Tracking errors in augmented reality [Elektronische Ressource] / Martin A. Bauer

technische_universitat_munchen

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178 pages

English

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Informatik

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Publié par	technische_universitat_munchen
Publié le	01 janvier 2007
Nombre de lectures	27
Langue	English
Poids de l'ouvrage	21 Mo

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Technische Universitat Munchen, Institut fur Informatik¨ ¨ ¨
Chair for Computer-Aided Medical Procedures & Augmented Reality
Tracking Errors
in Augmented Reality
Martin A. Bauer
Vollsta¨ndiger Abdruck der von der Fakulta¨t fu¨r Informatik der Technischen
Universita¨t Mu¨nchen zur Erlangung des akademischen Grades eines Doktors der
Naturwissenschaften (Dr.rer.nat.) genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr. Hans Michael Gerndt
Prufer der Dissertation: 1. Univ.-Prof. Gudrun J. Klinker, Ph.D.¨
2. Univ.-Prof. Nassir Navab, Ph.D.
Die Dissertation wurde am 23.04.2007 bei der Technischen Universit¨at Mu¨nchen
eingereicht und durch die Fakulta¨t fu¨r Informatik am 10.09.2007 angenommen.Abstract
Tracking is an important component of augmented reality systems. Track-
ing means to determine and follow the position and orientation of an object
with respect to some reference coordinate system over time. As with all phys-
ical measurements the estimate is aﬀected by errors. These errors propagate
through the chain of transformations from the tracking systems to the ﬁnal
application,until they appear as alignment errors in the augmented reality
application.
This thesis deals with the mathematical tools that are needed to estimate
theﬁnalresultingprecision,aswellaswithvisualizationconceptsforexploring
andunderstandingtrackingerrors.Thespecialcaseofopticaltrackingsystems
is further analyzed to provide useful initial estimates of the tracking error. A
set of examples ﬁnally shows the application of the error estimation in real-
world setups.Zusammenfassung
Das Tracking ist ein wichtiger Bestandteil von Anwendungen im Bereich der
Erweiterten Realitat¨ . Tracking bedeutet, die Position und Ausrichtung eines
Objektes relativ zu einem Referenzkoordinatensystem u¨ber einen gewissen
Zeitraum hin zu bestimmen und zu verfolgen. Wie jeder andere Messprozess
ist auch dieser Prozess fehlerbehaftet. Diese Messfehler pﬂanzen sich durch
die gesamte Kette der Transformationen von der Messung bis hin zur tatsac¨ h-
¨lichen Anwendung fort, wo sie dann als Fehler in der Uberlagerung sichtbar
werden.
Die vorliegende Arbeit befasst sich sowohl mit den mathematischen Meth-
oden,dienotigsindumdieendgultigeGenauigkeitabzuschatzen,alsauchmit¨ ¨ ¨
Visualisierungsmoglichkeiten um diese Fehler zu untersuchen und zu verste-¨
hen. Fur den Spezialfall eines optischen Trackingsystems wird ein Verfahren¨
zur Abschatzung des Messfehlers vorgestellt. Die Nutzlichkeit der prasen-¨ ¨ ¨
tierten Methoden wird anhand einer Reihe von Beispielen verdeutlicht.Contents
Part I Introduction
1 Augmented Reality ........................................ 3
1.1 History and Deﬁnition ................................... 4
1.2 Tracking Requirements for Augmented Reality .............. 5
1.2.1 Medical Applications .............................. 6
1.2.2 Industrial Applications............................. 6
1.2.3 Other Applications ................................ 7
1.3 Research Issues ......................................... 7
1.3.1 Tracking and Calibration........................... 7
1.3.2 Rendering and Interaction.......................... 8
1.3.3 Applications and Other Issues ...................... 8
2 Introduction to Ubiquitous Tracking....................... 11
2.1 Formal Description ...................................... 12
2.1.1 Descriptive Language .............................. 12
Static Measurements............................... 13
Dynamic Measurements ............................ 13
Direct Measurements .............................. 14
Inferred Measurements............................. 14
2.1.2 Visual Programming Toolbox ....................... 15
2.1.3 Automatic Pattern Search and Data-Flow Generation.. 15
2.2 Supporting Middleware .................................. 15
2.2.1 Studierstube / OpenTracker ........................ 15
2.2.2 DWARF ......................................... 16
2.2.3 OSGAR.......................................... 16
2.2.4 Ubitrack ......................................... 16
2.2.5 CampAR......................................... 17X Contents
Part II Modeling and Estimating Uncertain Transformations
3 Error Representations ..................................... 21
3.1 Accuracy and Precision .................................. 21
