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80
Catadioptric camera calibration using geometric invariants
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2004
"... Abstract—Central catadioptric cameras are imaging devices that use mirrors to enhance the field of view while preserving a single effective viewpoint. In this paper, we propose a novel method for the calibration of central catadioptric cameras using geometric invariants. Lines and spheres in space a ..."
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Cited by 47 (7 self)
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Abstract—Central catadioptric cameras are imaging devices that use mirrors to enhance the field of view while preserving a single effective viewpoint. In this paper, we propose a novel method for the calibration of central catadioptric cameras using geometric invariants. Lines and spheres in space are all projected into conics in the catadioptric image plane. We prove that the projection of a line can provide three invariants whereas the projection of a sphere can only provide two. From these invariants, constraint equations for the intrinsic parameters of catadioptric camera are derived. Therefore, there are two kinds of variants of this novel method. The first one uses projections of lines and the second one uses projections of spheres. In general, the projections of two lines or three spheres are sufficient to achieve catadioptric camera calibration. One important conclusion in this paper is that the method based on projections of spheres is more robust and has higher accuracy than that based on projections of lines. The performances of our method are demonstrated by both the results of simulations and experiments with real images. Index Terms—Camera calibration, catadioptric camera, geometric invariant, omnidirectional vision, panoramic vision. 1
A Toolbox for Easily Calibrating Omnidirectional Cameras
 In Proc. of the IEEE International Conference on Intelligent Systems, IROS06
, 2006
"... Abstract In this paper, we present a novel technique for calibrating central omnidirectional cameras. The proposed procedure is very fast and completely automatic, as the user is only asked to collect a few images of a checker board, and click on its corner points. In contrast with previous approac ..."
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Cited by 44 (3 self)
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Abstract In this paper, we present a novel technique for calibrating central omnidirectional cameras. The proposed procedure is very fast and completely automatic, as the user is only asked to collect a few images of a checker board, and click on its corner points. In contrast with previous approaches, this technique does not use any specific model of the omnidirectional sensor. It only assumes that the imaging function can be described by a Taylor series expansion whose coefficients are estimated by solving a fourstep leastsquares linear minimization problem, followed by a nonlinear refinement based on the maximum likelihood criterion. To validate the proposed technique, and evaluate its performance, we apply the calibration on both simulated and real data. Moreover, we show the calibration accuracy by projecting the color information of a calibrated camera on real 3D points extracted by a 3D sick laser range finder. Finally, we provide a Toolbox which implements the proposed calibration procedure. Index Terms – catadioptric, omnidirectional, camera, calibration, toolbox.
Visual Servoing/Tracking Using Central Catadioptric Images
 In Int. Symposium on Experimental Robotics, Advanced Robotics Series
, 2002
"... Visual control of robot motion may benefit from enhanced camera field of view. With traditional cameras the available fields of view are only enough to view a region around the observed object (for eyeinhand systems) or around the ende#ector (for independenteye systems). Central catadioptric sys ..."
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Cited by 36 (2 self)
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Visual control of robot motion may benefit from enhanced camera field of view. With traditional cameras the available fields of view are only enough to view a region around the observed object (for eyeinhand systems) or around the ende#ector (for independenteye systems). Central catadioptric systems have larger fields of view thus allowing the entire robot AND the surrounding objects to be imaged with a unique camera. Therefore, the whole robot's articulated mechanism can be observed and its joints can be tracked and controlled simultenously. This results in a new visual robot control concept where tracking and control are embedded together. Key to the understanding of both servoing and tracking is the central catadioptric Jacobian matrix linking the robot's joint velocities to image observations. In spite of a more complex projection matrix associated with catadioptric sensors, we study the catadioptric Jacobian matrix and we show that it does not introduce any additional singularity with respect to the traditional pinhole camera model. Experiments showing a rigid body being tracked with a catadioptric camera are described.
