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.

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.

Abstract – A new stereo vision tracker setup for virtual and augmented reality applications is presented in this paper. Performance, robustness and accuracy of the system are achieved under real-time constraints. The method is based on blobs extraction, two-dimensional prediction, the epipolar constraint and three-dimensional reconstruction. Experimental results using a stereo rig setup (equipped with IR capabilities) and retroreflective targets are presented to demonstrate the capabilities of our optical tracking system. The system tracks up to 25 independent targets at 30 Hz.

Edges, corners and vertices are strong and useful features in computer vision. This paper deals with the development of an efficient model based approach in order to detect and characterize precisely these important features. The key of our approach is first to propose some efficient models associated to each of these features and second to efficiently extract and characterize these features directly from the image. The models associated to each feature include a large number of intrinsic parameters (Grey level intensities, location, orientation of the line segments... ) but also an important parameter which is associated to the blurring effect due to the acquisition system. The important problem of the initialization phase in the minimization process is also considered and an original and efficient solution is proposed. In order to test and compare the reliability, the robustness and the efficiency of the different proposed approaches, a large number of experiments involving noisy...

One of the main problems to obtain a Euclidean 3D reconstruction from multiple views is the calibration of the camera. Explicit calibration is not always practical and has to be repeated regularly. Sometimes it is even impossible #i.e. for pictures taken by an unknown camera of an unknown scene#. The second possibility is to do auto-calibration.

Università degli studi di Verona We address the problem of autocalibration of a moving camera with unknown constant intrinsic parameters. Existing autocalibration techniques use numerical optimization algorithms whose convergence to the correct result cannot be guaranteed, in general. To address this problem, we have developed a method where an interval branchand-bound method is employed for numerical minimization. Thanks to the properties of Interval Analysis this method converges to the global solution with mathematical certainty and arbitrary accuracy, and the only input information it requires from the user are a set of point correspondences and a search box. The cost function is based on the Huang-Faugeras constraint of the fundamental matrix, and a closed form expression for its Jacobian and Hessian matrices is derived through matrix differential calculus. A recently proposed interval extension based on Bernstein polynomial forms has been investigated to speed up the search for the solution. Finally, experimental results on synthetic and real images are presented.

The problem of finding the collineation between two 3D projective reconstructions has been proved to be useful for a variety of tasks such as calibration of a stereo rig and 3D affine and/or Euclidean reconstruction. Moreover, such a collineation may well be viewed as a point transfer method between two image pairs with applications to visually guided robot control. Despite this potential, methods for properly estimating such a projective transformation have received little attention in the past. In this paper we describe linear, nonlinear, and robust methods for estimating this transformation. We test the numerical stability of these methods with respect to image noise, to the number of matched points, and as a function of the number of outliers. Finally, we devise a specialized technique for the case where 3D Euclidean coordinates are provided for a number of control points. c ○ 1999 Academic Press

Epipolar image analysis is a robust method for 3D scene depth reconstruction that uses all available views of an image sequence simultaneously. It is restricted to horizontal, linear, and equidistant camera movements. In this paper, we present a concept for an extension of epipolar image analysis to other camera configurations like, e.g., circular movements. Instead of searching for straight lines in the epipolar image, we explicitly compute the trajectories of particular points through the image cube. Variation of the unknown depth leads to different curves. From those, the one corresponding to the true depth is selected by evaluating color constancy along the curve. In order to handle occlusions correctly an explicit occlusion compatible ordering scheme is derived for the case of circular movements. To compensate the influence of perspective projection we introduce a depth corrected epipolar image analysis algorithm which we call image cube trajectory analysis (ICT).

Invariant representations are frequently used in computer vision algorithms to eliminate the effect of an unknown transformation of the data. These representations, however, depend on the order in which the features are considered in the computations. We introduce the class of projective/permutation p -invariants which are insensitive to the labeling of the feature set. A general method to compute the p -invariant of a point set (or of its dual) in the n-dimensional projective space is given. The one-to-one mapping between n 3 points and the components of their p -invariant representation makes it possible to design correspondence algorithms with superior tolerance to positional errors. An algorithm for coplanar points in projective correspondence is described as an application, and its performance is investigated. The use of p -invariants as an indexing tool in object recognition systems may also be of interest.

In this paper, we clarify some questions raised by the application of three-dimensional invariants of sets of points and lines from their images in a pair of weakly calibrated stereo images. The second part of this paper shows through experimental results some applications of the projective invariants obtained from a pair of images using the Grassmann-Cayley algebra.