3.2 Gaussian Error Distributions.............................. 22
3.2.1 Probability Density Functions....................... 23
One-dimensional Gaussian distribution............... 23
Multi-dimensional Gaussian Distributions ............ 23
3.2.2 Isocontours ....................................... 24
3.2.3 Matrix Norms .................................... 27
Positivity ................................. 27
Positive Homogeneity ...................... 27
Triangle Inequality......................... 27
Positive Deﬁniteness ....................... 27
Spectral Norm .................................... 28
Trace Norm ...................................... 28
3.3 Root Mean Square Error ................................. 29
3.4 Error Propagation for Gaussian Distributions ............... 30
3.4.1 Forward Propagation .............................. 30
3.4.2 Backward Propagation ............................. 32
3.4.3 Combination of Random Variables................... 32
Combination of Random Variables................... 32
Fusion of Random Variables ........................ 33
3.4.4 Monte-Carlo Simulation............................ 33
3.4.5 Unscented Transform .............................. 34
4 Representing Uncertain Transformations .................. 37
4.1 Representing Rigid Transformations ....................... 37
4.1.1 Homogeneous Matrix Representation ................ 37
4.1.2 Euler Angles...................................... 38
Conversion from Euler Angles to Rotation Matrix ..... 40
Conversion from Rotation Matrix to Euler Angles ..... 40
4.1.3 Axis Angle Representation ......................... 40
Conversion from Axis-Angle to Rotation Matrix....... 41
Conversion from Rotation Matrix to Axis-Angle....... 41
Composition of Rotations in Axis-Angle Representation 41
4.1.4 Quaternion Representation ......................... 41
Quaternion Addition............................... 42
Quaternion Multiplication .......................... 42
Unit Quaternions.................................. 42
Conjugate Quaternion ............................. 42
Representing Rotations ............................ 43
Representing Six-dimensional Transformations ........ 43
Interpolation ..................................... 43Contents XI
4.1.5 Dual Quaternions ................................. 44
4.1.6 Exponential Maps ................................. 44
4.2 Error Propagation ....................................... 44
4.2.1 Representing Gaussian Errors....................... 44
Choice of Error Representation...................... 45
4.2.2 Concatenation .................................... 47
Coordinate System Change ......................... 48
Propagation of an Error-free Transformation.......... 50
Concatenation of Two Transformations with Error..... 51
Nonlinear Error Propagation........................ 52
4.2.3 Inversion ......................................... 53
5 Statistical Models for the Accuracy of Optical Tracking
Systems ................................................... 57
5.1 Classiﬁcation of Errors ................................... 57
5.1.1 Dynamic Errors ................................... 57
5.1.2 Identiﬁcation and Visibility......................... 58
5.1.3 Systematic Errors ................................. 59
5.1.4 Random Noise .................................... 60
5.2 Measuring the Error ..................................... 60
5.2.1 Empirical Evaluations.............................. 60
5.2.2 Monte-Carlo Simulations ........................... 61
5.2.3 Analytical Models ................................. 61
5.3 n-Ocular Vision ......................................... 63
5.3.1 Terminology of Errors.............................. 64
Image Plane Error (IPE) ................... 64
Fiducial Location Error (FLE) .............. 65
Fiducial Registration Error (FRE) ........... 65
Marker Target Error (MTE) ................ 66
Target Registration Error (TRE) ............ 66
5.3.2 Estimating the FLE from IPE ...................... 66
Experimental Setup ............................... 67
Pinhole Camera Model............................. 67
Derivation of Covariance Formulas................... 68
Estimation of the Image Plane Error................. 69
Prediction for Common Setups...................... 70
Validation of the Model for Image Plane Error ........ 70
Additional Inﬂuences on the Plane Error ....... 72
Distance and Marker Size................... 73
Merging Markers and Partial Occlusions ...... 74
Anisotropic Behavior of Fiducials ............ 74
Using Residuals for Error Estimation......... 75
5.3.3 Estimating the MTE from FLE ..................... 76
5.3.4 Estimating the TRE directly from the IPE ........... 78
5.3.5 Estimating the TRE from MTE ..................... 78XII Contents
5.4 Monocular Vision ..................................