Multi-focal tensors as invariant differential forms

2016/10/13 by James C. Mathews, James Mathews, Mathews, James · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Engineering · Mathematics · #Advanced Vision and Imaging #Cell Image Analysis Techniques #Image Processing Techniques and Applications #math.AC #math.AG #msc:15A72 #msc:51A99

paper · pdf · doi:10.48550/arxiv.1610.04294

41 pages, 1 figure

arxiv created 2016/10/13 · arxiv updated 2016/10/17

Abstract

For each relative GL(V)-invariant tensor I∈ Λ^p₁+1V^\vee⊗ .. ⊗ Λ^p_n+1V^\vee we construct a GL(V)-invariant weighted differential form η on (ℙ V)ⁿ. Then η is expressed explicitly with respect to n-tuples of frames for tangent spaces at points of 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.

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