P.H.S. Torr and A. Zisserman. Robust computation and parametrization of multiple view relations. In 6th International Conference on Computer Vision, pages 727--732, Bombay, India, January 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and camera matrices or equivalently the projection operator, see e.g. 4] The piecewise planar segmentation is done by iteratively selecting the most likely plane using a random sampling technique. We use a multiple hypotheses version of RANSAC [8] modified in two ways. First, as described in [18], we maximize the likelihood of the plane instead of its support (the support of a plane is the set of points lying on it, up to a certain tolerance) Secondly, we devise a segmentation scheme inspired from [1] that allows for overlapping data segmentation, which is important in the piecewise ....

....testing for degeneracy. The set of points # that geometrically lie on #, up to a predefined tolerance, is then computed imagebased [1, 15] and the operation repeated. The dominant plane is the one that maximizes ##. The number of iterations is computed as indicated in e.g. 8] As proposed in [18] in the case of image points, it is possible to use another cost function instead of ## such as the likelihood Pr(S # of the plane hypothesis. Computing this likelihood from the plane equation is described in 5.3. Once the dominant plane has been estimated, points can be classified whether ....

P.H.S. Torr and A. Zisserman. Robust computation and parametrization of multiple view relation. In Proceedings of the 6th International Conference on Computer Vision, Bombay, India, 1998. 7

....separating the external causes from the intrinsic properties in the appearance of each object. This task, roughly equivalent to perceptual constancies in human visual perception, is currently an active research area in computer vision. Several papers in the special issue address this topic [1, 3, 6, 8]. Robust statistical methods were first adopted in computer vision to improve the performance of feature extraction algorithms at the bottom level of the vision hierarchy. These methods tolerate (at various degrees) the presence of data points not obeying the assumed model. Such points are ....

....of the minimally acceptable set of data points, and the amount of noise allowed to corrupt the model) it can be better adapted to complex data analysis situations. Both RANSAC and LMedS suffer from the same sensitivity problems, and improving the numerical behavior of RANSAC is addressed in [8]. Papers in the Special Issue The eight papers selected for the Robust Statistical Techniques in Image Understanding special issue of Computer Vision and Image Understanding offer a representative sample for the state of the art approaches in robust computer vision. Several common trends can be ....

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P. Torr and A. Zisserman. Robust computation and parametrization of multiple view relations. In this issue.

.... drawn by our sampler. The variation in values suggests that the sampler explores its domain effectively. ignore a significant percentage of the data. In comparison, the sampler quickly accretes all points consistent with its model, and so gives significantly more stable measurements (cf [25], which uses maximum likelihood to identify correspondences) 4 Bayesian motion segmentation We describe a segmentation method for the case of a maximum of two objects; extending this to the case of an unknown number of objects is straightforward. We require a new vector of discrete variables ....

P. Torr and A. Zisserman. Robust computation and parametrization of multiple view relations. In Int. Conf. on Computer Vision, pages 485--491, 1998.

....are the start points of our sampler) These reconstructions are guaranteed to ignore large error points but will ignore a significant percentage of the data. In comparison, the sampler quickly accretes all points consistent with its model, and so gives significantly more stable measurements (cf Torr and Zisserman, 1998, which uses maximum likelihood to identify correspondences) Because the reconstruction is in some unknown scaled Euclidean frame, reconstructions are best compared by comparing angles subtended by corresponding triples of points, and by comparing distances between corresponding points scaled to ....

Torr, P. and Zisserman, A. 1998. Robust computation and parametrization of multiple view relations. In Int. Conf. on Computer Vision, pp. 485--491.

....from uncalibrated image sequences taken by a continuously moving camera. The complete procedure is briefly summarized below. 1. Extract and track feature image points in the sequence. 2. Estimate projective camera geometry and scene points, using fundamental matrices and trifocal tensors, [21, 6]. 2. Upgrade to a Euclidean reconstruction, cf. 16, 22] 3. Apply bundle adjustment, as described in the next sections. 4. Find dense correspondences across the views, triangulate and texture map in order to obtain a 3D model that can be used in a graphics environment, cf. 15, 13] 4. ....

P. Torr and A. Zisserman. Robust computation and parametrization of multiple view relations. In Int. Conf. Computer Vision, pages 727-- 732, Mumbai, India, 1998.

....method can recover the dominant affine motion robustly by using the Least Median of Squares (LMedS) method [11] and it can discard feature points that are regarded as outliers. Therefore a measurement matrix is stably decomposed into the motion and shape matrices. Although some previous works [17, 14] have also used robust statistics for recovering the epipolar geometry and the multiple projective view relation, they haven t described how to cope with the increasing computational cost as the number of frames increases. In this method, discarding outliers based on the LMedS criterion is ....

