Predicting the Accuracy of Optical Tracking Systems

92 Pages·2007·13.47 MB·English

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Technische Universita¨t Mu¨nchen
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Fakulta¨t fu¨r Informatik d d d d
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Diplomarbeit in Informatik
Predicting the Accuracy
of Optical Tracking Systems
Michael Schlegel
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Technische Universita¨t Mu¨nchen
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Fakulta¨t fu¨r Informatik d d d d
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Diplomarbeit in Informatik
Predicting the Accuracy
of Optical Tracking Systems
Michael Schlegel
Aufgabensteller: Univ-Prof. Gudrun Klinker, Ph.D.
Betreuer: Dipl.-Inform. Martin Bauer
Abgabedatum: 17. Juli 2006
Erkla¨rung
Ichversichere,dassichdieseAusarbeitungderDiplomarbeitselbststa¨ndigverfasstundnur
dieangegebenenQuellenundHilfsmittelverwendethabe.
Mu¨nchen,den17.Juli2006 MichaelSchlegel
Zusammenfassung
OptischesTrackingisteineTechnologie,umObjektemitHilfevonoptischenKamerasim
Raum fortlaufend zu lokalisieren. Dabei kann sowohl die Position als auch die ra¨umliche
Lage bestimmt werden. Optisches Tracking ist eine Basistechnologie, welche in vielen Ein-
satzgebieten zur Anwendung kommt. So wird optisches Tracking in der Augmented Reality
benutzt,umrealeObjektemitvirtuellerInformationzuerweitern(engl.:toaugment).Damit
dieseErweiterungenandenrichtigenStellenplatziertwerdenko¨nnen,mu¨ssenPositionund
BlickrichtungdesBeobachtersundvondenzuerweiterndenObjektenbestimmtwerden.
Um nun eine genaue Verbindung zwischen realen und virtuellen Objekten herzustellen,
istesno¨tigdieMessungensopra¨zisewiemo¨glichdurchzufu¨hren,aberleidertrittselbstbei
gro¨sster Sorgfalt und Pra¨zision immer Rauschen in den Messungen auf. Dieses Rauschen
inderKamerakannzubetra¨chtlichenFehlerninderLokalisierungvonObjektenfu¨hren.In
bestimmten Anwendungen ist es no¨tig diese Fehler im Voraus und zur Laufzeit zu kennen
undzuminimieren.AlsBeispielseihierdieMedizintechnikgenannt.
Das Rauschen in den Messungen kann durch ein gaußsches Fehlermodell beschrieben
werden.ZusammenmitdemFehlerfortpflanzungsgesetzerlaubtesbeispielsweiseausdem
RauschenaufderBildebenederKameraaufdiePositionsungenauigkeitschliessen.
Mit Hilfe des gaußschen Fehlermodells und des Fehlerfortpflanzungsgesetzes wurden
mehrere mathematische Modelle entwickelt, die es erlauben ausgehend von Rauschfehlern
in einzelnen Messungen auf die zu erwartende Gesamtungenauigkeit zu schliessen. Das
Fehlerfortpflanzungsgesetz wird in dieser Arbeit sehr ha¨ufig benutzt, unter anderem auch
fu¨rdiezweientwickeltenAnwendungen,welchedieGenauigkeiteneinesTrackingsystems
mitmehralszweiKamerasquantitativscha¨tzenko¨nnen.
Abstract
Opticaltrackingisatechnologytolocalizethepositionandrotationofobjectscontinuously
withthehelpofopticalcameras. Opticaltrackingisabasictechnologyusedinmanyapplica-
tions. Inaugmentedrealitytrackingisusedtoaugmentrealobjectswithvirtualinformations.
To place the informations at the right position, the position and viewing direction of the
observerandthatoftheobjectstobeaugmentedhavetobedetermined.
To determine an accurate relationship between real and virtual object it is necessary to
take the measurements as precise as possible, but even with great diligence and precision
noisy errors will be inevitable. The noise on the image plane of the camera can lead to
considerable errors in the localization of the objects. In certain applications for instance in
themedicaldomainitisnecessarytoknowandminimizetheseerrorsaprioriandatruntime
The noise in measurements can be described with a gaussian error model. Together with
theerrorpropagationrulesitisforinstancepossibletopredictthelocalizationerror.
