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Proceedings of the 11th International Conference on Pattern Recognition, pp.
163-167, 1992
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Model-based calibration of a range camera
Beraldin, Jean-Angelo; Rioux, Marc; Blais, François; Godin, Guy; Baribeau,
R.
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CCD
Lens
Fixed
Mirror
Scanning
Mirror
Source
Fixed
Mirror
3 0 Abstract 1 IntroductionAuto-synchronizedscanning geometry.
Model-based Calibration of a Range Camera
NRC33205-11thIAPRInt.Conf.onPat.Rec.(1992),
163-J.-A. Beraldin,M. Rioux,F.Blais and G.Go din R.Barib eau
Institutefor Information Technology Canadian Conservation Institute
National ResearchCouncilCanada Departmentof Communications
Ottawa,Ontario, Canada K1A 0R6 Ottawa,Ontario, Canada K1A 0C8
Thispaperintroducesanewmethodforthe
calibra-tion of arange camerabased on activetriangulation.
The technique is based on a model derived from the
geometry of the synchronized scanner. From known
positionsofacalibrationbar,alogisticequationis
t-ted with values of spot positions read from a linear
position detector at anumber of angularpositions of
ascanningmirror. Furthermore,withanapproximate
form of the general equations describing the
geome-try, aseriesof designguidelinesarederivedtohelpa
designer conduct a preliminary study of a particular
rangecamera. Experimentalresultsdemonstratingthe
techniquearefoundtocomparefavorablywith
theoret-icalpredictions.
etal.
etal.
Among the many techniques prop osed to extract
three-dimensionalinformationfromascene,active
tri-angulation is used in applications such as automatic
welding, measurement and repro duction of objects,
and insp ection of printed circuit b oards [1]{[2]. An
innovative approach, based on triangulation using a
synchronizedscanningscheme,wasintro ducedby
Ri-oux [3] to allow very large elds of view with small
triangulation angles without compromising on
reso-lution. With smaller triangulation angles, a
reduc-tion of shadowe ects is inherently achieved.
Imple-mentationofthistriangulationtechniquebyan
auto-synchronizedscannerapproachgivesaconsiderable
re-ductionintheopticalheadsizecomparedtostandard
triangulation metho ds. A 3-D pro le of a surface is
captured by scanning a laser b eam onto a scene by
wayof an oscillatingmirror, collectingthelightthat
isscattered bythesceneinsynchronismwiththe
pro-jection mirror,and, nally,fo cusing this lightonto a
linearp ositiondetector. Figure1depictsthe
synchro-nization e ect pro duced by the double-sided mirror.
This measurement pro cess yields two quantities p er
sampling interval: one is for the angular p osition of
themirrorandoneforthep ositionofthelasersp oton
thep ositiondetector. Owingtotheshap eoftheco
or-dinatesystemspannedbythesevariables(seeFig.2a),
the resultantimagesarenot compatiblewiththe
co-ordinate systems used by mostgeometric image
pro-cessingalgorithms. Are-mappingofthesevariablesto
Figure1:
amorecommonco ordinatesystemlikearectangular
systemistherefore required.
In[4],Bumbaca presentametho dtocalibrate
arange nderwithoutassuminganyre-mapping
func-tionforthegeometric distortions. Theauthorsusea
calibrationbarcomp osedoftwosections: onehavinga
uniformre ectance andonemadeofequidistant
du-cial markings of lower re ectance , painted onto the
surface. By constructing two tables fromknown p
o-sitions of this calibration bar in space, they correct
range and longitudinal distortions. The calibration
barismovedso asto de nearectangularco ordinate
systemas depicted inFig.2b. Withoutknowledgeof
the distortion laws, any attempt to reduce the
ran-dom noise present in the raw data to reveal the
re-mappingfunctioninherentin therange nder
geom-etryb ecomes anart. Archibald [5] prop ose
re-placingoneofthetablesbyaseriesoflinearequations
ttedfromdatatakenalongthescanangles(Fig.2c).
It isthe goalof this articleto extend the metho d
presented in [4]{[5]and to intro duce anovel metho d
and pro cedure for the calibration of a range
cam-era based up on the synchronization principle.
a)
constant p
constant azimuth
c)
constant azimuth
constant range
b)
constant range
longitudinal contours
β
-γ / 2
z
x
β
-γ
θ
β
p
P
θ
R
(θ)
o
R
(θ)
-R
(θ)
p
f
e
d
a
b
c
S
T
τ
γ
γ−τ
O
P ro jectio n
axis
D etectio n
axis
f
o
Len s
p lan e
p p p p 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 A OAC OED OBC OFD D E F DE DF2 Range camera equations
01 01 1 1 01 1 1 01 01 01 01 1 1 01 1 01 1 01 1 01 1 01 1 p X Z p R R R R P P p p R R p R R p P P f f f f f X ;Z ~ ~ X X X X Z Z Z Z P P p xp; X P X X P p z p; Z P Z Z P p R R p p P R
Ane representations: (a) constant p and
azimuth contours, (b) rectangular, (c) constant range
and azimuth.
