Results 11  20
of
113
Relative Affine Structure: Canonical Model for 3D from 2D Geometry and Applications
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1996
"... We propose an affine framework for perspective views, captured by a single extremely simple equation based on a viewercentered 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, projec ..."
Abstract

Cited by 57 (9 self)
 Add to MetaCart
We propose an affine framework for perspective views, captured by a single extremely simple equation based on a viewercentered 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.
Robust structure from motion using motion parallax
 Proc. 2nd European Conference on Computer Vision
, 1992
"... We present an efficient and geometrically intuitive algorithm to reliably interpret the image velocities of moving objects in 3D. It is well known that under weak perspective the image motion of points on a plane can be characterised by an affine transformation. We show that the relative image motio ..."
Abstract

Cited by 55 (10 self)
 Add to MetaCart
We present an efficient and geometrically intuitive algorithm to reliably interpret the image velocities of moving objects in 3D. It is well known that under weak perspective the image motion of points on a plane can be characterised by an affine transformation. We show that the relative image motion of a nearby noncoplanar point and its projection on the plane is equivalent to motion parallax and because it is independent of viewer rotations it is a reliable geometric cue to 3D shape and viewer/object motion In particular we show how to interpret the motion parallax vector of noncoplanar points (and contours) and the curl, divergence and deformation components of the affine transformation (defined by the three points or a closedcontour of the plane) in order to recover the projection of the axis of rotation of a moving object; the change in relative position of the object; the rotation about the ray; the tilt of the surface and a one parameter family of solutions for the slant as a function of the magnitude of the rotation of the object. The latter is a manifestation of the basâ€“relief ambiguity. These measurements, although representing an incomplete solution to structure from motion, are the only subset of structure and motion parameters which can be reliably extracted from two views when perspective effects are small. We present a realtime example in which the 3D visual interpretation of hand gestures or a handheld object is used as part of a manmachine interface. This is an alternative to the Polhemus coil instrumented Dataglove commonly used in sensing manual gestures. 1
Stratification of 3D vision: Projective, affine, and metric representations
"... In this article we provide a conceptual framework in which to think of the relationships between the threedimensional structure of the physical space and the geometric properties of a set of cameras which provide pictures from which measurements can be made. We usually think of the physical space a ..."
Abstract

Cited by 49 (5 self)
 Add to MetaCart
In this article we provide a conceptual framework in which to think of the relationships between the threedimensional structure of the physical space and the geometric properties of a set of cameras which provide pictures from which measurements can be made. We usually think of the physical space as being embedded in a threedimensional euclidean space where measurements of lengths and angles do make sense. It turns out that for artificial systems, such as robots, this is not a mandatory viewpoint and that it is sometimes sufficient to think of the physical space as being embedded in an affine or even projective space. The question then arises of how to relate these models to image measurements and to geometric properties of sets of cameras. We show that in the case of two cameras, a stereo rig, the projective structure of the world can be recovered as soon as the epipolar geometry of the stereo rig is known and that this geometry is summarized by a single 3 3 matrix, which we called the fundamental matrix [1, 2]. The affine structure can then be recovered if we add to this information a projective transformation between the two images which is induced by the plane at infinity. Finally, the euclidean structure (up to a similitude) can be recovered if we add to these two elements the knowledge of two conics (one for each camera) which are the images of the absolute conic, a circle of radius p;1 in the plane at in nity. In all three cases we showhowthe threedimensional information can be recovered directly from the images without explicitely reconstructing the scene structure. This defines a natural hierarchy of geometric structures, a set of three strata, that we overlay onthephysical world and which we show to be recoverable by simple procedures relying on two items, the physical space itself together with possibly, but not necessarily, some a priori information about it, and some voluntary motions of the set of cameras.
Invariants of Six Points and Projective Reconstruction from Three Uncalibrated Images
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1995
"... There are three projective invariants of a set of six points in general position in space. It is well known that these invariants cannot be recovered from one image, however an invariant relationship does exist between space invariants and image invariants. This invariant relationship is first deriv ..."
Abstract

Cited by 47 (16 self)
 Add to MetaCart
There are three projective invariants of a set of six points in general position in space. It is well known that these invariants cannot be recovered from one image, however an invariant relationship does exist between space invariants and image invariants. This invariant relationship is first derived for a single image. Then this invariant relationship is used to derive the space invariants, when multiple images are available. This paper establishes that the minimum number of images for computing these invariants is three, and the computation of invariants of six points from three images can have as many as three solutions. Algorithms are presented for computing these invariants in closed form. The accuracy and stability with respect to image noise, selection of the triplets of images and distance between viewing positions are studied both through real and simulated images. Applications of these invariants are also presented. Both the results of Faugeras [1] and Hartley et al. [2] for...
Computing MatchedEpipolar Projections
"... This paper gives a new method for image rectification, the process of resampling pairs of stereo images taken from widely differing viewpoints in order to produce a pair of "matched epipolar projections". These are projections in which the epipolar lines run parallel with the xaxis and consequently ..."
Abstract

Cited by 41 (6 self)
 Add to MetaCart
This paper gives a new method for image rectification, the process of resampling pairs of stereo images taken from widely differing viewpoints in order to produce a pair of "matched epipolar projections". These are projections in which the epipolar lines run parallel with the xaxis and consequently, disparities between the images are in the xdirection only. The method is based on an examination of the essential matrix of LonguetHiggins which describes the epipolar geometry of the image pair. The approach taken is consistent with that recently advocated strongly by Faugeras ([1]) of av oiding camera calibration. The paper uses methods of projective geometry to define a matrix called the "epipolar transformation matrix" used to determine a pair of 2D projective transforms to be applied to the two images in order to match the epipolar lines. The advantages include the simplicity of the 2D projective transformation which allows very fast resampling as well as subsequent simplification in the identification of matched points and scene reconstruction.
Qualitative Egomotion
, 1995
"... Due to the aperture problem, the only motion measurement in images, whose computation does not require any assumptions about the scene in view, is normal flowthe projection of image motion on the gradient direction. In this paper we show how a monocular observer can estimate its 3D motion relativ ..."
Abstract

