Two images of a single scene/object are related by the epipolar geometry, which can be described by a 3×3 singular matrix called the essential matrix if images' internal parameters are known, or the fundamental matrix otherwise. It captures all geometric information contained in two images, and its determination is very important in many applications such as scene modeling and vehicle navigation. This paper gives an introduction to the epipolar geometry, and provides a complete review of the current techniques for estimating the fundamental matrix and its uncertainty. A well-founded measure is proposed to compare these techniques. Projective reconstruction is also reviewed. The software which we have developed for this review is available on the Internet.

In this paper we analyze in some detail the geometry of a pair of cameras, i.e. a stereo rig. Contrarily to what has been done in the past and is still done currently, for example in stereo or motion analysis, we do not assume that the intrinsic parameters of the cameras are known (coordinates of the principal points, pixels aspect ratio and focal lengths). This is important for two reasons. First, it is more realistic in applications where these parameters may vary according to the task (active vision). Second, the general case considered here, captures all the relevant information that is necessary for establishing correspondences between two pairs of images. This information is fundamentally projective and is hidden in a confusing manner in the commonly used formalism of the Essential matrix introduced by Longuet-Higgins [40]. This paper clarifies the projective nature of the correspondence problem in stereo and shows that the epipolar geometry can be summarized in one 3 \Theta 3 ma...

We describe a new method for camera autocalibration and scaled Euclidean structure and motion, from three or more views taken by a moving camera with fixed but unknown intrinsic parameters. The motion constancy of these is used to rectify an initial projective reconstruction. Euclidean scene structure is formulated in terms of the absolute quadric — the singular dual 3D quadric ( rank 3 matrix) giving the Euclidean dot-product between plane normals. This is equivalent to the traditional absolute conic but simpler to use. It encodes both affine and Euclidean structure, and projects very simply to the dual absolute image conic which encodes camera calibration. Requiring the projection to be constant gives a bilinear constraint between the absolute quadric and image conic, from which both can be recovered nonlinearly from images, or quasi-linearly from. Calibration and Euclidean structure follow easily. The nonlinear method is stabler, faster, more accurate and more general than the quasi-linear one. It is based on a general constrained optimization technique — sequential quadratic programming — that may well be useful in other vision problems.

We explore the geometric and algebraic relations that exist between correspondences of points and lines in an arbitrary number of images. We propose to use the formalism of the Grassmann-Cayley algebra as the simplest way to make both geometric and algebraic statements in a very synthetic and effective way (i.e. allowing actual computation if needed). We have a fairly complete picture of the situation in the case of points: there are only three types of algebraic relations which are satisfied by the coordinates of the images of a 3-D point: bilinear relations arising when we consider pairs of images among the N and which are the well-known epipolar constraints, trilinear relations arising when we consider triples of images among the N , and quadrilinear relations arising when we consider four-tuples of images among the N . In the case of lines, we show how the traditional perspective projection equation can be suitably generalized and that in the case of three images there exist two in...

In the general case, a trilinear relationship between three perspective views is shown to exist. The trilinearity result is shown to be of much practical use in visual recognition by alignment --- yielding a direct reprojection method that cuts through the computations of camera transformation, scene structure and epipolar geometry. Moreover, the direct method is linear and sets a new lower theoretical bound on the minimal number of points that are required for a linear solution for the task of reprojection. The proof of the central result may be of further interest as it demonstrates certain regularities across homographies of the plane and introduces new view invariants. Experiments on simulated and real image data were conducted, including a comparative analysis with epipolar intersection and the linear combination methods, with results indicating a greater degree of robustness in practice and a higher level of performance in re-projection tasks. Keywords--- Visual Recognition, Al...

A structure from motion algorithm is described which recovers structure and camera position, modulo a projective ambiguity. Camera calibration is not required, and camera parameters such as focal length can be altered freely during motion. The structure is updated sequentially over an image sequence, in contrast to schemes which employ a batch process. A specialisation of the algorithm to recover structure and camera position modulo an affine transformation is described, together with a method to periodically update the affine coordinate frame to prevent drift over time. We describe the constraint used to obtain this specialisation. Structure is recovered from image corners detected and matched automatically and reliably in real image sequences. Results are shown for reference objects and indoor environments, and accuracy of recovered structure is fully evaluated and compared for a number of reconstruction schemes. A specific application of the work is demonstrated -- affine structure is used to compute free space maps enabling navigation through unstructured environments and avoidance of obstacles. The path planning involves only affine constructions.

This paper describes a theory and a practical algorithm for the autocalibration of a moving projective camera, from views of a planar scene. The unknown camera calibration, and (up to scale) the unknown scene geometry and camera motion are recovered from the hypothesis that the camera’s internal parameters remain constant during the motion. This work extends the various existing methods for non-planar autocalibration to a practically common situation in which it is not possible to bootstrap the calibration from an intermediate projective reconstruction. It also extends Hartley’s method for the internal calibration of a rotating camera, to allow camera translation and to provide 3D as well as calibration information. The basic constraint is that the projections of orthogonal direction vectors (points at infinity) in the plane must be orthogonal in the calibrated camera frame of each image. Abstractly, since the two circular points of the 3D plane (representing its Euclidean structure) lie on the 3D absolute conic, their projections into each image must lie on the absolute conic’s image (representing the camera calibration). The resulting numerical algorithm optimizes this constraint over all circular points and projective calibration parameters, using the inter-image homographies as a projective scene representation.

We describe a method for determining affine and metric calibration of a camera with unchanging internal parameters undergoing planar motion. It is shown that affine calibration is recovered uniquely, and metric calibration up to a two fold ambiguity. The novel

. We address the problem of estimating three-dimensional motion, and structure from motion with an uncalibrated moving camera. We show that point correspondences between three images, and the fundamental matrices computed from these point correspondences, are sufficient to recover the internal orientation of the camera (its calibration), the motion parameters, and to compute coherent perspective projection matrices which enable us to reconstruct 3-D structure up to a similarity. In contrast with other methods, no calibration object with a known 3-D shape is needed, and no limitations are put upon the unknown motions to be performed or the parameters to be recovered, as long as they define a projective camera. The theory of the method, which is based on the constraint that the observed points are part of a static scene, thus allowing us to link the intrinsic parameters and the fundamental matrix via the absolute conic, is first detailed. Several algorithms are then presented, and their ...

We describe a method to determine affine and metric calibration for a stereo rig. The method does not involve the use of calibration objects or special motions, but simply a single general motion of the rig with fixed parameters (i.e. camera parameters and relative orientation of the camera pair). The novel aspects of this work are: first, relating the distinguished objects of Euclidean geometry to fixed entities of a Euclidean transformation matrix; second, showing that these fixed entities are accessible from the conjugate Euclidean transformation arising from the projective transformation of the structure under a motion of the fixed stereo rig; third, a robust and automatic implementation of the method. Results are included of affine and metric calibration and structure recovery using images of real scenes. 1 Introduction It is known that from two uncalibrated views, 3D structure can be recovered modulo a projectivity of 3space [5, 8]. Furthermore, given three or more views acquir...