Fax12158989145Emailaddressgreveramipgupennedu10773142seefrontmatter2004ElsevierIncAllrightsreserveddoi101016jcviu200405002 ComputerVisionandImageUnderstanding952004317 ID: 607944
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ThedeadreckoningsigneddistancetransformGeorgeJ.GreveraMedicalImageProcessingGroup,DepartmentofRadiology,UniversityofPennsylvania,4thFloorBlockleyHall,423GuardianDrive,Philadelphia,PA19104-6021,USAReceived3February2003;accepted17May2004Availableonline7July2004Considerabinaryimagecontainingoneormoreobjects.Asigneddistancetransformas-signstoeachpixel(voxel,etc.),bothinsideandoutsideofanyobjects,theminimumdistancefromthatpixeltothenearestpixelontheborderofanobject.Byconvention,thesignoftheassigneddistancevalueindicateswhetherornotthepointiswithinsomeobject(positive)oroutsideofallobjects(negative).Overtheyears,manydierentalgorithmshavebeenproposedtocalculatethedistancetransformofanimage.Thesealgorithmsoftentradeaccuracyforef-ciency,exhibitvaryingdegreesofconceptualcomplexity,andsomerequireparallelproces-sors.Onealgorithminparticular,theChamferdistance[J.ACM15(1968)600,Comput.Vis.Graph.ImageProcess.34(1986)344],hasbeenanalyzedforaccuracy,isrelativelye-cient,requiresnospecialcomputinghardware,andisconceptuallystraightforward.Itisun-derstandably,therefore,quitepopularandwidelyused.WepresentastraightforwardmodicationtotheChamferdistancetransformalgorithmthatallowsittoproducemoreac-curateresultswithoutincreasingthewindowsize.WecallthisnewalgorithmDeadReckoningasitislooselybasedontheconceptofcontinualmeasurementsandcoursecorrectionthatwasemployedbyoceangoingvesselnavigationinthepast.WecompareDeadReckoningwithawidevarietyofotherdistancetransformalgorithmsbasedontheChamferdistancealgorithmforbothaccuracyandspeed,anddemonstratethatDeadReckoningproducesmoreaccurateresultswithcomparableeciency.2004ElsevierInc.Allrightsreserved.Signeddistancetransform;Chamferdistance;Euclideandistance Fax:1-215-898-9145.E-mailaddress:grevera@mipg.upenn.edu.1077-3142/$-seefrontmatter2004ElsevierInc.Allrightsreserved.doi:10.1016/j.cviu.2004.05.002 ComputerVisionandImageUnderstanding95(2004)317 333 www.elsevier.com/locate/cviu 1.IntroductionGivenabinaryimageconsistingofoneormoreobjectsanda(possiblydisjoint)background,wedeneasigneddistancetransformasatransformthatassignstoev-erypoint(toboththoseinobjectsaswellasthoseinthebackground)theminimumdistancefromthatparticularpointtothenearestpointontheborderofanobject.Thesignoftheassigneddistancevalueindicateswhetherthepointiseitherinside(positive)oroutside(negative)objects.Manydistancetransformalgorithmshavebeenproposedwith[18]and[25]mostlikelybeingtheearliest.Ingeneral,distancetransformalgorithmsexhibitvaryingdegreesofaccuracyoftheresult,computa-tionalcomplexity,hardwarerequirements(suchasparallelprocessors),andconcep-tualcomplexityofthealgorithmsthemselves.In[7],theauthorproposedanalgorithmthatproducesextremelyaccurateresultsbypropagatingvectorswhichap-proximatethedistanceof2Dimagesbysweepingthroughthedataanumberoftimesbypropagatingalocalmaskinamannersimilartoconvolution.In[1],theau-thorpresentedtheChamferdistancealgorithm(CDA)thatpropagatesscalar,inte-gervaluestoecientlyandaccuratelycalculatethedistancetransformof2Dand3Dimages(againinamannersimilartoconvolution).Borgefors[1]alsopresentedanerroranalysisfortheCDAforvariousneighborhoodsizesandintegervalues.Morerecently[2],ananalysisof3Ddistancetransformsemploying33neighbor-hoodsoflocaldistanceswaspresented.In[3],ananalysisofthe2DChamferdis-tancealgorithmusing33,55,andlargerneighborhoodsemployingbothintegerandrealvalueswaspresented.Marchand-MailletandSharaiha[26]alsopres-entananalysisofChamferdistanceusingtopologicalorderasopposedtotheap-proximationtotheEuclideandistanceastheevaluationcriteria.BecauseoftheconceptualeleganceoftheCDAandbecauseofitswidespreadpopularity,itsim-provementisthemotivationforthiswork.Ofcourse,distancetransformsoutsideoftheChamferfamilyalsohavebeenpre-sented.AtechniquefromArticialIntelligence,namelyA*heuristicsearch,hasbeenusedasthebasisforadistancetransformalgorithm[27].Amultiplepassalgorithmusingwindowsofvariouscongurations(alongthelinesof[7]andotherrasterscan-ningalgorithmssuchastheCDA)waspresentedin[28]and[34].Amethodofdis-tanceassignmentcalledorderedpropagationwaspresentedin[29].ThebasisofthisalgorithmandotherssuchasA*(usedin[27])istopropagatedistancebetweenpix-elscanberepresentedasnodesinagraph.Thesealgorithmstypicallyemploysortedliststoorderthepropagationamongthegraphnodes.GuanandMa[30]andEggers[31]employlistsaswell.In[35],theauthorspresentfouralgorithmstoperformtheexact,Euclidean,-dimensionaldistancetransformviatheserialcompositionof-dimensionallters.Algorithmsfortheecientcomputationofdistancetransformsusingparallelarchitecturesarepresentedin[32]and[36].In[36],theauthorspresentanalgorithmthatconsistsoftwophaseswitheachphaseconsistingofbothafor-wardscanandabackwardscan.Intherstphase,columnsarescanned;inthesec-ondphase,rowsarescanned.Theynotethatsincethescanningofaparticularcolumn(orrow)isindependentofthescanningoftheothercolumns(orrows),eachcolumn(row)maybescannedindependently(i.e.,inparallel).AdistancetransformG.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 employingagraphsearchalgorithmisalsopresentedin[33].DistancetransformscontinuetobeinterestingandhavealsobeenthefocusofatleastonerecentPh.D.dissertation[4].Sincetheearlyformulationofdistancetransformalgorithms[18,25],applicationsemployingdistancetransformshavebecomewidespread.Forexample,distancetransformshavebeenusedforskeletonizationofimages[6,14,19,21].Distancetrans-formsarealsousefulforthe(shape-based)interpolationofbothbinaryimages[11,15]aswellasgrayimagedata[8].In[12],theauthorsemploydistancetransforminformationinmultidimensionalimageregistration.Anecientraytracingalgo-rithmalsoemploysdistancetransforminformation[13].Distancetransformshavealsobeenshowntobeusefulincalculatingthemedialaxistransformwith[16,17]em-ployingtheChamferdistancealgorithmspecically.Inadditiontotheusefulnessofdistancetransformsfortheinterpolationof3Dgraymedicalimagedata[9,10],theyhavealsobeenusedfortheautomaticclassicationofplantcells[22]andformea-suringcellwalls[24].TheChamferdistancewasalsoemployedinamethodtochar-acterizespinalcordatrophy[20].Becausedistancetransformsareapplicabletosuchawidevarietyofproblems,itisimportanttodevelopaccurateandecientdistancetransformalgorithms.Theoutlineofthispaperisasfollows.First,wepresentafewdenitionsfromdig-italtopology[23]thatwillbeusefulfordevelopingtheframeworkofthealgorithm.ThenwepresenttheChamferdistancealgorithmandcompareandcontrastitwiththenewDeadReckoningmethod.Toevaluatethisnewmethod,wecompareitwiththeChamferdistancealgorithmemployingvariouswindowsizes(33,55,and7)andtypes(Chamfer,cityblock,chessboard,andEuclideanwitha33win-dow).Anevaluationframeworkisthenpresented.Thisframeworkevaluatestheal-gorithmswithrespecttoexecutiontime,aquantitativeevaluation,andaqualitativeevaluation.Thequalitativeevaluationdemonstratesthepresenceofpolygonaliso-contoursinalloftheChamfer-baseddistancetransformalgorithmsexceptDeadReckoning.Givenknownandrandombinaryimages,thequantitativeevaluationdemonstratesthattheDeadReckoningalgorithmproducesthemostaccurateresultswithrespecttotheactual,exactEuclideandistanceassignment.2.DenitionsandmethodsWerstpresenttheCDAforcompletenessandtodemonstratethesimilaritiesanddierencesbetweenitandtheDeadReckoningalgorithm(DRA).NotethatCDAhasbeenandDRAmaybeextendedtodistancetransformsinhigherdimen-sionalspacesbutwerestrictourdiscussionandanalysisto2Dforsimplicity.Acom-pleteoutlineoftheCDAappearsinFig.1.A2Dbinaryinputimage,,havingcolumnsandrowsisgivenasinputtotheCDA.Sinceisabinaryimage,foragivenpoint,,either0indicatingapointoutsideofanyobject1indicatingapointwithinanobject.Notethattheinputimagemaycon-sistofmorethanoneobject(andweassumethatitcontainsatleastoneobject).Fur-thermore,weassumethatnoobjectextendstotheborderof.Moreformally,G.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 .(Ifthisisnotthecase,wesimplyembedofsizeofsize2).Theoutputwillbeagrayimage,,alsoofsizewherethevalueassignedtoapointintheoutputimagerepresentsthedistancefromthatpointinthebinaryimagetothenearestpointontheborderofanobject.Usingterminologyfromdigitaltopology[23]wecallaborderpointanelementoftheimmediateinterior,,i^½ Fig.1.OriginalBorgeforsChamferdistancealgorithmusinga33window.Typically,3and4.Thesectionsappearinginboldintherstandsecondpasswouldbemodiedtoaccommodatelargerwindows.(Wenotethatthetwosetsofinitializationloopsmaybecombinedintoasingleloopforamoreecientimplementation.)G.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 Thisindicatesthatforanypointtobeanelementoftheimmediateinterioritmustbeinsomeobjectandatleastoneofits4-connectedneighbors(i.e.,twoadjacentpixelswitheithercoordinatesorcoordinatesthatdierbyexactlyone)mustbeoutsideofanyobject.Similarly,wecallaborderpointanelementoftheimme-diateexterior,,i^½Thealgorithmproceedsasfollows.Similarto[7],theCDAinitiallyassignsforallpoints.Thenextstepistoassign0toallpointsbelongingtoeitherorthe.Wenotethatsomedistancetransformalgorithmsincluding[7]re-strictthedenitionofborderpointtoelementsofonly.Ouralgorithmeasilyaccommodatesthis(viaonelineintheinitializationofthestepinFigs.1and3).If,however,borderpointsconsistofofIE,theresultingdistancetransformexhibitsthepropertyofsymmetryundercomplement.Thismeansthatadistancevalueas-signedtoapixelwillbethesameregardlessofwhetherornotthepixelisinsideoroutsideofanyobjects.Toillustratethispoint,considerabinaryimageanditscomplement,where0if1otherwiseAdistancetransformwiththesymmetryundercomplementpropertyproducesthesameresultsregardlessofwhetherisusedasinput.Again,asillustratedinFigs.1and3,ouralgorithmeasilyaccommodatesbothdenitions.Thentwopasses,oneforwardandonebackward,aremadethroughtheimagedata.Eachpassemployslocal(typically33,55,or77)neighborhoodopera-tions(roughlyanalogoustoconvolution)inwhichoneattemptstominimizethecur-rentdistancevalueassignedtothepixel,,atthecenterofthewindowbycomparingthecurrentdistancevaluewiththedistancevaluesassignedtoitsneigh-,plusthedistancefromtothegivenneighbor,,asspeciedinthewindow.VariouswindowsareshowninFig.2.Forexample,considertheforwardpass33window,,asshowninFig.2.Letindicatethevalueofthecenterofthe33windowwhichiscenteredoverthe.Letbethecurrentdistancetoaborderpoint.Thealgorithmthenmakesthefollowingassignment:Theforwardwindowismovedthroughinaforwardpass.Thenabackwardwindowismovedthroughinabackwardpasstopropagateminimumdistancesthroughout.Asin[15]and[34],weadopttheconventionof0indicatingapointwithinanobject,i.e.,1and0indicatingapointoutsideofanyobject(G.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 Borgeforscleverlydemonstrated:(i)usingasmallwindowandpropagatingdis-tanceinthismannerintroduceserrorsintheassigneddistancevaluesevenifdoubleprecisionoatingpointisusedtorepresentdistancevalues,(ii)theseerrorsmaybeminimizedbyusingvaluesotherthan1andforthedistancesbetweenneighbor-ingpixels,andsurprisingly,(iii)usingintegerwindowvaluessuchas3and4yieldsmoreaccurateresultsthanusingwindowvaluesof1andanddoessowithmuchbetterperformance,and(iv)largerwindowswithappropriatevaluesminimizeerrorsevenfurtheratincreasedcomputationalcost.Although,asinouralgorithmaswell, Fig.2.VariouswindowsusedbytheChamferDistancealgorithm.Euclidean33isincludedforcom-parison,asisthe33windowemployedbytheDeadReckoningalgorithm.indicatesthecenterofthewindow;-indicatesapointthatisnotused.G.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 thecomputationalcomplexityremainsthesame,,evenforlargerandlargerwindowsizes.TheDRAontheotherhandisastraightforwardmodicationtotheCDAthat,employingequalsizedwindows,producesmoreaccurateresultsataslightlyin-creasedcomputationalcost.Furthermore,DRAusingonlya33windowtypically Fig.3.TheDeadReckoningalgorithm(usingonlya33window).SectionsinboldindicateareasthatdierfromtheChamferDistancealgorithm.(Wenotethatthetwosetsofinitializationloopsmaybecom-binedintoasingleloopforamoreecientimplementation.)G.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 producesmoreaccurateresults(seeTable5)thanCDAwitha77window(withsimilarexecutiontimes,seeTable2).Inadditiontowhichforagivenpoint,,istheminimumdistancefromtothenearestborderpoint,theDRAin-troducesanadditionaldatastructure,,whichisusedtoindicatetheactualbordersuchthatthatIEanddðx;yÞisminimumsimilartothatem-ployedDanielssonin[7].(Danielsson[7]in4SEDemploys3minimizationiterationsinboththeforwardandbackwardpasses.OurmethodasintheCDA[1,25]employsonly1ineachpass).NotethatastheCDAprogresses,maybeupdatedmanytimes.IntheDRA,eachtimethatisupdated,isupdatedaswell.WenotethattheorderinwhichthestatementsinFig.3areevaluatedmayinuencetheassignmentofandsubsequently,thevalueassignedto.Regardless,ourresultsdemonstratethatouralgorithmremainsmoreaccurateusingonlya3neighborhoodthanCDAusinga77neighborhood.AlthoughtheDRAemploysa3(orlarger)windowtoguidetheupdate/minimizationofdistanceprocessas Fig.3.(G.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 doesCDA,theactualvaluesassignedtoarenotthesameasCDAasshowninEq.(3).Letdenotethecomponentof,anddenotethecomponentof.DRAusesinsteadtheactualEuclideandistancefromthebordertothepointatthecenterofthewindowasshowninEq.