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178
Catadioptric Camera Calibration
 IEEE International Conference on Computer Vision
, 1998
"... AbstractÐCatadioptric sensors refer to the combination of lensbased devices and reflective surfaces. These systems are useful because they may have a field of view which is greater than hemispherical, providing the ability to simultaneously view in any direction. Configurations which have a unique ..."
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Cited by 120 (3 self)
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AbstractÐCatadioptric sensors refer to the combination of lensbased devices and reflective surfaces. These systems are useful because they may have a field of view which is greater than hemispherical, providing the ability to simultaneously view in any direction. Configurations which have a unique effective viewpoint are of primary interest, among these is the case where the reflective surface is a parabolic mirror and the camera is such that it induces an orthographic projection and which we call paracatadiotpric. We present an algorithm for the calibration of such a device using only the images of lines in space. In fact, we show that we may obtain all of the intrinsic parameters from the images of only three lines and that this is possible without any metric information. We propose a closedform solution for focal length, image center, and aspect ratio for skewless cameras and a polynomial root solution in the presence of skew. We also give a method for determining the orientation of a plane containing two sets of parallel lines from one uncalibrated view. Such an orientation recovery enables a rectification which is impossible to achieve in the case of a single uncalibrated view taken by a conventional camera. We study the performance of the algorithm in simulated setups and compare results on real images with an approach based on the image of the mirror's bounding circle. Index TermsÐOmnidirectional vision, panoramic vision, catadioptric camera, vanishing points, calibration. æ 1
Geometric Properties of Central Catadioptric Line Images and their Application in Calibration
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2005
"... Abstract—In central catadioptric systems, lines in a scene are projected to conic curves in the image. This work studies the geometry of the central catadioptric projection of lines and its use in calibration. It is shown that the conic curves where the lines are mapped possess several projective in ..."
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Cited by 80 (9 self)
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Abstract—In central catadioptric systems, lines in a scene are projected to conic curves in the image. This work studies the geometry of the central catadioptric projection of lines and its use in calibration. It is shown that the conic curves where the lines are mapped possess several projective invariant properties. From these properties, it follows that any central catadioptric system can be fully calibrated from an image of three or more lines. The image of the absolute conic, the relative pose between the camera and the mirror, and the shape of the reflective surface can be recovered using a geometric construction based on the conic loci where the lines are projected. This result is valid for any central catadioptric system and generalizes previous results for paracatadioptric sensors. Moreover, it is proven that systems with a hyperbolic/elliptical mirror can be calibrated from the image of two lines. If both the shape and the pose of the mirror are known, then two line images are enough to determine the image of the absolute conic encoding the camera’s intrinsic parameters. The sensitivity to errors is evaluated and the approach is used to calibrate a real camera. Index Terms—Catadioptric, omnidirectional vision, projective geometry, lines, calibration. 1
P.: Single view point omnidirectional camera calibration from planar grids. ICRA
, 2007
"... Abstract — This paper presents a flexible approach for calibrating omnidirectional single viewpoint sensors from planar grids. Current approaches in the field are either based on theoretical properties and do not take into account important factors such as misalignment or cameralens distortion or o ..."
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Cited by 59 (4 self)
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Abstract — This paper presents a flexible approach for calibrating omnidirectional single viewpoint sensors from planar grids. Current approaches in the field are either based on theoretical properties and do not take into account important factors such as misalignment or cameralens distortion or overparametrised which leads to minimisation problems that are difficult to solve. Recent techniques based on polynomial approximations lead to impractical calibration methods. Our model is based on an exact theoretical projection function to which we add well identified parameters to model realworld errors. This leads to a full methodology from the initialisation of the intrinsic parameters to the general calibration. We also discuss the validity of the approach for fisheye and spherical models. An implementation of the method is available as opensource software on the author’s Web page. We validate the approach with the calibration of parabolic, hyperbolic, wideangle and spherical sensors. I.
Structure and Motion from Uncalibrated Catadioptric Views
 In Proc. CVPR
, 2001
"... In this paper we present a new algorithm for structure from motion from point correspondences in images taken from uncalibrated catadioptric cameras with parabolic mirrors. We assume that the unknown intrinsic parameters are three: the combined focal length of the mirror and lens and the intersectio ..."
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Cited by 57 (5 self)
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In this paper we present a new algorithm for structure from motion from point correspondences in images taken from uncalibrated catadioptric cameras with parabolic mirrors. We assume that the unknown intrinsic parameters are three: the combined focal length of the mirror and lens and the intersection of the optical axis with the image. We introduce a new representation for images of points and lines in catadioptric images which we call the circle space. This circle space includes imaginary circles, one of which is the image of the absolute conic. We formulate the epipolar constraint in this space and establish a new 4 &times; 4 catadioptric fundamental matrix. We show that the image of the absolute conic belongs to the kernel of this matrix. This enables us to prove that Euclidean reconstruction is feasible from two views with constant parameters and from three views with varying parameters. In both cases, it is one less than the number of views necessary with perspective cameras.
