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David Suter - One of the best experts on this subject based on the ideXlab platform.
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Homography estimation and heteroscedastic noise - a first order perturbation analysis
2020Co-Authors: Pei Chen, David SuterAbstract:It is well known that one can collect the coecients of the homgraphies between two views into a large, rank deficient matrix. In principle, such an observation implies that one can refine the accuracy of the estimates of the homography coecients by exploiting the rank constraint. However, the straightforward approach suggested by this observation is impractical because it requires many Homographies and it also does not take into account correlations between the errors in the coecients. In a companion paper (4), we show how to jointly estimate multiple (but a realistic number of) Homographies over 2 views. By studying the special structure of the homography, we show that it is possible to calculate the dimension 4 subspace of the Homographies from 3 planes (and, in principle, with even two planes). This contradicts what seems to be the accepted situation regarding the exploitation of the rank-4 constraint amongst Homographies: that more than 4 planes are needed to calculate and exploit the dimension 4 subspace. Practical issues arise because the homography coecients, before rank-constrained refinement, are themselves estimates whose noise covariances need to be characterised and accounted for. In this paper, we develop a statistical analysis allowing for estimation of the covariance matrices required for the calculation of low-rank "denoised" Homographies.
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rank constraints for Homographies over two views revisiting the rank four constraint
International Journal of Computer Vision, 2009Co-Authors: Pei Chen, David SuterAbstract:It is well known that one can collect the coefficients of five (or more) Homographies between two views into a large, rank deficient matrix. In principle, this implies that one can refine the accuracy of the estimates of the homography coefficients by exploiting the rank constraint. However, the standard rank-projection approach is impractical for two different reasons: it requires many Homographies to even score a modest gain; and, secondly, correlations between the errors in the coefficients will lead to poor estimates. In this paper we study these problems and provide solutions to each. Firstly, we show that the matrices of the homography coefficients can be recast into two parts, each consistent with ranks of only one. This immediately establishes the prospect of realistically (that is, with as few as only three or four Homographies) exploiting the redundancies of the Homographies over two views. We also tackle the remaining issue: correlated coefficients. We compare our approach with the "gold standard"; that is, non-linear bundle adjustment (initialized from the ground truth estimate--the ideal initialization). The results confirm our theory and show one can implement rank-constrained projection and come close to the gold standard in effectiveness. Indeed, our algorithm (by itself), or our algorithm further refined by a bundle adjustment stage; may be a practical algorithm: providing generally better results than the "standard" DLT (direct linear transformation) algorithm, and even better than the bundle adjustment result with the DLT result as the starting point. Our unoptimized version has roughly the same cost as bundle adjustment and yet can generally produce close to the "gold standard" estimate (as illustrated by comparison with bundle adjustment initialized from the ground truth). Independent of the merits or otherwise of our algorithm, we have illuminated why the naive approach of direct rank-projection is relatively doomed to failure. Moreover, in revealing that there are further rank constraints, not previously known; we have added to the understanding of these issues, and this may pave the way for further improvements.
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Simultaneously Estimating the Fundamental Matrix and Homographies
IEEE Transactions on Robotics, 2009Co-Authors: Pei Chen, David SuterAbstract:The estimation of the fundamental matrix (FM) and/or one or more Homographies between two views is of great interest for a number of computer vision and robotics tasks. We consider the joint estimation of the FM and one or more Homographies. Given point matches between two views (and assuming rigid geometry of the camera-scene displacement), it is well known that all of the matched points satisfy the epipolar constraint that is usually characterized by the FM. Subsets of these point matches may also obey a constraint characterized by a homography (all matches in the subset coming from three-dimensional (3-D) points lying on a 3-D plane). The estimations of Homographies and the FM are well-studied problems, and therefore, the (separate) estimation of the FM, or the homography matrices, can be considered as effectively solved problems with mature algorithms. However, the Homographies and FM are not independent of each other: therefore, separate estimation of each is likely to be suboptimal. In this paper, we propose to simultaneously estimate the FM and Homographies by employing the compatibility constraint between them. This is done by first concentrating on a set of parameters that (jointly) parameterize the entire set of Homographies and FM (simultaneously) and that also implicitly enforce the compatibility between the estimates of each set. We then derive a reduced form with the purpose of improving the speed. We propose a solution method in which the Sampson error for the FM and Homographies is minimized by the Levenberg-Marquardt (LM) algorithm. Experiments show that the gains can be compared with separate estimates (the FM and/or the Homographies).
Pascal Morin - One of the best experts on this subject based on the ideXlab platform.
