Algebraic Properties of Multilinear Constraints
Author
Summary, in English
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.
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.
Department/s
- Mathematics (Faculty of Engineering)
- Mathematical Imaging Group
- Algebra
Publishing year
1997
Language
English
Pages
1135-1162
Publication/Series
Mathematical Methods in the Applied Sciences
Volume
20
Issue
13
Links
Document type
Journal article
Publisher
John Wiley & Sons Inc.
Topic
- Computer Vision and Robotics (Autonomous Systems)
- Mathematics
Status
Published
Research group
- Mathematical Imaging Group
- Algebra
ISBN/ISSN/Other
- ISSN: 1099-1476