Issues on the geometry of central catadioptric image formation
 In CVPR
, 2001
"... An imaging system with a single effective viewpoint is called a central projection system. The conventional perspective camera is an example of a central projection system. Systems using mirrors to enhance the field of view while keeping a unique center of projection are also examples of central pro ..."
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Cited by 50 (3 self)
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An imaging system with a single effective viewpoint is called a central projection system. The conventional perspective camera is an example of a central projection system. Systems using mirrors to enhance the field of view while keeping a unique center of projection are also examples of central projection systems. Perspective image formation can be described by a linear model with well known properties. In general central catadioptric imaging the mapping between points in the world and in the image is highly nonlinear. This paper establishes a general model for central catadioptric image formation made up of three functions: a linear function mapping the world into an oriented projective plane, a nonlinear transformation between two oriented projective planes, and a collineation in the plane. The model is used to study issues in the projection of lines. The equations and geometric properties of general catadioptric imaging of lines are derived. The application of the results in autocalibration of central catadioptric systems and reconstruction are discussed. A method to calibrate the system using three line images is presented. 1.
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
Imagebased visual servoing for nonholonomic mobile robots using epipolar geometry
 IEEE Transactions on Robotics
, 2007
"... Abstract—We present an imagebased visual servoing strategy for driving a nonholonomic mobile robot equipped with a pinhole camera toward a desired configuration. The proposed approach, which exploits the epipolar geometry defined by the current and desired camera views, does not need any knowledge ..."
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Cited by 36 (3 self)
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Abstract—We present an imagebased visual servoing strategy for driving a nonholonomic mobile robot equipped with a pinhole camera toward a desired configuration. The proposed approach, which exploits the epipolar geometry defined by the current and desired camera views, does not need any knowledge of the 3D scene geometry. The control scheme is divided into two steps. In the first, using an approximate input–output linearizing feedback, the epipoles are zeroed so as to align the robot with the goal. Feature points are then used in the second translational step to reach the desired configuration. Asymptotic convergence to the desired configuration is proven, both in the calibrated and partially calibrated case. Simulation and experimental results show the effectiveness of the proposed control scheme. Index Terms—Epipolar geometry, imagebased visual servoing (IBVS), input–output feedback linearization, nonholonomic mobile robots. I.
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.
Robust Scene Reconstruction from an Omnidirectional Vision System
, 2002
"... In this paper we present an efficient multibaseline stereo algorithm for panoramic image data. We derive a parameterization of epipolar curves in terms of inverse depth. As a result the search for image correspondences across multiple images can be performed efficiently. Furthermore, depth estimate ..."
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Cited by 36 (1 self)
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In this paper we present an efficient multibaseline stereo algorithm for panoramic image data. We derive a parameterization of epipolar curves in terms of inverse depth. As a result the search for image correspondences across multiple images can be performed efficiently. Furthermore, depth estimates are obtained directly thus bypassing the need to perform explicit stereoscopic triangulation. We apply our method to obtain a 3D reconstruction of an environment from a set of panoramic images. The images are acquired by a single omnidirectional vision sensor mounted on top of our mobile robot during navigation. Experimental results demonstrate the effectiveness of our approach.
Formation Control of Nonholonomic Mobile Robots with Omnidirectional Visual Servoing and Motion Segmentation
 In IEEE International Conference on Robotics and Automation
, 2003
"... We consider the problem of having a team of nonholonomic mobile robots follow a desired leaderfollower formation using omnidirectional vision. By specifying the desired formation in the image plane, we translate the control problem into a separate visual servoing task for each follower. We use a ra ..."
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Cited by 36 (3 self)
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We consider the problem of having a team of nonholonomic mobile robots follow a desired leaderfollower formation using omnidirectional vision. By specifying the desired formation in the image plane, we translate the control problem into a separate visual servoing task for each follower. We use a rank constraint on the omnidirectional optical flows across multiple frames to estimate the position and velocities of the leaders in the image plane of each follower. We show that the direct feedbacklinearization of the leaderfollower dynamics suffers from degenerate configurations due to the nonholonomic constraints of the robots and the nonlinearity of the omnidirectional projection model. We therefore design a nonlinear tracking controller that avoids such degenerate configurations, while preserving the formation inputtostate stability. Our control law naturally incorporates collision avoidance by exploiting the geometry of omnidirectional cameras. We present simulations and experiments evaluating our omnidirectional visionbased formation control scheme.