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When is one pinhole camera image equal to some other pinhole camera image?

Giorgio Ottaviani, Rekha R. Thomas

Abstract

Generically, one expects the images of two different point sets, in two different (projective) cameras, to be different. However, it can happen that the images are the same up to a projective transformation which is an instance of ill-posedness in computer vision. We prove that the images can become projectively equivalent only for point pairs with at most seven elements. In each case, we give explicit descriptions of the Zariski closure of the locus of camera centers which we call the centers-variety. To do this we use classical invariant theory and the geometry of moduli spaces of ordered points in the projective plane. The most involved case is that of seven points which uses a natural parametrization of the Goepel variety.

When is one pinhole camera image equal to some other pinhole camera image?

Abstract

Generically, one expects the images of two different point sets, in two different (projective) cameras, to be different. However, it can happen that the images are the same up to a projective transformation which is an instance of ill-posedness in computer vision. We prove that the images can become projectively equivalent only for point pairs with at most seven elements. In each case, we give explicit descriptions of the Zariski closure of the locus of camera centers which we call the centers-variety. To do this we use classical invariant theory and the geometry of moduli spaces of ordered points in the projective plane. The most involved case is that of seven points which uses a natural parametrization of the Goepel variety.
Paper Structure (17 sections, 19 theorems, 69 equations)

This paper contains 17 sections, 19 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Moduli spaces of ordered points in $\mathbb{P}^d$
  3. The GIT construction of the moduli space $\mathcal{M}_d^n$
  4. The moduli space $\mathcal{M}_2^5$
  5. The moduli space $\mathcal{M}_2^6$
  6. The moduli space $\mathcal{M}_2^7$
  7. Stability in $\mathcal{M}_d^n$
  8. Association
  9. A dimension bound on the centers-variety
  10. $n=5$
  11. $n=6$
  12. A strategy to find the centers-variety
  13. A computational proof of Theorem \ref{['thm:n=6locus']}
  14. $n=7$ and $n=8$
  15. The Goepel variety
  16. ...and 2 more sections

Key Result

Theorem 1

flatlandpaper Let $\mathcal{X} = \{x_1,\dots,x_n\} \subset \mathbb{P}^3$ and $\mathcal{Y} = \{y_1, \dots, y_n\} \subset \mathbb{P}^3$ be two sets of ordered points. There exists full-rank linear projections $A:\mathbb{P}^3\dashedrightarrow \mathbb{P}^{2}$ and $B:\mathbb{P}^3 \dashedrightarrow \mathb

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Corollary 7
  • Corollary 8
  • proof
  • Proposition 9
  • ...and 29 more