Algebraic Properties of Multilinear Constraints
Heyden, Anders LU
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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/788211
- author
- Heyden, Anders
LU
and Åström, Karl
LU
- organization
- publishing date
- 1997
- 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 Inc.
- 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
- date added to LUP
- 2016-04-01 11:38:40
- date last changed
- 2023-12-09 17:09:59
@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 & Sons Ltd.}},
author = {{Heyden, Anders and Åström, Karl}},
issn = {{1099-1476}},
language = {{eng}},
number = {{13}},
pages = {{1135--1162}},
publisher = {{John Wiley & Sons Inc.}},
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}},
doi = {{10.1002/(SICI)1099-1476(19970910)20:13<1135::AID-MMA908>3.0.CO;2-9}},
volume = {{20}},
year = {{1997}},
}