Matrix Formulas for Chow Forms
The following Macaulay2 files store the explicit formulas derived with
Gunnar Fløystad and Giorgio Ottaviani in our paper
The Chow Form of the Essential Variety in
Computer Vision.
Essential variety
The Chow form of the essential variety is a degree $10$ polynomial in the $84$ (dual) Plücker coordinates of $\text{Gr}(\mathbb{P}^{2}, \mathbb{P}^{8})$.
We show that it equals the Pfaffian of each of the $20 \times 20$ matrices below:
Corollary: Six point pairs $\{(x^{(i)},y^{(i)}) \in \mathbb{R}^{2} \times \mathbb{R}^{2} : i = 1, \ldots, 6\}$ are mutually consistent via two calibrated cameras if and only if these are rank-deficient after the substitution here.
On noisy data, we look for a sufficiently small lowest singular value.
Rank $2$ symmetric $4 \times 4$ matrices
The Chow form of $\sigma_2(\nu_2(\mathbb{P}^3))$ is a degree $10$ polynomial in the $120$ (primal) Plücker coordinates of $\text{Gr}(\mathbb{P}^{2}, \mathbb{P}^{9})$.
We show that it equals the Pfaffian of each of the $20 \times 20$ matrices below:
Corollary: A net $\langle A, B, C \rangle$ of $4 \times 4$ symmetric matrices contains a rank $2$ point if and only if these are rank-deficient after the substitution here.