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 propose and analyze several methods to estimate a rigid transformation from a set of 3-D matched points or matched frames, which are important features in geometric algorithms. We also develop tools to predict and verify the accuracy of these estimations. The theoretical contributions are: an intrinsic model of noise for transformations based on composition rather than addition; a unified formalism for the estimation of both the rigid transformation and its covariance matrix for points or frames correspondences, and a statistical validation method to verify the error estimation, which applies even when no "ground truth" is available. We analyze and demonstrate on synthetic data that our scheme is well behaved. The practical contribution of the paper is the validation of our transformation estimation method in the case of 3-D medical images, which shows that an accuracy of the registration far below the size of a voxel can be achieved, and in the case of protein substructure matching, where frame features drastically improve both selectivity and complexity. 1.

The Errors-in-Variables (EIV) model from statistics is often employed in computer vision though only rarely under this name. In an EIV model all the measurements are corrupted by noise while the a priori information is captured with a nonlinear constraint among the true (unknown) values of these measurements. To estimate the model parameters and the uncorrupted data, the constraint can be linearized, i.e., embedded in a higher dimensional space. We show that linearization introduces data-dependent (heteroscedastic) noise and propose an iterative procedure, the heteroscedastic EIV (HEIV) estimator to obtain consistent estimates in the most general, multivariate case. Analytical expressions for the covariances of the parameter estimates and corrected data points, a generic method for the enforcement of ancillary constraints arising from the underlying geometry are also given. The HEIV estimator minimizes the first order approximation of the geometric distances between the measurements and the true data points, and thus can be a substitute for the widely used LevenbergMarquardt based direct solution of the original, nonlinear problem. The HEIV estimator has however the advantage of a weaker dependence on the initial solution and a faster convergence. In comparison to Kanatani's renormalization paradigm (an earlier solution of the same problem) the HEIV estimator has more solid theoretical foundations which translate into better numerical behavior. We show that the HEIV estimator can provide an accurate solution to most 3D vision estimation tasks, and illustrate its performance through two case studies: calibration and the estimation of the fundamental matrix.

In this paper, we address the problem of the recovery of the Euclidean geometry of a scene from a sequence of images without any prior knowledge either about the parameters of the cameras, or about the motion of the camera(s). We do not require any knowledge of the absolute coordinates of some control points in the scene to achieve this goal. Using various computer vision tools, we establish correspondences between images and recover the epipolar geometry of the set of images, from which we show how to compute the complete set of perspective projection matrices for each camera position. These being known, we proceed to reconstruct the scene. This reconstruction is defined up to an unknown projective transformation (i.e. is parameterized with 15 arbitrary parameters). Next we show how to go from this reconstruction to a more constrained class of reconstructions, defined up to an unknown affine transformation (i.e. parameterized with 12 arbitrary parameters) by exploiting known geometr...

A requirement of a visual measurement device is that both measurements and their uncertainties can be determined. This paper develops an uncertainty analysis which includes both the errors in image localization and the uncertainty in the imaging transformation. The matrix representing the imaging transformation is estimated from image-toworld point correspondences. A general expression is derived for the covariance of this matrix. This expression is valid if the matrix is over determined and also if the minimum number of correspondences are used. A bound on the errors of the first order approximations involved is also derived. Armed with this covariance result the uncertainty of any measurement can be predicted, and furthermore the distribution of correspondences can be chosen to achieve a particular bound on the uncertainty. Examples are given of measurements such as distance and parallelism for several applications. These include indoor scenes and architectural measurements. Key word...

We describe a linear-time algorithm that recovers absolute camera orientations and positions, along with uncertainty estimates, for networks of terrestrial image nodes spanning hundreds of meters in outdoor urban scenes. The algorithm produces pose estimates globally consistent to roughly 0.1 # (2 milliradians) and 5 centimeters on average, or about four pixels of epipolar alignment.

We present a new method for the detection of multiple solutions or degeneracy when estimating the Fundamental Matrix, with specific emphasis on robustness to data contamination (mismatches). The Fundamental Matrix encapsulates all the information on camera motion and internal parameters available from image feature correspondences between two views. It is often used as a first step in structure from motion algorithms. If the set of correspondences is degenerate, then this structure cannot be accurately recovered and many solutions explain the data equally well. It is essential that we are alerted to such eventualities. As current feature matchers are very prone to mismatching the degeneracy detection method must also be robust to outliers. In this paper a definition of degeneracy is given and all two view non-degenerate and degenerate cases are catalogued in a logical way by introducing the language of varieties from algebraic geometry. It is then shown how each of the cases can be ro...

Epipolar geometry is a key point in computer vision and the fundamental matrix estimation is the only way to compute it. This article is a fresh look in the subject that overview classic and latest presented methods of fundamental matrix estimation which have been classified into linear methods, iterative methods and robust methods. All of these methods have been programmed and their accuracy analyzed in synthetic and real images. A summary including experimental results and algorithmic details is given and the whole code is available in Internet.

The paper deals with the registration and modeling of multiple 3-D profile maps acquired from different viewpoints by light striping. We analyze the propagation of measurement and calibration errors to the registration parameters and further to the reconstructed model of the scene consisting of planar patches. The analysis is performed for two strategies. In the first strategy, the maps are registered simultaneously using the Levenberg-Marquardt method to update the registration parameters and the model computed afterwards while in the second one, the maps are registered sequentially against the model reconstructed up till then using the method of unit quaternions to update the registration parameters. Our statistical analysis thus combines the registration and modeling steps, and in registration, we determine the corresponding points either on the parametric domains of the maps or as closest points in 3-D using the information from the parametric domain to restrict the search for the ...

Self-Calibration from Image Sequences This thesis develops new algorithms to obtain the calibration parameters of a camera using only information contained in an image sequence, with the objective of using the camera calibration to compute a Euclidean reconstruction. This problem is known as selfcalibration. The motivation for this work is to allow the Euclidean reconstruction of a scene using only a pre-recorded image sequence where no information is available on the camera or the objects in the scene. The approach used is to utilise known motion constraints, which are common for cameras mounted on mobile vehicles or robotic arms, to simplify the algebraic complexity of the self-calibration problem. The algorithms are designed to be easily extendible to use multiple images rather than the minimum number of three required for self-calibration. The uncertainty of the parameters are also computed to give a measure of confidence in the camera calibration. The first method uses a pure camera translation to allow the problem to be stratified by