R. I. Hartley. Computation of the quadrifocal tensor. In Proc. 5th European Conf. on Computer Vision, Freiburg, volume I, pages 20--35, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....among the four views: c) 3 (d) 2 Case (a) leads to trivial equations (always zero) Cases (b) and (c) lead to bifocal and trifocal relations respectively, whereas case (d) gives quadrifocal relations. Quadrifocal tensors for perspective views are dealt with e.g. in [8, 16]. More than four views. With five views, the joint matrix is of size 15 9. Obviously, there is no minor of size 9 that contains at least two rows per image. Hence, there are no multi linear matching constraints between five views (or more) that can not be represented using bifocal, trifocal ....

R.I. Hartley, "Computation of the quadrifocal tensor," Proceedings of the European Conference on Computer Vision, Freiburg, Germany, pp. 20--35, Vol. I, 1998.

....structure in a least squares sense. This optimal method is of course computationally expensive and inherits the problems of initialisation and convergence from the minimisation techniques used. On the other hand, the algebraic methods do have fast, linear solutions for two [11] three [6] and four [3] views. Some other linear methods for multi frame motion estimation are [9] 8] where structure and motion are solved for simultaneously using a factorisation technique based on using rank constraints. However the estimation accuracy of such methods is significantly reduced by the presence of ....

Hartley, R., "Computation of the Quadrifocal Tensor", Proceedings of the 5th European Conference on Computer Vision. pp.20-35, 1998.

....the tensors are known, points and lines can be transferred between views without explicit 3D reconstruction. The tensors are overparametrized, however, a fact that somewhat hampers their estimation. Worse still, the non linear constraints existing between the tensor elements are only partly known [13, 6]. In the case of affine cameras, the corresponding multifocal relations have not been completely uncovered [17, 21, 14, 15] The compact representation by registered affine tensors obtained from inhomogeneous 2 3 projection matrices, and used successfully to develop relations between finite ....

R. I. Hartley. Computation of the quadrifocal tensor. In Proc. ECCV'98, LNCS 1406, pages 20--35, Springer.

....of points and lines in three views in a common framework. Its relevance to tracking is obvious since it may be used to transfer a fixation point in two views into the third by means of the relation in equation (3) below. To calculate the tensor we have adopted the approach advocated by Hartley in [7, 9, 8] to minimize algebraic error while imposing the constraints. Minimizing algebraic rather than geometric error allows faster calculation without much of a penalty on accuracy. These methods involve linear operations followed by a refinement of the solution using a non linear optimization over a ....

....In contrast with the trifocal tensor, the constraints on the the quadrifocal tensor take an unwieldy form and are not, in fact, completely understood. We therefore opt instead for an adaptation of the method for calculating the reduced projective quadrifocal tensor due to Heyden [12] and Hartley [8]. We will show that the affine tensor turns out to have considerable computational advantages over its projective counterpart, although a minimum of three point correspondences is still required. To achieve transfer we extract two trifocal tensors from the quadrifocal tensor: if the left and ....

R. I. Hartley. Computation of the quadrifocal tensor. In Proc. ECCV '98, LNCS 1406/1407, vol 1, pages 20--35.

....6 point correspondences between the four images [15] Furthermore, it is proven in [22] that the quadrifocal tensor Q has rank 9. In fact, the entries Q ijk of Q satisfy 51 non linear algebraic relations, in addition to the scale ambiguity, such that Q actually only has 29 degrees of freedom [19]. Due to the presence of noise in the images, the tensor Q computed linearly from pointcorrespondences between the images most certainly will not satify these non linear relations. Moreover, it is clear that one cannot ignore the 51 non linear relations in the 81 entries of Q and hope to get ....

....relations. Moreover, it is clear that one cannot ignore the 51 non linear relations in the 81 entries of Q and hope to get reasonable results. Imposing these relations in the computation of Q results in non linear criteria. A practical and accurate algorithm for the compution of Q is given in [19]. The quadrifocal constraints mentioned in Table 2 express geometric incidence relations between between the image features in the four views. As in the trifocal case, these quadrifocal constraints can also be converted to transfer equations, predicting the position of an image feature in one ....

R.I. Hartley, Computation of the quadrifocal tensor, pp. I.20 -- I.35, in : H. Burkhardt and B. Neumann (eds.), Computer Vision --- ECCV'98, Lecture Notes in Computer Science, Vol. 1406, Springer-Verlag, Berlin, 1998.

....was not acquired simultaneously as the multi image geometric constraints were not commonly applied. A precise elaboration of multiple view geometry led to a new view on stereovision. Partial contributions for two, three, and four images followed. The unified treatment of the subject is provided in [6, 7, 4]. The new theoretical tools open the door to processing of large number of images and, subsequently, to qualitatively better results than before. We present a method which applies trifocal relations to obtain projective reconstruction from n 4 views. The proposed method comes from the algorithm ....

