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Exploiting polar symmetry in designing equivariant observers for vision-based motion estimation

Tarek Bouazza, Robert Mahony, Tarek Hamel

TL;DR

A continuous-time filter is designed that exploits the same perspective by using the epipolar constraint to define pseudo-measurements, and a novel polar symmetry on the pose of the camera is proposed that makes these measurements equivariant.

Abstract

Accurately estimating camera motion from image sequences poses a significant challenge in computer vision and robotics. Many computer vision methods first compute the essential matrix associated with a motion and then extract orientation and normalized translation as inputs to pose estimation, reconstructing the scene scale (that is unobservable in the epipolar construction) from separate information. In this paper, we design a continuous-time filter that exploits the same perspective by using the epipolar constraint to define pseudo-measurements. We propose a novel polar symmetry on the pose of the camera that makes these measurements equivariant. This allows us to apply recent results from equivariant systems theory to estimating pose. We provide a novel explicit persistence of excitation condition to characterize observability of the full pose, ensuring reconstruction of the scale parameter that is not directly observable in the epipolar construction.

Exploiting polar symmetry in designing equivariant observers for vision-based motion estimation

TL;DR

A continuous-time filter is designed that exploits the same perspective by using the epipolar constraint to define pseudo-measurements, and a novel polar symmetry on the pose of the camera is proposed that makes these measurements equivariant.

Abstract

Accurately estimating camera motion from image sequences poses a significant challenge in computer vision and robotics. Many computer vision methods first compute the essential matrix associated with a motion and then extract orientation and normalized translation as inputs to pose estimation, reconstructing the scene scale (that is unobservable in the epipolar construction) from separate information. In this paper, we design a continuous-time filter that exploits the same perspective by using the epipolar constraint to define pseudo-measurements. We propose a novel polar symmetry on the pose of the camera that makes these measurements equivariant. This allows us to apply recent results from equivariant systems theory to estimating pose. We provide a novel explicit persistence of excitation condition to characterize observability of the full pose, ensuring reconstruction of the scale parameter that is not directly observable in the epipolar construction.
Paper Structure (14 sections, 8 theorems, 50 equations, 2 figures)

This paper contains 14 sections, 8 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mathematical notation
  4. Uniform observability of linear time-varying systems
  5. Background on epipolar geometry
  6. Problem formulation
  7. Symmetry and equivariance
  8. Lift of the kinematics to the Lie algebra
  9. Equivariant observer design
  10. Origin choice and local coordinates
  11. Observability and stability analysis
  12. Simulation results
  13. Conclusions
  14. Proofs

Key Result

Lemma 1

If there exists a matrix-valued function $M(\cdot)$ of dimension $(p \times n)$$(p \geq 1)$ composed of row vectors of $N_0 = C, N_k = N_{k-1}A + \dot{N}_{k-1}, k = 1, \dots$, such that for some positive numbers $\bar{\delta}, \bar{\mu}$ and $\forall t \geq 0$ then $W(t, t + \delta)$ satisfies condition eqn:positivegramian.

Figures (2)

  • Figure 1: Estimation errors on the position direction and orientation.
  • Figure 2: Estimation error on the position range and the Lyapunov function value during motion phases: (a) in green, (b) in red and (c) in blue.

Theorems & Definitions (17)

  • Definition 1: Uniform observability
  • Lemma 1: see morin2017uniform
  • Proposition 1
  • Proposition 2
  • Proposition 3: Equivariance
  • Proposition 4
  • Proposition 5: Equivariant lift
  • Lemma 2
  • proof
  • Theorem 1
  • ...and 7 more