Algebraic Properties of Multilinear Constraints
(1997) In Mathematical Methods in the Applied Sciences 20(13). p.11351162 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, Vn, to work with is the image of P3 in P2 x P2 x ... x P2 under a corresponding product of projections, (A(1) x A(2) x ... x A(m)).
Another descriptor, the variety Vb, is the one generated by all bilinear forms between pairs of views, which consists of all points in P2 x P2 x ... x P2 where all bilinear forms vanish. Yet another descriptor, the variety Vt, is the variety generated by all trilinear forms between triplets of views. It has been shown that when m = 3, Vb is a reducible variety with one component corresponding to Vt 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, Vn, to work with is the image of P3 in P2 x P2 x ... x P2 under a corresponding product of projections, (A(1) x A(2) x ... x A(m)).
Another descriptor, the variety Vb, is the one generated by all bilinear forms between pairs of views, which consists of all points in P2 x P2 x ... x P2 where all bilinear forms vanish. Yet another descriptor, the variety Vt, is the variety generated by all trilinear forms between triplets of views. It has been shown that when m = 3, Vb is a reducible variety with one component corresponding to Vt and another corresponding to the trifocal plane.
Furthermore, when m = 3, Vt is generated by the three bilinearities and one trilinearity, when m = 4, Vt is generated by the six bilinearities and when m greater than or equal to 4, Vt can be generated by the ((m)(2)) bilinearities. This shows that four images is the generic case in the algebraic setting, because Vt 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
 10991476
 DOI
 10.1002/(SICI)10991476(19970910)20:13<1135::AIDMMA908>3.0.CO;29
 language
 English
 LU publication?
 yes
 id
 064e4342ee7347a693b65f5ea02b31be (old id 788211)
 alternative location
 http://www3.interscience.wiley.com/cgibin/fulltext/8078/PDFSTART
 date added to LUP
 20160401 11:38:40
 date last changed
 20231209 17:09:59
@article{064e4342ee7347a693b65f5ea02b31be, 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, Vn, to work with is the image of P3 in P2 x P2 x ... x P2 under a corresponding product of projections, (A(1) x A(2) x ... x A(m)).<br/><br> Another descriptor, the variety Vb, is the one generated by all bilinear forms between pairs of views, which consists of all points in P2 x P2 x ... x P2 where all bilinear forms vanish. Yet another descriptor, the variety Vt, is the variety generated by all trilinear forms between triplets of views. It has been shown that when m = 3, Vb is a reducible variety with one component corresponding to Vt and another corresponding to the trifocal plane.<br/><br> <br/><br> Furthermore, when m = 3, Vt is generated by the three bilinearities and one trilinearity, when m = 4, Vt is generated by the six bilinearities and when m greater than or equal to 4, Vt can be generated by the ((m)(2)) bilinearities. This shows that four images is the generic case in the algebraic setting, because Vt 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 = {{10991476}}, language = {{eng}}, number = {{13}}, pages = {{11351162}}, 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)10991476(19970910)20:13<1135::AIDMMA908>3.0.CO;29}}, doi = {{10.1002/(SICI)10991476(19970910)20:13<1135::AIDMMA908>3.0.CO;29}}, volume = {{20}}, year = {{1997}}, }