....numerical search in the ideal optimization technique. At present, the methods based on the linearization are known only for bifocal, trifocal or quadrifocal constraints. Thus, they can be used for the projective reconstruction either from two or three or four images. For detailed description see [3, 4, 5, 12]. Naturally, it is tempting to use already existing effective algorithms for projective reconstruction from two, three or four views at first and then only join the obtained classes somehow. However, this method fails due to inaccuracies in measured data and in numerical computations. In this ....

R. I. Hartley. Computation of the quadrifocal tensor. In ECCV-98, volume I, pages 20--35. Springer Verlag, 1998.

....data. Two, three, or more images can be involved. The minimum number of point correspondences is 7. The line correspondences can be used too. The algorithms for a projective reconstruction from 2 or 3 images are wellknown, see [4, 12, 13, 5, 1] The example of algorithms for 4 views is given in [3], for n 3 views is in [9] ii) Recovery of motion parameters from projective structure using the intrinsic parameters the parallelism. 3 3 Conclusion It is possible to employ the parallelism of occluding contours of a cylinder to establish the correspondence even in an environment where ....

R. I. Hartley. Computation of the quadrifocal tensor. In Proc. 5th European Conf. Computer Vision, volume I, pages 20-35. Springer Verlag, 1998.

....are varying in the amount of involved views and image points and in the type of applied constraint. Most of them are based on so called multiple view tensors, see [7, 16] For two views it is a fundemental matrix [18] for three views it is a trifocal and for four views a quadrifocal tenosr, see [6, 4, 5]. These methods are applicable only for a concrete number of images (2,3, or 4) The estimates can be enunciated in linear form and unlimited amount of point correspondences can be involved theoretically. In some cases, when the minimal parameterization of tensors is required, a non linear ....

....be enunciated in linear form and unlimited amount of point correspondences can be involved theoretically. In some cases, when the minimal parameterization of tensors is required, a non linear constraint must be used (particularly for trifocal and quadrifocal case) This is the subject of papers [5, 14]. The other possibility are the methods based on projective six point invariant introduced by Quan v [13] Only six correspondences of three or more views can involve the incident estimate in these methods. The minimal parameterization of geometry is used. Example of such a method is described in ....

R. I. Hartley. Computation of the quadrifocal tensor. In Proc. 5th European Conf. Computer Vision, volume I, pages 20-35. Springer Verlag, 1998.

....the tensors are known, points and lines can be transferred between views without explicit 3D reconstruction. The tensors are overparametrized, however, a fact that somewhat hampers their estimation. Worse still, the non linear constraints existing between the tensor elements are only partly known [13, 6]. In the case of affine cameras, the corresponding multifocal relations have not been completely uncovered [17, 21, 14, 15] The compact representation by registered affine tensors obtained from inhomogeneous 2 3 projection matrices, and used successfully to develop relations between finite ....

R. I. Hartley. Computation of the quadrifocal tensor. In Proc. ECCV'98, LNCS 1406, pages 20--35, Springer.

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R. Hartley. Computation of the quadrifocal tensor. In Proc. ECCV, LNCS 1406, pages 20--35. Springer-Verlag, 1998.

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R. I. Hartley. Computation of the quadrifocal tensor. In Proc. 5th European Conf. on Computer Vision, Freiburg, volume I, pages 20--35, 1998.

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R. I. Hartley. Computation of the quadrifocal tensor. In Proc. ECCV '98, LNCS 1406/1407, vol 1, pages 20--35.

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R. I. Hartley. Computation of the quadrifocal tensor. In Proc. 5th European Conf. on Computer Vision, Freiburg, volume I, pages 20--35, 1998.

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R. I. Hartley. Computation of the quadrifocal tensor. In Proc. ECCV '98, LNCS 1406/1407, vol 1, pages 20--35.

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Hartley, R. I.: Computation of the quadrifocal tensor. ECCVA 98, 1998, pp. 20--35.

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R.I. Hartley. Computation of the quadrifocal tensor. In ECCV98, pages 20--35, 1998.

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Hartley, R., "Computation of the Quadrifocal Tensor ", Proceedings of the 5th European Conference on Computer Vision. pp.20-35, 1998.

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Hartley, R., "Computation of the Quadrifocal Tensor", In Proceedings of 5th European conference on Computer Vision, ECCV'98, Springer Verlag, pp 20-35.

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R.I. Hartley. Computation of the quadrifocal tensor. In Proceedings of the European Conference on Computer Vision, pages 20#35, Freiburg, Germany, June 1998.

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