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Catadioptric Projective Geometry
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2001
"... Catadioptric sensors are devices which utilize mirrors and lenses to form a projection onto the image plane of a camera. Central catadioptric sensors are the class of these devices having a single effective viewpoint. In this paper, we propose a unifying model for the projective geometry induced by ..."
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Cited by 116 (16 self)
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Catadioptric sensors are devices which utilize mirrors and lenses to form a projection onto the image plane of a camera. Central catadioptric sensors are the class of these devices having a single effective viewpoint. In this paper, we propose a unifying model for the projective geometry induced by these devices and we study its properties as well as its practical implications. We show that a central catadioptric projection is equivalent to a twostep mapping via the sphere. The second step is equivalent to a stereographic projection in the case of parabolic mirrors. Conventional lensbased perspective cameras are also central catadioptric devices with a virtual planar mirror and are, thus, covered by the unifying model. We prove that for each catadioptric projection there exists a dual catadioptric projection based on the duality between points and line images (conics). It turns out that planar and parabolic mirrors build a dual catadioptric projection pair. As a practical example we describe a procedure to estimate focal length and image center from a single view of lines in arbitrary position for a parabolic catadioptric system.
Epipolar Geometry for Central Catadioptric Cameras
, 2002
"... Central catadioptric cameras are cameras which combine lenses and mirrors to capture a very wide field of view with a central projection. In this paper we extend the classical epipolar geometry of perspective cameras to all central catadioptric cameras. Epipolar geometry is formulated as the geometr ..."
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Cited by 85 (5 self)
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Central catadioptric cameras are cameras which combine lenses and mirrors to capture a very wide field of view with a central projection. In this paper we extend the classical epipolar geometry of perspective cameras to all central catadioptric cameras. Epipolar geometry is formulated as the geometry of corresponding rays in a threedimensional space. Using the model of image formation of central catadioptric cameras, the constraint on corresponding image points is then derived. It is shown that the corresponding points lie on epipolar conics. In addition, the shape of the conics for all types of central catadioptric cameras is classified. Finally, the theory is verified by experiments with real central catadioptric cameras.
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
Catadioptric SelfCalibration
, 2000
"... We have assembled astandH460 movable system that can capture long sequences ofomnid ectional images (up to 1,500 images at 6.7 Hzand a resolution of 1140 1030). The goal of this system is to reconstruct complex large environments, such as an entire floor of a buildH4 from the captured images on ..."
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Cited by 73 (0 self)
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We have assembled astandH460 movable system that can capture long sequences ofomnid ectional images (up to 1,500 images at 6.7 Hzand a resolution of 1140 1030). The goal of this system is to reconstruct complex large environments, such as an entire floor of a buildH4 from the captured images only. In this paper, wead ess the important issue of how to calibrate such a system. Our method uses images of the environment to calibrate the camera, without the use of a y specia ca93fl68900 pa93fl6 knowledge ofca08G motion, or knowledge of scene geometry. It uses the consistency of pairwise tracked point features across a sequence based on the characteristics of catad4H35 imaging. We also show how the projection equation for this catad0H30 camera can be formulated to be equivalent to that of a typical rectilinear perspective camera with just a simple transformation. 1 Introduction The visua63fl07'9 as modeling ofla00 environments is increa06DG' becominga aoming32 e proposition, due tof...
Structure from Motion with Wide Circular Field of View Cameras
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2006
"... Abstract—This paper presents a method for fully automatic and robust estimation of twoview geometry, autocalibration, and 3D metric reconstruction from point correspondences in images taken by cameras with wide circular field of view. We focus on cameras which have more than 180 field of view and f ..."
