# Lund University Publications

## LUND UNIVERSITY LIBRARIES

### A Common Framework for Multiple-View Tensors

(1998) Computer Vision - ECCV'98 5th European Conference on Computer Vision 1. p.3-19
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 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... (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 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 (Less)
author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
computational geometry, computer vision, constraint theory, estimation theory, polynomial matrices, tensors
host publication
[Host publication title missing]
volume
1
pages
3 - 19
publisher
Springer
conference name
Computer Vision - ECCV'98 5th European Conference on Computer Vision
conference location
Freiburg, Germany
conference dates
1998-06-02 - 1998-06-06
external identifiers
• scopus:84957616787
ISBN
3 540 64569 1
language
English
LU publication?
yes
id
2016-04-04 10:23:56
date last changed
2023-11-15 22:12:56
```@inproceedings{29bad306-cc4b-4453-9be7-b1838c70d762,
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 four-dimensional linear manifold in &amp;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&lt;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}},
author       = {{Heyden, Anders}},
booktitle    = {{[Host publication title missing]}},
isbn         = {{3 540 64569 1}},
keywords     = {{computational geometry; computer vision; constraint theory; estimation theory; polynomial matrices; tensors}},
language     = {{eng}},
pages        = {{3--19}},
publisher    = {{Springer}},
title        = {{A Common Framework for Multiple-View Tensors}},
volume       = {{1}},
year         = {{1998}},
}

```