Algebraic Varieties in Multiple View Geometry
Heyden, Anders LU
and Åström, Karl
LU
(1996)
Proceedings of Fourth European Conference on Computer Vision. ECCV '96
2.
p.671-682
- Abstract
- In this paper we will investigate the different algebraic varieties and ideals that can be generated from multiple view geometry with uncalibrated cameras. The natural descriptor, Vn, is the image of p3 in P2×p2×...×p2 under n different projections. However, we will show that Vn is not a variety. Another descriptor, the variety Vb, is generated by all bilinear forms between pairs of views and consists of all points in p2×p2×...×p2 where all bilinear forms vanish. Yet another descriptor, the variety, Vt, is the variety generated by all trilinear forms between triplets of views. We will show that when n=3, Vt is a reducible variety with one component corresponding to Vb and another corresponding to the trifocal plane. In ideal theoretic... (More)
- In this paper we will investigate the different algebraic varieties and ideals that can be generated from multiple view geometry with uncalibrated cameras. The natural descriptor, Vn, is the image of p3 in P2×p2×...×p2 under n different projections. However, we will show that Vn is not a variety. Another descriptor, the variety Vb, is generated by all bilinear forms between pairs of views and consists of all points in p2×p2×...×p2 where all bilinear forms vanish. Yet another descriptor, the variety, Vt, is the variety generated by all trilinear forms between triplets of views. We will show that when n=3, Vt is a reducible variety with one component corresponding to Vb and another corresponding to the trifocal plane. In ideal theoretic terms this is called a primary decomposition. This settles the discussion on the connection between the bilinearities and the trilinearities. Furthermore, we will show that when n=3, Vt is generated by the three bilinearities and one trilinearity and when n⩾4, Vt is generated by the (2n) bilinearities. This shows that four images is the generic case in the algebraic setting, because Vt can be generated by just bilinearities (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/787283
- author
- Heyden, Anders
LU
and Åström, Karl
LU
- organization
- publishing date
- 1996
- type
- Chapter in Book/Report/Conference proceeding
- publication status
- published
- subject
- keywords
- algebra, computer vision, image sequences, polynomials, multiple view geometry, uncalibrated cameras, natural descriptor, bilinear forms, trilinear forms, triplets of views, trifocal plane, ideal theoretic terms, primary decomposition, algebraic setting, polynomial equations
- host publication
- Computer Vision - ECCV '96. 4th Eurpean Conference on Computer Proceedings
- editor
- Cipolla, R. and Buxton, B.
- volume
- 2
- pages
- 671 - 682
- publisher
- Springer
- conference name
- Proceedings of Fourth European Conference on Computer Vision. ECCV '96
- conference location
- Cambridge, United Kingdom
- conference dates
- 1996-04-14 - 1996-04-18
- external identifiers
-
- scopus:84957874673
- ISBN
- 3 540 61123 1
- language
- English
- LU publication?
- yes
- id
- e36823ac-2ffa-45e9-835b-22b0236d8054 (old id 787283)
- date added to LUP
- 2016-04-04 11:42:12
- date last changed
- 2023-11-30 16:49:34
@inproceedings{e36823ac-2ffa-45e9-835b-22b0236d8054,
abstract = {{In this paper we will investigate the different algebraic varieties and ideals that can be generated from multiple view geometry with uncalibrated cameras. The natural descriptor, Vn, is the image of p3 in P2×p2×...×p2 under n different projections. However, we will show that Vn is not a variety. Another descriptor, the variety Vb, is generated by all bilinear forms between pairs of views and consists of all points in p2×p2×...×p2 where all bilinear forms vanish. Yet another descriptor, the variety, Vt, is the variety generated by all trilinear forms between triplets of views. We will show that when n=3, Vt is a reducible variety with one component corresponding to Vb and another corresponding to the trifocal plane. In ideal theoretic terms this is called a primary decomposition. This settles the discussion on the connection between the bilinearities and the trilinearities. Furthermore, we will show that when n=3, Vt is generated by the three bilinearities and one trilinearity and when n⩾4, Vt is generated by the (2n) bilinearities. This shows that four images is the generic case in the algebraic setting, because Vt can be generated by just bilinearities}},
author = {{Heyden, Anders and Åström, Karl}},
booktitle = {{Computer Vision - ECCV '96. 4th Eurpean Conference on Computer Proceedings}},
editor = {{Cipolla, R. and Buxton, B.}},
isbn = {{3 540 61123 1}},
keywords = {{algebra; computer vision; image sequences; polynomials; multiple view geometry; uncalibrated cameras; natural descriptor; bilinear forms; trilinear forms; triplets of views; trifocal plane; ideal theoretic terms; primary decomposition; algebraic setting; polynomial equations}},
language = {{eng}},
pages = {{671--682}},
publisher = {{Springer}},
title = {{Algebraic Varieties in Multiple View Geometry}},
volume = {{2}},
year = {{1996}},
}