O. Faugeras, "Stratification of three-dimensional vision: Projective, affine and metric representations," J. Opt. Soc. Amer., vol. 12, pp. 465--484, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....k are any two lines intersecting p 00 . Since the free indices are ; each in the range 1, 2, we have four trilinear equations (unique up to linear combinations) For more details, please refer to [9] 10] 4] In the sequel, it will become useful to represent the 3 fundamental matrix [6] [3] as embedded into a 3 3 tensor as follows [1] F li ; where F li is the Fundamental matrix and is the cross product tensor. This can be verified as follows: p i j r k z p 0l 0; 3 where s; r are any two lines coincident with p 0 . Finally, a triplet of images will ....

O.D. Faugeras, Stratification of Three-Dimensional Vision: Projective, Affine and Metric Representations, J. the Optical Soc. Amer., vol. 12, no. 3, pp. 465484, 1995.

....tion map is given simply by: X rx x z Figure 2: The orthographic camera. v = y ry Uy where (Ux, uy) t is an arbitrary central point in the camera s image plane, and rx and ry are the width and height respectively of the pixels. 3 Projective Reconstruction from Two Images Faugeras [1, 2] describes a method of reconstructing a scene up to a collineation. For his method, he requires two images of a scene and a method of corresponding points between images t. Once point correspondence have been established, he uses at least eight correspondences to find the fundamental matrix, and ....

.... When image coordinates are normalized so that the principal point is at the origin, the coordinates of epipole in the first camera are proportional to the the coordinates of epipole in the second camera: One way to find the locations of epipoles (and thus find f2) is to use a method presented in [2]. If the fundamental matrix F is known (which maps points on the first retinal plane to lines in the second) then et must be in the null space of F, i.e. Fet = 0. This is because et, Ct, and C2 must all be collinear. However, it is not necessary to employ this method. Pollefeys describes a way ....

Olivier Faugeras. Stratification of three-dimensional vision: projective, affine, and metric representations. Journal of the Optical Society of America A, 12(3), 1995.

.... of a rigid stereo rig [17] 83] This can be done for the stereo camera undergoing various types of rigid motion, among them ground plane motions [11] 3] with respective degenerate situations [14] The methods are closely related to the stratification hierarchy of geometric ambient spaces [21]. Among the special projective motions, pure translations and pure rotations with its axis in a general position actually a general ground plane motion is a non coordinate free instance of a pure rotation have so far been neglected. Finally, they have been studied in depth by the author ....

....inverse of a homography from the right onto a projection matrix transforms the cameras between the three ambient spaces. Applying these homographies from the left onto a reconstruction upgrades the latter to an ambient space that is on a higher level in the so called stratification hierarchy , [21]. On the projective level of this hierarchy, only coincidence and collinearity of geometric entities, i.e. points, planes, etc. are defined, as well as their cross ratios. On the a#ne level, this is upgraded to the notion of parallelism of lines and planes, and to length ratios of parallel ....

O. D. Faugeras. Stratification of three-dimensional vision: Projective, a#ne and metric representations. Journal of Optical Society of America, 12:465 -- 484, 1995.

....x . xn 2 =xn l = 0. The absolute conic 2 is the absolute quadric of p3. is a proper virtual conic in the ideal plane whose position uniquely defines the Euclidean structure of 3 space. The calibration of a camera is equivalent to determining the image o of 1 2, respectively, its dual o [7,11]. From the relation o KK T, the cali bration matrix K can uniquely be recovered by Cholesky decomposition [15] 3. Problem formulation We consider a sequence of n views, generally taken from different positions and with different orientations. The focal lengths for the views may all be ....

O. Faugeras, Stratification of three-dimensional vision: projective, affine and metric representations, Journal of the Optical Society of America A 12 (1995) 465-484.

....[11] In both cases the solution of a set of non linear equations is required. The stratified approach first upgrades the projective representation to an affine representation by solving for the homographies through the plane at infinity and then upgrades the affine representation to a metric one [3, 8]. It was pointed by several authors (e.g. 6, 7] that the first stage, the affine calibration, is the most challenging in stratified methods. By using a stereo rig, a stable solution for both affine and metric calibration can be computed linearly from two or more images from each camera ( 7] ....

O.D. Faugeras. Stratification of three-dimensional vision: projective, afflne and metric representations. Journal of the Optical Society of America, 12(3):465-484, 1995.

