Investigation of the Cross-Ratios Method for Point-of-Gaze Estimation

The cross-ratios method for point-of-gaze (PoG) estimation uses the invariance property of cross-ratios in projective transformations. The inherent causes of the subject-dependent PoG estimation bias exhibited by this method have not been well characterized in the literature. Using a model of the eye and the components of a system (camera, light sources) that estimates PoG, a theoretical framework for the cross-ratios method is developed. The analysis of the cross-ratios method within this framework shows that the subject-dependent estimation bias is caused mainly by: 1) the angular deviation of the visual axis from the optic axis and 2) the fact that the virtual image of the pupil center is not coplanar with the virtual images of the light sources that illuminate the eye (corneal reflections). The theoretical framework provides a closed-form analytical expression that predicts the estimation bias as a function of subject-specific eye parameters. The theoretical framework also provides a clear physical interpretation for an existing empirically derived two-step procedure that compensates for the estimation bias and shows that the first step of this procedure is equivalent to moving the corneal reflections to a new plane that minimizes the distance from this plane to the virtual image of the pupil center.