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.

This paper presents a linear algorithm for recovering 3D affine shape and motion from line correspondences with uncalibrated affine cameras. The algorithm requires a minimum of seven line correspondences over three views. The key idea is the introduction of a one-dimensional projective camera. This converts 3D affine reconstruction of "line directions" into 2D projective reconstruction of "points". In addition, a line-based factorisation method is also proposed to handle redundant views. Experimental results both on simulated and real image sequences validate the robustness and the accuracy of the algorithm.

In this paper a new method for Euclidean reconstruction from sequences of images taken by uncalibrated cameras, with constant intrinsic parameters, is described. Our approach leads to a variant of the so called Kruppa equations. It is shown that it is possible to calculate the intrinsic parameters as well as the Euclidean reconstruction from at least three images. The novelty of our approach is that we build our calculation on a projective reconstruction, obtained without the assumption on constant intrinsic parameters. This assumption simplifies the analysis, because a projective reconstruction is already obtained and we need ‘only’ to find the correct Euclidean reconstruction among all possible projective reconstructions.

This paper disc#274# the basic role of the trifoc al tensor insc#37 rec# nstr uc#r# n from three views. This 3 3 tensor plays a role in the analysis of sc#422 from three views analogous to the role played by the fundamental matrix in the two-viewc ase. In partic ular, the trifoc al tensor may bec omputed by a linear algorithm from a set of 13 linec orrespondenc#3 in three views. It is further shown in this paper, that the trifoc al tensor is essentially identic## to a set ofc oe#c#99 ts introduc#5 by Shashua toe#ec# point transfer in the three viewc##22 This observation means that the 13-line algorithm may be extended to allow for thec omputation of the trifoc al tensor given any mixture of su#c#36 tly many line and pointc orrespondenc#9# From the trifoc al tensor thec amera matric## of the images may be c#25371#- and the sc#35 may berec#31#-41562# For unrelatedunc# libratedc ameras, this rec# nstr uc#r# n will be unique up to projec#939# y. Thus, projec#61 e rec#376#-39162 of a set of lines and points may bec#40940 out linearly from three views.

this paper we develop geometric relationships between the residual (planar) parallax displacements of pairs of points. These geometric relationships address the problem of 3D scene analysis even in difficult conditions, i.e., when the epipole estimation is ill-conditioned, when there is a small number of parallax vectors, and in the presence of moving objects. We show how these relationships can be applied to each of the three problems outlined at the beginning of this section. Moreover, the use of the parallax constraints derived here provides a continuum between "2D algorithms" and the "3D algorithms" for each of the problems mentioned above. The geometric relationships presented in this work are expressed in terms of residual parallax displacements of points after canceling a planar homography

This paper presents a new framework for analyzing the geometry of multiple 3D scene points from multiple uncalibrated images, based on decomposing the projection of these points on the images into two stages: (i) the projection of the scene points onto a (real or virtual) physical reference planar surface in the scene; this creates a virtual "image" on the reference plane, and (ii) the re-projection of the virtual image onto the actual image plane of the camera. The positions of the virtual image points are directly related to the 3D locations of the scene points and the camera centers relative to the reference plane alone. All dependency on the internal camera calibration parameters and the orientation of the camera are folded into homographies relating each image plane to the reference plane. Bi-linear and tri-linear constraints involving multiple points and views are given a concrete physical interpretation in terms of geometric relations on the physical reference plane. In particu...

The topic of representation, recovery and manipulation of three-dimensional (3D) scenes from two-dimensional (2D) images thereof, provides a fertile ground for both intellectual theoretically inclined questions related to the algebra and geometry of the problem and to practical applications such as Visual Recognition, Animation and View Synthesis, recovery of scene structure and camera ego-motion, object detection and tracking, multi-sensor alignment, etc. The basic materials have been known since the turn of the century, but the full scope of the problem has been under intensive study since 1992, first on the algebra of two views and then on the algebra of multiple views leading to a relatively mature understanding of what is known as "multilinear matching constraints", and the "trilinear tensor" of three or more views. The purpose of this paper is, first and foremost, to provide a coherent framework for expressing the ideas behind the analysis of multiple views. Seco...

We show how to use the Grassmann-Cayley algebra to model systems of one, two and three cameras. We start with a brief introduction of the Grassmann-Cayley or double algebra and proceed to demonstrate its use for modeling systems of cameras. In the case of three cameras, we give a new interpretation of the trifocal tensors and study in detail some of the constraints that they satisfy. In particular we prove that simple subsets of those constraints characterize the trifocal tensors, in other words, we give the algebraic equations of the manifold of trifocal tensors.

We present in this paper an algorithm for determining 3D motion and structure from correspondences of line segments between two perspective images. To our knowledge, this paper is the first investigation on use of line segments in motion and structure from motion. Classical methods use their geometric abstraction, namely straight lines, but then three images are necessary. We show in this paper that two views are in general sufficient when using line segments. The assumption we use is that two matched line segments contain the projection of a common part of the corresponding line segment in space. Indeed, this is what we use to match line segments between different views. Both synthetic and real data have been used to test the proposed algorithm, and excellent results have been obtained with real data containing about one hundred line segments. The results are comparable with those obtained with calibrated stereo.

The paper deals with the structure--motion problem for images of point configurations taken by uncalibrated cameras. Using a parametrisation by affine shape and kinetic depth, a complete and explicit characterisation of the imaging geometry is given, including the shape of the object configuration and the positions of the cameras relative to the scene. No epipolar geometry is used. It is shown that not only the projective but also the affine structure of the scene can be recovered when knowing the relative placement of five of the camera centres (four if they are coplanar). Variational algorithms for reconstruction and motion are presented, thus avoiding numerically unstable solving of algebraic equations. Any number of points in any number of images can be treated simultaneously and uniformly, without preselection of reference points. The performances of the algorithms are illustrated on simulations and experiments. Keywords: Depth, shape, reconstruction, motion, invariants, proximit...