Cited by 36 (15 self)
 Add to MetaCart
Due to the aperture problem, the only motion measurement in images, whose computation does not require any assumptions about the scene in view, is normal flowthe projection of image motion on the gradient direction. In this paper we show how a monocular observer can estimate its 3D motion relative to the scene by using normal flow measurements in a global and qualitative way. The problem is addressed through a search technique. By checking constraints imposed by 3D motion parameters on the normal flow field, the possible space of solutions is gradually reduced. In the four modules that comprise the solution, constraints of increasing restriction are considered, culminating in testing every single normal flow value for its consistency with a set of motion parameters. The fact that motion is rigid defines geometric relations between certain values of the normal flow field. The selected values form patterns in the image plane that are dependent on only some of the motion parameters. These patterns, which are determined by the signs of the normal flow values, are searched for in order to find the axes of translation and rotation. The third rotational component is computed from normal flow vectors that are only due to rotational motion. Finally, by looking at the complete data set, all solutions that cannot give rise to the given normal flow field are discarded from the solution space.
Uncalibrated Euclidean reconstruction: a review
, 1999
"... This paper provides a review on techniques for computing a threedimensional model of a scene from a single moving camera, with unconstrained motion and unknown parameters. In the classical approach, called autocalibration or selfcalibration, camera motion and parameters are recovered first, using ..."
Abstract

Cited by 35 (8 self)
 Add to MetaCart
This paper provides a review on techniques for computing a threedimensional model of a scene from a single moving camera, with unconstrained motion and unknown parameters. In the classical approach, called autocalibration or selfcalibration, camera motion and parameters are recovered first, using rigidity; then structure is easily computed. Recently, new methods based on the idea of stratification have been proposed. They upgrade the projective structure, achievable from correspondences only, to the Euclidean structure, by exploiting all the available constraints.
A Common Framework for Kinetic Depth, Reconstruction and Motion for Deformable Objects
 ECCV'94, Lecture notes in Computer Science, Vol
, 1994
"... . In this paper, problems related to depth, reconstruction and motion from a pair of projective images are studied under weak assumptions. Only relative information within each image is used, nothing about their interrelations or about camera calibration. Objects in the scene may be deformed between ..."
Abstract

Cited by 34 (2 self)
 Add to MetaCart
. In this paper, problems related to depth, reconstruction and motion from a pair of projective images are studied under weak assumptions. Only relative information within each image is used, nothing about their interrelations or about camera calibration. Objects in the scene may be deformed between the imaging instants, provided that the deformations can be described locally by affine transformations. It is shown how the problems can be treated by a common method, based on a novel interpretation of a theorem in projective geometry of M. Chasles, and the notion of "affine shape". No epipolar geometry is used. The method also enables the computation of the "depth flow", i.e. a relative velocity in the direction of the ray of sight. Keywords: Depth, shape, reconstruction, motion, invariants. 1 Introduction Central problems in computer vision are concerned with reconstruction and recovery of motion from image pairs. A number of algorithms exist, in general based on iterative numerical t...
Relative orientation revisited
 Journal of the Optical Society of America A
, 1991
"... Relative Orientation is the recovery of the position and orientation of one imaging system relative to another from correspondences between five or more ray pairs. It is one of four core problems in photogrammetry and is of central importance in binocular stereo, as well as in long range motion visi ..."
Abstract

Cited by 34 (1 self)
 Add to MetaCart
Relative Orientation is the recovery of the position and orientation of one imaging system relative to another from correspondences between five or more ray pairs. It is one of four core problems in photogrammetry and is of central importance in binocular stereo, as well as in long range motion vision. While five ray correspondences are sufficient to yield a finite number of solutions, more than five correspondences are used in practice to ensure an accurate solution using least squares methods. Most iterative schemes for minimizing the sum of squares of weighted errors require a good guess as a starting value. The author has previously published a method that finds the best solution without requiring an initial guess. In this paper an even simpler method is presented that utilizes the representation of rotations by unit quaternions. 1.
See also: ``Relative Orientation,''
{\it International Journal of Computer Vision},
Vol.~4, No.~1, pp.~5978, January 1990.
Geometry and Algebra of Multiple Projective Transformations
, 1995
"... In this thesis several dioeerent cases of reconstruction of 3D objects from a number of 2D images, obtained by projective transformations, are considered. Firstly, the case where the images are taken by uncalibrated cameras, making it possible to reconstruct the object up to projective transformatio ..."
Abstract

Cited by 32 (8 self)
 Add to MetaCart
In this thesis several dioeerent cases of reconstruction of 3D objects from a number of 2D images, obtained by projective transformations, are considered. Firstly, the case where the images are taken by uncalibrated cameras, making it possible to reconstruct the object up to projective transformations, is described. The minimal cases of two images of seven points and three images of six points are solved, giving threefold solutions in both cases. Then linear methods for the cases where more points or more images are available are given, using multilinear constraints, based on a canonical representation of the multiple view geometry. The case of a continuous stream of images is also treated, giving multilinear constraints on the image coordinates and their derivatives. Secondly, the algebraic properties of the multilinear functions and the ideals generated by them are investigated. The main result is that the ideal generated by the bilinearities for three views have a primary decomposit...