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Institut F¨Ur Informatik Und Praktische Mathematik Christian-Albrechts-Universit¨At Kiel

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Institut F¨Ur Informatik Und Praktische Mathematik Christian-Albrechts-Universit¨At Kiel INSTITUT FUR¨ INFORMATIK UND PRAKTISCHE MATHEMATIK Pose Estimation Revisited Bodo Rosenhahn Bericht Nr. 0308 September 2003 CHRISTIAN-ALBRECHTS-UNIVERSITAT¨ KIEL Institut f¨ur Informatik und Praktische Mathematik der Christian-Albrechts-Universit¨at zu Kiel Olshausenstr. 40 D – 24098 Kiel Pose Estimation Revisited Bodo Rosenhahn Bericht Nr. 0308 September 2003 e-mail: [email protected] “Dieser Bericht gibt den Inhalt der Dissertation wieder, die der Verfasser im April 2003 bei der Technischen Fakult¨at der Christian-Albrechts-Universit¨at zu Kiel eingereicht hat. Datum der Disputation: 19. August 2003.” 1. Gutachter Prof. G. Sommer (Kiel) 2. Gutachter Prof. D. Betten (Kiel) 3. Gutachter Dr. L. Dorst (Amsterdam) Datum der m¨undlichen Pr¨ufung: 19.08.2003 ABSTRACT The presented thesis deals with the 2D-3D pose estimation problem. Pose estimation means to estimate the relative position and orientation of a 3D ob- ject with respect to a reference camera system. The main focus concentrates on the geometric modeling and application of the pose problem. To deal with the different geometric spaces (Euclidean, affine and projective ones), a homogeneous model for conformal geometry is applied in the geometric algebra framework. It allows for a compact and linear modeling of the pose scenario. In the chosen embedding of the pose problem, a rigid body motion is represented as an orthogonal transformation whose parameters can be es- timated efficiently in the corresponding Lie algebra. In addition, the chosen algebraic embedding allows the modeling of extended features derived from sphere concepts in contrast to point concepts used in classical vector cal- culus. For pose estimation, 3D object models are further treated two-fold, feature based and free-form based: While the feature based pose scenarios provide constraint equations to link different image and object entities, the free-form approach for pose estimation is achieved by applying extracted im- age silhouettes from objects on 3D free-form contours modeled by 3D Fourier descriptors. In conformal geometric algebra an extended scenario is derived, which deals beside point features with higher-order features such as lines, planes, circles, spheres, kinematic chains or cycloidal curves. This scenario is extended to general free-form contours by interpreting contours generated with 3D Fourier descriptors as n-times nested cycloidal curves. The intro- duced method for shape modeling links signal theory, geometry and kine- matics and is applied advantageously for 2D-3D silhouette based free-form pose estimation. The experiments show the real-time capability and noise stability of the algorithms. Experiments of a running navigation system with visual self-localization are also presented. ii ACKNOWLEDGMENT This thesis was developed with the help of several researchers. Without them the thesis would not be in this form. The interaction with other researchers during conferences and exchange projects gave me the motivation to continue with my work and to look forward and extend my research. I can not mention everyone who assisted me during my work, but I would like to acknowledge the most important persons. First of all, I appreciate my supervisor Prof. Dr. Gerald Sommer for the interesting topic of research, for giving me hints during problem solving and for supporting my conference visits and exchange stays. I will never forget the long discussions, sometimes extending to 3 or 4 hours. Furthermore, I acknowledge the support and help of all members of the Cognitive Systems Group, University Kiel. My special thanks go to Christian Perwass with whom I had endless discussions and who gave me private lessons in geometric algebra, signal theory and stochastics. He influenced my results fundamentally, since he firstly formalized the Fourier descriptors in conformal geometric algebra and explained the basics of stereographic projections to me. I further acknowledge my colleagues Vladimir Banarer, Sven Buchholz, Oliver Granert, Martin Krause, Nils Siebel and Yohannes Kassahun. I am also thankful to our technical staff, Henrik Schmidt and Gerd Diesner and our secretary Francoise Maillard. They have been helpful all the time and helped to avoid some catastrophes during talks or presentations. I also thank my students for their work. Especially Oliver Granert, Daniel Grest and Andreas Bunten, with whom I had long discussions about various topics. They helped me to learn more about mathematics, programming and robotics. I liked their autonomous style of working, with lots of creative ideas and visions. I am also grateful for all the discussions I had with other scientists, es- pecially