Anterior eye segment surgeries are usually video-recorded. If we are able to efficiently analyze surgical videos in real-time, new decision support tools will emerge. The main anatomical landmarks in these videos are the pupil boundaries and the limbus, but segmenting them is challenging due to the variety of colors and textures in the pupil, the iris, the sclera and the lids. In this paper, we present a solution to reliably normalize the center and the scale in videos, without explicitly segmenting these landmarks. First, a robust solution to track the pupil center is presented: it uses the fact that the pupil boundaries, the limbus and the sclera / lid interface are concentric. Second, a solution to estimate the zoom level is presented: it relies on the illumination pattern reflected on the cornea. The proposed solution was assessed in a dataset of 186 real-live cataract surgery videos. The distance between the true and estimated pupil centers was equal to 8.0 ± 6.9% of the limbus radius. The correlation between the estimated zoom level and the true limbus size in images was high: R = 0.834.