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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 113 (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.
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 61 (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 58 (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 56 (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.
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 45 (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 minimal solution to the autocalibration of radial distortion
, 2007
"... Epipolar geometry and relative camera pose computation are examples of tasks which can be formulated as minimal problems and solved from a minimal number of image points. Finding the solution leads to solving systems of algebraic equations. Often, these systems are not trivial and therefore special ..."
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Cited by 39 (12 self)
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Epipolar geometry and relative camera pose computation are examples of tasks which can be formulated as minimal problems and solved from a minimal number of image points. Finding the solution leads to solving systems of algebraic equations. Often, these systems are not trivial and therefore special algorithms have to be designed to achieve numerical robustness and computational efficiency. In this paper we provide a solution to the problem of estimating radial distortion and epipolar geometry from eight correspondences in two images. Unlike previous algorithms, which were able to solve the problem from nine correspondences only, we enforce the determinant of the fundamental matrix be zero. This leads to a system of eight quadratic and one cubic equation in nine variables. We simplify the system by eliminating six of these variables. Then, we solve the system by finding eigenvectors of an action matrix of a suitably chosen polynomial. We show how to construct the action matrix without computing complete Gröbner basis, which provides an efficient and robust solver. The quality of the solver is demonstrated on synthetic and real data. 1.
Mixing catadioptric and perspective cameras
 in: Workshop on Omnidirectional Vision
, 2002
"... We analyze relations that exist between multiple views of a static scene, where the views can be taken by any mixture of paracatadioptric, perspective or affine cameras. Concretely, we introduce the notion of fundamental matrix, trifocal and quadrifocal tensors for the different possible combinatio ..."
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Cited by 38 (15 self)
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We analyze relations that exist between multiple views of a static scene, where the views can be taken by any mixture of paracatadioptric, perspective or affine cameras. Concretely, we introduce the notion of fundamental matrix, trifocal and quadrifocal tensors for the different possible combinations of these camera types. We also introduce the notion of plane homography for mixed image pairs. Generally speaking, these novel multiview relations may form the basis for the typical geometric computations like motion estimation, 3D reconstruction or (self) calibration. A few novel algorithms illustrating some of these aspects, are described, especially concerning what we call calibration transfer, using fundamental matrices, and selfcalibration from plane homographies. 1.