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

TL;DR

This paper explores the geometric conditions causing rank deficiency in a matrix associated with point correspondences in two-view geometry, revealing connections to classical algebraic geometry and improving understanding of reconstruction algorithms.

Contribution

It characterizes rank deficiency for matrices with 7 to 9 points in terms of geometric configurations, extending previous work for up to 6 points.

Findings

Characterization of rank deficiency for 7-9 points.

Connections between algebraic geometry and computer vision.

Insights into the conditioning of reconstruction algorithms.

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.