A Plane Measuring DeviceWord文档格式.docx
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A Plane Measuring DeviceWord文档格式.docx
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ErrorAnalysis;
ProjectiveGeometry.
1Introduction
Theaimoftheworkinthispaperistobeabletomakemeasurementsofworldplanesfromtheirperspectiveimages,andaccuratelypredicttheuncertaintyofthesemeasurements.Thusacamerabecomesameasurementdevice.Exampleapplicationsofthisdeviceincludemeasurementofinteriorscenessuchaswallsorfloorsforfurnitureplacementandinteriordesignpurposes,andarchitecturalmeasurements,wherethesizeandpositionofwindows,doorsetcaredetermined.
Thecameramodelforperspectiveimagesofplanesiswellknown[13]:
pointsontheworldplanearemappedtopointsontheimageplanebyaplanetoplanehomography,alsoknownasaplaneprojectivetransformation.Ahomography
30November1998
PreprintsubmittedtoElsevierPreprint
isdescribedbya3x3matrixH.Oncethismatrixisdeterminedtheback-projectionofanimagepointtoapointontheworldplaneisstraightforward.ThedistancebetweentwopointsontheworldplaneissimplycomputedfromtheEuclideandistancebetweentheirback-projectedimages.
However,ameasurementisoflittleuseunlessitsaccuracyisknown.Estimatingtheaccuracy(oruncertainty)requiresapropertreatmentofthesourcesoferror,notjusttheerrorinselectingtheimagepointsbutalsotheerrorsinthehomographymatrixitself.Thehomographymatrixerrorarisesfromthepositionerrorsofthepointcorrespondencesfromwhichthematrixiscomputed.
Inthispaperwemakethreenovelcontributions:
first,itisshowninsection4thatfirstorderuncertaintyanalysisissufficientfortypicalimagingarrangements.Thisisachievedbydevelopingtheanalysistosecondorderandobtainingaboundonthetruncationerror.Second,itisshownthatthefirstorderanalysisisexactfortheaffinepartofthehomography,andthatanapproximationisonlyinvolvedforthenon-linearpart.Third,insection5anexpressionisobtainedforthecovarianceoftheestimatedHmatrixbyusingfirstordermatrixperturbationtheory.
Theuncertaintyanalysisdevelopedherebuildsonandextendspreviousanalysisoftheuncertaintyinrelationsestimatedfromhomogeneousequations,forexamplehomographies[11]andepipolargeometry[4,5].Itextendstheseresultsbecauseitcoversthecaseswherethematrixisexactlydeterminedandthecasewherethematrixisover-determined,bytheworld-imagecorrespondences.Furthermore,theanlysisisnotadverselyaffectedwhentheestimationmatrixisnearsingular.Thisisexplainedinmoredetailinsection5.ThecorrectnessoftheuncertaintypredictionshasbeenextensivelytestedbothbyMonteCarlosimulationandbynumerousexperimentsonrealimages.
Sections6and7describehowtheuncertaintyanalysisisappliedtoparticularmeasurementstakingaccountofthecumulativeeffectsofdifferenterrorsources,includingtheimagepointlocalizationandthehomographymatrixcovariance.
Section8givesexamplesofpredictinguncertaintiesandachievingaspecifieduncertaintybyvaryingthenumberanddistributionofcorrespondences.Bothinteriorandarchitecturalmeasurementexamplesarecovered.
Fig.1.PlaneCameraModel:
apointXontheworldplaneisimagedasx.EuclideancoordinatesXYandxyareusedfortheworldandimageplanes,respectively.Cisthecameracentre.
2Thecameramodel
Wedescribeherethecameramodelwhichconsistsofcentralprojectionspecialisedtoplanes.
Figure1showstheimagingprocess.Thenotationusedisthatpointsontheworldplanearerepresentedbyuppercasevectors,X,andtheircorrespondingimagesarerepresentedbylowercasevectorsx,wherexandXarehomogeneous3-vectors,X=(X,F,l)Tandx=(x,y,l)T.Underperspectiveprojectioncorrespondingpointsarerelatedby[9,13]:
X=Hx
(1)
whereHisa3x3homogeneousmatrix,and“=”isequalityuptoscale.Thecameramodeliscompletelyspecifiedoncethematrixisdetermined.Thematrixcanbecomputedfromtherelativepositioningofthetwoplanesandcameracentre.However,itcanalsobecomputeddirectlyfromimagetoworldpointcorrespondences.Thiscomputationisdescribedinthenextsection.
3Computingtheplanetoplanehomography
Fromequation
(1)eachimagetoworldpointcorrespondenceprovidestwoequationslinearintheHmatrixelements.Theyare
h\\X+h12y+ft13—h^\xX+h32yX+h^Xh2iX+h22y+h23=h^ixY+flz2yY+^33^
Forncorrespondencesweobtainasystemof2nequationin8unknowns.Ifn=4thenanexactsolutionisobtained.Otherwise,ifn>
4,thematrixisoverdetermined,andfornon-perfectdataHisestimatedbyasuitableminimisationscheme.
