Algebraic Properties of Multilinear Constraints

Heyden, Anders; Åström, Karl (1997). Algebraic Properties of Multilinear Constraints. Mathematical Methods in the Applied Sciences, 20, (13), 1135 - 1162
| Published | English
Heyden, Anders ; Åström, Karl
Mathematics (Faculty of Engineering)
Mathematical Imaging Group
Research Group:
Mathematical Imaging Group
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.
Computer Vision and Robotics (Autonomous Systems) ; Mathematics
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