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Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices

Timothy Duff, Viktor Korotynskiy, Anton Leykin, Tomas Pajdla

Abstract

We characterize the variety of compatible fundamental matrix triples by computing its multidegree and multihomogeneous vanishing ideal. This answers the first interesting case of a question recently posed by Bråtelund and Rydell. Our result improves upon previously discovered sets of algebraic constraints in the geometric computer vision literature, which are all incomplete (as they do \emph{not} generate the vanishing ideal) and sometimes make restrictive assumptions about how a matrix triple should be scaled. Our discussion touches more broadly on generalized compatibility varieties, whose multihomogeneous vanishing ideals are much less well understood. One of our key new discoveries is a simple set of quartic constraints vanishing on compatible fundamental matrix triples. These quartics are also significant in the setting of essential matrices: together with some previously known constraints, we show that they locally cut out the variety of compatible essential matrix triples.

Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices

Abstract

We characterize the variety of compatible fundamental matrix triples by computing its multidegree and multihomogeneous vanishing ideal. This answers the first interesting case of a question recently posed by Bråtelund and Rydell. Our result improves upon previously discovered sets of algebraic constraints in the geometric computer vision literature, which are all incomplete (as they do \emph{not} generate the vanishing ideal) and sometimes make restrictive assumptions about how a matrix triple should be scaled. Our discussion touches more broadly on generalized compatibility varieties, whose multihomogeneous vanishing ideals are much less well understood. One of our key new discoveries is a simple set of quartic constraints vanishing on compatible fundamental matrix triples. These quartics are also significant in the setting of essential matrices: together with some previously known constraints, we show that they locally cut out the variety of compatible essential matrix triples.
Paper Structure (10 sections, 6 theorems, 65 equations, 2 figures)

This paper contains 10 sections, 6 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Previous work
  4. Fundamentals
  5. New Quartic Constraints
  6. Interpolation with symmetry
  7. Geometric interpretation of the new quartic constraints.
  8. Comparing quartics with previously-known constraints
  9. Proofs of Main Results
  10. Open Problems

Key Result

Theorem 1

The variety $Y_\mathcal{F}$ has dimension $18$. Its multidegree function is supported on the lattice simplex with values shown in fig:multidegrees (left). The $\mathbb{Z}^3$-homogeneous vanishing ideal $\mathcal{I} (Y_\mathcal{F})$ has an explicit set of minimal generators in total degree $3$--$7$, as tallied below:

Figures (2)

  • Figure 1: Multidegrees of $Y_\mathcal{F}$ (left) and $Y_\mathcal{E}$ (right). Values for $Y_{\mathcal{F}}$ and $\mathop{\mathrm{mdeg}}\nolimits_{\mathcal{E}} (5,5,1)=\mathop{\mathrm{mdeg}}\nolimits_{\mathcal{E}} (5,1,5)=\mathop{\mathrm{mdeg}}\nolimits_{\mathcal{E}} (1,5,5)=400$ are from \ref{['thm:uncalibrated-ideal']} and \ref{['prop:essential-multidegree']}, respectively. The remaining values for $Y_\mathcal{E}$ are obtained by numerical monodromy heuristics.
  • Figure 2: Geometry of the quartics \ref{['eq:quartics']}: each epipolar line $l_i=F_{ij}e_{jk}$ belongs to the pencil of epipolar lines $\mathop{\mathrm{im}}\nolimits (F_{ik})$.

Theorems & Definitions (23)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Proposition 1
  • Remark 3
  • Proposition 2
  • proof
  • Remark 4
  • ...and 13 more