P. Torr and A. Zisserman. Robust computation and parametrization of multiple view relations. In Proc. ICCV98, 727--732, 1998.

.... streams of images, modeled as projective mappings from three dimensional space, and to efficient techniques for the reconstruction of the Euclidean structure of the three dimensional scene (cf. Spetsakis and Aloimonos 1990] Faugeras and Robert 1994] Luong and Faugeras 1996] Hartley 1997] [Torr and Zisserman 1998], Triggs 1998] Avidan and Shashua 1998] Csurka et al. 1998] This goes along with the development of matching techniques for image sequences which may be dense or sparse (cf. Horn and Schunck 1998] Fleet and Jepson 1990] Beardsley et al. 1996] Here Photogrammetry and Computer Vision ....

Torr, P.; Zisserman, A. (1998): Robust Computation and Parametrization of Multiple View Relations. In: in Proc. of ICCV, 1998.

....to Pt (t) drawn by our sampler. The variation in values suggests that the sampler explores its domain effectively. ignore a significant percentage of the data. In comparison, the sampler quickly accretes all points consistent with its model, and so gives significantly more stable measurements (cf [25], which uses maximum likelihood to identify correspondences) 4 Bayesian motion segmentation We describe a segmentation method for the case of a maximum of two objects; extending this to the case of an unknown number of objects is straightforward. We require a new vector of discrete variables t, ....

P. Torr and A. Zisserman. Robust computation and parametrization of multiple view relations. In Int. Conf. on Computer Vision, pages 485--491, 1998.

.... feature tracking segmentation algorithm is due to Torr [104] which we adapt to our ends. 4.4. 1 Estimation The problem of how best to estimate the fundamental matrix has been solved, in the sense of a maximum likelihood estimate assuming Gaussian image plane noise in the feature locations [44, 106], but the optimal estimate is computationally expensive. Accurate and efficient linear algorithms are available [44, 59] but in the very simple case of pure translation a locally optimal algorithm is available. The case of pure translation 4.4 Motion segmentation 56 1) Fit straight lines to ....

P. H. S. Torr and A. Zisserman. Robust computation and parametrization of multiple view relations. In ICCV6. IEEE, 1998.

....2 i is a minimum is the maximum likelihood estimate of the relation (fundamental matrix, or projectivity) Hartley and Sturm [12] show how e, x and x 0 may be found as the solution of a degree 6 polynomial. A computationally efficient first order approximation to these is given in Torr et al. [32, 34, 35]. The above derivation assumes that the errors are Gaussian, often however features are mismatched and the error on m is not Gaussian. Thus the error is modeled as a mixture model of Gaussian and uniform distribution: Pr(e) fl 1 p 2oe 2 exp( Gamma e 2 2oe 2 ) 1 Gamma fl) 1 ....

....inliers score nothing and each outlier scores a constant penalty. Thus the higher T 2 is the more solutions with equal values of C tending to poor estimation e.g. if T were sufficiently large then all solutions would have the same cost as all the matches would be inliers. In Torr and Zisserman [34] it was shown that at no extra cost this undesirable situation can be remedied. Rather than minimizing C a new cost function can be minimized C 2 = X i ae 2 Gamma e 2 i Delta (16) 8 TORR AND ZISSERMAN where the robust error term ae 2 is ae 2 (e 2 ) ae e 2 e 2 T 2 T 2 ....

P. H. S. Torr and A. Zisserman. Robust computation and parametrization of multiple view relations. In U Desai, editor, ICCV6, pages 727--732. Narosa Publishing House, 1998.

....is the maximum likelihood estimate of the relation (fundamental matrix, or projectivity) The optimally estimated correspondence m and its error l may be obtained as the solution of a high order polynomial equation. A computationally efficient first order approximation to these is given in Torr Zisserman (1998). If the type of relation R is unknown then we cannot use maximum likelihood estimation to decide the form of R as the most general model will always be most likely i.e. have lowest L. Fisher was aware of the limitations of maximum likelihood estimation and admits the possibility of a wider form ....

P. H. S. Torr and A Zisserman. Robust computation and parametrization of multiple view relations. OUEL Report, accepted to ICCV98, 1998.

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P.H.S. Torr and A. Zisserman. Robust computation and parametrization of multiple view relations. In 6th International Conference on Computer Vision, pages 727--732, Bombay, India, January 1998.

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P.H.S. Torr and A. Zisserman. Robust computation and parametrization of multiple view relations. In 6th International Conference on Computer Vision, pages 727--732, Bombay, India, January 1998.

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