With the help of the gaussian error model and the error propagation rules several math-
ematical models are developed which allow to predict the expected uncertainty based on
noise errors. The error propagation is widely used in this thesis eg. for the two developed
applicationswhichcanquantitativelypredicttheaccuracyofanopticaltrackingsystemwith
morethantwocameras.
Contents
1 Introduction 1
1.1 OpticalTracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 AugmentedReality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 GoalsofthisThesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.1 Accuracyofn-ocularTrackingSystems . . . . . . . . . . . . . . . . . . 3
1.4.2 AccuracyofMarkerTargets . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Tracking Systems 8
2.1 CriteriaforTrackingPerformance . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 OpticalTrackers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 InertialTrackers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 MagneticTrackers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 AcousticTrackers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 MechanicalTrackers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.7 GlobalNavigationSatelliteTrackers . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Mathematical Basics 14
3.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.2 EulerAngles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.3 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1 Errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.2 Multi-DimensionalRandomVariable . . . . . . . . . . . . . . . . . . . 16
3.2.3 ProbabilityDensityFunctions . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.4 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.5 VarianceandCovariance . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.6 ProbabilityDensityFunctions . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 InterpretationofCovariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3.1 QuadraticForms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3.2 SurfaceofConstantProbability . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 ErrorPropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4.1 ForwardPropagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4.2 BackwardPropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 CameraModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5.1 PinholeCameraModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
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Contents
4 Errors And Accuracy 28
4.1 KindsOfErrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.1 SystematicErrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.2 RandomNoise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 RepresentationofAccuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.1 MultiDimensionalNormalDistribution . . . . . . . . . . . . . . . . . . 36
4.2.2 RootMeanSquare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Accuracy of Optical Tracking Systems 41
5.1 N-OcularSystemDetectingSingleFeaturePoint . . . . . . . . . . . . . . . . . 41
5.1.1 SurveyofN-OcularTrackingSystems . . . . . . . . . . . . . . . . . . . 44
5.1.2 DerivationofCovarianceFormulas . . . . . . . . . . . . . . . . . . . . 44
5.1.3 ExperimentalEstimationofErrors . . . . . . . . . . . . . . . . . . . . . 46
5.2 MonocularTrackingSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2.1 TheoreticalAccuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.2 VisualizationofUncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 Accuracy of Marker Targets 57
6.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.2 UncertaintyintheCentroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3 PointOfInterest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.4 ValidationofthePoseAccuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.4.1 RotationalUncertainy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.4.2 PositionalUncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.5 OneUncertaintyforallFeatures . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.6 VisualizationofthePoseUncertainty . . . . . . . . . . . . . . . . . . . . . . . . 64
6.6.1 PositionalUncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.6.2 RotationalUncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.6.3 CovariancealongaPath . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7 Tools 67
7.1 MathematicaToolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.2 CovarianceFieldViewer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.3 PoseUncertaintyVisualizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
8 Future Work and Conclusions 75
8.1 FutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.1.1 MonocularAccuracyPrediction . . . . . . . . . . . . . . . . . . . . . . 75
8.1.2 Volumerendering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.1.3 FieldCovarianceMeasureSystem . . . . . . . . . . . . . . . . . . . . . 75
8.1.4 UncertaintyintheTargetGeometry . . . . . . . . . . . . . . . . . . . . 76
8.1.5 Real-timeEstimationofAccuracy . . . . . . . . . . . . . . . . . . . . . 76
8.1.6 ConsiderUncertaintyofExtrinsicandIntrinsicParameters . . . . . . . 77
8.1.7 Occlusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
8.1.8 AreasorVolumesofInterest . . . . . . . . . . . . . . . . . . . . . . . . 78
8.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
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Contents
Bibliography 80
iii
1 Introduction
Inthischapterashortoverviewoverthecontextandthegoalsofthisdiplomathesiswillbe
given.
1.1 Optical Tracking
Optical tracking is a technology to localize the position and rotation, also called pose, of
objects continuously with the help of optical cameras. Therefore optical tracking systems
aretakingpicturesandextractingfeaturesfromthepictureswhichallowthedetermination
of the pose. Without placing artificial markers on the object this is a challenging task, so
mostoptical trackingsystem usedistinctive markersto easethe featureextraction fromthe
image.
