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.

Recently, different approaches for uncalibrated stereo have been suggested which permit projective reconstructions from multiple views. These use weak calibration which is represented by the epipolar geometry, and so we require no knowledge of the intrinsic or extrinsic camera parameters. In this paper we consider projective reconstructions from pairs of views, and compare a number of the available methods. Projective stereo algorithms can be categorized by the way in which the 3D coordinates are computed. The first class is similar to traditional stereo algorithms in that the 3D world geometry is made explicit; the initial phase of the processing always involves the estimation of the camera matrices from which the 3D coordinates are computed. We show how the camera matrices can be computed either from point correspondences, or how they are constrained by the fundamental matrices. The second class of algorithms are based on implicit image measurements which are used to compute project...

In this paper, we describe a method for calibrating a stereo pair of cameras using general or planar motions. The method consists of upgrading a 3D projective representation to affine and to Euclidean without any knowledge, neither about the motion parameters nor about the 3D layout. We investigate the algebraic properties relating projective representation to the plane at infinity and to the intrinsic camera parameters when the camera pair is considered as a moving rigid body. We show that all the computations can be carried out using standard linear resolutions techniques. An error analysis reveals the relative importance of the various steps of the calibration process: projective-to-affine and affine-to-metric upgrades. Extensive experiments performed with calibrated and natural data confirm the error analysis as well as the sensitivity study performed with simulated data.

The classical approach to motion and structure estimation problem from two perspective projections consists of two stages: (i) using the 8-point algorithm to estimate the 9 essential parameters defined up to a scale factor, which is a linear estimation problem; (ii) refining the motion estimation based on some statistically optimal criteria, which is a nonlinear estimation problem on a five-dimensional space. Unfortunately, the results obtained using this approach are often not satisfactory, especially when the motion is small or when the observed points are close to a degenerate surface (e.g. plane). The problem is that the second stage is very sensitive to the initial guess, and that it is very difficult to obtain a precise initial estimate from the first stage. This is because we perform a projection of a set of quantities which are estimated in a space of 8 dimensions, much higher than that of the real space which is five-dimensional. We propose in this paper a novel approach by introducing...