Practical Range Camera Calibration

READ THESE TERMS AND CONDITIONS CAREFULLY BEFORE USING THIS WEBSITE.

https://nrc-publications.canada.ca/eng/copyright

Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la

première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n’arrivez

pas à les repérer, communiquez avec nous à [email protected].

Questions? Contact the NRC Publications Archive team at

[email protected]. If you wish to email the authors directly, please see the

first page of the publication for their contact information.

NRC Publications Archive

Archives des publications du CNRC

This publication could be one of several versions: author’s original, accepted manuscript or the publisher’s version. /

La version de cette publication peut être l’une des suivantes : la version prépublication de l’auteur, la version

acceptée du manuscrit ou la version de l’éditeur.

Access and use of this website and the material on it are subject to the Terms and Conditions set forth at

Practical Range Camera Calibration

Beraldin, Jean-Angelo; El-Hakim, Sabry; Cournoyer, Luc

https://publications-cnrc.canada.ca/fra/droits

L’accès à ce site Web et l’utilisation de son contenu sont assujettis aux conditions présentées dans le site

LISEZ CES CONDITIONS ATTENTIVEMENT AVANT D’UTILISER CE SITE WEB.

NRC Publications Record / Notice d'Archives des publications de CNRC:

https://nrc-publications.canada.ca/eng/view/object/?id=5b241a5c-e0fc-49ff-8eb7-5dbca1434e30

https://publications-cnrc.canada.ca/fra/voir/objet/?id=5b241a5c-e0fc-49ff-8eb7-5dbca1434e30

1 2 2 ;; x y z     Abstract o o 128 3 o o

Practical Range Camera Calibration

1 Introduction

NRC35064{SPIE{Vol.2067VideometricsI I(1993),pp.21{31.

J.-A. Beraldin,S.F.El-Hakim,andL.Cournoyer InstituteforInformationTechnology NationalResearch CouncilofCanada

Ottawa,CanadaK1A 0R6

This pap erpresents a calibration pro cedure adapted to a rangecamera intended for space applications. The range camera, which is based up on an auto-synchronized triangulationscheme, can measure objects from ab out 0.5 m to 100m. Theeld of viewis30 30 . Objects situated at distancesb eyond 10mcan b e measuredwith thehelp of co op erative targets. Such alarge volumeof measurementpresents signicantchallenges to aprecise calibration. A two-step metho dologyisprop osed. Intherststep,theclose-rangevolume(from0.5mto 1.5m)iscalibratedusing anarrayoftargetsp ositionedatknownlo cationsintheeldofviewoftherangecamera. Alargenumb eroftargetsare evenlyspacedinthateldofviewb ecausethisistheregionofhighestprecision. Inthesecondstep,severaltargetsare p ositioned atdistances greater than1.5 mwiththehelp of anaccurate theo doliteand electronic distancemeasuring device. Thissecondstepwillnotb ediscussedfullyhere. Theinternalandexternalparametersofamo deloftherange cameraareextracted withaniterativenonlinearsimultaneousleast-squares adjustmentmetho d. Exp erimentalresults obtainedfor aclose-range calibrationsuggest aprecision alongthe , and axes of200 m,200 m, and250 m, resp ectively,and abiasoflessthan100 minalldirections.

Inmanydisciplinesofscience,thereisonethatiscertainlycommontoall,thatis,thescience ofmeasurement,or metrology. Asoneseeks tomeasureaphysicalquantity(variable)eithertoinform,e.g.,temp erature,topredict ,e.g., weatherpatterns,ortogivesomeindicationofthevalidityofatheory,themostaccuratedataarealwayssought. One thingmustrstb edened,thatis,whatonemeansbymostaccurate. Forsome,itmaymeananabsolutemeasurement takenincomparisontosomepredenedstandard. Forothers, itmayjust b earelative measurement. Thepro cess by whicha measurablequantity isobtainedshould b e as free as p ossible of systematicerrors (bias) andshould provide the highestlevelof precision (spreadcentered aroundanaverage value). Usually, these characteristics arealso given withsomelevelofcondence,e.g.,condenceintervalsorprobabilitydensityfunctions.