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Estimation of Homography Dynamics on the Special Linear Group
Visual Servoing via Advanced Numerical Methods, 2020Co-Authors: Ezio Malis, Tarek Hamel, Robert Mahony, Pascal MorinAbstract:During the last decade, a number of highly successful visual servo control and real-time image processing algorithms have been proposed that use the homography between images of a planar scene as the primary measurement. The performance of the algorithms depends directly on the quality of the homography estimates obtained, and these estimates must be computed in real-time. In this chapter, we exploit the special linear Lie group structure of the set of all Homographies to develop an on-line dynamic observer that provides smoothed estimates of a sequence of Homographies and their relative velocities. The proposed observer is easy to implement and computationally undemanding. Furthermore, it is straightforward to tune the observer gains and excellent results are obtained for test sequences of simulation and real-world data.
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Feature-based Recursive Observer Design for Homography Estimation and its Application to Image Stabilization
Asian Journal of Control, 2019Co-Authors: Minh-duc Hua, Tarek Hamel, Robert Mahony, Jochen Trumpf, Pascal MorinAbstract:This paper presents a new algorithm for online estimation of a sequence of Homographies applicable to image sequences obtained from robotic vehicles equipped with vision sensors. The approach taken exploits the underlying Special Linear group structure of the set of Homographies along with gyroscope measurements and direct point-feature correspondences between images to develop temporal filter for the homography estimate. Theoretical analysis and experimental results are provided to demonstrate the robustness of the proposed algorithm. The experimental results show excellent performance and robustness even in the case of very fast camera motions (relative to frame rate) and severe occlusions.
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Feature-based Recursive Observer Design for Homography Estimation
arXiv: Computer Vision and Pattern Recognition, 2016Co-Authors: Jochen Trumpf, Tarek Hamel, Robert Mahony, Pascal MorinAbstract:This paper presents a new algorithm for online estimation of a sequence of Homographies applicable to image sequences obtained from robotic vehicles equipped with vision sensors. The approach taken exploits the underlying Special Linear group structure of the set of Homographies along with gyroscope measurements and direct point-feature correspondences between images to develop temporal filter for the homography estimate. Theoretical analysis and experimental results are provided to demonstrate the robustness of the proposed algorithm. The experimental results show excellent performance even in the case of very fast camera motion (relative to frame rate), severe occlusion, and in the presence of specular reflections.
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CDC-ECE - Homography estimation on the Special Linear group based on direct point correspondence
IEEE Conference on Decision and Control and European Control Conference, 2011Co-Authors: Tarek Hamel, Robert Mahony, Jochen Trumpf, Pascal MorinAbstract:This paper considers the question of obtaining a high quality estimate of a time-varying sequence of image Homographies using point correspondences from an image sequence without requiring explicit computation of the individual Homographies between any two given images. The approach uses the representation of a homography as an element of the Special Linear group and defines a nonlinear observer directly on this structure. We assume, either that the group velocity of the homography sequence is known, or more realistically, that the Homographies are generated by rigid-body motion of a camera viewing a planar surface, and that the angular velocity of the camera is known.
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homography estimation on the special linear group based on direct point correspondence
Conference on Decision and Control, 2011Co-Authors: Tarek Hamel, Robert Mahony, Jochen Trumpf, Pascal MorinAbstract:This paper considers the question of obtaining a high quality estimate of a time-varying sequence of image Homographies using point correspondences from an image sequence without requiring explicit computation of the individual Homographies between any two given images. The approach uses the representation of a homography as an element of the Special Linear group and defines a nonlinear observer directly on this structure. We assume, either that the group velocity of the homography sequence is known, or more realistically, that the Homographies are generated by rigid-body motion of a camera viewing a planar surface, and that the angular velocity of the camera is known.
Robert Laganière - One of the best experts on this subject based on the ideXlab platform.
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Projective rectification of image triplets
Signal Image and Video Processing, 2010Co-Authors: Robert Laganière, Florian KangniAbstract:This paper describes a method for image rectification of a trinocular setup. The rectification method used is an extension of a recent approach based on the fundamental matrix to generate the correcting Homographies in the case of a stereo pair. The extended method uses the fact that the triplet of images can be treated as two pairs and that Homographies are projections of the different images planes onto new planes. Rectification thus becomes a matter of deciding which plane will be the common one and what transformation or homography is to be applied to each image.