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Cited by 65 (7 self)
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Abstract—This paper presents a method for fully automatic and robust estimation of twoview geometry, autocalibration, and 3D metric reconstruction from point correspondences in images taken by cameras with wide circular field of view. We focus on cameras which have more than 180 field of view and for which the standard perspective camera model is not sufficient, e.g., the cameras equipped with circular fisheye lenses Nikon FCE8 (183), Sigma 8mmf4EX (180), or with curved conical mirrors. We assume a circular field of view and axially symmetric image projection to autocalibrate the cameras. Many wide field of view cameras can still be modeled by the central projection followed by a nonlinear image mapping. Examples are the abovementioned fisheye lenses and properly assembled catadioptric cameras with conical mirrors. We show that epipolar geometry of these cameras can be estimated from a small number of correspondences by solving a polynomial eigenvalue problem. This allows the use of efficient RANSAC robust estimation to find the image projection model, the epipolar geometry, and the selection of true point correspondences from tentative correspondences contaminated by mismatches. Real catadioptric cameras are often slightly noncentral. We show that the proposed autocalibration with approximate central models is usually good enough to get correct point correspondences which can be used with accurate noncentral models in a bundle adjustment to obtain accurate 3D scene reconstruction. Noncentral camera models are dealt with and results are shown for catadioptric cameras with parabolic and spherical mirrors. Index Terms—Omnidirectional vision, fisheye lens, catadioptric camera, autocalibration. 1
Caustics of Catadioptric Cameras
 In Proc. International Conference on Computer Vision
, 2001
"... Conventional vision systems and algorithms assume the camera to have a single viewpoint. However, sensors need not always maintain a single viewpoint. For instance, an incorrectly aligned system could cause nonsingle viewpoints. Also, systems could be designed to specifically deviate from a single ..."
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Cited by 62 (10 self)
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Conventional vision systems and algorithms assume the camera to have a single viewpoint. However, sensors need not always maintain a single viewpoint. For instance, an incorrectly aligned system could cause nonsingle viewpoints. Also, systems could be designed to specifically deviate from a single viewpoint to tradeoff image characteristics such as resolution and field of view. In these cases, the locus of viewpoints forms what is called a caustic. In this paper, we present an indepth analysis of caustics of catadioptric cameras with conic reflectors. Properties of caustics with respect to field of view and resolution are presented. Finally, we present ways to calibrate conic catadioptric systems and estimate their caustics from known camera motion.
A flexible technique for accurate omnidirectional camera calibration and structure from motion
 In Proc. of IEEE Intl. Conf. of Vision Systems
, 2006
"... In this paper, we present a flexible new technique for single viewpoint omnidirectional camera calibration. The proposed method only requires the camera to observe a planar pattern shown at a few different orientations. Either the camera or the planar pattern can be freely moved. No a priori knowled ..."
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Cited by 60 (12 self)
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In this paper, we present a flexible new technique for single viewpoint omnidirectional camera calibration. The proposed method only requires the camera to observe a planar pattern shown at a few different orientations. Either the camera or the planar pattern can be freely moved. No a priori knowledge of the motion is required, nor a specific model of the omnidirectional sensor. The only assumption is that the image projection function can be described by a Taylor series expansion whose coefficients are estimated by solving a twostep leastsquares linear minimization problem. To test the proposed technique, we calibrated a panoramic camera having a field of view greater than 200° in the vertical direction, and we obtained very good results. To investigate the accuracy of the calibration, we also used the estimated omnicamera model in a structure from motion experiment. We obtained a 3D metric reconstruction of a scene from two highly distorted omnidirectional images by using image correspondences only. Compared with classical techniques, which rely on a specific parametric model of the omnidirectional camera, the proposed procedure is independent of the sensor, easy to use, and flexible. 1.
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.
Plenoptic Stitching: A Scalable Method for Reconstructing 3D Interactive Walkthroughs
, 2001
"... Interactive walkthrough applications require detailed 3D models to give users a sense of immersion in an environment. Traditionally these models are built using computeraided design tools to define geometry and material properties. But creating detailed models is timeconsuming and it is also diffi ..."
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Cited by 54 (4 self)
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Interactive walkthrough applications require detailed 3D models to give users a sense of immersion in an environment. Traditionally these models are built using computeraided design tools to define geometry and material properties. But creating detailed models is timeconsuming and it is also difficult to reproduce all geometric and photometric subtleties of realworld scenes. Computer vision attempts to alleviate this problem by extracting geometry and photogrammetry from images of the realworld scenes. However, these models are still limited in the amount of detail they recover.
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.