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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.
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
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.
A Noniterative Method for Correcting Lens Distortion from NinePoint Correspondences
 In Proc. OmniVision’05, ICCVworkshop,2005
, 2005
"... This paper presents a new method for calibrating and correcting large radial distortion. It makes use of a number of image point correspondences from two views only. No knowledge of the scene structures, nor camera intrinsic parameters, is required. By using two singularity conditions, the method su ..."
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Cited by 33 (1 self)
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This paper presents a new method for calibrating and correcting large radial distortion. It makes use of a number of image point correspondences from two views only. No knowledge of the scene structures, nor camera intrinsic parameters, is required. By using two singularity conditions, the method successfully decouples the estimation of the radial distortion from the estimation of fundamental matrix. The solution technique is basically noniterative, it thereby does not need any initial guess, with no risk of local minima. It also proposes a kernelvoting scheme (instead of the conventional RANSAC scheme). The result is shown to be reliable and robust to noise. In addition, the method is easy to implement. 1
Simultaneously calibrating catadioptric camera and detecting line features using Hough transform
 in 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2005. (IROS
, 2005
"... Abstract A line in space is projected to a conic in a central catadioptric image, and such a conic is called a line image. This paper proposes a novel approach to calibrating catadioptric camera and detecting line images simultaneously by using Hough transform. Previous approaches to catadioptric c ..."
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Cited by 8 (0 self)
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Abstract A line in space is projected to a conic in a central catadioptric image, and such a conic is called a line image. This paper proposes a novel approach to calibrating catadioptric camera and detecting line images simultaneously by using Hough transform. Previous approaches to catadioptric cameras calibration employ the traditional conic detecting or fitting methods for line images, and then use these recovered conics to estimate the intrinsic parameters based on some properties of line images. However, the type of a line image can be line, circle, ellipse, hyperbola or parabola, and in general only a small arc of the conic is visible in the image, which brings novel challenges for conic detection and fitting where traditional conic detecting and fitting methods may fail. As we know, the accuracy of the estimated intrinsic parameters highly depends on the accuracy of the extracted conics. The main contribution of this work is we show that all line images from catadioptric cameras with the same intrinsic parameters must belong to a family of conics with only two degreeoffreedom, and such a family is called a line image family. Therefore, we present a novel special Hough transform for line images detection which ensures that all detected conics must belong to a line image family related to certain intrinsic parameters. For all possible values of the unknown intrinsic parameters, the line image special Hough transform are performed. The one with the highest confidence is chosen as the estimated values for these unknown intrinsic parameters, and the corresponding results of line image detection are chosen as the estimated values for line images. In order to make the searching process more efficient, the hierarchical approaches are employed in this paper. The validity of our proposed approach is illustrated by experiments. Index Terms – Camera calibration, omnidirectional camera, Hough transform, feature extraction, line features.
Scene structure recovery from a single omnidirectional image
 in ICCV Workshops
"... This work tackles the problem of recovering the structure of a scene from a single image. The goal is to interpret automatically the image to obtain the spatial layout of the scene. In essence, the method proposed classifies the environment as floor or walls and their relative positions. Instead of ..."
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Cited by 3 (3 self)
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This work tackles the problem of recovering the structure of a scene from a single image. The goal is to interpret automatically the image to obtain the spatial layout of the scene. In essence, the method proposed classifies the environment as floor or walls and their relative positions. Instead of using standard cameras for solving this particular task, our work is novel in using omnidirectional vision, which is advantageous as it captures in a single image the whole surrounding structure. We also consider manmade indoor scenes, where geometric relationships like parallelism and orthogonality are common. Our contribution is a new method for recovering the scene layout by using extracted line segments from a single omnidirectional image. Collection of lines and geometric constraints provide sufficient information to generate a set of possible scene structures. We also create a map of orientations in the image to test these hypotheses and select the one with the best fitting as the resultant structure. 1.
Experiments in Visualbased Navigation with an Omnidirectional Camera
 ICAR2001 Workshop on Omnidirectional Vision Applied to Robotic Orientation and Nondestructive Testing
, 2001
"... This paper overviews experiments in autonomous visualbased navigation undertaken at the Instituto de Sistemas e Rob\'otica. By considering the precise nature of the robot's task, we specify a navigation method which fulfills the environmental and localization requirements best suited to a ..."
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Cited by 1 (0 self)
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This paper overviews experiments in autonomous visualbased navigation undertaken at the Instituto de Sistemas e Rob\'otica. By considering the precise nature of the robot's task, we specify a navigation method which fulfills the environmental and localization requirements best suited to achieving this task. Ongoing research into task specification using 3D models, along with improvements to our topological navigation method are presented. We show how to build 3D models from images obtained with an omnidirectional camera equipped with a spherical mirror, despite the fact that it does not have a single projection centre.
Toward Robot Perception through Omnidirectional Vision
"... “My dear Miss Glory, Robots are not people. They are mechanically more perfect than we are, they have an astounding intellectual capacity...” From the play R.U.R. (Rossum’s Universal Robots) by Karel Capek, 1920. 1 ..."
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“My dear Miss Glory, Robots are not people. They are mechanically more perfect than we are, they have an astounding intellectual capacity...” From the play R.U.R. (Rossum’s Universal Robots) by Karel Capek, 1920. 1
Fitting Conics to Paracatadioptric Projections of
"... The paracatadioptric camera is one of the most popular panoramic systems currently available in the market. It provides a wide field of view by combining a parabolic shaped mirror with a camera inducing an orthographic projection. Previous work proved that the paracatadioptric projection of a line i ..."
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The paracatadioptric camera is one of the most popular panoramic systems currently available in the market. It provides a wide field of view by combining a parabolic shaped mirror with a camera inducing an orthographic projection. Previous work proved that the paracatadioptric projection of a line is a conic curve, and that the sensor can be fully calibrated from the image of three or more lines. However, the estimation of the conic curves where the lines are projected is hard to accomplish because of the partial occlusion. In general only a small arc of the conic is visible in the image, and conventional conic fitting techniques are unable to accurately estimate the curve. The present work provides methods to overcome this problem. We show that in uncalibrated paracatadioptric views a set of conic curves is a set of line projections if and only if certain properties are verified. These properties are used to constrain the search space and correctly estimate the curves. The conic fitting is solved naturally by an eigensystem whenever the camera is skewless and the aspect ratio is known. For the general situation the line projections are estimated using nonlinear optimization. The set of paracatadioptric lines is used in a geometric construction to determine the camera parameters and calibrate the system. We also propose an algorithm to estimate the conic locus corresponding to a line projection in a calibrated paracatadioptric image. It is proved that the set of all line projections is a hyperplane in the space of conic curves. Since the position of the hyperplane depends only on the sensor parameters, the accuracy of the estimation can be improved by constraining the search to conics lying in this subspace. We show that the fitting problem can be solved by an eigensystem, which leads to a robust and computationally efficient method for paracatadioptric line estimation.