A Common Framework for Multiple View Tensors

Abstract

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 IR^3 _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 16 n − 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.