Imagebased Visual Servoing with Central Catadioptric Camera
 Int. J. Robot. Res
"... Abstract — This paper presents an epipolar based visual servoing for mobile robots equipped with a panoramic camera. The proposed visual servoing is based on the epipolar geometry and exploits the autoepipolar property, a special configuration for the epipoles which occurs when the desired and the ..."
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Cited by 23 (3 self)
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Abstract — This paper presents an epipolar based visual servoing for mobile robots equipped with a panoramic camera. The proposed visual servoing is based on the epipolar geometry and exploits the autoepipolar property, a special configuration for the epipoles which occurs when the desired and the current views undergo a pure translation. This occurrence is detectable observing when the biosculating mirror conics cointersect at the two epipoles. The autoepipolar condition enables our controller to retrieve the equal orientation between target and current camera. Translation is performed by exploiting the epipoles. Simulated experiments and Lyapunovbased stability analysis demonstrate the parametric robustness of the proposed method. I.
Paracatadioptric Camera Calibration Using Lines
 Proc. IEEE Int’l Conf. Computer Vision
, 2003
"... Paracatadioptric sensors combine a parabolic shaped mirror and a camera inducing an orthographic projection. Such a configuration provides a wide field of view while keeping a single effective viewpoint. Previous work in central catadioptric sensors proved that a line projects into a conic curve a ..."
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Cited by 21 (3 self)
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Paracatadioptric sensors combine a parabolic shaped mirror and a camera inducing an orthographic projection. Such a configuration provides a wide field of view while keeping a single effective viewpoint. Previous work in central catadioptric sensors proved that a line projects into a conic curve and that three line images are enough to calibrate the system. However the estimation of the conic curves where lines are mapped is hard to accomplish. In general only a small arc of the conic is visible in the image and conventional conic fitting techniques are unable to correctly estimate the curve. The present work shows that a set of conic curves corresponds to paracatadioptric line images if, and only if, certain properties are verified. These properties are used to constraint the search space and correctly estimate the curves. The accurate estimation of a minimum of three line images allows the complete calibration of the paracatadioptric camera. If the camera is skewless and the aspect ratio is known then the conic fitting problem is solved naturally by an eigensystem. For the general situation the conic curves are estimated using nonlinear optimization. 1
A comparison of two camera configurations for opticflow based navigation of a uav through urban canyons
 In EEE/RSJ International Conference on Intelligent Robots and Systems
, 2004
"... Abstract — We present a comparison of two camera configurations for avoiding obstacles in 3Dspace using optic flow. The two configurations were developed for use on an autonomous helicopter, with the aim of enabling it to fly in environments with tall obstacles (e.g. urban canyons). The comparison ..."
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Cited by 20 (3 self)
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Abstract — We present a comparison of two camera configurations for avoiding obstacles in 3Dspace using optic flow. The two configurations were developed for use on an autonomous helicopter, with the aim of enabling it to fly in environments with tall obstacles (e.g. urban canyons). The comparison is made based on real data captured from two sidewayslooking cameras and an omnidirectional camera mounted onboard an autonomous helicopter. Optic flow information from the images is used to determine the relative distance to obstacles on each side of the helicopter. We show that on average, both camera configurations are equally effective and that they can be used to tell which of the canyon walls is closer with an accuracy of 74%. It is noted that each configuration is however more effective under certain conditions, and so a suitable hybrid approach is suggested. We also show that there is a linear relationship between the optic flow ratios and the position of the helicopter with respect to the center of the canyon. We use this relationship to develop a proportional control strategy for flying the helicopter along the Voronoi line between buildings. I.
Central Catadioptric Visual Servoing from 3D Straight Lines
 Computer Systems Science&Eng.,7
, 2004
"... In this paper we consider the problem of controlling a robotic system using the projection of 3D lines in the image plane of central catadioptric systems. Most of the effort in visual servoing are devoted to points, only few works have investigated the use of lines in visual servoing with traditiona ..."
