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Algebraic Properties of Multilinear Constraints

Heyden, Anders LU and Åström, Karl LU orcid (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:
author
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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)
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http://www3.interscience.wiley.com/cgi-bin/fulltext/8078/PDFSTART
date added to LUP
2016-04-01 11:38:40
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2022-02-25 19:13:14
@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 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}},
}