Algebraic Varieties in Multiple View Geometry

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, V _n, is the image of $\mathcal{P}^3$ in $\mathcal{P}^2 \times \mathcal{P}^2 \times \cdot \cdot \cdot \times \mathcal{P}^2$ under n different projections. However, we will show that V _n is not a variety. Another descriptor, the variety V _b, is generated by all bilinear forms between pairs of views and consists of all points in $\mathcal{P}^2 \times \mathcal{P}^2 \times \cdot \cdot \cdot \times \mathcal{P}^2$ where all bilinear forms vanish. Yet another descriptor, the variety, V _t, is the variety generated by all trilinear forms between triplets of views. We will show that when n =3, V _t is a reducible variety with one component corresponding to V _b 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, V _t is generated by the three bilinearities and one trilinearity and when n ≥4, V _t is generated by the ( _2 ^n ) bilinearities. This shows that four images is the generic case in the algebraic setting, because V _t can be generated by just bilinearities.