As atechnology that encompasses manydisciplines of the physical sciences, digital3-D imagingmustalso rely up on metrology. Amongthe manynoncontact techniques prop osed to extract3-D informationfrom ascene, active triangulation is used in applications as diverse as measurement and repro duction of objects, insp ection of printed circuitb oards,andautomaticrob otwelding . Aninnovativeapproach,basedontriangulationusingasynchronized scanningscheme,was intro duced byRioux to allowverylargeeldsofviewwithsmalltriangulationangles,without compromisingonprecision. A3-Dsurfacemapiscapturedbyscanningalaserb eamontoascenealongtwoorthogonal directions. Owing to the shap e of the co ordinate system spanned by the measured variables, the resultant images are not compatiblewithco ordinate systems used by mostgeometric image pro cessing algorithms. An unambiguous transformation(one-to-one)or,inother words,are-mappingof thesevariablestoamorecommonco ordinate system like a rectangular system is therefore required. The pro cess of measuring the original variables to a high level of condence andthedeterminationof there-mappingfunctionisknownascameracalibration.

This pap er presents a calibration pro cedure adapted to a range camera intended for space applications. With this camera, p osition and orientation of objects can b e measured when they are withina eld of view of 30 30 and with corresp ondingrange fromab out0.5 mto 100m. Such a largevolumeof measurementpresents signicant challenges to aprecise calibration. A two-step metho dologyis prop osed to calibratethis rangecamera at close and far range. The details of the range camera are presented in Section 2. Section 3 highlightsthe advantages of the synchronized scanningschemeoverconventionaltriangulation. Itincludesananalyticaltreatmentofthesynchronized geometry,wheretheprojective transformationequationsofthisopticalarrangementforsingleanddual scanaxesare presented. These equations are used in Section 4 as designto ols to characterize the spatialprecision and establish

a)

Known

Geometrical

Feature

or

Target

Tracking F.O.V. (~ 1°)

130 Measurements / s

(Position / Attitude)

Searching F.O.V. (~ 30°)

1 - 10 s Search Time

Laser

Scanner

High

Precision

Pointing

b)

3.1 Conventional Active Triangulation

2 Laser Range Camera for Space Applications

3 Range Camera Equations

Rangecamera: (a)photograph,(b) real-timetracking ofsatellites. Figure1:

a calibration pro cedure. The precision and accuracy assessments of the range cameraand the prop osed calibration metho daredescrib edinSection5. Exp erimentalresultsobtainedfromaclose-rangecalibrationarepresented. Finally, adiscussion followsinSection 6ofseveral advantages andlimitationsoftheanalysis and oftheprop osed technique. Conclusions app earinSection7.

ThelaserrangecameraisshowninFigure1a. Thisrangecameracanop erateineitheravariableresolutionmo de orarastertyp emo de. Inthevariableresolutionmo de,asillustratedinFigure1b,thelaserscannertrackstargetsand geometricalfeaturesofobjectslo catedwithinaeldofviewof30 30 andwithcorresp ondingrangefromab out0.5m to 100m. For objects lo catedat distances greater than 10 m,co op erative targetson their surfaces are required for go o dsignal-to-noiseratio . Therangecamerausestwohigh-sp eedgalvanometerstosteerthelaserb eamtoanyspatial lo cation withinthe eld of viewof the camera. A compact and versatile dual-galvanometer controllerwas designed sp ecically forthis task. Each axiscan b e addressed by morethan16 bits witha step resp onse of 18 s over wide angles eventhoughthemirrorsarequitelarge. Thisincrease insp eed andversatilityover theoriginalmanufactured controllerallowsthegenerationofLissajouspatternsfortrackingof objects atarefresh rateofupto 130Hz.

The raster mo de is used primarily for the measurement of registered range and intensity information of large stationaryobjects . Figure2showsashadedimageofaquarter-scalemo del(attheNationalResearchCouncil)ofthe cargo bayof theSpace ShuttleOrbiter. The scalemo delmeasures4.33mby1.42 mby0.6 m. Itwasdigitized toa resolutionof2048 4096.

