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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:

First matrix

Second matrix

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:

First matrix

Second matrix

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.

Matrix Formulas for Chow Forms

Essential variety

Rank 22 symmetric 4×44 \times 4 matrices

Rank $2$ symmetric $4 \times 4$ matrices