T. Vieville, C. Zeller, and L. Robert. Using collineations to compute motion and structure in an uncalibrated image sequence. International Journal of Computer Vision, 20(3):213--242, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....camera motion, linear subspace constraints apply also to the homographies of a single plane across multiple views (Section 6) Different video related applications can benefit from such multiview constraints. For example, many algorithms based on planar homographies (e.g. 18] 21] 11] [28], 17] or on planar homologies (e.g. 23] 4] rely on accurate precomputation of these homographies (or homologies) However, the image region corresponding to a planar surface may be small. In such cases, the homography estimation tends to be highly inaccurate [25] i.e. when applied to ....

T. Vieville, C. Zeller, and L. Robert, "Using Collineations to Compute Motion and Structure in an Uncalibrated Image Sequence," Int'l J. Computer Vision, vol. 20, pp. 213-242, 1996.

.... over multiple ( 4) views (Section 3) This constraint is then extended to a constraint on homographies of multiple planes across multiple views (Section 4) Algorithms for 3D analysis whicharebasedonthe use of multiple homographies (in scenes with multiple planes)have been suggested (e.g. [8, 9,13,7]) Most of these algorithms rely on accurate precomputation of the homographies. However, in scenes containing multiple planes, the image region corresponding to each plane may be small. In such cases, the homography estimation tends to be highly inaccurate [11] i.e, when applied to small image ....

T. Vieville, C.Zeller, and L.Robert. Using collineations to compute motion and structure in an uncalibrated image sequence. International Journal of Computer Vision, 20:213-- 242, 1996.

....is aligned with the left image (a) Note that all points on the tabletop are aligned while those above the table (such as the top surface of the book) are displaced by a residual parallax. homography H , and a set of residual motion vectors, known as a planar parallax field. It has been shown [4, 13] that for a plane of equation n T X = d in the coordinate system of the first camera, H is of the form H = C[Rd tn T ]C ;1 where C is the camera calibration matrix (assumed to be the same for each camera) and R is the rotation and t the translation between the cameras. Thus H encapsulates ....

....epipole for this image pair has been independently computed to be at position (3187, 31) i.e. far to the right and just above the top of the image. rotation, image warping and parallax have been used to simplify egomotion estimation, and to detect independently moving objects in a scene [8] In [4, 13] planar parallax is linked to projective depth, which with camera calibration information can be related to euclidean structure. 3 Model refinement algorithm 3.1 Overview This section details each step of our algorithm for applying planar parallax to model refinement from multiple uncalibrated ....

T. Vieville, C. Zeller, and L. Robert. Using collineations to compute motion and structure in an uncalibrated image sequence. International Journal of Computer Vision, 20(3):213--242, 1996. BMVC99 82

....and hence the epipolar geometry of the camera configuration. However in practice for nearly planar scenes the parallax vectors are small, and estimation of their common point of intersection is unstable. However the planar parallax decomposition has been applied to many problems of computer vision [26, 20, 32]; in particular it complements more general algorithms for which 2D structure is a degenerate case, and greatly simplifies the correspondence problem. We now review some systems which use simpler approximations to scene structure such as the planar parallax decomposition to reduce the complexity ....

T. Vieville, C. Zeller, and L. Robert. Using collineations to compute motion and structure in an uncalibrated image sequence. International Journal of Computer Vision, 20(3):213--242, 1996.

.... be also estimated from an homography matrix related to a reference plane on the target [12, 35] The homography matrix can be estimated jointly to the epipole using, for example, the algorithms presented in [3, 16] or after the epipole has been found [29] if more than two views are available, see [19, 31]) It will be shown in this paper that the motion parameters estimation is more robust from an homography matrix than from the fundamental matrix, especially when the epipole is not dened in the image (for example, if the motion is a pure rotation or if the target is planar [22] Since the ....

T. Vieville, C. Zeller, and L. Robert. Using collineations to compute motion and structure in an uncalibrated image sequence. IJCV, 20(3):213 242, 1996.

....(given by a) allows to compute the Euclidean structure up to an affine transformation, that is an affine reconstruction. From affine to Euclidean. Another useful observation is, if H # is known and the intrinsic parameters are constant, the intrinsic parameter matrix A can easily be computed [8,14,18,48]. Let us consider the case of two cameras. If A # # A# then H # is exactly known (with the right scale) since det#H # ##det#ARA #1 ##1# #32# From Eq. 17) we obtain R # A ##1 H # A# and, since RR T # I# it is easy to obtain: H # KH T # # K #33# where K # AA T is the ....

