Multi-focal tensors as invariant differential forms

Abstract

For each relative $\mathrm{GL}(V)$-invariant tensor $I\in {\Lambda }^{p_1+1}{V}^{\vee }\otimes ..\otimes {\Lambda }^{p_n+1}{V}^{\vee }$ we construct a $\mathrm{GL}(V)$-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.

41 pages, 1 figure

References in corpus (1)

• The ideal of the trifocal variety

Cited by in corpus (2)

• Trifocal Relative Pose from Lines at Points and its Efficient Solution
• Minimal Problems for the Calibrated Trifocal Variety