsinuencethemodeledcovari-anceofthecorrespondingtargetvalues.Inthismodel,eachkernelmatrixiisinternallyrepresentedbyitseigenvectorsandeigenvalues.PaciorekandSchervishbuildahierarchicalmodelbyplacingadditionalGaussianprocesspriorsonthesekernelparametersandsolvetheintegrationusingMarkov-ChainMonteCarlosampling.Whilethemodelpresentedin[7]providesaexibleandgeneralframework,itis,asalsonotedbytheauthors,computationallydemandingandclearlynotfeasiblefortherealworldterraindatasetsthatweareaimingforinthiswork.Asaconsequence,weproposetomodelthekernelmatricesinEquation(5)asindependentrandomvariablesthatareinitializedwiththelearnedkernelofthecorrespondingstationarymodelandtheniterativelyadaptedtothelocalstructureofthegiventerraindata.Concretely,weassigntoeveryinputlocationxifromthetrainingsetDalocalkernelmatrixi,whichinturnisrepresentedbyoneorientationparameterandtwoscaleparametersforthelengthoftheaxes.Giventheseparameters,theevaluationofEquation(5)isstraightforward.Inthefollowingsection,wewilldiscussindetail,howthekernelmatricesicanbeadaptedtothelocalstructureoftheterrain.C.LocalKernelAdaptationTheproblemofadaptingsmoothingkernelstolocalstruc-turehasbeenwellstudiedinthecomputervisioncommunity.Itisthereforenotsurprisingthat,althoughimageprocessingalgorithmsaretypicallyrestrictedtodenseanduniformlydistributeddata,wecanusendingsfromthateldasaninspirationforourterrainadaptationtask.Indeed,Midden-dorfandNagel[6]presentatechniqueforiterativekerneladaptationinthecontextofopticalowestimationinimagesequences.Theirapproachbuildsontheconceptofthesocalledgrey-valuestructuretensor(GST),whichcapturesthelocalstructureofanimageorimagesequencebybuildingthelocallyweightedouterproductofgrey-valuegradientsintheneighborhoodofthegivenimagelocation.Analogouslytotheirwork,wedenetheelevationstructuretensor(EST)foragivenlocationxiasEST(xi):= ry(ry)T(xi);(6)wherey(x)denotestheterrainelevationatalocationxand
standsfortheoperatorthatbuildsalocallyweightedaverageofitsargumentaccordingtothekerneli.Fortwo-dimensionalxi,Equation(6)calculatesthelocallyweightedaverageoftheouterproductofry=(@y @x1;@y @x2)T.Thislocalelevationderivativecanbeestimateddirectlyfromtherawelevationsamplesintheneighborhoodofthegiveninputlocationxi.Wecopewiththenoisestemmingfromtherawdatabyaveragingovertheterraingradientsinthelocalneighborhood.Equation(6)yieldsatensor,representableasa22real-valuedmatrix,whichdescribeshowtheterrainelevationchangesinthelocalneighborhoodoflocationxi.Togetanintuition,whatEST(xi)encodesandhowthiscanguidetheadaptationofthelocalkerneli,considerthefollowingsituations.Let1and2denotetheeigenvaluesofEST(xi)andbetheorientationangleofthersteigenvector.Ifxiislocatedinaatpartoftheterrain,theelevationgradients ryaresmallintheneighborhoodofxi.ThisresultsintwoequallysmalleigenvaluesofEST(xi).Incontrast,ifxiwaslocatedinanascendingpartoftheterrain,thersteigenvalueofEST(xi)wouldbeclearlygreaterthanthesecondoneandtheorientationwouldpointtowardsthestrongestascent.IntuitivelyandasdiscussedinmoredetailbyMiddendorfandNagel[6],thekernelidescribingtheextentofthelocalenvironmentofxishouldbesettotheinverseofEST(xi).Inthisway,atareasarepopulatedbylarge,isotropickernels,whilesharpedgeshavelong,thinkernelsorientedalongtheedgedirections.Cornerstructures,havingstrongelevationgradientsinalldimensions,resultinrelativelysmalllocalkernels.Topreventunrealisticallylargekernels,MiddendorfandNageldescribehowthisinversioncanbeboundedtoyieldkernels,whosestandarddeviationsliebetweengivenvaluesminandmax.Basedontheirndings,wegivethreeconcretelocaladaptationrulesthathavebeencomparedinourexperimentalevaluation.Tosimplifynotation,weintroduce