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...

This work is in the context of motion and stereo analysis. It presents a new uni ed representation which will be useful when dealing with multiple views in the case of uncalibrated cameras. Several levels of information might be considered, depending on the availability of information. Among other things, an algebraic description of the epipolar geometry of N views is introduced, as well as a framework for camera self-calibration, calibration updating, and structure from motion in an image sequence taken by a camera which is zooming and moving at the same time. We show how a special decomposition of a set of two or three general projection matrices, called canonical enables us to build geometric descriptions for a system of cameras which are invariant with respect to a given group of transformations. These representations are minimal and capture completely the properties of each level of description considered: Euclidean (in the context of calibration, and in the context of structure from motion, which we distinguish clearly), a ne, and projective, that we also relate to each other. In the last case, a new decomposition of the well-known fundamental matrix is obtained. Dependencies, which appear when three or more views are available, are studied in the context of the canonic decomposition, and new composition formulas are established. The theory is illustrated by tutorial examples with real images.

Leonard McMillan Jr. An Image-Based Approach to Three-Dimensional Computer Graphics (Under the direction of Gary Bishop) The conventional approach to three-dimensional computer graphics produces images from geometric scene descriptions by simulating the interaction of light with matter. My research explores an alternative approach that replaces the geometric scene description with perspective images and replaces the simulation process with data interpolation. I derive an image-warping equation that maps the visible points in a reference image to their correct positions in any desired view. This mapping from reference image to desired image is determined by the center-of-projection and pinhole-camera model of the two images and by a generalized disparity value associated with each point in the reference image. This generalized disparity value, which represents the structure of the scene, can be determined from point correspondences between multiple reference images. The image-warpi...

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...

: This paper discusses the problem of predicting image features in an image from image features in two other images and the epipolar geometry between the three images. We adopt the most general camera model of perpective projection and show that a point can be predicted in the third image as a bilinear function of its images in the first two cameras, that the tangents to three corresponding curves are related by a trilinear function, and that the curvature of a curve in the third image is a linear function of the curvatures at the corresponding points in the other two images. Our analysis relies heavily on the use of the fundamental matrix which has been recently introduced [7] and on the properties of a special plane which we call the trifocal plane. We thus answer completely the following question: given two views of an object, what would a third view look like? the question and its answer bear upon several areas of computer vision, stereo, motion analysis, and model-based object re...

: In this report, we address the problem of the prediction of new views of a given scene from existing weakly or fully calibrated views called reference views. Our method does not make use of a three-dimensional model of the scene, but of the existing relations between the images. The new views are represented in the reference views by a viewpoint and a retinal plane, i.e. by four points which can be chosen interactively. From this representation and from the constraints between the images, we derive an algorithm to predict the new views. We discuss the advantages of this method compared to the commonly used scheme : 3-D reconstruction-projection. We show some experimental results with synthetic and real data. Key-words: 3-D scene representation, multi-view stereo, image synthesis (R'esum'e : tsvp) This work was partially supported by DRET contract No 91-815/DRET/EAR and by the EEC under Esprit project 6448, Viva Unite de recherche INRIA Sophia-Antipolis 2004 route des Lucioles, BP 9...

A method for computing the 3D camera motion #the ego-motion# in a static scene is introduced, which is based on computing the 2D image motion of a single image region directly from image intensities. The computed image motion of this image region is used to register the images so that the detected image region appears stationary. The resulting displacement #eld for the entire scene between the registered frames is affected only by the 3D translation of the camera. After canceling the e#ects of the camera rotation by using such 2D image registration, the 3D camera translation is computed by #nding the focus-of-expansion in the translation-only set of registered frames. This step is followed by computing the camera rotation to complete the computation of the ego-motion.

We address the problem of reconstructing 3D space in a projective framework from two or more views, and the problem of artificially generating novel views of the scene from two given views (re-projection). We describe an invariance relation which provides a new description of structure, we call projective depth, which is captured by a single equation relating image point correspondences across two or more views and the homographies of two arbitrary virtual planes. The framework is based on knowledge of correspondence of features across views, is linear, extremely simple, and the computations of structure readily extends to over-determination using multiple views. Experimental results demonstrate a high degree of accuracy in both tasks - reconstruction and re-projection. Keywords---Visual Recognition, 3D Reconstruction from 2D Views, Projective Geometry, Algebraic and Geometric Invariants. I. Introduction The geometric relation between objects (or scenes) in the world and their imag...

We propose an affine framework for perspective views, captured by a single extremely simple equation based on a viewer-centered invariant we call relative affine structure. Via a number of corollaries of our main results we show that our framework unifies previous work -- including Euclidean, projective and affine -- in a natural and simple way, and introduces new, extremely simple, algorithms for the tasks of reconstruction from multiple views, recognition by alignment, and certain image coding applications.

We propose an affine framework for perspective views, captured by a single extremely simple equation based on a viewer-centered invariant we call relative affine structure. Via a number of corollaries of our main results we show that our framework unifies previous work --- including Euclidean, projective and affine --- in a natural and simple way. Finally, the main results were applied to a real image sequence for purpose of 3D reconstruction from 2D views. 1 Introduction The introduction of affine and projective tools into the field of computer vision have brought increased activity in the fields of structure from motion and recognition by alignment in the recent few years. The emerging realization is that non-metric information, although weaker than the information provided by depth maps and rigid camera geometries, is nonetheless useful in the sense that the framework may provide simpler algorithms, camera calibration is not required, more freedom in picture-taking is allowed --- ...