Un-folded geometry.
Figure 2:
try from which amo del is de ned. These equations
areused asdesignto olstocharacterizethevolumeof
measurement and the spatial resolution, and to give
some guidelines for a precise calibration of a range
camera. Some exp erimental results obtainedfrom a
range nder intended for space applications are
pre-sented. Finally,adiscussionofseveraladvantagesand
limitationsoftheanalysisandtheprop osedtechnique
follows.
Figure 3showsthegeometry usedforthe
triangu-lation where the projection and collection axes have
b een unfolded. Thedotted lines depictthe static
ge-ometry,i.e.,for( =0). Here, thescanningangle is
measuredfromthesedottedlines. Most
triangulation-basedrange nderstakeadvantageoftheScheimp ug
optical arrangement [6]. The equations relating the
sp ot p osition to thelo cationof ap ointonthe
pro-jection axis can b e found from Fig. 3 in the
follow-ingmanner. Here,thescanningmirrorhasb een
tem-p orarilyremovedandapinholemo delforthelenshas
b een assumed. A rectangular co ordinate system ha
b een lo cated onthe axisjoining the equivalentp
osi-tionof theresp ective pivotoftheprojectionand
col-lection axes. These twop ositions are represe nted by
largecirclesonthe axis. The axisextendsfroma
p ointmidwayb etween them towards in nity. Sup
er-imp osedonthe gureistheequivalentgeometryfora
synchronizedrotationoftheprojectionandcollection
axesbyanangle .
Thesynchronized geometryimpliesthat,forasp ot
p osition = 0 (p oint on Fig.3), the acute angle
b etween the projection and collection paths is equal
to aconstant . From this, all the other angles can
b einferred. Twosetsofsimilartriangles {
and, { can b e identi ed. From these, the
followingrelation isextracted,
( ) ( )
( ) ( )
= (1)
where is the sp ot p osition on the detector
(detec-tionaxis),e.g.,CCD,andforagivenscannerangle ,
( ) is the distance ( ) along the projection axis
corresp ondingto , ( )isthelo cationofthe
van-ishing p ointon the projection axis,and ( ) is the
lo cation corresp onding to = 0. is the lo cation
Figure3: detectionaxis: = sin( ) cos( ) = ( ) sin( ) (2)
where isthefo callengthofthelens, isthee ective
distanceof thep ositiondetector to theimaginglens,
isthetiltangleofthep ositiondetector,and isthe
triangulationangle.
Thetransformationof eq. (1) to an( )
repre-sentationiscomputedfromthefactthattwop oints
and andathirdp oint b elongtothesamestraight
lineifthevectors and arelinearlydep endent.
Hence, ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) = (3)
Theab ove equationsaredecomp osableinb oth
or-thogonaldirections,i.e.,
( )= ( )+ ( ) ( ) (4) ( )= ( )+ ( ) ( ) (5)
These linear fractional equations, also known
as logistic equations, emphasize the nature of the
Scheimp ug geometry, that is, the limitingresp onse
( = ) for as it approaches (collection
pathparallelto thep osition detector) and ( = )
01 01 x z x p z p p 0 0 0 0 0 0 0 0 0 0 0 ' p ' ' 0 0 o o p 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2
3 Proposed calibration method
x z p 2 x p z p x z p 3.3.1 Table construction: 3.1 Volume of view 3.2 Spatial resolution
3.3 Improvedcalibrati on metho d
X T S = Z T S = X T Z T S T ; ; g ; h ; : p p f D D p x g p; z h p; p x;z ; p @g @p @g @ @h @p @h @ g h p p p p p u v u v p ( ) =
cos( 2 )+ cos( )sin( 2 )
cos ( ) (6) ( ) = (sin( 2 )+sin( )) cos ( ) cos( )cos ( 2 ) cos( ) (7) ( ) = sin(2 ) sin( ) (8) ( ) = cos(2 )+cos( ) sin( ) (9)
where is the distanceb etween thelens and the
ef-fective p osition of the collectionaxis pivot and is
halfthedistanceb etweentheprojectionandcollection
pivots. These equations constitute the basis for the
derivationofthedesignto olsandcalibrationmetho d
presentedinSection3.
