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74
A unifying theory for central panoramic systems and practical implications
 In ECCV
, 2000
"... Abstract. Omnidirectional vision systems can provide panoramic alertness in surveillance, improve navigational capabilities, and produce panoramic images for multimedia. Catadioptric realizations of omnidirectional vision combine reflective surfaces and lenses. A particular class of them, the centra ..."
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Cited by 178 (5 self)
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Abstract. Omnidirectional vision systems can provide panoramic alertness in surveillance, improve navigational capabilities, and produce panoramic images for multimedia. Catadioptric realizations of omnidirectional vision combine reflective surfaces and lenses. A particular class of them, the central panoramic systems, preserve the uniqueness of the projection viewpoint. In fact, every central projection system including the well known perspective projection on a plane falls into this category. In this paper, we provide a unifying theory for all central catadioptric systems. We show that all of them are isomorphic to projective mappings from the sphere to a plane with a projection center on the perpendicular to the plane. Subcases are the stereographic projection equivalent to parabolic projection and the central planar projection equivalent to every conventional camera. We define a duality among projections of points and lines as well as among different mappings. This unification is novel and has a a significant impact on the 3D interpretation of images. We present new invariances inherent in parabolic projections and a unifying calibration scheme from one view. We describe the implied advantages of catadioptric systems and explain why images arising in central catadioptric systems contain more information than images from conventional cameras. One example is that intrinsic calibration from a single view is possible for parabolic catadioptric systems given only three lines. Another example is metric rectification using only affine information about the scene. 1
Visionbased Navigation and Environmental Representations with an Omnidd Camera
 IEEE Transactions on Robotics and Automation
, 2000
"... This paper proposes a method for the visualbased navigation of a mobile robot in indoor environments, using a single omnidirectional (catadioptric) camera. The geometry of the catadioptric sensor and the method used to obtain a bird's eye (orthographic) view of the ground plane are presented. ..."
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Cited by 123 (17 self)
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This paper proposes a method for the visualbased navigation of a mobile robot in indoor environments, using a single omnidirectional (catadioptric) camera. The geometry of the catadioptric sensor and the method used to obtain a bird's eye (orthographic) view of the ground plane are presented. This representation significantly simplifies the so to navigation protiok by eliminating any perspective effects. The nature of each navigation task is taken into account when designing the required navigation skills and environmental representation. We propose two main navigation mo dalities: Topological Navigation and Visual Path Following. To po lok Navigatio is used fo traveling lo distances and do es no require knowledge of the exact position of the robot but rather, a qualitative position of the took map. The navigation process combines appearance based methods and visual servorv up oso environmental features. Visual Path Following is required for local, very precise navigation fo e.g.do o traversal,do cking. The robot is contro to fo w a prespecified p...
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
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.
The Space of All Stereo Images
, 2001
"... A theory of stereo image formation is presented that enables a complete classification of all possible stereo views, including nonperspective varieties. Towards this end, the notion of epipolar geometry is generalized to apply to multiperspective images. It is shown that any stereo pair must consis ..."
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Cited by 85 (2 self)
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A theory of stereo image formation is presented that enables a complete classification of all possible stereo views, including nonperspective varieties. Towards this end, the notion of epipolar geometry is generalized to apply to multiperspective images. It is shown that any stereo pair must consist of rays lying on one of three varieties of quadric surfaces. A unified representation is developed to model all classes of stereo views, based on the concept of a quadric view. The benefits include a unified treatment of projection and triangulation operations for all stereo views. The framework is applied to derive new types of stereo image representations with unusual and useful properties.
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
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.