Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlinear constructive observer design for direct homography estimation

Tarek Bouazza, Pieter Van Goor, Robert Mahony, Tarek Hamel

TL;DR

This work tackles direct, image-based estimation of planar homographies by formulating images as maps on the sphere and exploiting an induced ${\\mathbf{SL}}(3)$ action on image maps. It proposes a nonlinear observer with state $H \in {\\mathbf{SL}}(3)$ and group error $E=\hat{H}H^{-1}$, minimized via a direct image error $\\mathcal{F}(E)=\frac{1}{2}\|I^e-\\mathring{I}\|^2$, and proves local asymptotic stability using a Lyapunov argument with $\\dot{\\mathcal{F}}(E)=-\\frac{1}{k_{\\Delta}}|\\Delta|^2$. The approach is extended to partly known velocity by augmenting the state with $\\hat{\\Gamma}$ and a joint cost $\\mathcal{F}^*$, showing stability under slow motion assumptions. Simulations on real images demonstrate rapid convergence of the homography estimate and image mismatch, supporting practical viability for direct, geometry-informed visual estimation.

Abstract

Feature-based homography estimation approaches rely on extensive image processing for feature extraction and matching, and do not adequately account for the information provided by the image. Therefore, developing efficient direct techniques to extract the homography from images is essential. This paper presents a novel nonlinear direct homography observer that exploits the Lie group structure of $\mathbf{SL}(3)$ and its action on the space of image maps. Theoretical analysis demonstrates local asymptotic convergence of the observer. The observer design is also extended for partial measurements of velocity under the assumption that the unknown component is constant or slowly time-varying. Finally, simulation results demonstrate the performance of the proposed solutions on real images.

Nonlinear constructive observer design for direct homography estimation

TL;DR

This work tackles direct, image-based estimation of planar homographies by formulating images as maps on the sphere and exploiting an induced action on image maps. It proposes a nonlinear observer with state and group error , minimized via a direct image error , and proves local asymptotic stability using a Lyapunov argument with . The approach is extended to partly known velocity by augmenting the state with and a joint cost , showing stability under slow motion assumptions. Simulations on real images demonstrate rapid convergence of the homography estimate and image mismatch, supporting practical viability for direct, geometry-informed visual estimation.

Abstract

Feature-based homography estimation approaches rely on extensive image processing for feature extraction and matching, and do not adequately account for the information provided by the image. Therefore, developing efficient direct techniques to extract the homography from images is essential. This paper presents a novel nonlinear direct homography observer that exploits the Lie group structure of and its action on the space of image maps. Theoretical analysis demonstrates local asymptotic convergence of the observer. The observer design is also extended for partial measurements of velocity under the assumption that the unknown component is constant or slowly time-varying. Finally, simulation results demonstrate the performance of the proposed solutions on real images.
Paper Structure (11 sections, 5 theorems, 52 equations, 3 figures)

This paper contains 11 sections, 5 theorems, 52 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary material
  3. Notation
  4. Special Linear Group ${\mathbf{SL}}(3)$
  5. Definition of Homography
  6. Images as Maps on the Sphere
  7. Observer design
  8. Homography Observer Design
  9. Observer design with partially known velocity
  10. Simulation results
  11. Conclusion

Key Result

Proposition 1

The function $\phi: {\mathbf{SL}}(3) \times {\mathbb S}^2 \mapsto {\mathbb S}^2$ given by is a right group action of ${\mathbf{SL}}(3)$ on ${\mathbb S}^2$.

Figures (3)

  • Figure 1: Time evolution of the homography estimation error (blue) and image intensity error divided by the number of pixels (red) of the first simulation.
  • Figure 2: Reference image $\mathring{\mathcal{I}}$, warped image $\mathcal{I}^e$ and difference image $\tilde{\mathcal{I}}$ at different time intervals.
  • Figure 3: Time evolution of the homography estimation error (blue), partial velocity estimation error (black) and image intensity error (red) of the second simulation.

Theorems & Definitions (11)

  • Remark 1
  • Proposition 1: hua2020nonlinear
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • ...and 1 more