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A Computer Vision Problem in Flatland

Sameer Agarwal, Erin Connelly, Annalisa Crannell, Timothy Duff, Rekha R. Thomas

TL;DR

The paper investigates when two labeled point sets in the plane share a common image under flatland projection to a line, and characterizes the loci of projection centers that realize this equivalence. It shows that two $n$-point sets have the same $\mathbb{P}^1$-image if and only if they arise from projecting a single configuration in $\mathbb{P}^3$, bridging a 2D two-view problem with a 3D reconstruction perspective. The authors develop invariant-theoretic and algebraic-geometry machinery—SL$(2)$-invariants of $n$ points in $\mathbb{P}^1$, the camera-centers variety, and explicit geometric constructions—to describe the camera loci for $n=4$ to $7$, revealing explicit loci: conics for $n=4$, intersecting conics and a degree-5 Cremona map for $n=5$, cubic curves for $n=6$, and a finite set of three centers for $n=7$. They also discuss connections to the fundamental matrix and classical Cremona transformations, and they show nonexistence for $n>7$, highlighting a sharp boundary. The work provides explicit algebraic descriptions and constructive procedures, linking projective geometry, invariant theory, and computer-vision concepts like epipolar geometry and 3D reconstruction.

Abstract

When is it possible to project two sets of labeled points lying in a pair of projective planes to the same image on a projective line? We give a complete answer to this question and describe the loci of the projection centers that enable a common image. In particular, we find that there exists a solution to this problem if and only if these two sets are themselves images of a common pointset in projective space.

A Computer Vision Problem in Flatland

TL;DR

The paper investigates when two labeled point sets in the plane share a common image under flatland projection to a line, and characterizes the loci of projection centers that realize this equivalence. It shows that two -point sets have the same -image if and only if they arise from projecting a single configuration in , bridging a 2D two-view problem with a 3D reconstruction perspective. The authors develop invariant-theoretic and algebraic-geometry machinery—SL-invariants of points in , the camera-centers variety, and explicit geometric constructions—to describe the camera loci for to , revealing explicit loci: conics for , intersecting conics and a degree-5 Cremona map for , cubic curves for , and a finite set of three centers for . They also discuss connections to the fundamental matrix and classical Cremona transformations, and they show nonexistence for , highlighting a sharp boundary. The work provides explicit algebraic descriptions and constructive procedures, linking projective geometry, invariant theory, and computer-vision concepts like epipolar geometry and 3D reconstruction.

Abstract

When is it possible to project two sets of labeled points lying in a pair of projective planes to the same image on a projective line? We give a complete answer to this question and describe the loci of the projection centers that enable a common image. In particular, we find that there exists a solution to this problem if and only if these two sets are themselves images of a common pointset in projective space.
Paper Structure (7 sections, 27 theorems, 95 equations, 9 figures)

This paper contains 7 sections, 27 theorems, 95 equations, 9 figures.

Table of Contents

  1. Answer to Question I
  2. Invariant theory of labeled points in $\mathbb{P}^1$
  3. The camera centers variety
  4. Camera loci when $n=4$
  5. Camera loci when $n=5$
  6. Camera loci when $n=6$
  7. Camera loci when $n=7$

Key Result

Theorem 1

For generic $\mathcal{X}$ and $\mathcal{Y}$ in $\mathbb{P}^2$, the loci of centers $(a,b) \in \mathbb{P}^2 \times \mathbb{P}^2$ in Question II satisfy the following:

Figures (9)

  • Figure 1: A geometric interpretation of Theorem \ref{['thm-many-cameras']}.
  • Figure 2: These figures show several examples of the Plücker relation being used to "uncross" the two (bold) edges in a bracket monomial on $6$ variables. .
  • Figure 3: The six non-crossing graphs of degree $(2,2,2,2,2)$ on the pentagon.
  • Figure 4: A geometric construction that operationalizes Theorem \ref{['thm:n=4 loci invariants']}.
  • Figure 5: If $\ell_x\cap\bar{\ell}_x = L\in \ell_y$, then $\ell_x$ and $\bar{\ell}_x$ both construct the same point $b$.
  • ...and 4 more figures

Theorems & Definitions (55)

  • Theorem 1: Loci Theorem
  • Remark 1
  • Theorem 2
  • Lemma 3
  • proof
  • proof : Proof of \ref{['thm:twocam-existence-general']}
  • Theorem 4
  • proof
  • Theorem 5
  • Theorem 6
  • ...and 45 more