Once the equations describing the geometry are
known,one canestimatesomeofthemostimp ortant
characteristicsofaparticulardesign.
Usually one b egins with agiven eld of view and
resolution for a particular application and uses the
static geometry ( = 0) to evaluate those numb ers.
At this p oint, laser b eam propagation must also b e
considered. Then, eqs. (4)-(9) are computed for a
particularscanwidthandp ositiondetectorsize. This
pro cessisrep eateduntilthesystemsp eci cationshave
b een met.
Twometho ds can b e used to estimate the
resolu-tion. The rst, given b elow, is the result of the
ap-plicationto ( )ofthe lawof propagation oferrors
due to the measurement of ( ) through eqs. (4)
and (5). The second metho d is the actual
calcula-tion of thejointdensity functionof = ( 2) and
= ( 2). This metho d allows for a full
charac-terization ofthetworandomvariables and taken
jointly. Thisresult willb erep orted separately.
Blais [7] has designed a galvanometer controller
that achieves a p eak-to-p eak error of 1 part in 5000
on a uni-directionalscan for an optical angle of 30 ,
i.e., 00001 . The measurementof isin
prac-ticelimitedbythelasersp eckleimpingingontheCCD
p osition detector. Barib eau and Rioux [8] predicted
that such noise b ehaves like a Gaussian pro cess and
theestimated rms uctuationof determinedbythe
noise isapproximately = 1 2 cos ( ) (10)
where is the wavelength of thelaser source and
fect of the noise generated in the electronic circuits
andthequantizationnoiseofthep eakdetectoronthe
measurementof areswamp edbysp ecklenoise.
Assumingthefunctions = ( ) and = ( )
have nosudden jumpsinthedomainaroundamean
value and ,thenthemeans( )andthevariances
( )can b eestimated intermsofthemean,
vari-ance,andcovarianceoftherandomvariables and2.
Theanalysisoftheopticalarrangementtogetherwith
the fact that the errors asso ciated with the physical
measurement of and are Gaussian random
pro-cesses and are not related lead one to assume that
theyareuncorrelated. Therefore,
+ (11)
+ (12)
wherethefunctions and and theirderivatives are
evaluated at = and = , and and are
the variances of the sp ot, and the angular scanning
measurement,resp ect ively.
As demonstrated in[4]
and[5],twotablescanb econstructedfromknownp
o-sitionsofacalibration barinspace. A go o d
calibra-tionpro cedure shouldthusallow and tob e
deter-minedaccuratelyforallcalibrationplaneand ducial
markings. Both metho ds require that one table b e
generatedfrommeasurementsof ducialmarkingson
acalibrationbar. Thesemarkingsmustb erecognized
by interpreting the brightness image of the bar. As
of yet, no mo delhas b een prop osed to quantify the
e ectof thelightdistributiononthemeasurementof
these ducialmarkings. For now, only the
measure-mentsof and have b eensucientlycharacterized.
Also,b ecauseamo deldescribingthesynchronized
ge-ometryintermsofthese samevariablesis now
avail-able(Section2),acalibrationtechniquethat requires
onlythemeasurementsof and along known p
osi-tionsofacalibration bar,withoutanyinterpretation
of the brightness image, would b e highly desirable.
Moreover,thedesignequations canb e used
advanta-geouslytodeterminethenumb erofcalibrationplanes
andtheirlo cationforlo ok-uptableconstruction.
One can think of an arrangement where a
cali-bration bar is moved in such a way as to de ne an
obliqueCartesian co ordinate systemas illustratedin
Fig. 4. This arrangement represents a variation of
the one prop osed in [9]. One table is created from
constant displacements of the bar in the direction
and another from constant displacements. These
displacementsare not necessarily known exactly but
sucientlytomeetthetargetedcalibrationaccuracy.
Then, these two tables are transformed such that
and b ecomeafunctionof b oth and . Ina
prac-tical situation,an obliqueco ordinate systemis much
unwarp-co
n
st
an
t
v
co
n
st
an
t
u
h o rizo n tal
referen ce
3 -D
C am era
0 1 X N i i i i i i i 2 =1 2 o o o 3 3 3 u v u p pu; p u N v p u;v = P x z > 3.3.2 Mo del-based tting: a a aObliqueCartesianco ordinatesystem.
4 Applications
5 Discussion
6 Conclusion
Figure 4:
de ned alongsomegivenhorizoncanb ecomputedby
measuringawedgeplacedatdi erentp ositionsinthe
eld ofview. Arectangularco ordinate systemshould
yieldaconstantangleforthewedge. Theexact
spac-ingb etweenhorizontalandverticallinesarecomputed
with areference object.
