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 two-step mapping via the sphere. The second step is equivalent to a stereographic projection in the case of parabolic mirrors. Conventional lens-based 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.

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 × 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.

Abstract—This paper presents a method for fully automatic and robust estimation of two-view 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 fish-eye lenses Nikon FC-E8 (183), Sigma 8mm-f4-EX (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 above-mentioned fish-eye 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, fish-eye lens, catadioptric camera, autocalibration. 1

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 non-iterative, it thereby does not need any initial guess, with no risk of local minima. It also proposes a kernel-voting 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

This paper overviews experiments in autonomous visual-based 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. On-going 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.

Localization of 3D lines from single images using off-axis catadioptric cameras The reconstruction of 3D scenes constituted by straight lines can find many applications both in Computer vision and in Mobile robotics. Most approaches to 3D reconstruction analyze several images, which need the determination of the correspondences between the used images. This paper studies the localization of straight lines in 3D space from single 2D images, acquired by a noncentral catadioptric camera. In general, using a noncentral camera, the viewing rays starting from the points of a straight line constitute a nonplanar surface: if this non-planar surface only contains a finite number of straight line, other than the viewing rays, then the straight line can be localized (up to a finite number of solutions). In a previous work we dealt with line localization using an axial-symmetric catadioptric camera. In this paper the axial-symmetry constraint is relaxed, since the camera is placed at a generic position relative to an axial-symmetric mirror. The geometrical conditions for line localization are derived for off-axis catadioptric cameras based on a conic mirror, and some preliminary experimental results are presented. 1.

The reconstruction of 3D scenes constituted by straight lines can find many applications both in Computer vision and in Mobile robotics. Most of the approaches to this problem involve either stereo-vision or the analysis of a sequence of images taken from different viewpoints: in both cases, the solution of the correspondence problem is required. This paper studies the localization of straight lines in 3D space from single 2D images, acquired by a catadioptric camera. In general, using a noncentral camera, the viewing rays starting from the points of a straight line constitute a non-planar surface: if this non-planar surface only contains one straight line, other than the viewing rays, then the straight line can univocally be localized. Some conditions for the univocal localization of straight lines are derived. A simple technique for the straight line localization is presented, and some preliminary experimental results are discussed.

pour obtenir le grade de Habilitation à diriger des Recherches de l’INSTITUT NATIONAL POLYTECHNIQUE DE GRENOBLE