Abstract
We 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 let a group act on them and look at the different orbits. In this setting the multiple view geometry can be viewed as a fourdimensional linear manifold in ℛ3m, 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 socalled quadratic prelations, 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 is 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 16nn(n1)/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
Details
Authors 

Organisations 

Research areas and keywords 
 computational geometry, computer vision, constraint theory, estimation theory, polynomial matrices, tensors

Original language  English 

Title of host publication  [Host publication title missing] 

Publisher  Springer 

Pages  319 

Volume  1 

ISBN (Print)  3 540 64569 1 

Publication status  Published  1998 

Publication category  Research 

Peerreviewed  No 

Event  Computer Vision  ECCV'98 5th European Conference on Computer Vision  Freiburg, Germany Duration: 1998 Jun 2 → 1998 Jun 6 

Conference  Computer Vision  ECCV'98 5th European Conference on Computer Vision 

Country  Germany 

City  Freiburg 

Period  1998/06/02 → 1998/06/06 