....easy to obtain: H # KH T # # K #33# where K # AA T is the Kruppa coefficients matrix. As Eq. 33) is an equality between 3 3 symmetric matrices, we obtain a linear system of six equations in the five unknown k 1 , k 2 , k 3 , k 4 , k 5 . In fact, only four equations are independent [14,48], hence at least three views (with constant intrinsic parameters) are required to obtain an over constrained linear system, which can be easily solved with a linear least squares technique. Note that two views would be sufficient under the usual assumption that the image reference frame is ....

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T. Vie ville, C. Zeller, L. Robert, Using collineations to compute motion and structure in an uncalibrated image sequence, International Journal of Computer Vision 20 (3) (1996) 213--242.

.... over multiple ( 4) views (Section 3) This constraint is then extended to a constraint on homographies of multiple planes across multiple views (Section 4) Algorithms for 3D analysis which are based on the use of multiple homographies (in scenes with multiple planes) have been suggested (e.g. [8, 9, 13, 7]) Most of these algorithms rely on accurate precomputation of the homographies. However, in scenes containing multiple planes, the image region corresponding to each plane may be small. In such cases, the homography estimation tends to be highly inaccurate [11] i.e, when applied to small image ....

T. Vieville, C.Zeller, and L.Robert. Using collineations to compute motion and structure in an uncalibrated image sequence. International Journal of Computer Vision, 20:213-- 242, 1996.

....the layer sprites. For the experiments presented in this paper, we set wE , i.e. we reconstructed the sprites in the coordinate system of the first camera. Using the computed homographies, we found the best plane estimate for each layer using a Euclidean structure from motion algorithm [33]. The results of applying these steps to the MPEG flower garden sequence are shown in Figure 3. Figures 3(a) and (b) show the first and last image in the subsequence we used (the first nine even images) Figure 3(c) shows the initial pixel labeling into seven layers. Figures 3(d) and (e) show the ....

T. Vieville, C. Zeller, and L. Robert. Using collineations to compute motion and structure in an uncalibrated image sequence. IJCV, 20(3):213--242, 1996.

....general situation. For examples of other work dealing with the ego motion of a calibrated camera see [10, 11] A contrasting approach is given by Beardsley et al. 5] involving the computation of projective and affine (rather than Euclidean) structure from motion. Other related papers include [1 4, 15, 19, 23, 26, 27]. 2 Differential epipolar equation and a cubic constraint Consider a camera with an associated coordinate frame such that the origin of the frame coincides with the camera s optical centre, two basis vectors span the focal plane, and the other basis vector passes through the optical axis. Suppose ....

T. Vieville, C. Zeller, and L. Robert, Using collineations to compute motion and structure in an uncalibrated image sequence, International Journal of Computer Vision 20 (1996), no. 3, 213--242.

.... 1 l u il n T t l d = 0, or w 1 il = n T R 1 l V 1 l u il (d n T t l ) d 1 l n T R 1 l V 1 l u il , where d l = d n T t l is the distance of camera center l (t l ) to the plane (n, d) Substituting w 1 il into (3) and multiplying through by d l , we obtain [VZR96] u ik # (H # kl d 1 l e kl n T R 1 l V 1 l )u il . 4) Letting n l = V T l R l n be the plane normal in the lth camera s (scaled) coordinate system, we see that the homography induced by the plane can be written as H kl # H # kl d 1 l e kl n T l (5) i.e. it is very similar in ....

T. Vieville, C. Zeller, and L. Robert. Using collineations to compute motion and structure in an uncalibrated image sequence. IJCV, 20(3):213--242, 1996.

....as defined in Equation (7) For the experiments presented in this paper, we set Q l = P 1 , i.e. we reconstruct the sprites in the coordinate system of the first camera. Using these homographies, we find the best plane estimate for each layer using a Euclidean structure from motion algorithm [40]. The results of applying these steps to the MPEG flower garden sequence are shown in Figure 8. Figures 8(a) and (b) show the first and last image in the subsequence we used (the first seven even images) Figure 8(c) shows the initial pixel labeling into seven layers. Figures 8(d) and (e) show ....