kk=(12),k=1;2andthere-parameterizationiRT1002R1(7)where1and2scaleinorthogonaldirectionsandRisarotationmatrixspeciedbytheorientationangle.1)DirectInverseAdaptation:iEST(xi)12)BoundedLinearAdaptation:k k2min+(1 k)2max;k=1;23)BoundedInverseAdaptation:k2max2min k2max+(1 k)2min;k=1;2Thetwoboundedadaptationprocedurespreventunrealisti-callysmallandlargekernels.TheBoundedInversestronglyfavorsthelargereigenvaluedimensionandproducesmorepronouncedkernels(largerdifferencebetweensemiaxes)whiletheBoundedLinearLineartendstoproducemorebalancedandlargerkernels.ThisiswhyBoundedLinearperformsbetterinthepresenceofsparsedataasitislessvulnerabletoovertting.Inthiswork,theboundsminandmaxareesti-matedempirically.Wearecurrentlyworkingondeterminingoptimalvalueswithrespecttothemarginaldatalikelihood.Sofar,wehavedescribedhowtoperformonelocaladaptationstepforanarbitrarykerneli.Asthecompletelearningandadaptationprocedure,whichissummarizedinAlgorithm1,weproposetoassigntoeachinputlocationxiofthetrainingsetDakernelmatrixi,whichisinitializedwithaglobalparametervector,thatinturnhasbeenlearnedusingstandardGPlearningwiththecorrespondingstationarycovariancefunction.Thelocalkernelsaretheniterativelyadaptedtotheelevationstructureofthegiventerraindatasetuntiltheirparametershaveconverged.Toquicklyadaptthekernelsatlocationswheretheregressionerrorishigh(relativetothegiventrainingdataset),weproposetomaketheadaptationspeedforeachidependentonthelocaldatatdf(xi),whichisthenormalizedobservationlikelihoodofthecorrespondingyifromthetrainingsetrelativetothecurrentpredictivedistribution(seeEquation(III-A)),andthekernelcomplexityapproximatedasi=1=jij.Bothquantitiesareusedtoformalearningrateparametercalculatedbymeansofamodiedsigmoidfunction,isigmoid(df(xi)
i;),wheretheadditionalparametersaredeterminedempirically.Intuitively,wegetahighadaptationspeedwhenthedata-trelativetothekernelsizeissmall.Algorithm1summarizestheadaptationprocedure. Algorithm1LocalKernelAdaptation Learnglobalparametersforthestationarysquaredexpo-nentialcovariancefunction.Initializealllocalkernelsiwith.whilenotconvergeddoforallidoEstimatethelocallearningrateiEstimateEST(xi)accordingtoiiADAPT(EST(xi))iii+(1i)iendforendwhile IV.EXPERIMENTALEVALUATIONThegoalsoftheexperimentalevaluationpresentedinthissectionare(a)toshowthatourterrainmodelingapproachisindeedapplicabletorealdatasets,(b)thatourmodelisabletoremovenoisewhileatthesametimepreservingimportantstructuralfeatures,and(c)thatourmodelyieldsmoreaccurateandrobustelevationpredictionsatsparselysampledinputlocationsthananalternativeapproachtothisproblem.Asanevaluationmetric,weusethemeansquarederrorMSE(X)=1 mPmi=1(yiyi)2ofpredictedelevationsyirelativetogroundtruthelevationsyionasetofinputlocationsXfxigmi=1.A.EvaluationonArticialTerrainDataTherstsetofexperimentswasdesignedtoquantifythebenetsoflocalkerneladaptationandtocomparethethreedifferentadaptationrules.Asatestscenario,wetookthearticialterraindatasetdepictedinFigure2consistingof441datapoints,whichcontainsuniformregionsaswellassharpedgesandcorners,whicharehardtoadapttolocally.Note,forexample,thattheedgebetweenthelowestandthesecondlowestplateauhasacurvatureandthatthreedifferentheightlevelscanbefoundinthelocalneighborhoodofthecornerinthemiddleofthediagram.Wesetmin=0:001andmax=5:0fortheboundedadaptationrules.Togeneratetrainingdatasetsforthedifferentexperimentsreportedonhere,weaddedwhitenoiseofavaryingstan-darddeviationtothetrueterrainelevationsandrandomlyremovedaportionofthesamplestobeabletoassessthemodel'spredictiveabilities.Figure4visualizesacompleteadaptationprocessforthecaseofadatasetgeneratedusinganoiserateof=0:3. (a)Alldatapointsgiven. (b)15%ofthedatapointsremoved. (c)30%ofthedatapointsremoved.Fig.3.PredictionaccuracyforthescenariodepictedinFigure4with(a)alldatapointsavailable,(b)15%ofthedata-pointsrandomlyremovedand(c)30%randomlyremoved.Eachgureplotsthemeansquarederrorofelevationpredictionsforavaryinglevelofaddedwhitenoise.Thevaluesareaveragedover10independentrunsperconguration.(Inthecaseof(c),theerrorofDirectInversewasalwaysgreaterthan4.0). (a)Terrain (b)ConvergenceFig.2.Anarticialterraindatasetusedintheexperimentalevaluation,thatexhibitsseverallocalfeaturesthatarehardtoadaptto(a).Testdatasetsaregeneratedbyaddingwhitenoiseandrandomlyremovingaportionofthedatapoints.Themeansquarederror(MSE)ofpredictedelevationsconvergeswithanincreasingnumberofadaptationsteps(b).Iteration0givestheMSEforthelearnedstandardGP.Valuesareaveragedovertenindependentruns.Onaverage,asingleiterationperruntook44secondsonthisdata-setusingaPCwitha2.8GHzCPUand2GBofRAM.Figures4(c)-4(f)showtheresultsofstandardGPregressionwhichplacesthesamekernelsatallinputlocations.Whilethisleadstogoodsmoothingperformanceinhomogeneousregions,thediscontinuitieswithinthemaparealsosmoothedascanbeseenfromtheabsoluteerrorsinthethirdcolumn.Consequently,thoselocationsgetassignedahighlearningrate,seerightcolumn,usedforlocalkerneladaption.TherstadaptationstepleadstotheresultsdepictedinFigures4(g)-4(j).Itisclearlyvisible,thatthestepsandcornersarenowbetterrepresentedbytheregressionmodel.Thishasbeenachievedbyadaptingthekernelstothelocalstructure,seetherstcolumnofthisrow.Note,howthekernelsizesandorientationsreectthecorrespondingterrainproperties.Kernelsareorientedalongdiscontinuitiesandaresmallinareasofstronglyvaryingelevation.Incontrast,theyhavebeenkeptrelativelylargeinhomogeneousregions.Afterthreeiterations,theregressionmodelhasadaptedtothediscontinu-itiesaccuratelywhilestillde-noisingthehomogeneousregions(Figures4(k)-4(n)).Note,thatafterthisiteration,thelocallearningrateshaveallsettledatlowvalues.Figure2givestheconvergencebehaviorofourapproachusingtheBoundedLinearadaptationruleintermsofthemeansquaredpredictionerrorfordifferentamountsofpointsremovedfromthenoisydataset.Afteratmost6iterations,theerrorshavesettledclosetotheirnalvalue.Inadifferentsetofexperiments,weinvestigatedthepre-dictionperformanceofourapproachforallthreeadaptationrulespresentedinSectionIII-C.Forthisexperiment,weaddedwhitenoiseofavaryingnoiseleveltothearticialterraingiveninFigure2.ThediagramsinFigure3givetheresultsfordifferentamountsofpointsremovedfromthenoisydataset.Whennopointsareremovedfromthetestset,theBoundedInverseadaptationruleperformsbestforsmallnoisevalues.Forlargenoisevalues,BoundedLinearandDirectInverseachievebetterresults.Inthecaseof15%and30%datapointsremoved,DirectInverseandBoundedInversearenotcompetitive.Incontrast,BoundedLinearstillachievesverygoodresultsforallnoiselevels.Thus,BoundedLinearproducesreliablepredictionsforalltestednoiseratesanddatadensities.Thisndingwassupportedbyexperimentsonotherrealdatasetsnotpresentedhere.B.EvaluationonRealTerrainDataInordertodemonstratetheusefulnessofourapproachonrealdatasets,weacquiredasetof3Dscansofasceneusingamobilerobotequippedwithalaserrangender,seeFigure5(a).Wecomparedourpredictionresultstoanapproachfromtheroboticsliterature[4]thathasbeenappliedsuccessfullytotheproblemof3-dimensionallymappingurbanareas.WeemployedtheBoundedLinearadaptationprocedureforourlearningalgorithmwherewesetmin=0:25andmax=4:0.Figure5givestheresultsofthisexperiment.Anobstacle,inthiscaseaperson,isplacedinfrontoftherobotandthusoccludestheslopedterrainbehind.WeevaluatedourapproachforthesituationdepictedinthegureaswellasforthreesimilaronesandcompareditspredictionaccuracytotheapproachofFr¨uhetal.