Utility of theoretical Hirschberg ratio for gaze position calibration

Gaze position is calibrated in first Purkinje image-based eye trackers using the population-average Hirschberg ratio (HR) that is prone to inaccuracies or using the individual's HR that is cumbersome to obtain empirically. This study investigated (1) the agreement between HR calculated theoretically from the individual's corneal curvature and anterior chamber (AC) depth and those obtained empirically and (2) the contribution of corneal curvature and AC depth in the intersubject variance of the two HRs.

Twenty-four subjects (mean ± SD age, 23.6 ± 3.5 years) fixated monocularly on a light-emitting diode array spanning ±24 degrees of horizontal or vertical gaze angle, in 4-degree steps, at 95 cm viewing distance. Empirical HR was determined using a custom-designed infrared eye tracker as the magnitude of separation between Purkinje image position and entrance pupil center per unit change in angular eccentricity. Theoretical HR was calculated from the subject's corneal curvature and AC depth using the model of Brodie (1987).

Empirical and theoretical HRs for horizontal and vertical gaze directions were well correlated (r ≥ 0.83) and not significantly different from each other (p ≥ 0.23; mean difference [±95% limits of agreement], -0.35 [0.85 to -1.55] degrees/mm for horizontal HR and -0.16 [1.01 to -1.33] degrees/mm for vertical HR). Corneal curvature and AC depth together accounted for greater than or equal to 80% and greater than or equal to 91% of intersubject variance in empirical and theoretical HR, respectively (p < 0.001). Hirschberg ratios changed at -2.3 to -2.8 degrees/mm per millimeter change in corneal curvature and at 2.0 to 2.4 degrees/mm per millimeter change in AC depth.

Theoretical HR calculated from the individual's corneal curvature and AC depth can be used in lieu of the empirical HR for gaze position calibration to within approximately 2 degrees/mm of accuracy. Gaze position accuracy significantly improves by using the theoretical HR, relative to the population-average HR. Corneal curvature and AC depth combined explain the majority of intersubject variability in HR.