# 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.

## Keywords

Bilinear Form Algebraic Variety Primary Decomposition Epipolar Line Natural Descriptor## References

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