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Critical configurations for two projective views, a new approach

Martin Bråtelund

TL;DR

This work addresses when structure-from-motion reconstructions with two projective cameras fail to be unique by providing a complete algebraic classification of maximal critical configurations. It leverages a blow-up geometry and the multi-view framework to show that all such configurations lie on ruled quadric surfaces and are organized by the pullback of bilinear forms, namely the fundamental form $F_P$. The authors establish a precise correspondence between conjugate configurations via maps between quadrics, distinguishing cases where the quadric is smooth, a cone, or a union of two planes, and they provide explicit descriptions of epipolar lines and the birational conjugation map. The results recover and extend prior two-view classifications, supply explicit conjugacy structures, and lay the groundwork for extending to more views (as discussed in a companion paper) and for understanding reconstruction stability in practice.

Abstract

The problem of structure from motion is concerned with recovering 3-dimensional structure of an object from a set of 2-dimensional images. Generally, all information can be uniquely recovered if enough images and image points are provided, but there are certain cases where unique recovery is impossible; these are called critical configurations. In this paper we use an algebraic approach to study the critical configurations for two projective cameras. We show that all critical configurations lie on quadric surfaces, and classify exactly which quadrics constitute a critical configuration. The paper also describes the relation between the different reconstructions when unique reconstruction is impossible.

Critical configurations for two projective views, a new approach

TL;DR

This work addresses when structure-from-motion reconstructions with two projective cameras fail to be unique by providing a complete algebraic classification of maximal critical configurations. It leverages a blow-up geometry and the multi-view framework to show that all such configurations lie on ruled quadric surfaces and are organized by the pullback of bilinear forms, namely the fundamental form . The authors establish a precise correspondence between conjugate configurations via maps between quadrics, distinguishing cases where the quadric is smooth, a cone, or a union of two planes, and they provide explicit descriptions of epipolar lines and the birational conjugation map. The results recover and extend prior two-view classifications, supply explicit conjugacy structures, and lay the groundwork for extending to more views (as discussed in a companion paper) and for understanding reconstruction stability in practice.

Abstract

The problem of structure from motion is concerned with recovering 3-dimensional structure of an object from a set of 2-dimensional images. Generally, all information can be uniquely recovered if enough images and image points are provided, but there are certain cases where unique recovery is impossible; these are called critical configurations. In this paper we use an algebraic approach to study the critical configurations for two projective cameras. We show that all critical configurations lie on quadric surfaces, and classify exactly which quadrics constitute a critical configuration. The paper also describes the relation between the different reconstructions when unique reconstruction is impossible.
Paper Structure (11 sections, 17 theorems, 31 equations, 4 figures, 1 table)

This paper contains 11 sections, 17 theorems, 31 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. The one-view case
  4. The two-view case
  5. The multi-view ideal
  6. The multi-view variety
  7. Trivial critical configurations
  8. Critical configurations for two views
  9. Maps between quadrics
  10. Epipolar lines
  11. Maps between quadrics

Key Result

Proposition 2.12

Let $\textbf{P}$ and $\textbf{Q}$ be two (different) $n$-tuples of cameras, and let $X$ and $Y$ be their respective sets of critical points. Then $(\textbf{P},X)$ is a critical configuration, with $(\textbf{Q},Y)$ as its conjugate. Furthermore, $(\textbf{P},X)$ is maximal with respect to $\textbf{Q}

Figures (4)

  • Figure 1: All types of real ruled quadrics.
  • Figure 2: Illustration of the blow-downs of the non-trivial critical configurations for two views. The two marked with crosses are not critical.
  • Figure 3: Illustration of the map taking a point to its conjugate. The smooth quadric has two conjugates (original in the middle), the left is the one we get if we take the lines in the "A" family to be the epipolar lines, whereas the one on the right is the one we get if we choose "B". In both cases the lines in the family with the epipolar lines are preserved, whereas the lines in the other family are mapped to conic curves.
  • Figure 4: Illustration of the map taking a point to its conjugate. The line spanned by the camera centers (right) maps to the vertex on the cone. The other lines in this family map to conics. Lines in the other family are preserved.

Theorems & Definitions (50)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 40 more