Multi-focal tensors as invariant differential forms

Abstract

For each relative $\operatorname{GL}(V)$-invariant tensor $I\in \Lambda^{p_1+1}V^{\vee}\otimes .. \otimes \Lambda^{p_n+1}V^{\vee}$ we construct a $\operatorname{GL}(V)$-invariant weighted differential form $\eta$ on $(\mathbb{P} V)^{n}$. Then $\eta$ is expressed explicitly with respect to $n$-tuples of frames for tangent spaces at points of $\mathbb{P} V$ to obtain elements of a different tensor space. For certain invariants $I$, the resulting elements are shown to be the multi-focal tensors appearing in the machine vision literature (Demazure 1988, Longuet-Higgins 1981, Luong 1992, Faugeras 1993, Faugeras and Luong 2001, Hartley and Zisserman 2003). This generalizes the 3 multi-focal varieties known in dimension $\dim V = 4$ to an infinite collection of special tensor subvarieties. We use this framework to exhibit a new system of degree 4 polynomial equations, reminiscent of the braid relation, satisfied by the Euclidean trifocal variety.