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Three-DimensionalAbsoluteOrientatione)f

StereoModelsUsingDigitalElevation

DanRosenholmandKenner!Torlegard

control whereX o,

Yo,and2

0 aretranslations,misthescalefactor,and theorthonormalrotationmatrixis

INTRODUCTION

T also from

FORMULATION

elevationofapointwithplanecoordinatesX,Y intheground themeasuredmodelcoordinates(x ij,

YwZij)p,wherex,y,zare

mulation,theindiceswillnotfurtherbeused. as (2) (4) m+mJ+mRm r32=sin(t)cos(a)+cos(t)sin(T})sin(a) r 13 =sin(T}) r 23
=-sin(t)cos(T}) r33=cos(t)cos(T}) respectively.Linearization aroundg=T}=a=I)andm=1,aspe tionsystemforabsoluteorientation:

Thisformulationdescribes

howdifferentialchangesinthe

2+Ll2=2

0 +m(Y31x+Y32Y--Y33z) +Ll2 0 xLlT}+YA£+zLlm(3) coordinatesintheOEM:2=I(X,Y)

Ll2=dlLlX+dlLlY

dX dY aroundg=T}=a=Oandm=1,

LlX=dX

o +xdm-yda+zdT}

LlY=dY

o +Xda+ydm-zdg(5) vation systems,i.e.,theOEMandthestereomodel:

A=2 -2

0- m(Y 31
x+Y 32

Y+'33z)

(1) m+mRm

Therotationelementsareherechosentobe

r ll cos(T})cos(a) r 21
cos(t)sin(a)+sin(t)sin( r31 sin(t)sin(;a)-cos(t)sin( r'2=---cos(T})sin(a) r22=cos(t)cos(a)-sin(t)sin(T})sin(a) d2 0- xdT}+Ydg+zdm -:{(dX o +xdm-yda+zdT}) -(dY o +xda+ydm-zdt) (6)

©1988AmericanSocietyforPhotogrammetry

andRemoteSensing vorable.This meansthateachpointinthisexampleshouldbe cerningslopeasindicatedinFigure1.

THEINVESTIGATIONMETHODS

parativetest inStockholm, wheretheworkreportedinthispaperwasper formed.Threeof thesixareasusedinthecomparativetestwere ure (Figure4).Foreachtestarea threemodelsmeasuredbythe aspossible(given thenumbers''1''inthetables),onewith of polationin differenttest areasalsohaddifferentnumbersofgridpoints (Table1).

Thecalculationsinthisinvestigation

wereperformedasab

FIG..1.Anexampleofthenec

essaryelevationandslopein formation intheOEM. matevaluesofthetransformationareused. derivativesare thechangeinZintheOEMcorrespondingtoa thecomputationofthefirstderivatives. Thisisdonemainlyfor apractical derivativeswillgiveerrorsin thedesignmatrix.Thediscrete firstderivativein xisthusinthisinvestigationcomputedas (7) pendingontheactualOEMstructure. The

6aresolvedfor

bytheleast-squaresmethod.Therotationpa untilconvergence.Aftereachiteration,all pointsinthestereo each tion tance,thescalefactorbyless than0.05percent,andtherotations astatisticaltestof thejointsignificanceoftheunknownsinthe adjustment. vationscanbe computedbyclassicalerrorpropagation.

WHATSLOPEANDELEVATIONINFORMATIONARE

NEEDED?

orientation.InFigure1 thediscussionisillustrated.Inallseven thatwillbedeterminedisshownforeachpoint.

First,

minationofdZ o.

TherotationsaroundtheX-andtheY-axes(dg

three togetherwithatranslationinplane(saydY o ),determinedby directions.Thenexttranslationinplane(dX o) needsonefurther nally thez-axisimpossibleisneeded(da). slopes)are proximationsof therealslopes.Iftheslopeissmall,thenthe •dE jda jdm dX D

ApPROXIMATEMEANSLOPESINXandy.

#Grid

PointsGridAreaSizeSlope%

TestAreainx

inySpacinginxinyinxiny

Bohuslan643519.81m1248m674m6.3 6.6Stockholm

46 4511.98m539m527m5.67.4Siihnstetten

1042011.38m1172m216m14.525.7

gardedasgivencontrol).

Asinitialvaluesnotranslations(X

o =Yo=2 0 ),norotations valuesisnotinvestigated.

RESULTSANDDISCUSSION

ACOMPARISONTOTHERESULTSOFTHEISPRS-TEST

intheISPRS-test. thelevelingdoneintheISPRS-test. inthecomparativetestcorrespondsto2 0, and17,whileX o,

FROMTHEISPRS-TESTAREGIV:N.

ThisInvestiga-

tionTheISPRS-test (RMS)

InitialGrossE.

