(updated on Aug. 10, 2002; a typo in Appendix B) (last updated on Aug. 13, 2008; a typo in Section 3.3)

We propose a flexible new technique to easily calibrate a camera. It only requires the camera to observe a planar pattern shown at a few (at least two) different orientations. Either the camera or the planar pattern can be freely moved. The motion need not be known. Radial lens distortion is modeled. The proposed procedure consists of a closed-form solution, followed by a nonlinear refinement based on the maximum likelihood criterion. Both computer simulation and real data have been used to test the proposed technique, and very good results have been obtained. Compared with classical techniques which use expensive equipment such as two or three orthogonal planes, the proposed technique is easy to use and flexible. It advances 3D computer vision one step from laboratory environments to real world use. The corresponding software is available from the author’s Web page.

This work is in the context of motion and stereo analysis. It presents a new uni ed representation which will be useful when dealing with multiple views in the case of uncalibrated cameras. Several levels of information might be considered, depending on the availability of information. Among other things, an algebraic description of the epipolar geometry of N views is introduced, as well as a framework for camera self-calibration, calibration updating, and structure from motion in an image sequence taken by a camera which is zooming and moving at the same time. We show how a special decomposition of a set of two or three general projection matrices, called canonical enables us to build geometric descriptions for a system of cameras which are invariant with respect to a given group of transformations. These representations are minimal and capture completely the properties of each level of description considered: Euclidean (in the context of calibration, and in the context of structure from motion, which we distinguish clearly), a ne, and projective, that we also relate to each other. In the last case, a new decomposition of the well-known fundamental matrix is obtained. Dependencies, which appear when three or more views are available, are studied in the context of the canonic decomposition, and new composition formulas are established. The theory is illustrated by tutorial examples with real images.

We describe how 3D affine measurements may be computed from a single perspective view of a scene given only minimal geometric information determined from the image. This minimal information is typically the vanishing line of a reference plane, and a vanishing point for a direction not parallel to the plane. It is shown that affine scene structure may then be determined from the image, without knowledge of the camera's internal calibration (e.g. focal length), nor of the explicit relation between camera and world (pose). In particular, we show how to (i) compute the distance between planes parallel to the reference plane (up to a common scale factor); (ii) compute area and length ratios on any plane parallel to the reference plane; (iii) determine the camera's (viewer's) location. Simple geometric derivations are given for these results. We also develop an algebraic representation which unifies the three types of measurement and, amongst other advantages, permits a first order error pr...

We describe the geometry, constraints and algorithmic implementation for metric rectification of planes. The rectification allows metric properties, such as angles and length ratios, to be measured on the world plane from a perspective image. The novel contributions are: first, that in a stratified context the various forms of providing metric information, which include a known angle, two equal though unknown angles, and a known length ratio; can all be represented as circular constraints on the parameters of an affine transformation of the plane --- this provides a simple and uniform framework for integrating constraints; second, direct rectification from right angles in the plane; third, it is shown that metric rectification enables calibration of the internal camera parameters; fourth, vanishing points are estimated using a Maximum Likelihood estimator; fifth, an algorithm for automatic rectification. Examples are given for a number of images, and applications demonstrated for textu...

. We present a linear rectification algorithm for general, unconstrained stereo rigs. The algorithm takes the two perspective projection matrices of the original cameras, and computes a pair of rectifying projection matrices. It is compact (22-line MATLAB code) and easily reproducible. To maximize usefulness, we have made its implementation available on the WWW. We report tests proving the correct behavior of our method, as well as the negligible decrease of the accuracy of 3-D reconstruction performed from the rectified images directly. Key words: stereo, epipolar geometry, rectification 1 Introduction and motivations Given a pair of stereo images, rectification determines a transformation of each image plane such that pairs of conjugate epipolar lines become collinear and parallel to one of the image axes (usually the horizontal one). The rectified images can be thought of as acquired by a new stereo rig, obtained by rotating the original cameras. The important advantage of rectif...

Abstract. In this paper we describe a flexible approach for determining the relative orientation of the camera with respect to the scene. The main premise of the approach is the fact that in man-made environments, the majority of lines is aligned with the principal orthogonal directions of the world coordinate frame. We exploit this observation towards efficient detection and estimation of vanishing points, which provide strong constraints on camera parameters and relative orientation of the camera with respect to the scene. By combining efficient image processing techniques in the line detection and initialization stage we demonstrate that simultaneous grouping and estimation of vanishing directions can be achieved in the absence of internal parameters of the camera. Constraints between vanishing points are then used for partial calibration and relative rotation estimation. The algorithm has been tested in a variety of indoors and outdoors scenes and its efficiency and automation makes it amenable for implementation on robotic platforms. Key words: Vanishing point estimation, relative orientation, calibration using vanishing points, vision guided mobile and aerial robots. 1

A man-made environment is characterized by a lot of parallel lines and a lot of orthogonal edges. In this article, a new method for detecting the three mutual orthogonal directions of such an environment is presented. Since realtime performance is not necessary for architectural application, like building reconstruction, a computationally more intensive approach was chosen. On the other hand, our approach is more rigorous than existing techniques, since the information given by the condition of three mutual orthogonal directions in the scene is identified and incorporated. Since knowledge about the camera geometry can be deduced from the vanishing points of three mutual orthogonal directions, we use this knowledge to reject falsely detected vanishing points. Results are presented from interpreting outdoor scenes of buildings. Key words Vanishing points, vanishing lines, geometric constraints, architecture, camera calibration

We present a simple approach to combining scene and auto-calibration constraints for the calibration of cameras from single views and stereo pairs. Calibration constraints are provided by imaged scene structure, such as vanishing points of orthogonal directions, or rectified planes. In addition, constraints are available from the nature of the cameras and the motion between views. We formulate these constraints in terms of the geometry of the imaged absolute conic and its relationship to pole-polar pairs and the imaged circular points of planes. Three significant advantages result: first, constraints from scene features, camera characteristics and auto-calibration constraints provide linear equations in the elements of the image of the absolute conic. This means that constraints may easily be combined, and their solution is straightforward. Second, the degeneracies that occur when constraints are not independent may be easily identified. Lastly, the constraints from scene planes and i...

We present an approach for 3D reconstruction of objects from a single image. Obviously, constraints on the 3D structure are needed to perform this task. Our approach is based on user-provided coplanarity, perpendicularity and parallelism constraints. These are used to calibrate the image and perform 3D reconstruction. The method is described in detail and results are provided. 1