ThebasicgeometricalprincipleofopticaltriangulationisshowninFigure3a. Thelightb eamgeneratedbythelaseris de ectedbyamirrorandscannedontheobject. Acamera,comp osedofalensandap ositionsensitive photo detector,

 z z p p x x 0 0 0 0 2 0 0 0 x;z z df p f  x z  p  d f f z  p  z f d   p z f d d d P f

measuresthelo cationoftheimageoftheilluminatedp ointontheobject. Bysimpletrigonometry,the co ordinate oftheilluminatedp ointontheobject inarectangularco ordinate systemis calculated. FromFigure3a,

=

+ tan( )

(1)

and

= tan( ) (2)

where isthep ositionoftheimagedsp otonthep ositiondetector, isthede ectionangleofthelaserb eam, isthe separationb etweenthelensandthelasersource,and istheeectivedistanceb etweenthep ositiondetectorandthe lens. isrelatedtothefo callengthofthelens.

Toshowsome ofthe limitationsof this triangulationmetho d, let us approximatethe standard deviation of the error in , ,asafunctionof only. Thelawofpropagation oferrors gives

= (3)

where is thestandard deviation of theerror in themeasurementof . Theerror inthe estimate of is therefore inversely prop ortional to b oth theseparation b etween the laser and the p osition detector and the eective p osition of thelens,but directly prop ortionalto thesquareofthe distance. Unfortunately, and cannot b emade aslarge as desired. is limited mainlyby the mechanical structure of the optical setup and by shadow eects. As seen in Figure3b, mustb ekeptsmalltominimizetheseeects .

Intheconventionaltriangulationgeometry,theeldofview8 ofthesensor,wherethelightb eamcanb escanned overthewholeeld ofview,isgivenapproximatelyby

8 2arctan( 2

Laser

Object

Deflector

Position

Detector

P

p

X

Z

Lens

d

Mirror

(X,Z)

θ

Φ

_x

f

o

S

_D

OBJECT

a)

b)

3.3 Single Scan Axis Case

Activetriangulationfrom : (a)basicprinciple, (b)shadoweects. Figure3:

where is thelength of thep osition detector. Therefore,in theconventionalsetup, a compromise b etween eld of view,precisionofthe3-Dmeasurement,andshadoweectsmustb e considered.

Large elds of view with smalltriangulation angles can b e obtained using synchronized scanner techniques without sacricing precision. With smaller triangulation angles, a reduction of shadow eects is inherently achieved. The intentistosynchronizetheprojectionofthelightsp otwithitsdetection. Asdepicted inFigure4a,theinstantaneous eld of view ofthe p osition detector followsthesp ot as it scans thescene. The fo callength of the lensis therefore relatedonlytothedesired depthofeld ormeasurementrange. Rioux intro ducedaninnovativeapproach,based on suchasynchronizedscanningscheme. Implementationofthistriangulationtechniquebyanauto-synchronized scanner approach allows a considerable reduction in the optical head size compared to conventional triangulation metho ds (Figure4b).

Figure 5depicts schematicallythebasic comp onentsofadual-axiscamera. Withsuch acamera,one captures a 3-Dsurface mapbyscanningalaserb eamontoascene by way oftwooscillatingmirrors,collectingthelightthat is scattered bythesceneinsynchronismwiththeprojectionmirrors,and,nally,fo cusingthislightontoalinearp osition detector, e.g., photo dio de array, charge-coupled device, or lateral eect photo dio de. The image acquisition pro cess yields threequantitiesp er samplinginterval: twoarefor theangularp ositionof themirrorsandonefor thep osition ofthelasersp otonthep ositiondetector. Theprojectivetransformationequationsofthisopticalarrangementfollow.

