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A Geometric Approach to the Trifocal Tensor

A Geometric Approach to the Trifocal Tensor,10.1007/s10851-010-0216-4,Journal of Mathematical Imaging and Vision,Alberto Alzati,Alfonso Tortora

A Geometric Approach to the Trifocal Tensor   (Citations: 1)
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In geometric computer vision the trifocal tensors are 3×3×3 tensors T by whose means three different camera views of the same scene are related to each other. In this paper we find two different sets of constraints, in the entries of T, that must be satisfied by trifocal tensors. The first set gives exactly the (closure of the) trifocal locus, i.e. all trifocal tensors, but it is very big. The second set, although not complete and still very big, has the property that it is possible to extract from it a set of only eight equations that are generically complete, i.e. for a generic choice of T, they suffice to decide whether T is indeed trifocal. Note that 8 is the codimension of the trifocal locus in its ambient space.
Journal: Journal of Mathematical Imaging and Vision - JMIV , vol. 38, no. 3, pp. 159-170, 2010
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    • ...We give an account of the results, with a broad outline of the arguments leading thereto, referring to our forthcoming paper [1] for the details of the proofs...
    • ...Proposition 2.3 ([1]) The entries of the trifocal tensor T =[ t jk i ], associated to the matrices M, M � , M �� ,a re...
    • ...[1]) to be equivalent to three equations of degree 4 in the coefficients of Y (y), hence of degree 12...
    • ...Theorem 3.1 ([1]) A complete set of constraints for the trifocal locus Θ is given by 10 equations of degree three, 20 equations of degree 9 and 6 equations of degree 12 on the 27 entries of a generic 3 × 3 × 3 tensor...
    • ...This is a little trickier to translate into algebraic relations; in [1] we show that it implies the following algebraic relations...
    • ...Summing up: Theorem 4.1 ([1]) Let Ω be the subvariety of P 26 defined by equations (5)—(8), then the trifocal locus Θ is an irreducible component, of maximal dimension, of the variety Ω ...
    • ...Theorem 5.1 ([1]) Let I be the ideal generated by the eight polynomials (5), (9) and (10), then it defines the trifocal locus Θ , outside the hypersurface F(t)G(t )= 0...

    Alberto Alzatiet al. Constraints for the Trifocal Tensor

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