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Homography-Based Positioning and Planar Motion Recovery Wadenbäck, Mårten 2017 Document Version: Publisher's PDF, also known as Version of record Link to publication Citation for published version (APA): Wadenbäck, M. (2017). Homography-Based Positioning and Planar Motion Recovery. Lund University. Total number of authors: 1 General rights Unless other specific re-use rights are stated the following general rights apply: Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Read more about Creative commons licenses: https://creativecommons.org/licenses/ Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. LUND UNIVERSITY PO Box 117 221 00 Lund +46 46-222 00 00 – Centrum SCientiarum mathematiCarum – Homography-Based Positioning and Planar Motion Recovery mårten wadenbäck Lund University Faculty of Engineering Centre for Mathematical Sciences Mathematics Homography-Based Positioning and Planar Motion Recovery Mårten Wadenbäck ACADEMIC THESIS which, by due permission of the Faculty of Engineering at Lund Univer- sity, will be publicly defended on Friday the 7th of April 2017, at 13:15 in lecture hall MH:Hörmander, Matematikhuset, Sölvegatan 18, Lund, for the degree of Doctor of Philosophy in Engineering. Faculty opponent Dr. Juho Kannala, Aalto University, Finland Organisation Document name LUND UNIVERSITY DOCTORAL THESIS IN Centre for Mathematical Sciences MATHEMATICAL SCIENCES Box 118 Date of defence SE-221 00 Lund 2017-04-07 Sweden Sponsoring organisation Author(s) Mårten Wadenbäck Title and subtitle Homography-Based Positioning and Planar Motion Recovery Abstract Planar motion is an important and frequently occurring situation in mobile robotics applications. This thesis concerns estimation of ego-motion and pose of a single downwards oriented camera under the assumptions of planar motion and known internal camera parameters. The so called essential matrix (or its uncalibrated counterpart, the fundamental matrix) is frequently used in computer vision applications to compute a recon- struction in 3D of the camera locations and the observed scene. However, if the observed points are expected to lie on a plane – e.g. the ground plane – this makes the determination of these matrices an ill-posed problem. Instead, methods based on homographies are better suited to this situation. One section of this thesis is concerned with the extraction of the camera pose and ego-motion from such homographies. We present both a direct SVD-based method and an iterative method, which both solve this problem. The iterative method is extended to allow simultaneous determination of the camera tiltfrom several homographies obeying the same planar motion model. This extension improves the robustness of the original method, and it provides consistent tilt estimates for the frames that are used for the estimation. The methods are evaluated using experiments on both real and synthetic data. Another part of the thesis deals with the problem of computing the homographies from point corres- pondences. By using conventional homography estimation methods for this, the resulting homography is of a too general class and is not guaranteed to be compatible with the planar motion assumption. For this reason, we enforce the planar motion model at the homography estimation stage with the help of a new DOKUMENTDATABLAD enl SIS 61 41homography 21 solver using a number of polynomial constraints on the entries of the homography matrix. In addition to giving a homography of the right type, this method uses only 2.5 point correspondences instead of the conventional four, which is good e.g. when used in a RANSAC framework for outlier removal. Keywords Homography, Planar Motion, SLAM, Motion Estimation Classification system and/or index terms (if any) Supplementary bibliographical information Language English ISSN and key title ISBN 1404-0034 978-91-7753-153-1 (printed) 978-91-7753-154-8 (electronic) Recipient’s notes Number of pages Price xvi+115 Peppercorn Security classification I, the undersigned, being copyright owner of the abstract of the above-mentioned thesis, hereby grant to all reference sources the permission to publish and disseminate the abstract of the above-mentioned thesis. Signature Date 2017-03-08 HOMOGRAPHY-BASED POSITIONING AND PLANAR MOTION RECOVERY MÅRTEN WADENBÄCK Faculty of Engineering Centre for Mathematical Sciences Mathematics Cover image: Küchensee seen from Ratzeburg Kurpark on the 30th of April 2015 Mathematics Centre for Mathematical Sciences Lund University Box 118 SE-221 00 Lund Sweden http://www.maths.lu.se/ Doctoral Theses in Mathematical Sciences 2017:3 ISSN 1404-0034 ISBN 978-91-7753-153-1 (printed) ISBN 978-91-7753-154-8 (electronic) LUTFMA-1064-2017 © Mårten Wadenbäck, 2017 Printed in Sweden by MediaTryck, Lund 2017 Abstract Planar motion is an important and frequently occurring situation in mo- bile robotics applications. This thesis concerns estimation of ego-motion and pose of a single downwards oriented camera under the assumptions of planar motion and known internal camera parameters. The so called essen- tial matrix (or its uncalibrated counterpart, the fundamental matrix) is fre- quently used in computer vision applications to compute a reconstruction in 3D of the camera locations and the observed scene. However, if the ob- served points are expected to lie on a plane – e.g. the ground plane – this makes the determination of these matrices an ill-posed problem. Instead, methods based on homographies are better suited to this situation. One section of this thesis is concerned with the extraction of the camera pose and ego-motion from such homographies. We present both a direct SVD-based method and an iterative method, which both solve this prob- lem. The iterative method is extended to allow simultaneous determination of the camera tilt from several homographies obeying the same planar mo- tion model. This extension improves the robustness of the original method, and it provides consistent tilt estimates for the frames that are used for the estimation. The methods are evaluated using experiments on both real and synthetic data. Another part of the thesis deals with the problem of computing the homographies from point correspondences. By using conventional homo- graphy estimation methods for this, the resulting homography is of a too general class and is not guaranteed to be compatible with the planar motion assumption. For this reason, we enforce the planar motion model at the ho- mography estimation stage with the help of a new homography solver using a number of polynomial constraints on the entries of the homography mat- rix. In addition to giving a homography of the right type, this method uses only 2.5 point correspondences instead of the conventional four, which is good e.g. when used in a RANSAC framework for outlier removal. iii iv Popular Science Summary The introduction of the robotic arm in the early 1960s was an important step towards automation of industry. Much of the noisy, dangerous, and repetitious labour at assembly lines and blast furnaces could now be per- formed by a robotic operator instead of a human. The robots were very often individually calibrated and programmed each to perform their spe- cific task – tasks which were now performed faster than ever, and without risking fatigue or injuries. However, despite the increased productivity, the improved safety, and the standardised output, the robots lacked crucial skills which the earlier human operators possessed in great measure – flexibility and adaptability. In contrast to their human counterparts, the robots were highly stationary and fixated to their workspace, often bolted to the floor or at best moving along short rails, and even minor changes to their task specification required a complete reprogramming and thus time offline. Over the past half century, efforts to endow robots with flexibility and adaptability have been major themes in robotics research. The work in this thesis is a part of those efforts, and deals with an important sub-problem called Simultaneous Localisation and Mapping (SLAM). Algorithms for the SLAM problem use data which the robot acquire from its sensors (sonar, laser range finders, cameras, wheel encoders, …) to determine and keep track of the surrounding environment. In other words, the robot has to create a model of its surroundings (the mapping part), and use it to de- termine its own position (the localisation part) relative to the model. This type of algorithms is necessary in order to enable mobile robots to move autonomously – that is, without a human operator actively controlling the robot. Only in the last two or three decades have cameras become a realistic choice of sensor to use for robotic navigation. There are three major reasons for this. First, digital cameras have become available, and they have gone through a revolution in terms of both reduced price
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