Figure6ashowsthegeometryused forthetriangulationwhere theprojectionandcollectionaxeshave b eenunfolded. The dottedlines depictthe staticgeometry, i.e.,for =0. Theoptical angle is measuredfromthese dottedlines. Sup erimp osedonthisgureistheequivalentgeometryforasynchronizedrotationoftheprojectionandcollectionaxes byanangle . Collinearityequationsrelatethesp otp osition tothelo cationofap ointontheprojectionaxis. Here, the scanningmirrors have b een temp orarily removed and apinholemo delfor the collectinglens has b een assumed. Furthermore,itisassumed thatthescanningmirroristhinand atandthatthereisnowobblepresent intheaxisof rotation. Arectangularco ordinatesystemhasb eenlo catedontheaxisjoiningtheequivalentp ositionoftheresp ec tive pivotoftheprojectionandcollectionaxes. Thesetwop ositionsarerepresente d bylargecirclesonthe axis. The axisextends fromap ointmidwayb etween themtowardinnity.

Thesynchronized geometryimpliesthat, forasp otp osition =0(p oint on thep ositiondetectorof Fig.6a), the acuteangleb etween theprojectionandcollectionpathsis equalto aconstant . From this,all otheranglescan

θ

θ

Object

Z

Laser

Position

Detector

Lens

(X,Z)

p

X

f

_o

a)

CCD

Lens

Fixed

Mirror

Scanning

Mirror

Source

Fixed

Mirror

b)

D E F DE DF 01 01 1 1 01 1 1 1 01 01 01 01 01 01 1 1 01 1 01 1 01 1 01 1

Synchronize d scanner approach: (a)basicprinciple, (b) auto-synchronization.

R  R  R  R  P P p  R  R  p R  R  p P P f  f f  f f  pR P R  R  R  R  X ;Z ~ ~ X  X  X  X  Z  Z  Z  Z  P P p x p; X  P X  X  P p z p; Z  P Z  Z  P p Figure 4:

b e inferred. Twosetsof similartriangles, { and { ,can b eidentied. Fromthese, thefollowing relationisextracted,

( ) ( ) ( ) ( )

= (5)

where foragivenopticalangle , ( ) isthedistance ( )alongtheprojectionaxiscorresp onding to , ( ) is thelo cationofthevanishingp ointontheprojectionaxis,and ( )isthelo cationcorresp ondingto =0. isthe lo cationofthevanishingp ointonthedetectionaxis:

= sin( ) cos( ) = ( ) sin( ) (6)

where isthefo callengthofthelens, istheeectivedistanceofthep ositiondetectortotheimaginglens, isthe tiltangleofthep ositiondetectorfoundaccordingtotheScheimp ugcondition,and isthetriangulationangle.

Usingthesameterminologyasinphotogrammetry,(5)isdesignatedastheprojective transformationequationfor thisactive triangulationmetho d. Thecollineartransformationequationisfoundbyinverting(5), thatis

( )= 1

( ) ( ) ( ) ( )

(7)

Thetransformationofeq. (5)toan representation iscomputedfromthefact that threep oints , and b elongtothesamestraightlineifthevectors and arelinearlydep endent. Hence,

( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) = (8)

Theequationisdecomp osableinb oth orthogonaldirections,i.e.,

( )= ( )+ ( ) ( ) (9) ( )= ( )+ ( ) ( ) (10)

0.5 m

_Position

Detector

Laser

Y-Axis

Scanner

X-Axis

Scanner

Collecting

Lens

3.4 Dual Scan Axis Case

Dual-axissynchronized scanner.

X  T   S  =   Z  S  =   T     X  T  Z  T  S T x;y ;z p; ; x p; ; x p; y p; ; z p; h = h  h  h z p; ; z p; h = h  h  h Figure5:

Theco ordinatesofp oints , ,and are

( ) =

cos ( 2 )+ cos ( )sin( 2 ) cos( ) (11) ( ) = cos ( )cos( 2 ) cos ( ) (sin( 2 )+sin( )) cos ( ) (12) ( ) = sin(2 ) sin( ) (13) ( ) = cos(2 )+cos( ) sin( ) (14)

where isthedistanceb etweenthelensandtheeectivep ositionofthecollectionaxispivotand ishalfthedistance b etween theprojectionandcollectionpivots.

