Matrix Formulas for Chow Forms

The following Macaulay2 files store the explicit formulas derived with Gunnar Fl√łystad 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.