A common framework for multiple view tensors
In this paper, we will introduce a common framework for the definition and operations on the different multiple view tensors. The novelty of the proposed formulation is to not fix any parameters of the camera matrices, but instead letting a group act on them and look at the different orbits. In this setting the multiple view geometry can be viewed as a four-dimensional linear manifold in IR3 m, where m denotes the number of images. The Grassman coordinates of this manifold are the epipoles, the components of the fundamental matrices, the components of the trifocal tensor and the components of the quadfocal tensor. All relations between these Grassman coordinates can be expressed using the so called quadratic p-relations, which are quadratic polynomials in the Grassman coordinates. Using this formulation it is evident that the multiple view geometry is described by four different kinds of projective invariants; the epipoles, the fundamental matrices, the trifocal tensors and the quadfocal tensors.
As an application of this formalism it will be shown how the multiple view geometry can be calculated from the fundamental matrix for two views, from the trifocal tensor for three views and from the quadfocal tensor for four views. As a byproduct, we show how to calculate the fundamental matrices from a trifocal tensor, as well as how to calculate the trifocal tensors from a quadfocal tensor. It is, furthermore, shown that, in general, n < 6 corresponding points in four images gives 16n − n(n − 1)/2 linearly independent constraints on the quadfocal tensor and that 6 corresponding points can be used to estimate the tensor components linearly. Finally, it is shown that the rank of the trifocal tensor is 4 and that the rank of the quadfocal tensor is 9.
KeywordsIEEE Computer Society Tensor Component Fundamental Matrix Multiple View Fundamental Matrice
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- 1.O. Faugeras and B. Mourrain. About the correspondence of points between n images. In IEEE Workshop on Representation of Visual Scenes, MIT, Boston, MA, pages 37–44. IEEE Computer Society Press, 1995.Google Scholar
- 2.O. Paugeras and B. Mourrain. On the geometry and algebra of the point and line correspondences between n images. In Proc. 5th Int. Conf. on Computer Vision, MIT, Boston, MA, pages 951–956. IEEE Computer Society Press, 1995.Google Scholar
- 3.O. Faugeras and T. Papadopoulo. Grassman-cayley algebra for modeling systems of cameras and the algebraic equations of the manifold of trifocal tensors. Technical Report 3225, Institut national de rechereche en informatique et en automatique, july 1997.Google Scholar
- 4.O. Faugeras and T. Papadopoulo. A nonlinear method for estimating the projective geometry of three views. In Proc. 6th Int. Conf. on Computer Vision, Mumbai, India, 1998. to appear.Google Scholar
- 5.B. Gelbaum. Linear Algebra: Basic, Practice and Theory. North-Holand, 1989.Google Scholar
- 6.R. Hartley. A linear method for reconstruction from points and lines. In Proc. 5th Int. Conf. on Computer Vision, MIT, Boston, MA, pages 882–887. IEEE Computer Society Press, 1995.Google Scholar
- 9.A. Heyden and K. åström. Algebraic varieties in multiple view geometry. In B. Buxton and R. Cipolla, editors, Proc. 4th European Conf. on Computer Vision, Cambridge, UK, volume 1065 of Lecture notes in Computer Science, pages 671–682. Springer-Verlag, 1996.Google Scholar
- 11.W. V. D. Hodge and D. Pedoe. Methods of Algebraic Geometry. Cambridge University Press, 1947.Google Scholar
- 12.Q.-T. Luong and T. Vieville. Canonic representations for the geometries of multiple projective views. In J.-O. Eklund, editor, Proc. 4th European Conf. on Computer Vision, Cambridge, UK, volume 800 of Lecture Notes in Computer Science, pages 589–599. Springer-Verlag, 1994.Google Scholar
- 13.A. Shashua. Trilinearity in visual recognition by alignment. In J.-O. Eklund, editor, Proc. 4th European Conf. on Computer Vision, Cambridge, UK, volume 800 of Lecture Notes in Computer Science, pages 479–484. Springer-Verlag, 1994.Google Scholar
- 14.A. Shashua and S. Maybank. Degenerate n point configurations of three views: Do critical surfaces exist? Technical Report TR 96-19, Hebrew University of Jerusalem, 1996.Google Scholar
- 15.A. Shashua and M. Werman. On the trilinear tensor of three perspective views and its associated tensor. In Proc. 5th Int. Conf. on Computer Vision, MIT, Boston, MA, pages 920–925. IEEE Computer Society Press, 1995.Google Scholar
- 16.M. E. Spetsakis and J. Aloimonos. A unified theory of structure from motion. In Proc. DARPA IU Workshop, pages 271–283, 1990.Google Scholar
- 17.B. Triggs. Matching constraints and the joint image. In Proc. 5th Int. Conf. on Computer Vision, MIT, Boston, MA, pages 338–343. IEEE Computer Society Press, 1995.Google Scholar
- 18.A. Zisserman. A users guide to the trifocal tensor. Draft, july 1996.Google Scholar