The second scan axisis implementedas depicted inFigure 6b with agalvanometer driven mirror. Again, it is assumedthatthescanningmirroristhinand atandnowobbleispresentintheaxisofrotation. Moreover,thesecond scan axismirrorhasb een mountedorthogonallyto therst mirror( -axisscanner). According totheconvention set outinFigure6bandaftersomemanipulations,theequationsofap oint( )inthecameraeldofviewasafunction of( )are

( ) = ( ) (15)

( ) = ( ( ) cos( 2) )sin(2 ) cos (2 )+ (16)

C

B

p

γ

γ−τ

vo

vi

P

β

β

β

τ

Detection

Axis

Projection

Axis

f

o

0

S

Plane

Lens

Z

R

_p

(θ)

γ

θ

θ

γ/

2

T

A

D

E

F

R

o

(θ)

R

(θ)

X

a)

Target

CCD

Laser

Fixed Mirror

Y - Axis Scanner

Fixed Mirror

Collecting Lens

Z

O

Y

X

X - Axis Scanner

h

y

h

x

h

z

b)

x p; ;z p;  p; ;    p p   f D   D

4 Description of the Calibration Procedure

Optical arrangement: (a) geometry un-folded along the x-axis scanner, (b) schematic represen tation of the dual-axis case.

Figure 6:

wheretheco ordinatepair ( ) ( )isthep ositionofap ointthatwouldb emeasuredwitharangecamerahaving only asinglescan axis,i.e.,(9)-(10)and is themechanicalangleof the -axisscanner. These analyticalequations arethe projective transformationequations ofthesynchronized geometry inarectangular co ordinate systemand for the dualscan axiscase. They formthebasis forthederivation ofthedesignto ols and calibrationmetho dpresented inthefollowingsection.

Theequationsthatdescrib e thegeometryofthecameraareused asdesignto olstocharacterizethespatialprecision. Theapplicationofthe lawofpropagation oferrors to (15)-(17)is usedto estimatetheprecision as afunctionof the standard deviation asso ciated to themeasurementof ( ). Another metho disthe actualcalculation ofthe joint densityfunction. This metho d,althoughnumericallycomplex,allowsfor afullcharacterizationof thethree random variables , ,and takenjointly. For thepurp oseof thisdesign, such alevelofsophisticationis notrequired. The lawofpropagationoferrors issucient.

This range camerauses twogalvanometer to drive each mirror. The p ointing precision was measured b etween 15 mand 47mand theworst case for anopticalangle of30 wasab out60 rad. Themeasurementof is inpractice limitedbythe lasersp eckleimpingingonthep ositiondetector. Barib eau andRioux predicted that such noise b ehaveslikeaGaussian pro cessand theestimated uctuationof determinedbythat noiseisapproximately

= 1

2

cos ( )

(18)

where isthewavelengthofthelasersourceand isthelensdiameter. Inawell-designedsystem,whenenoughlight iscollectedfromthescene,theeect ofthenoisegeneratedintheelectroniccircuitsandthequantizationnoiseofthe

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Expected Precision (mm)

σ

_z

σ

_y

σ

_x

Distance from Range Camera along Z axis (m)

Distance from Range Camera along Z axis (m)

Expected Precision

σ

z

(mm)

1

10

100

0.1

1

10

100

1000

a)

b)

4.2 Proposed Calibration Method

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 5 3

Exp ectedrangecameraprecision   ,  , : m: (a)close-range,(b)far-range.

p  :  x f p; ; y g p; ; z h p; ; p   x;y ;z  ; ; p    @f @p  @f @  @f @   @g @p  @g @  @g @   @h @p  @h @  @h @  f g h p p        x;y ;z Z 

Figure7: = 60 rad = 60 rad = 31

p eakdetector onthemeasurementof are swamp edbysp ecklenoise. Withthecurrent cameradesignthepredicted valueis 31 m.

