The ideal of the trifocal variety

TL;DR

The paper determines the full defining ideal $I(X)$ of the trifocal variety $X$, arising from trifocal tensors tied to triples of cameras, by combining representation theory, symbolic computation, and numerical algebraic geometry. It proves that $I(X)$ is minimally generated by $10$ cubics, $81$ quintics, and $1980$ sextics, and shows $X$ is irreducible of degree $297$; it also gives an effective test for recognizing trifocal tensors based on $\operatorname{P\text{-}Rank}$ and $\operatorname{F\text{-}Rank}$. The analysis unites orbit-closure classifications (Nurmiev), subspace and rank varieties, and a careful primary-decomposition argument to identify four components in the relevant zero sets and to prove $J=I(X)$. The results provide a concrete, computable algebraic description of the trifocal variety and a practical criterion for trifocal tensors with potential applications in multiview geometry and computer vision.

Abstract

Techniques from representation theory, symbolic computational algebra, and numerical algebraic geometry are used to find the minimal generators of the ideal of the trifocal variety. An effective test for determining whether a given tensor is a trifocal tensor is also given.

Figures (2)

• Figure 1: The trifocal tensor as a map $\mathbb{P}^2 \times \mathbb{P}^2 \rightarrow \mathbb{P}^2$
• Figure 2: A poset diagram for orbit closures in $\operatorname{Rank}_{C}^{2}$

Theorems & Definitions (20)

• Theorem 1.1
• Remark 3.1
• Remark 3.2
• Proposition 5.1
• Proposition 5.2
• proof
• Remark 7.1
• Proposition 7.2
• proof
• Proposition 7.3