A Common Framework for Multiple View Tensors

In this paper, we will introduce a common framework for the deenition and operations on the diierent multiple view tensors. The novelty of the proposed formulation is to not x any parameters of the camera matrices, but instead letting a group act on them and look at the diierent orbits. In this setting the multiple view geometry can be viewed as a four-dimensional linear manifold in IR 3m , 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 diierent kinds of projective invari-ants; the epipoles, the fundamental matrices, the trifocal tensors and the quadfocal tensors. Another interesting result is that this approach gives a natural scale between the diierent multiple view tensors, although they are homogeneous as entities of their own. Related works are 1, 3, 17] among others. 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 byprod-uct, 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, also shown in 14], and that the rank of the quadfocal tensor is 9. We hope that this new formalism can give researchers in computer vision a deeper insight into the multiple view geometry problems and the relations between diierent kinds of multiple view tensors. The results concerning the quadfocal tensors can be used to solve for structure and motion from at least 6 corresponding points in 4 images, using linear methods.