# Lund University Publications

## LUND UNIVERSITY LIBRARIES

### Algebraic Properties of Multilinear Constraints

and Åström, Karl LU (1997) In Mathematical Methods in the Applied Sciences 20(13). p.1135-1162
Abstract
In this paper the different algebraic varieties that can be generated from multiple view geometry with uncalibrated cameras have been investigated. The natural descriptor, V-n, to work with is the image of P-3 in P-2 x P-2 x ... x P-2 under a corresponding product of projections, (A(1) x A(2) x ... x A(m)).

Another descriptor, the variety V-b, is the one generated by all bilinear forms between pairs of views, which consists of all points in P-2 x P-2 x ... x P-2 where all bilinear forms vanish. Yet another descriptor, the variety V-t, is the variety generated by all trilinear forms between triplets of views. It has been shown that when m = 3, V-b is a reducible variety with one component corresponding to V-t and another... (More)
In this paper the different algebraic varieties that can be generated from multiple view geometry with uncalibrated cameras have been investigated. The natural descriptor, V-n, to work with is the image of P-3 in P-2 x P-2 x ... x P-2 under a corresponding product of projections, (A(1) x A(2) x ... x A(m)).

Another descriptor, the variety V-b, is the one generated by all bilinear forms between pairs of views, which consists of all points in P-2 x P-2 x ... x P-2 where all bilinear forms vanish. Yet another descriptor, the variety V-t, is the variety generated by all trilinear forms between triplets of views. It has been shown that when m = 3, V-b is a reducible variety with one component corresponding to V-t and another corresponding to the trifocal plane.

Furthermore, when m = 3, V-t is generated by the three bilinearities and one trilinearity, when m = 4, V-t is generated by the six bilinearities and when m greater than or equal to 4, V-t can be generated by the ((m)(2)) bilinearities. This shows that four images is the generic case in the algebraic setting, because V-t can be generated by just bilinearities. Furthermore, some of the bilinearities may be omitted when m greater than or equal to 5. (C) 1997 by B. G. Teubner Stuttgart - John Wiley & Sons Ltd. (Less)
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Mathematical Methods in the Applied Sciences
volume
20
issue
13
pages
1135 - 1162
publisher
John Wiley & Sons
external identifiers
• scopus:0031237868
ISSN
1099-1476
DOI
10.1002/(SICI)1099-1476(19970910)20:13<1135::AID-MMA908>3.0.CO;2-9
language
English
LU publication?
yes
id
064e4342-ee73-47a6-93b6-5f5ea02b31be (old id 788211)
alternative location
http://www3.interscience.wiley.com/cgi-bin/fulltext/8078/PDFSTART
2008-09-01 13:06:49
date last changed
2019-03-19 01:33:12
```@article{064e4342-ee73-47a6-93b6-5f5ea02b31be,
abstract     = {In this paper the different algebraic varieties that can be generated from multiple view geometry with uncalibrated cameras have been investigated. The natural descriptor, V-n, to work with is the image of P-3 in P-2 x P-2 x ... x P-2 under a corresponding product of projections, (A(1) x A(2) x ... x A(m)).<br/><br>
Another descriptor, the variety V-b, is the one generated by all bilinear forms between pairs of views, which consists of all points in P-2 x P-2 x ... x P-2 where all bilinear forms vanish. Yet another descriptor, the variety V-t, is the variety generated by all trilinear forms between triplets of views. It has been shown that when m = 3, V-b is a reducible variety with one component corresponding to V-t and another corresponding to the trifocal plane.<br/><br>
<br/><br>
Furthermore, when m = 3, V-t is generated by the three bilinearities and one trilinearity, when m = 4, V-t is generated by the six bilinearities and when m greater than or equal to 4, V-t can be generated by the ((m)(2)) bilinearities. This shows that four images is the generic case in the algebraic setting, because V-t can be generated by just bilinearities. Furthermore, some of the bilinearities may be omitted when m greater than or equal to 5. (C) 1997 by B. G. Teubner Stuttgart - John Wiley &amp; Sons Ltd.},
author       = {Heyden, Anders and Åström, Karl},
issn         = {1099-1476},
language     = {eng},
number       = {13},
pages        = {1135--1162},
publisher    = {John Wiley & Sons},
series       = {Mathematical Methods in the Applied Sciences},
title        = {Algebraic Properties of Multilinear Constraints},
url          = {http://dx.doi.org/10.1002/(SICI)1099-1476(19970910)20:13<1135::AID-MMA908>3.0.CO;2-9},
volume       = {20},
year         = {1997},
}

```