As the extraction of the features and the measurements in the camera can only be done
witherrors,theestimatedposewillalsobeerroneous. Thereareseveralcausesforerrorsfor
instancewhenthecameraproducesdistortedpictures. Somekindsoferrorscanbereduced
oreventeliminatedbutothersareinevitable. Thenoiseoccurringonthesensorofthecamera
isanexampleforsuchaninevitablesourceoferror.
The user of an optical tracking system desires accurate results and so the errors affecting
the estimation of the pose have to be reduced. In some applications, for instance in the
medicaldomainitisnecessarytoknowtheseerrorsaprioriandatruntime.
1.2 Augmented Reality
AugmentedRealityisatechniquewhichcombinestherealworldandthevirtualworld. The
userofanaugmentedrealitysystemcanseetherealworldwithvirtualobjectssuperimposed
upon or composited with the real world. This can be achieved by using head-mounted
displays(HMDs)whichallowtheusertoseetherealworldandtoaddadditionalcomputer
generated visual informations. A more general definition is given by Azuma [5, 6] where a
augmentedrealitysystemhastofulfillthefollowingcriteria:
1. Combinesrealandvirtual
2. Interactiveinrealtime
3. Registers(aligns)realandvirtualobjects
1
1 Introduction
Thefirstpointemphasizesthattherealandthevirtualworldhavetobecombined,which
excludespurevirtualapplications. Thesecondcriterionclaimsthatthesystemhastobein-
teractivefortheuser. Thethirdcriteriondemandsthatthevirtualobjectshavetobealigned
with the real world. As the third criterion is not easy to understand for those people who
never heard of augmented reality an example will be given. Caudell et al. are describing
an example for augmented reality in fabrication of an airplane [11]. An user has to drill a
hole in a workpiece and a HMD is used to visualize the location. So the user looks at the
workpiece and sees the exact 3D location indicated by a green arrow along with the drill
size and depth of the hole specified in a text window next to the arrow. As the user walks
around the workpiece the arrow will be kept aligned with the desired drill location. This
example fulfills Azuma’s criteria. First the reality is augmented with virtual objects, the
user can interact with the system and the virtual objects are aligned with real objects. The
alignmentofthevirtualobjectswithrealobjectsontheHMDrequiresseveralinformations.
FirsttheposeoftheHMDandsecondtheposeoftherealworldobjecttowhichthevirtual
object should be aligned have to be known. This relationship is called the registration and
the process of keeping this registration up to date is termed tracking. With help of the pose
datadeterminedfromthetrackingsystemthesystemcanaddthevirtualobjectattheright
positionandorientation,eveniftheusermoveshisheadoriswalkingaround.
More about augmented reality can be found in a comprehensive survey by Azuma et
al.[6].
1.3 Goals of this Thesis
The goal of this thesis is the development of a mathematical framework, which allows to
predicttheexpectedaccuracyofanopticaltrackingsystembasedonerrorscausedbynoise.
The noise error can be described with a gaussian error model. With the help of error
propagation rules it is possible to predict the expected error based on the noise in the mea-
surements. For example the features of the tracked object are measured with noise on the
sensorofthecameraintwodimensions. Whenusingmultiplecameras,allaffectedbynoise,
theestimatedspatialpositionoftheobjectisalsonoisy. Knowledgeofthenoiseinestimating
the2Dpositiononthesensorcanthebeusedtopredictthenoiseofthespatialposition.
The users of optical tracking systems want to know how accurate their specific tracking
system is in the working volume which is crucial for eg. in medical applications. Manufac-
turersofopticaltrackingsystemswanttopredicttheaccuracyofthetrackingsystemwithout
assessingrealsetups. Designersoruserswanttooptimizethesetupwithrespecttoaccuracy
andcosts. Forexampletheaccuracyofanopticaltrackingsystemcanbeincreasedmerelyby
adding more cameras, but this would also increase the monetary costs of the system. With
helptheofdevelopedframeworkdifferenttrackingsetupscanbeassessedbychangingthe
basicfeaturesofthesystemastherearethepositions, thekindandthequantityofcameras
used.
Thereforeaframeworkwasdevelopedforprovidingmathematicalprocedurestopredict
theexpectedaccuracyofdifferenttypesofopticaltrackingsystems,astherearen-ocularand
monocular tracking systems. The framework was developed in a computer algebra system
2