(4).Usingonlya33window,theDRAtypicallydeterminesamoreaccurateesti-mationoftheexactEuclideandistancewithintheframeworkoftheCDA.DetailsoftheDRAareshowninFig.3.3.ResultsanddiscussionWecomparedDRAwithotherdistancetransformalgorithms,CDA33,cityblock,chessboard,CDA55,CDA77,andEuclidean33,onthebasisofbothexecutionspeedandaccuracyoftheresultingdistancetransformsforknownimages.3.1.ExecutiontimesTodetermineexecutionspeeds,wecompiledandexecutedallprogramsona2GHzPentium4basedDellPrecision340with1GbofRAMrunningunderRedHatLinuxrelease7.1.AllalgorithmswereimplementedinC++andwerecompiledwithg++version2.96usingthe O3optionformaximumoptimizationforspeed.Testimagesconsistedofanumberofinputbinaryimagesofvarioussizescontainingasingleobjectpointatthecenterofeachimage.Forinputtestimagesofsizeslessthan50005000,executiontimeswereaveragedover100iterations.Forthe5000image,executiontimeswereaveragedover10iterations(wedeneaniterationtobeonecompleteexecutionofadistancetransformalgorithm).ExecutiontimesappearinTables1and2.Forthosedistancetransformwindowsthatuseanintegerrepresentationfordistance(viz.,CDA33,cityblock,chessboard,CDA5,andCDA77),anoptionalnormalizationstepmaybeperformedtoconvertnon-unitadjacentwindowvaluestounit1asshowninEq.(5).Thisallowsustocomparecalculateddistancevalueswithactual,knownEuclid-eandistancevaluesfortestimages. ExecutiontimeswithandwithoutthisconversionarereportedinTable1,andareincludedinallofthetimesinTable2.Wenotethatalthoughcityblockandchessboarduseanintegerrepresentation,theydonotrequirenormalizationsinceG.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 theyemployunitdistancesalready.Sincecityblockisthefastestwithorwithoutnormalization,thispointismoot.Itisthefastestbecauseourimplementationen-tirelyeliminatestheunnecessarycomputationofintheforwardandbackwardpasses,respectively.SinceCDA33,chessboard,andEuclidean33allemploy33windows,theyexhibitapproximatelythesameexecutiontimeseventhoughtheEuclidean33methodrepresentsdistancesusingoatingpoint.Actually,chessboardwasfasterthanCDA3andEuclidean33becausenormalizationwasnotrequired.AlsoCDA3 Table2Timeinsecondstoperformonecomplete2DdistancetransformforvariousimagesizesandalgorithmsImagesize256512512100010005000CDA330.010.020.092.30Cityblock0.010.020.082.01Chessboard0.010.020.092.34CDA550.010.030.123.02CDA770.010.050.184.42Euclidean330.010.020.092.22DeadReckoning330.010.050.153.94DeadReckoning770.020.080.266.864SED0.010.040.123.118SED0.010.040.153.888SED(improved)0.010.040.133.38Alltimesreportedfor33,cityblock,chessboard,55,and77includenormalizationofintegerdistancevaluestodoubles.Allinputimagesconsistedofasolitarypointatthecenter. Table1Resultsoftimingcomparisonforvariousdistancetransformalgorithmsappliedtotestimagesofsizes1000and5000Representa-Window10005000Nonormali-Nonormali-CDA33Integer330.090.062.301.50CityblockInteger330.080.052.011.19ChessboardInteger330.090.062.341.47CDA55Integer550.120.093.022.19CDA77Integer770.180.144.423.62Euclidean33Double330.092.22DeadReckoningDouble330.153.94DeadReckoningDouble770.266.864SEDDouble330.123.118SEDDouble330.153.888SED(improved)Double330.133.38Allinputimagesconsistedofasolitarypointatthecenter.G.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 wasslightlyfasterthanEuclidean33whichcanbeattributedtointegervs.oatingpointperformance.