Acalibrationmetho d
that triestoestablishasetoftablescanb ecomevery
cumb ersometo implementwhen the volume of
mea-surement is large. A technique based on tting a
mo del to some calibration p oints in space would b e
of considerable interest. A mo dellike theone
devel-op edinSection2,i.e.,(4)and(5)canb e ttedtothe
data measured at each sampled angle in the eld of
view.
We prop ose using a linear fractional equation to
calibratearangecameraforlarge elds ofview. The
chi-squaredmeritfunctionisprop osed inorderto
es-timate the six parameters required for each angular
p osition;three parameters are inthe direction and
threeinthe direction. Onewillminimizethe
follow-ingmeritfunctionforthe p ositions
( ) =
( )
(13)
where is the parameter vector, is the rms error
asso ciated with each measurement of , the are
known withinsystem tolerance (e.g., accurate
trans-lation stage), and is the numb er of data p oints.
The samemeritfunctioncan b e appliedto the p
o-sitions. According to theory [8], if the measurement
errorsonthe arenormallydistributed,then(13)will
give themaximumlikeliho o destimationofthose
pa-rameters. Relevant intervals of con dence for those
parameters can also b e estimated. Once the six
pa-rameters are found for a particular angular p osition
of theprojection mirror,thensimpleinversion ofthe
fractional equation can b e done. The unwarping of
the ( )oblique co ordinate systemto arectangular
A numb erof tests, some of which are repro duced
here, werecarried outto demonstratethecalibration
techniques and were found to compare favorably to
theoreticalpredictions. Inoneof thesetests, therms
uctuationofthelogisticequation ttingfor512
an-gular p ositions was found to b e approximately 1 30
pixel. In that sametest, areference wedge was
cali-bratedandtheresultingerrorb etween trueand
mea-surededges was wellwithin systemtolerance. In
an-other test, theestimated obtained from ttinga
logisticequationandthetheoreticalmo delweretried
for a large volume on a range camera intended for
space vision applications. It features random access
of anyp osition ina eld of viewof 30 by 30 [10].
Objects lo catedwithinarangeof0.5mto100mcan
b e detecte d. Figure5 showsthe image after
calibra-tion of a quarter-scale mo del (at National Research
Council) of the cargo bay of the Space Shuttle
Or-biter. Theclosest and farthest p oints in the eld of
viewwereat2.6mand4.5m,resp ec tively. Both
esti-matedandmeasuredresolutionofthecameraatthese
twolo cations were found to b e in (along width of
cargo)125 mand1.5mmand,in (alongthedepth
ofcargo)200 mand 4mm. Thisscalemo delofthe
cargobaymeasures4.33mby1.42mby0.6 m.
Thecriticalelementsofthiscalibrationmetho dare
thevalidityofthemo delandthenumb erandlo cation
ofthe calibrationtargets. The fractionalequation of
Section2resultedfromseveralassumptionsrelatedto
thenatureof theimaginglens andtheplanarityand
lo cation of themirrors. The scanning mirroris
con-sidered to b e in nitely thin. Also, the galvanometer
wobbleandthedi ractionofthelaserb eamwere
ne-glected. Obviously, any distortion pro duced by the
imaginglens could b e mo deled. But considering the
factthattheangular eld ofviewoftheimaginglens
inourrangecamerasrarelyexceeds10 forvolumesof
viewvarying b etween 1cm and100m ,the pinhole
assumptionapp earsreasonable.
This pap er has intro duced a new metho d for the
calibration of a range camera based on active
trian-gulation. A mo delwhichis derivedfromtheanalysis
ofthesynchronizedgeometry providesabasisforthe
metho d. From the analysis of the optical
arrange-ment,somepartialdesignguidelinesarepresente d to
assistintheinitialstage ofdesignofarangecamera.
Parameters like the size of the volume of view and
longitudinal and range resolution can b e computed.
Twocalibrationmetho ds arediscussed. One metho d
is based on lo ok-up table construction. The second
isaimedatrange ndersthatcanscanlargevolumes,
i.e., 1m . Alogisticequationderivedfromthe
anal-ysisis ttedwithvaluesofsp otp ositionsreadfroma
linearp ositiondetector at anumb erof angular p
osi-tionsofthescanningmirror. Anexp erimentinvolving
themeasurementofaquarter-scalemo delofthecargo
References
Acknowledgements
Though the analysis considered only laser range
ndersbasedup onthesynchronizedscannerapproach
other triangulationgeometries with di erent
require-mentscanb e accommodatedbyasimilaranalysis.
The authors wish to express their gratitude to
L. Cournoyer for his technical assistance provided in
the course of the exp eriments. Finally, the authors
wish to thank E. Kidd for her help inpreparing the
text.
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