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The ideal of the trifocal variety. (English) Zbl 1347.13011

This article deals with the following problem in multiview geometry. Consider that a camera consists on projecting from a point (focus) to a plane. If we have three fixed cameras we can construct the following application: given two lines \(l_1,l_2\) in the projecting planes of cameras 1 and 2, each one defines a plane upon linking it to its focus; intersect this with the projecting plane of camera 3, and you obtain another line \(l_3\). The application is then \((l_1,l_2)\mapsto l_3\), from \(\mathbb{P}^2\times\mathbb{P}^2\) to \(\mathbb{P}^2\). This is equivalent to a tensor, called a trifocal tensor. Now, via a different, algebro-geometric construction, one can define the trifocal variety. Then, this article provides minimal generators of the ideal of this variety. The main result is that the ideal is minimally generated by about 2100 polynomials of degrees 3 to 6. This improves, in a sense, a previous result of 2010 providing equations that cut out the variety set-theoretically, but that do not generate the ideal.
The strategy is: “to find equations in the ideal in the lowest degrees, next show that the equations we found cut out the variety set theoretically and thus define an ideal that agrees up to radical with the ideal we want, and then we show that the two ideals are actually equal.” Given that the variety is acted on by certain groups, techniques from representation theory are used to compute generators. Also, numerical primary decomposition has been used, via the Bertini software. Other techniques are used then to identify the computed components and prove that the polynomials calculated generate the ideal. Finally, “An effective test for determining whether a given tensor is a trifocal tensor is also given.”
The authors note that the output of many computations are included in the arXiv version of the article as ancillary files.

MSC:

13P99 Computational aspects and applications of commutative rings
14Q15 Computational aspects of higher-dimensional varieties
68W30 Symbolic computation and algebraic computation
15A69 Multilinear algebra, tensor calculus
15A72 Vector and tensor algebra, theory of invariants
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
13A50 Actions of groups on commutative rings; invariant theory
68T45 Machine vision and scene understanding

Software:

Macaulay2; Bertini
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Full Text: DOI arXiv

References:

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