....space [20] A plane in space intersects rroo in a line which intersects the absolute conic in two points. These two points are the circular points of that plane. The conic relevant to calibration is the image of the absolute conic w. It is simply related to the camera calibra tion by w = K n K 1 [6]. The calibration matrix K may be computed from w by Cholesky decomposition [8] so determining w in an image is equivalent to knowing the camera internal parameters. The vanishing line of a space plane intersects w in two points. These points are the imaged circular points of the plane they are ....

O. D. Faugeras. Stratification of three-dimensional vision: projecrive, affine, and metric representation. J. Opt. Soc. Am., A12:465 484, 1995.

....Hat) 7 va 7 va and H2D : 02 . 02 are a 3D projective transformation and a 2D projective trans formation, respectively. There is a hierarchy of such transformations obtained by successive specialisation of the projective transformation to affine, similarity, and Euclidean transformations [42]. This can be understood by noting that any nonsin [ h2] such that gular 4 x 4 or 3 x 3 homogeneous matrix H = h h4 HlHtha = ha (2.15) where A t denotes the Moore Penrose inverse of A, can be decomposed as I I . purely projectlye I I affine iP transformation transformation (2.16) and, ....

O. D. Faugeras. Stratification of three-dimensional vision: Projective, affine and metric representations. J. Opt. Soc. America A, 12(3):465-484, March 1995.

....of the equations, and in a practical and robust implementation. To reduce the ambiguity of reconstruction from projective to affine it is necessary to identify the plane at infinity, 1 , and to reduce further to a metric ambiguity the absolute conic Omega 1 on 1 must also be identified [4, 13]. Both 1 and Omega 1 are fixed entities under Euclidean motions of 3 space. The key idea in this paper is that these fixed entities can be accessed via fixed entities (points, lines, conics) in the image. To determine the fixed image entities we utilise geometric relations between images that ....

Faugeras, O. Stratification of three-dimensional vision: projective, affine, and metric representation. J. Opt. Soc. Am., A12:465--484, 1995.

....by (1) tracking regions and color fiducial points at frame rate, and (2) representing virtual objects so that their projection can be computed as a linear combination of the projection of the fiducial points. The resulting affine virtual object representation is a non Euclidean representation [21, 23 25] in which the coordinates of vertices on a virtual object are relative to an affine reference frame defined by the fiducial points (Fig. 2) Affine object representations have been a topic of active research in computer vision in the context of 3D reconstruction [21, 24, 26] and recognition [27] ....

O. Faugeras, "Stratification of three-dimensional vision: projective, affine, and metric representations, " J. Opt. Soc. Am. A, vol. 12, no. 3, pp. 465--484, 1995.

....step. This eliminates the need for complex partial reconstruction and merging operations (Curless and Levoy, 1996; Turk and Levoy, 1994) in which partial 3D shape information is extracted from subsets of the photographs (Narayanan et al. 1998; Kanade et al. 1995; Zhao and Mohr, 1996; Seales and Faugeras, 1995), and where global consistency with the entire set of photographs is not guaranteed for the final shape. 5. We describe an efficient multi sweep implementation of the Space Carving Algorithm that enables recovery of photo realistic 3D models from multiple photographs of real scenes, and exploits ....

....a later stage in the reconstruction process, when tight bounds on scene structure are available and where these constraints are used only to choose among shapes within the class of photo consistent reconstructions. This approach is similar in spirit to stratification approaches of shape recovery (Faugeras, 1995; Koenderink and van Doorn, 1991) where 3D shape is first recovered modulo an equivalence class of reconstructions and is then refined within that class at subsequent stages of processing. The remainder of this paper is structured as follows. Section 2 analyzes the constraints that a set of ....

Faugeras, O.: 1995, `Stratification of three-dimensional vision: projective, affine, and metric representations'. J. Opt. Soc. Am. A 12(3), 465--484.

....using the correspondence between images. The need of calibration object limits the application of such algorithm. Thus, more relaxing calibration algorithms have been actively researched. One of these methods is to use a priori knowledge about the scene other than the absolute 3 D coordinates. In [2], a stratified approach to recover the Euclidean geometry of a scene is presented using a priori information about the scene. They recovered the scene structure sequentially from the projective to the affine and finally to the Euclidean. In this letter, we propose a calibration algorithm, which ....

