Algebraic properties of multilinear constraints
Anders Heyden, Kalle Åström · Mathematical Methods in the Applied Sciences · 1997
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×ℙ2×…×ℙ2 under a corresponding product of projections, (A1×A2×…×Am). Another descriptor, the variety 𝒱b, is the one generated by all bilinear forms between pairs of views, which 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. 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⩾4, 𝒱t can be generated by the m(m-1)/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⩾5. © 1997 by B. G. Teubner Stuttgart – John Wiley & Sons Ltd.