Assuming the functions = ( ), = ( ), and = ( ) have no sudden jumps inthe domain aroundameanvalue , ,and allowsestimationofthemeans( )andthevariances ( )intermsofthe mean,variance,andcovarianceoftherandomvariables ,2,and8. Theanalysisoftheopticalarrangementtogether withthefact that theerrorsasso ciated withthephysicalmeasurementof , ,and are Gaussianrandompro cesses and are lo osely related leads one to assumethat they are uncorrelated. Therefore, thelawof propagation of errors gives

+ + (19)

+ + (20)

+ + (21)

where thefunctions , , and and theirderivatives are evaluated at = , = , and = , and , ,and are thevariances ofthe lasersp ot detection andthe p ositionof thescan mirrors, resp ectively. Figure 7a providesa graphoftheprecisionpredictedbytheequationsab ove. Atclose-range,i.e.,0.5mto1.5m,themaximalvalueofthe ( )precisionintheeldofviewwasplottedagainstthedistancefromthecamera, . Thedistancesaremeasured fromtheexit windowofthecamera. Figure 7bshowsonly fordistancesb etween 1mand100m.

For this project, an extension ofthe metho dpresented byBeraldin is used to calibratetherange camera. A calibrationbarisadequate tocalibrate3-Dvolumesintheorderof1m . Anyvolumeslargerthanthatrequireavery large calibrationbarand acarefulsetup. A two-step metho dologyisprop osed to alleviatetheshortcomings ofthat metho d. Intherststep,theclose-range volume(from0.5mto1.5m)iscalibratedusinga atglassplatecomp osed ofanarrayoftargets. Thisplateismadeupofevenlyspacedtargetsonagridof5 7. Thecenterofeachtargetwas previously measuredwithaco ordinate measuringmachine(CMM). Thetarget-background contrastshould b e high.

6 2 7 o o p; ; p; ;   p  : 5 Accuracy Assessment Resolution bias precision accuracy 5.1 Denitions

5.2 Close-Range Calibration and Evaluation

Here, alargenumb er oftargets isused inthis regionb ecause itis where thecamerahas itshighest rangeprecision. Inthesecondstep,it isprop osed touseasmallernumb eroftargets(the numb eroftargetsisaectedbythenumb er of parameters in themo del)p ositioned at distances greater than 1.5m with thehelp of an accurate theo dolite and electronicdistancemeasuringdevice. Beyond10m,theseinstrumentshaveangularanddistancemeasuringaccuracies 10timesb etterthanthisrangecamerathoughtheyhave verylowmeasurementratecomparedtothe18000samples p ersecondofthepresentsystem. Oncetheregisteredrangeandintensityimageshaveb eenacquiredforallthetargets, theedgesofthosetargetsareextracted tosub-pixelaccuracyusingamomentpreservingalgorithm. Thecenterofeach target (in termsof theco ordinates ( ))is thendetermined by ttingan ellipseto theedge p oints. An iterative nonlinearsimultaneousleast-squares adjustmentmetho dextracts theinternaland externalparametersofthecamera mo del,i.e.,(15)-(17).

Toremove mostresidual systematicerrors fromthecalibration,distortion parameters on( ) were included in themo del. Equations(15)-(17) require that and b e in angularunits and inunit of length. Unfortunately, this is not the case. The angular p osition of the mirrors are given as an output numb er from an analog-to-digital converterlo catedonthegalvanometercontrollercard. Thelasersp otp ositionisaninterp olatedpixelnumb er. Third-orderp olynomialsareincludedinthemo delforthemappingofthesedimensionlessquantitiestotheprop er variables. Twentyparameters mustb esolvedforinthecurrent rangecameramo del.

To establish standardized measures for vision systems p erformance, terminology for accuracy must b e clearly dened. Ideally, a vision system intended for metrologyshould provide the distribution of rep eated measurements taken on well-knownobjects and ducialmarks, the op erating conditions,all p ertinent systemparameters, and the sp ecictargetedapplication.Moreover,thevisionsystemshouldyielddimensionalmeasurementsthathavesymmetric error distributions,free ofsystematicerrors and artifacts. Considering that therequirements andsp ecicationsvary greatly fromoneapplicationto another,characterizing avisionsystemb ecomes anon-trivialtask . For thepurp ose ofthisexp eriment,thetermsresolution,bias,precision,andaccuracywillb edenedandusedinthecontextofdigital rangeimaging. Noattempthasb eenmadetofullycharacterize theerrordistributions.