[4],whoperformhorizontallinearinterpolationorthogonallytotherobot'sview.Thesescenariosusedareactuallyrathereasyonesfor[4],asthelargegapscanallbelledorthogonallyto (a)Testdataset(Noise:=0:3) (b)Localerrors (c)Localkernels(iter.0) (d)Regressionwithoutkerneladaptation (e)Localerrors (f)Learningrate (g)Localkernels(iter.1) (h)Regressionafterrstiteration (i)Localerrors (j)Learningrate (k)Localkernels(iter.3) (l)Regressionafterthirditeration (m)Localerrors (n)LearningrateFig.4.Thelocalkerneladaptationprocessonanarticialterraindataset:theoriginaldataset,depictedinFigure2,exhibitsseverallocalfeaturesthatarehardtoadaptto.Thetestdataset(a)wasgeneratedbyaddingwhitenoise,resultingintheerrorsshownin(b).Thesecondrowofdiagramsgivesinformationabouttheinitializationstateofouradaptationprocess,i.e.theresultsofstandardGPlearningandregression.Thefollowingtworowsdepicttheresultsofourapproachaftertherstandafterthethirdadaptationiterationrespectively.Intherstcolumnofthisgure,wevisualizethekerneldimensionsandorientationsafterthecorrespondingiteration.Thesecondcolumndepictsthepredictedmeansoftheregression.Thethirdcolumngivestheabsoluteerrorstotheknowngroundtruthelevationsandtheright-mostcolumngivestheresultinglearningratesforthenextadaptationstepresultingfromtheestimateddatalikelihoods. therobot'sview,whichisnotthecaseingeneral.Toestimatethekernelsatunseenlocations,webuiltaweightedaverageoverthelocalneighborhoodwithanisotropictwo-dimensionalGaussianwithastandarddeviationof3whichwehadfoundtoproducethebestresults.TableIgivestheresults.Inallfourcases,ourapproachachievedhigherpredictionaccuracies,reducingtheerrorsby30%to70%.Figure5(b)depictsthepredictionsofourapproachinoneofthesituations.IncontrasttoFr¨uhetal.,ourmodelisabletoalsogivethepredictiveuncertainties.ThesevariancesarelargestinthecenteroftheoccludedareaascanbeseeninFigure5(c).Inasecondreal-worldexperimentillustratedinFigure6,weinvestigatedtheabilityofourterrainmodelapproachtopreserveandpredictsharpdiscontinuitiesinrealterraindata.Wepositionedtherobotinfrontofarectangularstoneblocksuchthatthestraightedgesoftheblockrundiagonallytotherobot'slineofview.Apersonstoodinbetweentherobotandtheblock,therebyoccludingpartsoftheblockandoftheareainfrontofit.Thisscenarioisdepictedin6(a).Thetaskistorecoverthelinearstructureofthediscontinuityandlltheoccludedareaconsistentwiththesurroundingterrainelevationlevels.Theadaptationprocedureconvergedalreadyaftertwoiterations.Thelearnedkernelstructure,illustratedinFigure6(c),enablesthemodeltocorrectlyrepresentthestoneblocksascanbeseenfromthepredictedelevationsvisualizedin6(d).Thisgurealsoillustratestheuncertaintiesofthesepredictions,correspondingtothevariancesofthepredictivedistributions,bymeansoftwocontourlines.Thisindicatesthatamobilerobotwouldberelativelycertainabouttheblockstructurewithinthegapalthoughnothavingobserveditdirectly.Incontrast,itwouldbeawarethatitcannotrelyuponitsterrainmodelintheoccludedareasbeyondtheblocks:therearenoobservationswithinareasonabledistanceandthus,thepredictivevariancesarelarge.Toshowthatourapproachisapplicabletolarge,real-worldproblems,wehavetesteditonalargedata-setrecordedattheUniversityofFreiburgcampus1.Therawterraindatawaspreprocessed,corrected,andthenrepresentedinamulti-levelsurfacemapwithacellsizeof10cm10cm.Thescannedareaspansapproximately299by147meters.Forsimplicity,weonlyconsideredthelowestdata-pointsperlocation,i.e.,weremovedoverhangingstructuresliketreetopsorceilings.Theresultingtestsetconsistsof531,920data-points.Tospeedupcomputations,wesplitthismapinto542overlappingsub-maps.Thisispossiblewithoutlossofaccuracyaswecanassumecompactsupportforthelocalkernelsinvolvedinourcalculations(asthekernelsizesinourmodelarebounded).Werandomlyremoved20%ofthedata-pointspersub-map.Afullrunoverthecompletedata-settookabout50hours.Notethatthecomputationalcomplexitycanbereducedsubstantiallybyexploitingthesparsityofourmodel(duetotheboundedkernels)andbyintroducingadditionalsparsityusingapproximativemethods,e.g.,sparseGPs.TableIIgives1Additionalmaterialforthecampusexperimentcanbefoundathttp://www.informatik.uni-freiburg.de/plagem/rss07terReg Scenario LinearInterp.[4] AdaptedGP Improvement 1(Fig.5) 0.116 0.060 48.3% 2 0.058 0.040 31.0% 3 0.074 0.023 69.9% 4 0.079 0.038 51.9% TABLEIPREDICTIONPERFORMANCEINTERMSOFMSERELATIVETOASECOND,NOTOCCLUDEDSCAN. Adaptationprocedure MSE StandardGP 0.071 DirectInverse 0.103 BoundedLinear 0.062 BoundedInverse 0.059 TABLEIIPREDICTIONPERFORMANCEONALARGECAMPUSENVIRONMENT.theresultsofthisexperimentforthedifferentadaptationrules.TheBoundedLinearandtheBoundedInverseadaptationproceduresoutperformtheStandardGPmodelwherekernelsarenotadapted,whileDirectInverseisnotcompetitive.Togetherwiththeresultsoftheotherexperiments,thisleadstotheconclusionthatBoundedLinearisanadequatechoiceasanadaptationruleinsyntheticandreal-worldscenarios.V.CONCLUSIONSInthispaper,weproposeanadaptiveterrainmodelingapproachthatbalancessmoothingagainstthepreservationofstructuralfeatures.OurmethodusesGaussianprocesseswithnon-stationarycovariancefunctionstolocallyadapttothestructureoftheterraindata.Inexperimentsonsyntheticandrealdata,wedemonstratedthatouradaptationprocedureproducesreliablepredictionsinthepresenceofnoiseandisabletollgapsofdifferentsizes.Comparedtoastate-of-the-artapproachfromtheroboticsliteratureweachieveapredictionerrorreducedbyapproximately30%-70%.Inthefuture,weintendtoevaluateourapproachinonlinepathplanningapplicationsformobilerobots.Sinceourapproachretrievesterrainpropertiesintermsofkernels,itsapplicationtoterrainsegmentationispromising.AnotherdirectionoffurtherresearchareSLAMtechniqueswherethetrajectoryoftherobotisalsounknownandthemodelhastobeupdatedsequentially.Wealsointendtoevaluateourapproachontypicaltestcasesincomputervisionandtocompareitwiththealgorithmsofthiscommunity.Finally,weworkonananalyticalderivationforoptimalkernelsbasedsolelyondatalikelihoodsandmodelcomplexity.VI.ACKNOWLEDGMENTSTheauthorswouldliketothankKristianKerstingforthestimulatingdiscussionaswellasRudolphTriebelandPatrickPfaffforprovidingthecampusdata-setandtheirsourcecodeformulti-levelsurfacemaps.ThisworkhasbeensupportedbytheECundercontractnumberFP6-004250-CoSyandbytheGermanFederalMinistryofEducationandResearch(BMBF)undercontractnumber01IMEO1F(projectDESIRE). (a)Thersttestscenario (b)Observations(points)andpredictedmeans(lines) (c)Predictiveuncertain-ties(white:zero)Fig.5.Areal-worldscenario,whereapersonblockstherobot'sviewonaninhomogeneousandslopedterrain(a).Figure(b)givestherawdatapointsaswellasthepredictedmeansofouradaptednon-stationaryregressionmodel.Importantly,ourmodelalsoyieldsthepredictiveuncertaintiesforthepredictedelevationsasdepictedinFigure(c). 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