(Inourimplementation,thecalculationofinEuclidean3andinDRAonlyoccursonceandnotrepeatedly.)ThenextfastestwasCDAwitha5window,DRA33,CDAwitha77window,andDRA77whichwastheslowest.AlthoughDRA33usesoatingpointandtheextrastepsoftheDRA,itwasfasterthanCDAwitha77windowandnormalization(andonlyslightlyslowerthanCDAwitha77windowwithoutnormalization).Insummary,thealgorithmsfromfastesttoslowestwerecityblock,chessboard,CDA33,Eu-clidean33,CDA55,DRA33,CDA77,andDRA77(allincludingnormalizationifnecessary).3.2.TestimagesandquantitativeevaluationTonumericallyevaluatetheaccuracyofthevariousdistancetransforms,wede-nedanumberofinputbinaryimagesofvarioussizes.Thersttestconsistsofim-agesofsizes256256,512512,10001000,and50005000consistingofasingleobjectpointatthecenterofeachimage.Let()bethiscenterpoint.Then¼fð¼fð.Asmentionedpre-viously,initially0forallallIE(x0,y0).ThenforanypointtheimageweknowthattheactualEuclideandistance,,fromtotheboundarycanbecalculateddirectlyby,by,IEjfg:ð6ÞThisisthevaluethatshouldbeassignedbyanydistancetransformalgorithmforanypointinthisparticulartestimage.Toassesstheaccuracyofaparticulardistancetransformalgorithm,wecalculatetherootmeansquarederror(RMSE).ThequantitativeresultsarereportedinTable3,respectively,forvariousinputimagesizes.Fromthistable,onecanseethattheDRAwasthemostaccuratewithanRMSEof0.ThesecondandthirdmostaccuratewereCDAusinga77window Table3RootmeansquarederrorforaparticulardistancetransformfromtheknownEuclideandistanceforinputtestimagesconsistingofasinglepoint/objectatthecenteroftheimageImagesize256512512100010005000CDA333.827.6614.9874.97Cityblock34.8970.19137.49689.08Chessboard17.5835.1768.69343.45CDA550.992.003.9319.74CDA770.621.262.4812.50Euclidean335.7611.6622.90115.03DeadReckoning33141413DeadReckoning771414134SED0.390.390.390.398SED2228SED(improved)141413G.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 andCDAwitha55window,respectively.TheaccuracyofCDA77overCDA5overCDA33wasdemonstratedingeneralbyBorgeforsanalysisandthisanalysisdemonstratesitforaspecicimage.CDAwitha33windowhadthefourthbestRMSEwhileEuclideanwitha33windowhadthefthbestRMSE.Thesixthandseventh(least)accuratemethodswerechessboardandcityblock,respectively.Thisisnotsurprisingbecausechessboardemploysanincorrectunitdiagonaldistanceandcityblockneveremploysthediagonaldistancesoittendstooverestimatethem.Thesecondsetoftestimagesconsistsofaspeciccongurationofthreepointsthatareknowntoproduceerrorsfordistancetransformalgorithmsthatemploy33win-dows[4,5].Wegeneratedtestimagesofvarioussizes(3232to50005000)asfol-lows.Lettheinputimagebeofsizeandlet2.Thenwesetthefollowingpoints:and1.TheresultsareshowninTable4whichshowsthatforextremelysmallimages(3232to Table4RootmeansquarederrorforaparticulardistancetransformfromtheknownEuclideandistancefortheinputtestimageconsistingofthreeseparatepointsknowntobeparticularlyproblematic(seetext)Imagesize326464128128256256512512100010005000CDA330.420.881.843.767.6114.9375.92CDA770.060.130.290.601.242.4612.480.110.240.380.460.510.540.560.000.000.030.060.080.090.100.000.010.010.010.010.010.01 Table5AveragerootmeansquarederrorforaparticulardistancetransformfromtheknownEuclideandistancefor100inputtestimages(each256256)consisting1000randompointsAlgorithmAvgrmseCDA330.16Cityblock1.19Chessboard0.69CDA550.04CDA770.02Euclidean330.22DeadReckoning330.01DeadReckoning770.000034SED0.368SED0.358SED(improved)0.0008G.