Equivariant Observer Design on SL(3) for Image Intensity-Based Homography Estimation

TL;DR

This paper introduces a nonlinear observer on SL(3) for homography estimation that directly uses image intensities, providing a feature-free alternative with proven convergence and improved variants.

Contribution

It develops a novel observer on SL(3) leveraging full image data, with conditions for non-degeneracy, convergence analysis, and a second-order variant incorporating the Hessian.

Findings

The observer achieves local exponential convergence.

The second-order variant improves local convergence.

Simulation results validate the approach on real images.

Abstract

This paper addresses the problem of homography estimation using a nonlinear observer designed on the Lie group $\mathbf{SL}(3)$ that exploits the full image information through direct image registration. Unlike traditional feature-based methods, which rely on extensive feature extraction and matching, the proposed approach formulates an observer that minimises a cost function defined directly in terms of image pixel intensities. Explicit conditions ensuring the non-degeneracy of the cost function are derived, and a comprehensive analysis is conducted to characterise and generate degenerate (unobservable) image configurations. Theoretical results demonstrate local exponential convergence of the observer. To improve local convergence properties, a second-order observer variant is introduced by incorporating the Hessian of the cost function into the correction term. Simulation results demonstrate the performance of the proposed solutions on real images.