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The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products

Erin Connelly, Sameer Agarwal, Alperen Ergur, Rekha R. Thomas

Abstract

Given $k \leq 6$ 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 configurations $\{x_i\}$ and $\{y_i\}$. While this question comes from computer vision the answer relies on tools from classical algebraic geometry: For $k \leq 5$, the geometry of the rank-drop locus is characterized by cross-ratios and basic (projective) geometry of point configurations. For the case $k=6$ the rank-drop locus is captured by the classical theory of cubic surfaces.

The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products

Abstract

Given points , we characterize rank deficiency of the matrix with rows in terms of the geometry of the point configurations and . While this question comes from computer vision the answer relies on tools from classical algebraic geometry: For , the geometry of the rank-drop locus is characterized by cross-ratios and basic (projective) geometry of point configurations. For the case the rank-drop locus is captured by the classical theory of cubic surfaces.
Paper Structure (14 sections, 30 theorems, 92 equations, 3 figures)

This paper contains 14 sections, 30 theorems, 92 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work(WIP)
  3. Preprocessing the input
  4. Cross-ratios and brackets
  5. $k \leq 4$
  6. $k=5$
  7. $k=6$
  8. Algebraic Characterizations of Rank Deficiency
  9. Geometric Characterizations of Rank Deficiency
  10. Cubic surfaces and Schäfli double sixes
  11. Conic intersections and the bracket ideal $\mathcal{B}_{rackets}(Z_6)$
  12. The Cremona hexahedral form of a cubic surface
  13. The Proof of Theorem \ref{['thm: k=6 one side in a line']}
  14. Facts about cubic surfaces

Key Result

Lemma 2.1

Let $(x_i,y_i) \in \mathbb P^2 \times \mathbb P^2$ for $i=1, \ldots,k$ and let $H_1,H_2 \in \textup{PGL}(3)$ be homographies acting on $\mathbb{P}_x^2$ and $\mathbb{P}^2_y$ respectively (i.e., $(x,y)\mapsto (H_1x,H_2y)$). Then $Z_k = \left (x^\top_i \otimes y^\top_i\right )_{i=1}^k$ has the same ran

Figures (3)

  • Figure 1: For distinct points $p_1,\ldots,p_5 \in \mathbb P^2$, the planar cross-ratio $(p_1,p_2;p_3,p_4;p_5)$ equals the $4$ point cross-ratio $(a_1,a_2;a_3,a_4)$
  • Figure 2: We construct the projection with center $c$.
  • Figure 3: The point $y_6$ is the unique intersection point of the $5$ conics $C_i'$ in $\mathbb P^2_y$.

Theorems & Definitions (64)

  • Lemma 2.1
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • Example 1
  • Definition 3.4
  • ...and 54 more