# Lund University Publications

## LUND UNIVERSITY LIBRARIES

### Algebraic Varieties in Multiple View Geometry

and Åström, Karl LU (1996) Proceedings of Fourth European Conference on Computer Vision. ECCV '96 In Computer Vision - ECCV '96. 4th Eurpean Conference on Computer Proceedings 2. p.671-682
Abstract
In this paper we will investigate the different algebraic varieties and ideals that can be generated from multiple view geometry with uncalibrated cameras. The natural descriptor, Vn, is the image of p3 in P2×p2×...×p2 under n different projections. However, we will show that Vn is not a variety. Another descriptor, the variety Vb, is generated by all bilinear forms between pairs of views and consists of all points in p2×p2×...×p2 where all bilinear forms vanish. Yet another descriptor, the variety, Vt, is the variety generated by all trilinear forms between triplets of views. We will show that when n=3, Vt is a reducible variety with one component corresponding to Vb and another corresponding to the trifocal plane. In ideal theoretic... (More)
In this paper we will investigate the different algebraic varieties and ideals that can be generated from multiple view geometry with uncalibrated cameras. The natural descriptor, Vn, is the image of p3 in P2×p2×...×p2 under n different projections. However, we will show that Vn is not a variety. Another descriptor, the variety Vb, is generated by all bilinear forms between pairs of views and consists of all points in p2×p2×...×p2 where all bilinear forms vanish. Yet another descriptor, the variety, Vt, is the variety generated by all trilinear forms between triplets of views. We will show that when n=3, Vt is a reducible variety with one component corresponding to Vb and another corresponding to the trifocal plane. In ideal theoretic terms this is called a primary decomposition. This settles the discussion on the connection between the bilinearities and the trilinearities. Furthermore, we will show that when n=3, Vt is generated by the three bilinearities and one trilinearity and when n&ges;4, Vt is generated by the (2n) bilinearities. This shows that four images is the generic case in the algebraic setting, because Vt can be generated by just bilinearities (Less)
author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
algebra, computer vision, image sequences, polynomials, multiple view geometry, uncalibrated cameras, natural descriptor, bilinear forms, trilinear forms, triplets of views, trifocal plane, ideal theoretic terms, primary decomposition, algebraic setting, polynomial equations
in
Computer Vision - ECCV '96. 4th Eurpean Conference on Computer Proceedings
editor
and Buxton, B.
volume
2
pages
671 - 682
publisher
Springer
conference name
Proceedings of Fourth European Conference on Computer Vision. ECCV '96
external identifiers
• Scopus:84957874673
ISBN
3 540 61123 1
language
English
LU publication?
yes
id
e36823ac-2ffa-45e9-835b-22b0236d8054 (old id 787283)
2008-09-16 14:04:38
date last changed
2016-10-13 04:47:18
```@misc{e36823ac-2ffa-45e9-835b-22b0236d8054,
abstract     = {In this paper we will investigate the different algebraic varieties and ideals that can be generated from multiple view geometry with uncalibrated cameras. The natural descriptor, Vn, is the image of p3 in P2×p2×...×p2 under n different projections. However, we will show that Vn is not a variety. Another descriptor, the variety Vb, is generated by all bilinear forms between pairs of views and consists of all points in p2×p2×...×p2 where all bilinear forms vanish. Yet another descriptor, the variety, Vt, is the variety generated by all trilinear forms between triplets of views. We will show that when n=3, Vt is a reducible variety with one component corresponding to Vb and another corresponding to the trifocal plane. In ideal theoretic terms this is called a primary decomposition. This settles the discussion on the connection between the bilinearities and the trilinearities. Furthermore, we will show that when n=3, Vt is generated by the three bilinearities and one trilinearity and when n&amp;ges;4, Vt is generated by the (2n) bilinearities. This shows that four images is the generic case in the algebraic setting, because Vt can be generated by just bilinearities},
author       = {Heyden, Anders and Åström, Karl},
editor       = {Cipolla, R. and Buxton, B.},
isbn         = {3 540 61123 1},
keyword      = {algebra,computer vision,image sequences,polynomials,multiple view geometry,uncalibrated cameras,natural descriptor,bilinear forms,trilinear forms,triplets of views,trifocal plane,ideal theoretic terms,primary decomposition,algebraic setting,polynomial equations},
language     = {eng},
pages        = {671--682},
publisher    = {ARRAY(0x8792738)},
series       = {Computer Vision - ECCV '96. 4th Eurpean Conference on Computer Proceedings},
title        = {Algebraic Varieties in Multiple View Geometry},
volume       = {2},
year         = {1996},
}

```