Algebraic Properties of Multilinear Constraints

In this paper the diierent 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 IP 3 in IP 2 I P 2 I P 2 under a corresponding product of projections, (A1 A2 : : :Am). Another descriptor, the variety V b , is the one generated by all bilinear forms between pairs of views, which consists of all points in IP 2 IP 2 IP 2 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, V b 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 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.