ALGEBRAIC PROPERTIES OF MULTILINEAR CONSTRAINTS

In this paper the different algebraic varieties that can be generated from multiple view geometry with uncalibrated cameras have been invest igated. 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 t he one generated by all bilinear forms between pairs of views, which c onsists of all points in P-2 x P-2 x ... x P-2 where all bilinear form s vanish. Yet another descriptor, the variety V-t, is the variety gene rated by all trilinear forms between triplets of views. It has been sh own that when m = 3, V-b is a reducible variety with one component cor responding to V-t and another corresponding to the trifocal plane. Fur thermore, when m = 3, V-t is generated by the three bilinearities and one trilinearity, when m = 4, V-t is generated by the six bilinearitie s 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 biline arities. 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.