A Common Framework for Multiple-View Tensors

Research output: Chapter in Book/Report/Conference proceedingPaper in conference proceeding


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 four-dimensional linear manifold in &Rscr;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 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 by-product, 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


Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Mathematics


  • computational geometry, computer vision, constraint theory, estimation theory, polynomial matrices, tensors
Original languageEnglish
Title of host publication[Host publication title missing]
ISBN (Print)3 540 64569 1
Publication statusPublished - 1998
Publication categoryResearch
EventComputer Vision - ECCV'98 5th European Conference on Computer Vision - Freiburg, Germany
Duration: 1998 Jun 21998 Jun 6

Publication series



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