Lines, Quadrics, and Cremona Transformations in Two-View Geometry

TL;DR

The characterization of rank deficiency of the matrix Z_k, which arises in the conditioning of certain well-known reconstruction algorithms in computer vision, has surprising connections to classical algebraic geometry via the interplay of quadric surfaces, cubic curves and Cremona transformations.

Abstract

Given $7 \leq k \leq 9$ points $(x_i,y_i) \in \mathbb{P}^2 \times \mathbb{P}^2$, we characterize rank deficiency of the $k \times 9$ matrix $Z_k$ with rows $x_i^\top \otimes y_i^\top$, in terms of the geometry of the point sets $\{x_i\}$ and $\{y_i\}$. This problem arises in the conditioning of certain well-known reconstruction algorithms in computer vision, but has surprising connections to classical algebraic geometry via the interplay of quadric surfaces, cubic curves and Cremona transformations. The characterization of rank deficiency of $Z_k$, when $k \leq 6$, was completed in arXiv:2301.09826.

Figures (3)

• Figure 1: The cubic curves $C_x$ and $C_y$ from Example \ref{['ex: k=7 example 1']}, with $x_i$ and $y_i$ labeled.
• Figure 2: The cubic curves $C_x$ and $C_y$, with $x_7$ and $y_7$ highlighted.
• Figure 3: The cubic curves $C_x^{\hat{i}}$ and $C_y^{\hat{i}}$. The intersection points are exactly the three possible epipoles associated to the fundamental matrices.

Theorems & Definitions (76)

• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Lemma 2.5
• Lemma 2.6
• proof
• Lemma 2.7