Dieter Betten, Thomas B¨ulow, Leo Dorst, Ulrich Eckhardt, Michael Felsberg, Wolfang F¨orstner, David Hestenes, Reinhard Klette, Reinhard Koch, iv Norbert Kr¨uger, Volker Kr¨uger, Joan Lasenby, Hongbo Li, Josef Pauli, Jon Selig and Frank Sommen. I further acknowledge Silke Kleemann, Anke Kleemann, Christian Per- wass, Hongbo Li and Jon Selig for proofreading parts of the thesis. I also acknowledge the DFG Graduiertenkolleg No. 357 and the EC Grant IST-2001-3422 (VISATEC) for their financial support during my PhD stud- ies. Last, but not least, I thank my family, my parents Lothar and Regine Rosenhahn, my brothers Axel and Carsten Rosenhahn and my son Julian Rosenhahn. With special thanks I dedicate this thesis to my wife Anke Kleemann. CONTENTS 1. Introduction ............................. 1 1.1 Literature overview of pose estimation algorithms . .. 7 1.2 Thecontributionofthethesis . 13 2. Foundations of the 2D-3D pose estimation problem ...... 17 2.1 Thescenarioofposeestimation . 17 2.2 The stratification hierarchy as part of the pose estimation problem ............................. 19 2.3 Principles of solving the pose estimation problem . 21 3. Introduction to geometric algebras ................ 25 3.1 The Euclidean geometric algebra . 29 3.1.1 Representation of points, lines and planes in the Eu- clideangeometricalgebra. 31 3.1.2 Rotations and translations in the Euclidean space . 32 3.2 Theprojectivegeometricalgebra . 35 3.2.1 Representation of points, lines and planes in the pro- jectivegeometricalgebra . 35 3.3 Theconformalgeometricalgebra . 39 3.3.1 Stereographicprojections. 39 3.3.2 Definition of the conformal geometric algebra . 43 3.3.3 Geometric entities in conformal geometric algebra . 44 vi Contents 3.3.4 Conformaltransformations. 48 3.3.5 RigidmotionsinCGA . 50 3.3.6 Twistsandscrewtransformations . 51 4. Stratification of the pose estimation problem .......... 57 4.1 Relating entities in Euclidean, projective and conformal ge- ometry.............................. 58 4.1.1 Change of representations of geometric entities . 60 4.1.2 Pose constraints in conformal geometric algebra . 63 5. Pose constraints for point, line and plane correspondences . 65 5.1 Point-lineconstraint . 66 5.2 Point-planeconstraint . 68 5.3 Line-planeconstraint . 69 5.4 Constraint equations for pose estimation . 72 5.5 Numerical estimation of pose parameters . 74 5.5.1 Generating an example system of equations . 76 5.6 Experiments with pose estimation of rigid objects . 77 5.6.1 Adaptive use of pose estimation constraints . 81 6. Pose estimation for extended object concepts .......... 85 6.1 Pose estimation of kinematic chains . 85 6.1.1 Constraint equations of kinematic chains . 87 6.2 Circlesandspheres ....................... 88 6.2.1 The problem of tangentiality constraints . 88 6.2.2 Operational definition of circles and spheres using twists 91 6.2.3 The constraint equations of circles and spheres to lines 92 6.3 Cycloidalcurves......................... 93 6.3.1 Algebraiccurves .................... 94 Contents vii 6.3.2 Cycloidal curves in conformal geometric algebra . 96 6.3.3 Ray-tracing on cycloidal curves . 101 6.3.4 Constraint equations for pose estimation of cycloidal curves .......................... 104 6.4 Experiments for pose estimation of extended object concepts 105 6.4.1 Pose estimation experiments of kinematic chains . 105 6.4.2 Pose estimation experiments with circles and spheres 111 6.4.3 Pose estimation experiments with cycloidal curves . 112 7. Pose estimation of free-form objects ............... 119 7.1 Pose estimation constraints for free-form contours . ... 122 7.1.1 Multiplicative formalization of 3D Fourier descriptors 124 7.1.2 Coupling kinematic chains with Fourier descriptors . 125 7.2 Experiments on pose estimation of free-form contours . .. 126 7.2.1 The algorithm for pose estimation of free-form contours 126 7.2.2 The performance of the pose estimation algorithm . 130 7.2.3 Simultaneous pose estimation of multiple contours . 139 7.2.4 Pose estimation of deformable objects . 143 8. Conclusion .............................. 149 8.1 Further extensions, open problems and future work . 151 A. Basic Mathematics ......................... 153 A.1 Mathematicspaces . 153 A.2 LiegroupsandLiealgebras . 157 A.3 Thepinholecameramodel . 161 A.4 Signaltheory .......................... 164 A.4.1 Foundations. .. .. 164 A.4.2 Trigonometricinterpolation . 165 viii Contents A.4.3 The 1D discrete Fourier transformation . 165 B. Implemented modules of the navigation system ........ 169 B.1 Thetrackingproblem. 169 B.1.1 The tracking assumption . 170 B.1.2 Thematchingalgorithm . 171 B.1.3 Imagefeatureextraction . 175 B.2 Thehardwaresetup. 178 B.2.1 Behaviorbasedrobotics . 179 B.3 Thenavigationsystem . 180 B.4 Experiments with the navigation system . 182 Index ................................. 205 Chapter 1 INTRODUCTION Mobile
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