is the smallestspatialinterval that can b e displayed. With this rangecamera, theresolutions of the scanning axesand thesp ot p ositiondetection areeach 16bits. System isdened as thedierence b etween the mean of the measurementand the true valueof some ducialmark. The true valueof a measurement is, however, unknown andtheb est estimateonecould obtainis fromusing asup erior system. Measurementsprovidedby a well-acceptedgaugingtechniquesuchasaCMMservesasabaseforcomparison. Theterm isdenedasthedegree ofconformityamongaset ofobservationsofthesamequantity. Thespreador disp ersionofthemeasurementofthat quantityisanindicationofprecision.Whenthedistributionofthemeasurementsb ehaveslikeaGaussiandistribution, the standard deviation is used as ameasure ofprecision. For , b oth bias and precision mustb e givenas measureofsystemmetricp erformance. Noattemptshouldb emadetocombinethetwonumb ersintoasinglenumb er. Obviously,environmentalparametersliketemp erature,atmosphericpressure,humidity,andmechanicalstabilitymust b e sp ecied. Inthis exp eriment,theaccuracywasestimatedandveriedat atemp eratureof25 C 05 C.

Theresultsforaclose-rangecalibrationarerep ortedhere. Thesecondstepinthecalibrationpro cedurewillb ediscussed inaseparate pap er. The eld ofviewof thecamerawas digitizedto aresolutionof 512 512. The calibrationwas p erformedbyp ositioningthetargetarray0.7mand1.1mfromthecamera. Fortytargetsorcontrolp ointswereused for the solution(the minimumb eing seven). Theextracted parameters for themo del were then appliedto theraw data(b othrangeandintensity)obtainedforthesametargetarraylo catedatthreeotherlo cations: 0.9m,1.3m,and 1.5m. Notethatthelasttwop ositionsofthearrayareessentiallyextrap olateddata. Positions1.3mand1.5mcould notb eusedinthesolutionb ecausethetargetswereto osmall. Thistargetarraywas notsp ecicallydesigned forthis rangecamera. Futurecalibrationswillnotb e hinderedbytheselimitations.

p x y z 6 Discussion z :  ; ; p; ; x;y x y z z z 

Left, scatter plot of target p ositioning error for m. Upp er right, bias; Lower right, for 100targets at variousdistances fromcamera.

= 09

The center of each target (in terms of the co ordinates ( )) was also determined bytting an ellipse to the edge p ointsobtainedfromamomentpreserving algorithmappliedto theintensityimage. The co ordinates were thenfoundwith(15)-(17). Thebiasandprecisionalongthe and axesweredeterminedbycomparisontothevalues givenwiththetargetarraycerticate. The comp onentwasdeterminedbyttingaplanarpatchoneachtargetatall ve p ositionsofthetargetarray(ab out100targets). Theresultsonthebiasandprecision arepresentedinFigure8.

The prop osed calibration metho d allows for a rapid extraction of the internal and external parameters of the rangecamera. Atargetarrayisp ositionedataminimumoftwoknownlo cations(knownspacingb etween them)anda registeredrangeandintensityimageisacquired. Ittakeslessthan1min.onap ersonalcomputertosolveforthecamera parameters foratotalof40 targets. The techniquewas veriedforclose-range calibration. Theexp erimentalresults agreewiththepredictedones. The precisionis,infact,b etterthanpredicted,exceptp erhapswhenmeasurementsare extrap olatedoutsidethecalibratedvolume. Extrap olationoughttob eavoidedwiththecurrentmo delofthiscamera. The increased precision is explainedbytheoverestimate of . Inpractice, it islower thanthe valuepredicted by thesp ecklenoisemo del. Afutureexp erimentwillverifythecalibrationoftheremainingvolumeofmeasurementwith veryaccurate surveyingequipment.