ComputationRMSlT

o

InitialExclus.Level

Bohuslan11.320.83

1.311.30 1.03

Bohuslan21.621.48 1.62 1.50 1.43

Bohuslan32.442.18 2.35 2.252.15

Stockholm10.59 0.50 0.64 0.540.52

Stockholm21.290.951.32 1.25

1.11

Stockholm33.19d)3.23 3.23 3.12

Siihnstetten1

0.380.210.380.380.24

Siihnstetten20.48

0.410.480.42 0.38

Siihnstetten31.221.15 1.19 1.101.08

XoYo2 01)a

Computationmetremetre metremrnradmrad mrad

Bohuslan11.82.80.71.00170.6

-0.1-3.0

Bohuslan20.2-2.80.30.99920.40.7-0.1

Bohuslan33.7-3.30.80.99770.11.02.5

Stockholm11.40.50.0

1.0021-0.50.7-3.0

Stockholm2-1.01.10.31.0169-0.1-0.3-10.9

Siihnstetten1-0.30.5-0.41.00070.3 0.8-0.7

Siihnstetten20.0-0.20.41.0025-0.1 -0.1-1.1

Siihnstetten3-0.10.7 0.00.9979-0.4 -1.9-0.5

investigationcomparedtothe

ISPRS-test.

parametersare shown. (redundancy)inthis study,allratiosbetweenthecomputed fact participantsinthetest. absoluteorientation, andabsoluteorientationwithOEMsisto comparethe propagatedstandarddeviationsoftheunknown of (,TJ,X o,

Yo,and2

0 fromanabsoluteorientationinaKern andaccordingto the The operator.InTable6thecorresponding standarddeviationsfrom theabsoluteorientation withOEMsarepooledforeacharea.

Absoluteorientation

withOEMsgivesahigherprecisionof thejointheightparameters (2 0,

TJ,g).Thedifferenceishighly

significantconcerningthe2 0 parameter,whichinthiscase THE

ORIENTATIONWITHDEMs.

u Xo CY{U a

ComputationUometremetre metreU

m mradmrad mrad

BohusHin10.83 0.320.310.030.00090.20.11.0

BohusHin21.480.480.47 0.050.00120.20.11.3

BohusHin32.180.67 0.680.070.00170.30.22.0

Stockholm1

0.500.180.14 0.020.00080.10.10.8

S6hnstettentl0.210.04 0.05 0.020.00020.00.10.2

S6hnsteten20.410.07 0.09 0.040.00040.00.20.3

S6hnstetten31.150.200.26 0.100.00100.10.7 0.7

THE Uo uTI

Computationmetre metre metre metremradmrad

Bohuslan0.70.6 0.4 0.80.270.31

Stockholm0.30.2 0.20.40.090.16

S6hnstetten0.10.1 0.1 0.10.060.10

DEVIATIONSOFTHE

SAMEABSOLUTEORIENTATIONPARAMETERSASIN

TABLE2AFTERABSOLUTEORIENTATIONWITHDEMS.

Uo CYxoo U Yo CY TI

Computationmetremetremetremetremradmrad

Bohuslan1.600.510.510.050.20.1

Stockholm0.76 0.42 0.420.040.2 0.2

S6hnstetten0.720.120.160.060.10.4

is accuracyobtainedfromclassicalorientation oncontrolpoints.

MODELFIDELITY

1986).

does theproblems

Wehaveerrors

inthedesignmatrixbecauseoftworeasons. overa

Theobservations(theright

handside)arealsoaffected.The interpolationsinthe

OEM,performedbetweeneachiteration,

are fromtheestimationmodel thatisveryimportant. In distances, investigation.

CONCLUSIONS

andtheslopecharacteristicsoftheobject. In tional vation.Theaccuracyinplanimetryishighly dependentonthe

Thetypeofterrain,

willinfluencetheaccuracyofthesolution. The regulargrids ineitherthegroundOEMorinthestereomodel derivatives andoftheinterpolationintheOEM. allybeapproximated bytheunitmatrix,makingthecompu tationsverysimple. The asa pho graphicinformationcan thusbeusedforabsolute orientation. redundancy. soluteorientation. errorsin

1986b,1987)

ofphotogrammetryandforrobotvision.

ACKNOWLEDGMENTS

forgivingusaccesstotheir resultsfrommeasurementsinthe

NewSustainingMember

Rosenholm.

REFERENCES

eralizedModel beiderErzeugungDigitalerGelandemodelle.

InternationalArchives

of

RoyalInstituteofTechnology,Stockholm.

metry,Vol.XXVI,Comm.III,pp.573-587.

621-626.

ModelsusingDigitalElevationModels.

ProceedingsoftheACSM

ASPRSAnnual

ConventionPart4,pp.241-250.

430.
grammetria,Vol.41,No.1,pp.1-16. (Received

10June1988;accepted17June1988)

PlanGraphics,Inc.

Telephone502-223-1501

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