FAUGERAS, O.: `Stratification of three-dimensional vision: projective, affine, and metric representations', J. Opt. Soc. Am. A, 1995, 12, pp. 465-484

....Luong and Maybank [5] propose to solve the Kruppa equations from points matches in 3 images. However, this requires non linear resolution methods. An alternative solution consists to first recover affine structure and then solve for the camera calibration using this. This stratified approach [4] can be applied to a single camera motion [8] or to a stereo rig in motion [3] and requires no knowledge about the observed scene. Affine calibration has already been studied by many authors and amounts to recovering the equation of the plane at infinity, or equivalently the infinite homographies ....

O. Faugeras. Stratification of three-dimensional vision: Projective, affine and metric representations. Journal of the Optical Society of America, 12:465--484, 1995.

.... number of image correspondences (a minimum of 7; more in the presence of noise) in a nondegenerate configuration are needed for its estimation, and iterative methods are required in order to minimize the non linear criteria needed to properly characterize good solutions in the presence of noise [24]. Thus, second order distortions of the perceived space arise when ffl sufficient information is not available to compute the epipolar geometry, e.g. if too few points have been brought into correspondence. ffl the available correspondences form a degenerate configuration. ffl the system does ....

....geometry, e.g. if too few points have been brought into correspondence. ffl the available correspondences form a degenerate configuration. ffl the system does not have or chooses not to invest the computational resources necessary for the estimation of the epipolar geometry. Faugeras [24] introduces a stratification of three dimensional vision methods along the lines of Klein s hierarchy of geometries. He proposes a three layered structure which consists of projective, affine, and Euclidean layers. The projective structure of the world can be obtained if the system is weakly ....

[Article contains additional citation context not shown here]

O. Faugeras. Stratification of three-dimensional vision: Projective, affine, and metric representations. J. Opt. Soc. Am. A, 12:465--484, 1995.

....Faugeras, Luong and Maybank [5] proposed solving the Kruppa equations from point correspondences in 3 images. However, this requires non linear solution methods. An alternative is to first recover affine structure and then solve for the camera calibration from this. Such a stratified approach [4] can be applied to a single camera motion [1, 7, 9, 12, 14] or to a stereo rig in motion [2, 10, 20] and requires no knowledge of the observed scene. The stratified approach applied to the autocalibration of a stereo rig involves the computation of projective transformations of 3 D space, that is ....

O. Faugeras. Stratification of three-dimensional vision: Projective, affine and metric representations. Journal of the Optical Society of America, 12:465--484, 1995.

....3D space and in the image correspond to each other via this 1D projective camera model [21] Other cases will be discussed later. In this paper, we first introduce the concept of self calibration of a 1D projective camera by analogy to that of a 2D projective camera which is a very active topic [17, 12, 7, 13, 1, 29, 20] since the pioneering work of [18] It turns out that the theory of self calibration of 1D camera is considerably simpler than the corresponding one in 2D. It is essentially determined in a unique way by a linear 1 algorithm using the trifocal tensor of 1D cameras. After establishing this result, ....

O. Faugeras. Stratification of three-dimensional vision: Projective, affine and metric representations. Journal of the Optical Society of America, 12:465--484, 1995.

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O. Faugeras, "Stratification of three-dimensional vision: Projective, affine and metric representations," J. Opt. Soc. Amer., vol. 12, pp. 465--484, 1995.

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O. D. Faugeras. Stratification of three-dimensional vision: projective, affine and metric representations. J. Opt. Soc. Am. A, 12:465--484, 1995.

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O.D. Faugeras. Stratification of three-dimensional vision: projective, affine and metric representation. J. Opt. Soc. Am., (A12):465--484, 1995.

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Faugeras, O. Stratification of three-dimensional vision: projective, affine, and metric representation. J. Opt. Soc. Am.,A12, 1995.

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O. Faugeras, Stratification of three-dimensional vision: Projective, affine and metric representations, Journal of Optical Soc. of America A 12 (3) (1995) 465--484.

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O. Faugeras, "Stratification of three-dimensional vision: projective, affine, and metric representations," J. Opt. Soc. Am. A, vol. 12, no. 3, pp. 465--484, 1995.

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Faugeras, O. 1995. Stratification of three-dimensional vision: projective, affine and metric representations. Journal of Optical Society of America, 12:465--484.

No context found.

Faugeras, O. Stratification of three-dimensional vision: projective, affine, and metric representation. J. Opt. Soc. Am., A12:465--484, 1995.

No context found.

O.D. Faugeras. Stratification of three-dimensional vision: projective, affine and metric representations. Journal of the Optical Society of America, 12(3):465--484, 1995.

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O.D. Faugeras. Stratification of three-dimensional vision: projective, affine and metric representations. Journal of the Optical Society of America, 12(3):465--484, 1995.

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