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Cited by 18 (4 self)
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In this paper we consider the problem of controlling a robotic system using the projection of 3D lines in the image plane of central catadioptric systems. Most of the effort in visual servoing are devoted to points, only few works have investigated the use of lines in visual servoing with traditional cameras and none has explored the case of omnidirectional cameras. First a generic central catadioptric interaction matrix for the projection of 3D straight lines is derived from the projection model of an entire class of camera. Then an imagebased control law is designed and validated through simulation results.
General Imaging Geometry for Central Catadioptric Cameras
"... Abstract. Catadioptric cameras are a popular type of omnidirectional imaging system. Their imaging and multiview geometry has been extensively studied; epipolar geometry for instance, is geometrically speaking, well understood. However, the existence of a bilinear matching constraint and an associa ..."
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Cited by 13 (8 self)
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Abstract. Catadioptric cameras are a popular type of omnidirectional imaging system. Their imaging and multiview geometry has been extensively studied; epipolar geometry for instance, is geometrically speaking, well understood. However, the existence of a bilinear matching constraint and an associated fundamental matrix, has so far only been shown for the special case of paracatadioptric cameras (consisting of a paraboloidal mirror and an orthographic camera). The main goal of this work is to obtain such results for all central catadioptric cameras. Our main result is to show the existence of a general 15 × 15 fundamental matrix. This is based on and completed by a number of other results, e.g. the formulation of general catadioptric projection matrices and plane homographies. 1 Introduction and Previous Work The geometry of single and multiple images has been extensively studied in computer vision and photogrammetry [1]. The picture is rather complete for perspective cameras and many results have been obtained for other camera models too,
Omnidirectional Egomotion Estimation From Backprojection Flow
 IN OMNIVIS
, 2003
"... The current stateoftheart for egomotion estimation with omnidirectional cameras is to map the optical flow to the sphere and then apply egomotion algorithms for spherical projection. In this paper, we propose to backproject image points to a virtual curved retina that is intrinsic to the geometr ..."
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Cited by 11 (1 self)
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The current stateoftheart for egomotion estimation with omnidirectional cameras is to map the optical flow to the sphere and then apply egomotion algorithms for spherical projection. In this paper, we propose to backproject image points to a virtual curved retina that is intrinsic to the geometry of the central panoramic camera, and compute the optical flow on this retina: the socalled backprojection flow. We show that wellknown egomotion algorithms can be easily adapted to work with the backprojection flow. We present extensive simulation results showing that in the presence of noise, egomotion algorithms perform better by using backprojection flow when the camera translation is in the XY plane. Thus, the proposed method is preferable in applications where there is no Zaxis translation, such as ground robot navigation.
Reconstructing a 3d line from a single catadioptric image
 In Proceedings of the Third International Symposium on 3D Data Processing, Visualization, and Transmission
, 2006
"... This paper demonstrates that, for axial noncentral optical systems, the equation of a 3D line can be estimated using only four points extracted from a single image of the line. This result, which is a direct consequence of the lack of vantage point, follows from a classic result in enumerative geom ..."
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Cited by 9 (0 self)
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This paper demonstrates that, for axial noncentral optical systems, the equation of a 3D line can be estimated using only four points extracted from a single image of the line. This result, which is a direct consequence of the lack of vantage point, follows from a classic result in enumerative geometry: there are exactly two lines in 3space which intersect four given lines in general position. We present a simple algorithm to reconstruct the equation of a 3D line from four image points. This algorithm is based on computing the Singular Value Decomposition (SVD) of the matrix of Plücker coordinates of the four corresponding rays. We evaluate the conditions for which the reconstruction fails, such as when the four rays are nearly coplanar. Preliminary experimental results using a spherical catadioptric camera are presented. We conclude by discussing the limitations imposed by poor calibration and numerical errors on the proposed reconstruction algorithm. 1