The critical elements of this calibration metho d are the validity of the mo del and the numb er and lo cation of the calibrationtargets. Themo delpresented inSection 3resultedfromseveral assumptionsrelated to thenature of theimaginglensandtheplanarityand lo cationofthescanningmirrors. Thescanningmirrorswere consideredtob e innitely thinand mountedorthogonally. Also, thegalvanometerwobble andthe diractionof thelaser b eamwere neglected. Further renementstothemo delwillcomefromab etterunderstandingofthedistortionscreatedbythese assumptions. The next version of the target array willb e designed and build according to therequirementsof this rangecamera. Finally,environmentalconditionsaecting theoverallaccuracy andothersystemp erformancesuchas automaticextractionoffeaturesareeithernotyet denedortheireects havenotb eenfullydetermined.

x;y z    

8 Acknowledgements

References

This pap erintro duced anewmetho dforthecalibrationof arangecameraintendedforspace applications. Amo del based on the collinearity principle is derived for the synchronized geometry. It is used to extract the internal and external parameters of the rangecamerawith an iterative nonlinearsimultaneous least-squares adjustmentmetho d. Anassessmentoftheprecisionandaccuracyofthisrangecameraandtheprop osedcalibrationmetho dwereconducted inaeldofviewlo catedwithinarangeof0.5mto1.5m. Exp erimentalresults obtainedforacloserangecalibration suggest aprecision alongthe and axesof 200 m,200 m,and 250 m, resp ectively, and a biasof less than 100 minalldirections. These resultsare satisfactoryfor theintendedapplicationofthisrangecamera. Althougha laserrangenderbasedup onthesynchronizedscannerapproachwasconsidered,otheractivetriangulationgeometries withdierentrequirementscan b eaccommo datedbyasimilaranalysisandcalibrationpro cedure.

The evaluation of the p erformance of vision systems for metrology is challengingsince it is concerned with all asp ects of evaluation, i.e., image quality and detection succes s, and also involves the understanding of all system parameters andhowtheyinteract topro duce certainresults. Rigorousrangecameracalibrationisessentialforvision applications, ifone wantsto sp eed up thetransition of vision systems fromlab oratory to the spaceenvironmentor to thefactory o orwhere accurate measurementis required. An imp ortantfactoristhe lackofreliablemetho ds for predictingthep erformanceofaprop osed solutionforagivenapplication. It isalsoverydicultto mo delor predict thep erformancewithoutanearp erfect andstablecalibrationpro cess.

TheauthorswishtothanktheCanadianSpaceAgencyfortheirsupp ortalongthecourseofthisproject. Theauthors wish to express theirgratitudeto P.Boulanger whoprovided thesegmentationroutines usedto evaluatethecamera p erformanceandP.Amiraultforthetechnicalillustrations. Finally,theauthorswishtothankE.Kiddforherhelpin preparing thetext.

[1] M.Rioux,\ApplicationsofDigital3-DImaging,"CanadianConference onElectricalandComputerEngineering., Ottawa,Sept. 4{6,37.3.1{37.3.10(1990).

[2] P.J.Besl,\RangeImagingSensors,"Mach.VisionApplic.,1,127{152(1988).

[3] M.Rioux,\LaserRangeFinderbased onSynchronized Scanners,"Appl.Opt.,23,3837{3844(1984).

[4] F. Blais,J.-A. Beraldin,M.Rioux,R.A.Couvillon,and S.G.MacLean,\Developmentof aReal-Time Tracking Laser Range Scanner forSpace Applications," Workshop on Computer Vision forSpace Applications, Antib es, France,Sept. 22{24(1993)InPress.

[5] J.-A.Beraldin,R. Barib eau,M.Rioux,F.Blais,and G.Go din,\Mo del-BasedCalibrationof aRangeCamera", Pro c.of11th IAPRInt.Conf.onPatternRecognition,TheHague, TheNetherlands, Aug.30{Sept.3,163{167, (1992).

[6] R.Barib eauandM.Rioux,\In uenceofSp eckleonLaserRange Finders,"Appl.Opt.,30,2873{2878(1991).

[7] S.F.El-Hakim, \Applicationand Performance Evaluationof aVision-based AutomatedMeasurement System," Videometrics,SPIEPro ceedings, Vol.1820,181{195(1992).

[8] F. Blais, M. Rioux and J.-A. Beraldin, \Practical Considerations for a Design of a High Precision 3-D Laser ScannerSystem,"SPIEPro ceedings,Vol.959,225{246(1988).