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 Fig.4.Resultsofvariousdistancetransformalgorithmsappliedtoaninputtestimagecontaininganob-jectconsistingofasinglepointatthecenteroftheimage.Theresultofeachdistancetransform(leftineachpair)isthresholded(rightineachpair).Theresultofaperfectdistancetransformshouldbecircularforthisparticularinputimage. Fig.5.ComparisonofartifactspresentinCDA77andabsentinDRA.Althoughtheresultsappearsimilar,thisguredemonstratesthatartifactsarestillpresentinCDA7G.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 128)withthisspecicconguration,CDA77performsbetterthanDRA3butDRA33outperformsCDA77formorerealisticimagesizesof256to50005000.Sinceouralgorithmisreadilyadaptableto77windows, Fig.6.Distancetransformresults(leftineachpair)foranotherinputtestimageconsistingof3singlepointobjects,andthethresholdedversionillustratingartifactsinallG.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 333 weevaluatedDRA77aswell.TheresultsshowthatDRA77outperformedCDA77forallinputimagesizes.Thethirdandlastsetoftestimagesconsistsof100randomlygenerated256binaryimageswith1000randompointsineachimagesettoone.Theresultsofvar-iousdistancetransformalgorithmsarethenappliedtoeachoftheseimagesandeval-uatedintermsofRMSE.Table5reportstheaverageRMSEforthese100imagesforvariousalgorithms.DRA33andDRA77outperformalloftheothermethodswithDRA77onlyslightlybetterthanDRA33.3.QualitativeevaluationTodemonstratetheartifacts(linearsegmentsattheperipheryofthecirculariso-contour,introducedbythevariousdistancetransformalgorithmsandthelackofar-tifactsfortheDRA),consideragainthebinaryinputimageconsistingofasinglepoint/objectatthecenteroftheimage.Theresultinggreydistancetransform,viewedasanimage,shouldconsistofconcentric,circularisovaluecontoursthatarecen-teredaboutandradiatingfromthecenter.Furthermore,ifwechooseathresholdva-lueandapplyittothegraydistancetransformresult,acircularobjectcenteredaboutthecenterpointshouldresult.TheresultsofthisqualitativetestareshowninFig.4.AlldistancetransformalgorithmsDRA33(andDRA77)ex-hibitedartifacts.EventhoughartifactsappearabsentforCDA77,closerinspec-tionrevealsthattheyarestillpresentinCDA77andabsentinDRA33asshowninFig.5.Toillustratetheseartifactsfurther,consideradierentinputbinaryimagecon-sistingofthreeobjects,eachconsistingofasinglepoint/objectseparatedbysomenitedistances.Fig.6illustratestheresultsofapplyingCDA33,CDA7andDRA33tothistestimage.AgainthethresholdedCDA33and77bothexhibitartifactsbutartifactsareabsentinDRA34.ConclusionsWehavepresentedanewdistancetransformalgorithm,DeadReckoning,thatisastraightforwardmodicationofthewell-knownChamferdistancealgorithm.Wealsodevelopedandpresentedaframeworkbywhichonecanevaluatedistancetransformmethodswithregardtoboththeirquantitativeaswellasqualitativeas-pects.WedemonstratedthatthisnewalgorithmexecuteswithacomputationalcostapproximatelythesameasthemostaccurateChamferdistancealgorithm,CDA77,butismuchmoreaccurate.Inthefuture,weintendtoprovethatthisnewalgorithmisindeedadistancemetric.Wealsointendtocomparethisalgo-rithmwithotheralgorithms(suchasDijkstras)intermsofspeed,computationalcomplexity,andaccuracyusingatestingframeworkthatweareintheprocessofdeveloping.FreesourcecodeforDRAaswellasCDAandothermethodsmentionedinthispapermaybeobtainedfromhttp://www.mipg.upenn.edu/~grevera/codeG.J.Grevera/ComputerVisionandImageUnderstanding95(2004)317 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