A Common Framework for MultipleView Tensors
(1998) Computer Vision  ECCV'98 5th European Conference on Computer Vision In [Host publication title missing] 1. p.319 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... (More)
 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 (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/787271
 author
 Heyden, Anders ^{LU}
 organization
 publishing date
 1998
 type
 Chapter in Book/Report/Conference proceeding
 publication status
 published
 subject
 keywords
 computational geometry, computer vision, constraint theory, estimation theory, polynomial matrices, tensors
 in
 [Host publication title missing]
 volume
 1
 pages
 3  19
 publisher
 Springer
 conference name
 Computer Vision  ECCV'98 5th European Conference on Computer Vision
 external identifiers

 scopus:84957616787
 ISBN
 3 540 64569 1
 language
 English
 LU publication?
 yes
 id
 29bad306cc4b44539be7b1838c70d762 (old id 787271)
 date added to LUP
 20080331 13:04:44
 date last changed
 20180107 10:33:06
@inproceedings{29bad306cc4b44539be7b1838c70d762, 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 &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 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}, author = {Heyden, Anders}, booktitle = {[Host publication title missing]}, isbn = {3 540 64569 1}, keyword = {computational geometry,computer vision,constraint theory,estimation theory,polynomial matrices,tensors}, language = {eng}, pages = {319}, publisher = {Springer}, title = {A Common Framework for MultipleView Tensors}, volume = {1}, year = {1998}, }