Line multiview ideals

Abstract

We study the following problem in computer vision from the perspective of algebraic geometry: Using m pinhole cameras we take m pictures of a line in ${\mathbb{P}}^3$. This produces m lines in ${\mathbb{P}}^2$ and the question is which m-tuples of lines can arise that way. We are interested in polynomial equations and therefore study the complex Zariski closure of all such tuples of lines. The resulting algebraic variety is a subvariety of ${({\mathbb{P}}^2)}^m$ and is called line multiview variety. In this article, we study its ideal. We show that for generic cameras the ideal is generated by $3\times 3$-minors of a specific matrix. We also compute Gröbner bases and discuss to what extent our results carry over to the non-generic case.

Keywords:

• Generic translational cameras
• multiview ideal
• partially-symbolic multiview ideal

2020 Mathematics Subject Classification:

• 13P10
• 68T45

Acknowledgments

Part of this work was completed while the authors visited the Czech Institute of Informatics, Robotics, and Cybernetics as part of the Intelligent Machine Perception Project. We thank Tomas Pajdla for the invitation.

Notes

1 The proof is similar to the computation in [5, Section 3]

2 Recall (see e.g. [6, Section 2.7, Theorem 9]) that a polynomial ideal has a unique reduced Gröbner basis with respect to any monomial order. A Gröbner basis G is said to be reduced if every $g\in G$ has leading coefficient 1 and, for distinct $g,g^{\prime }\in G,$ the leading term ${\text{in}}_{<}(g)$ does not divide any term of $g^{\prime }.$

Additional information

Funding

The research of Elima Shehu and Paul Breiding was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Projektnummer 445466444. Tim Duff acknowledges support from an NSF Mathematical Sciences Postdoctoral Research Fellowship (DMS-2103310). Felix Rydell was supported by the Knut and Alice Wallenberg Foundation within their WASP (Wallenberg AI, Autonomous Systems and Software Program) AI/Math initiative. Lukas Gustafsson was supported by the VR Grant [NT:2018-03688].