ThecovarianceoftheestimatedHmatrixdependsbothontheerrorsinthepositionofthepointsusedforitscomputationandtheestimationmethod.TherearethreestandardmethodsforestimatingH:
3.1Non-homogeneouslinearsolution.
Oneofthe9matrixelementsisgivenafixedvalue,usuallyunity,andtheresultingsimultaneousequationsfortheother8elementsarethensolvedusingapseudo-inverse.Thisisthemostcommonlyusedmethod.Ithasthedisadvantagethatpoorestimatesareobtainedifthechosenelementshouldactuallyhavethevaluezero.
Themostfrequentassumptionish33=1.Thisisnotvalidifthevanishinglineoftheworldplaneintersectstheoriginoftheimagecoordinatesystem.
3.2Homogeneoussolution.
ThesolutionisobtainedusingSVD.Thisisthemethodusedhereandisexplainedinmoredetailbelow.Itdoesnothavethedisadvantageofthenon-homogeneousmethod.
3.3Non-lineargeometricsolution.
ThesummedEuclideandistancesbetweenthemeasuredandamappedpointisminimised.Thismethodhastheadvantage,overtheabovetwoalgebraicmethods,thatthequantityminimisedismeaningfulandcorrespondstotheerrorinvolvedinthemeasurement(similarminimizationsareusedtoestimatethefundamentalmatrixandtrifocaltensor[15,17]).Thereisnoclosedformsolutioninthiscaseandanumericalminimisationscheme,suchasLevenberg-Marquardt[10],isemployed.Usuallyaninitialsolutionisobtainedbymethod3.2,andthen“polished”usingthenon-linearone.
O
<
Fig.2.One-dimensionalCameraModel:
Thecameracentreisadistancef(thefocallength)fromtheimageline.Therayattheprincipalpointpisperpendiculartotheimageline,andintersectstheworldlineatP,withworldordinatet.Theanglebetweentheworldandimagelinesisto.
P
X
t
3.4Homogeneousestimationmethod
WritingtheHmatrixinvectorformash=(^11,h12,^13,^21,^22,h2s,ft31,^32?
h^)Tthehomogeneousequations
(1)fornpointsbecomeAh=0,withAthe2nx9matrix.Eachcomputationmatch(x^0X^)providestworowsoftheAmatrix:
1-XiXji—yiXi-Xji
2-XiYji—yjYi-Yji
Xiyi100000Xiyi
Thevectorhthatminimisesthealgebraicresiduals||Ah||,subjectto||h||=1,isgivenbytheeigenvectorofleasteigenvalueofATA.ThiseigenvectorcanbeobtaineddirectlyfromtheSVDofA.Inthecaseofn=4theresidualsarezeroandhisthenull-vectorofA.
4Firstandsecondorderuncertaintyanalysis
Toavoidunnecessarilycomplicatedalgebrathecomparisonbetweenfirstandsecondorderanalysisisdevelopedforalinetolinehomography.Theonedimensionalcaseillustratesalltheideasinvolved,andthealgebraicexpressionsareeasilyinterpreted.Thegeneralisationto3x3matricesisstraightforwardanddoesnotprovideanynewinsightshere.
Intheone-dimensionalcaseequation
(1)reducesto:
1,
_U
—«
2x2
H‘
whereH2x2isa2x2homographymatrix.Forthegeometryshowninfigure2thematrixisgivenby
tan(co)
as
ax+1
withparametersa=,蘭!
&
)—jtan(u))andfi
Underback-projectionanimagepointxmaps
h11x+h12
Thisnon-linearmapping(oninhomogeneouscoordinates)canbeexpandedinaTaylorseries.StatisticalmomentsofXjsuchasthevariance,arethencomputedintermsoftheTaylorcoefficientsandthemomentsofx[2,12].Itisassumedherethatthehomographyisexact(noerrors)andthemeasurementoftheimagetestpointxissubjecttoGaussiannoisewithstandarddeviationax.TheTaylorseriesisdevelopedaboutthepoint’smeanpositiondenotedasx.
ax+ta—at
—1z~~—
fiX+1(fiX+1):
H(a-fit),—、5
{X—x)——:
7Z^{X—x)
、)(_+I)3、)
4.lFirstorder.
IftheTaylorseriesistruncatedtofirstorderin(x—x)thenthemappingislinearised.ThevarianceofXthereforeis:
(a—fity(^x+1)^
a—fd(px+1):
(x—x)
E[(x—xY
o\=E[(X-X)2]^EandsinceE[(x—x)2]=a^
(2)
Usuall
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