  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.