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Projective Rectification Of Image Triplets From The Fundamental Matrix
2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings, 2006Co-Authors: Florian Kangni, Robert LaganièreAbstract:This paper describes a method for image rectification of a trinocular setup. The rectification method used is an extension of a recent approach based on the fundamental matrix to generate the correcting Homographies in the case of a stereo pair. The extended method uses the fact that the triplet of images can be treated as two pairs and that Homographies are simply projections of the different images planes onto new planes. Rectification thus becomes a matter of deciding which plane is the common one and what transformation or homography is to be applied to each image
M. Irani - One of the best experts on this subject based on the ideXlab platform.
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Multiview constraints on Homographies
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2002Co-Authors: L. Zeinik-manor, M. IraniAbstract:The image motion of a planar surface between two camera views is captured by a homography (a 2D projective transformation). The homography depends on the intrinsic and extrinsic camera parameters, as well as on the 3D plane parameters. While camera parameters vary across different views, the plane geometry remains the same. Based on this fact, we derive linear subspace constraints on the relative Homographies of multiple (/spl ges/ 2) planes across multiple views. The paper has three main contributions: 1) We show that the collection of all relative Homographies (homologies) of a pair of planes across multiple views, spans a 4-dimensional linear subspace. 2) We show how this constraint can be extended to the case of multiple planes across multiple views. 3) We show that, for some restricted cases of camera motion, linear subspace constraints apply also to the set of Homographies of a single plane across multiple views. All the results derived are true for uncalibrated cameras. The possible utility of these multiview constraints for improving homography estimation and for detecting nonrigid motions are also discussed.
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ICCV - Multi-view subspace constraints on Homographies
Proceedings of the Seventh IEEE International Conference on Computer Vision, 1999Co-Authors: L. Zelnik-manor, M. IraniAbstract:The motion of a planar surface between two camera views induces a homography. The homography depends on the camera intrinsic and extrinsic parameters, as well as on the 3D plane parameters. While camera parameters vary across different views, the plane geometry remains the same. Based on this fact, the paper derives linear subspace constraints on the relative motion of multiple (/spl ges/2) planes across multiple views. The paper has three main contributions. It shows that the collection of all relative Homographies of a pair of planes (homologies) across multiple views, spans a 4-dimensional linear subspace. It shows how this constraint can be extended to the case of multiple planes across multiple views. It suggests two potential application areas which can benefit from these constraints: the accuracy of homography estimation can be improved by enforcing the multi-view subspace constraints; and violations of these multi-view constraints can be used as a cue for moving object detection. All the results derived in this paper are true for uncalibrated cameras.
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Multi-view subspace constraints on Homographies
Proceedings of the Seventh IEEE International Conference on Computer Vision, 1999Co-Authors: L. Zelnik-manor, M. IraniAbstract:The motion of a planar surface between two camera views induces a homography. The homography depends on the camera intrinsic and extrinsic parameters, as well as on the 3D plane parameters. While camera parameters vary across different views, the plane geometry remains the same. Based on this fact, the paper derives linear subspace constraints on the relative motion of multiple (/spl ges/2) planes across multiple views. The paper has three main contributions. It shows that the collection of all relative Homographies of a pair of planes (homologies) across multiple views, spans a 4-dimensional linear subspace. It shows how this constraint can be extended to the case of multiple planes across multiple views. It suggests two potential application areas which can benefit from these constraints: the accuracy of homography estimation can be improved by enforcing the multi-view subspace constraints; and violations of these multi-view constraints can be used as a cue for moving object detection. All the results derived in this paper are true for uncalibrated cameras.
Anibal Ollero - One of the best experts on this subject based on the ideXlab platform.
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Vision-based multi-UAV position estimation
IEEE Robotics and Automation Magazine, 2006Co-Authors: Luís Merino, Anders Moe, Per Erik Forsén, Klas Nordberg, J. R. Martínez-de Dios, Fernando Caballero, Johan Wiklund, Anibal OlleroAbstract:This paper describes a method for vision-based unmanned aerial vehicle (UAV) motion estimation from multiple planar Homographies. The paper also describes the determination of the relative displacement between different UAVs employing techniques for blob feature extraction and matching. It then presents and shows experimental results of the application of the proposed technique to multi-UAV detection of forest fires
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robot position estimation based on Homographies of the ground plane
IFAC Proceedings Volumes, 2004Co-Authors: J Ferruz, Sebastian Hurtado, Anibal OlleroAbstract:Abstract This paper presents a method for visual robot position estimation in environments whose dominant structure is an approximately flat surface. A cluster-based image matching algorithm, able to work robustly in non-structured environments, is used to obtain point-to-point matching pairs. The matching results are then used by a position estimation algorithm based on the computation of Homographies that takes into account camera rotation to increase robustness against camera vibrations. An approximately constant and known height of the camera over the ground is assumed, which is used to recover the scale factor.