T. Vieville, C. Zeller, and L. Robert. Using collineations to compute motion and structure in an uncalibrated image sequence. International Journal of Computer Vision, 20(3):213--242, 1996.

....the layer sprites. For the experiments presented in this paper, we set Q l = P 1 , i.e. we reconstructed the sprites in the coordinate system of the first camera. Using the computed homographies, we found the best plane estimate for each layer using a Euclidean structure from motion algorithm [33]. The results of applying these steps to the MPEG flower garden sequence are shown in Figure 3. Figures 3(a) and (b) show the first and last image in the subsequence we used (the first nine even images) Figure 3(c) shows the initial pixel labeling into seven layers. Figures 3(d) and (e) show the ....

T. Vieville, C. Zeller, and L. Robert. Using collineations to compute motion and structure in an uncalibrated image sequence. IJCV, 20(3):213--242, 1996.

....to estimate a relevant reference plane [9, 12] The planar structure assumption greatly simplifies the problem, since the number of degrees of freedom no longer depends on the image size. Given point and or line correspondences, the discrete time motion problem has been solved by several authors [6, 7, 11, 13, 16]. For instantaneous representations, excellent work has been done in using multiscale estimation techniques to couple the flow and motion estimation problems to provide a direct method This document is GRASP Laboratory s Technical Report #410. Research supported by ARO grant DAAH04 96 1 0007, ....

T. Vieville, C. Zeller, and L. Robert. Using Collineations to Compute Motion and Structure in an Uncalibrated Image Sequence. International Journal of Computer Vision, 1995.

....techniques typically make the unrealistic assumption that the flow field is smooth. In many situations, a more plausible assumption is that of a rigid world. Given point and or line correspondences, the discrete time rigid motion problem has been studied and solved by a number of authors (e.g. [6, 7, 11, 14, 17]) For instantaneous representations, multi scale estimation techniques have been used to couple the flow and motion estimation problems to provide a direct method for planar surfaces [4, 8] These methods use the multi scale technique to capture large motions while significantly constraining the ....

T. Vieville, C. Zeller, and L. Robert. Using Collineations to Compute Motion and Structure in an Uncalibrated Image Sequence. IJCV, 1995. This article was processed using the L A T E X macro package with LLNCS style.

.... be also estimated from an homography matrix related to a reference plane on the target [12, 35] The homography matrix can be estimated jointly to the epipole using, for example, the algorithms presented in [3, 16] or after the epipole has been found [29] if more than two views are available, see [19, 31]) It will be shown in this paper that the motion parameters estimation is more robust from an homography matrix than from the fundamental matrix, especially when the epipole is not defined in the image (for example, if the motion is a pure rotation or if the target is planar [22] Since the ....

T. Vieville, C. Zeller, and L. Robert. Using collineations to compute motion and structure in an uncalibrated image sequence. IJCV, 20(3):213--242, 1996.

.... it is possible, for a rather large class of real scenes, to infer the collineation of the plane at infinity, i.e. the Q matrix, considering points at the horizon , that is with a negligible depth [37] This segmentation process, although not rigorous is quite efficient and is discussed elsewhere [40]. Structure from motion using the Qs representation. In order to increase our understanding of the potentialities of this representation, let us explicit a very useful, but oversimple result. The formula is obtained from equation (2) very easily : Proposition 1 Using the Qs representation, the ....

....by two normal correspondences between two points of the line. We can summarize this result as follow : Proposition 3 The retinal motion of a line is entirely determined by the normal displacement of two points of this line. This result is very important for implementations as already stress in [40] for other estimations, and will be used in the implementation proposed in this paper. 3.2 Motion of lines in three views Let us now study the problem of computing the motion parameters S ij , s ij and q ijk , considering points and or lines correspondences. In this section we are going to ....

T. Vi'eville, C. Zeller, and L. Robert. Using collineations to compute motion and structure in an uncalibrated image sequence, 1994. Accepted after review.

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T. Vieville, C. Zeller, and L. Robert. Using collineations to compute motion and structure in an uncalibrated image sequence. International Journal of Computer Vision, 20(3):213--242, 1995.

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T. Vieville, C. Zeller, and L. Robert. Using Collineations to Compute Motion and Structure in an Uncalibrated Image Sequence. IJCV, 1995. This article was processed using the L A T E X macro package with LLNCS style. 8

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