Fig 1.
The geometry of the Eye Vergence Angle (EVA) while fixating on a near and far target.
A closer target is associated with a larger EVA, and a farther target is associated with a smaller EVA.
Fig 2.
(a) The visual stimulus was a Landolt C resolving to a fixation cross, resulting in a four-alternative forced choice visual discrimination task.
(b) Pupil Labs eye tracker custom mounted on a Microsoft HoloLens 2 display. (c) Real stimuli presented on monitors at four different depths. (d) Virtual stimuli presented at the same depth and position for the AR and VR environments. Note that the captured image does not reflect the actual perceptual experience of the user. (e) Participant performing the experiment in the real environment. (f) Participant performing the experiment in the VR environment.
Table 1.
The experiment placed targets at 4 different depths, resulting in 12 different vergence eye movements. Measurements are expressed in meters (m) and diopters (D).
Fig 3.
Eye Vergence Angle (EVA) calculation and an example of unprocessed eye tracking data.
(a) The participant needs to rotate the head and eyes in different amounts to focus on objects at different distances, such as 0.25 m and 4.0 m (b). EVA is calculated from the 3D gaze direction vectors of the left and right eye. (c) An example of unprocessed eye tracking data at 0.25 m (4.0 D) and 4.0 m (0.25 D) for a participant. The vertical dotted line at 0 seconds shows fixation onset, which occurs approximately 250 ms after stimulus onset, and before the participant pressed the button. This example shows the difference in EVA values for the experiment’s two extreme depths (near and far) in the real, AR, and VR environments. For EVA analysis, a time window between 1 and 2 seconds after the fixation onset was used.
Fig 4.
The Eye Vergence Angle (EVA) as predicted by (a) environment and eye movement end depth in meters, and (b) as predicted by eye movement end depth in .
When expressed in diopters (D), the geometry of the eye vergence angle (Fig 1) results in a linear relationship between EVA and eye movement end depth. This can be seen in (b), where a linear model is fitted to the data, with the parameters of intercept (a), slope (b), and percentage of explained variation (R2).
Fig 5.
The EVA as predicted by eye movement end depth and participant (data from Fig 4b).
The linear model for each participant is added. Note the large variation in intercept (a) compared to slope (b).
Fig 6.
Predicting the eye vergence angle by end depth (Hypothesis H1) and environment (Hypothesis H2): reduction of the maximal model to the fitted model.
The reduction uses the technique described by Kuznetsova et al. [58]. The random effects section tests whether the participant term should be removed, but finds that the amount of described variation would be significantly reduced , and therefore the participant term remains in the fitted model. The fixed effects section finds that the Env:EndDepthD interaction term should be removed
, but that the main effects of ENV and EndDepthD should be retained
. For both the maximal and fixed models, the amount of explained variation (R2) is given, as well as the difference in explained variation between the models (dR2). The meaning of the columns: Eliminated: If 0, the term is retained; otherwise the order in which a term is eliminated. npar: The number of model parameters; less indicates a better (more parsimonious) model. logLik: The log-likelihood for the model. AIC: The Akaike Information Criterion for the model. Smaller is better. LRT: The likelihood ratio test statistic, which is chi-square distributed. DF: The degrees of freedom for the chi-square test. p(>Chisq): The p value for the chi-square test. SumSq, MeanSq, NumDF, DenDF, F.value: The sums of squares, mean square, numerator and denominator degrees of freedom, and F statistic of the F test. p(>F): The p value for the F test. This table is constructed using the R functions lme4::lmer, lmerTest::step, and MuMln::r.squaredGLMM. Degrees of freedom are calculated with the Satterhwaite method [58].
Fig 7.
Analysis of Eq 2, the fitted model testing the effect of end depth (Hypothesis H1) and environment (Hypothesis H2) on EVA.
The model is visualized in Figs 8 and 9. The fixed effects section shows that the effects of both end depth and environment are significant . Overall, the model explains
of the observed variation in the data, with
due to the random effects (participant intercept differences), and
due to the fixed effects (environment and end depth). For each level of environment, Real, AR, and VR, the contrasts section gives the distance of each intercept from the overall participant intercept (Fig 9). The contrasts section also tests the significance of each distance. This table is constructed using the R functions lme4::lmer, lmerTest::summary, lmerTest::anova, MuMln::r.squaredGLMM, emmeans::emmeans, and emmeans::contrast. Degrees of freedom are calculated with the Satterhwaite method [58].
Fig 8.
The EVA as predicted Eq 2, the linear mixed effects model.
The model parameters are shown: the global intercept (a_overall), slope (b), and the intercept offsets for each participant (a_part) and environment (a_Real, a_AR, a_VR). x marks the intercept of each participant, a_overall + a_part. Note the large variation in participant intercepts (a_part) compared to environment intercepts (a_Real, a_AR, a_VR). Also see Fig 9.
Fig 9.
A zoomed-in view of participant 11 from Fig 8.
Here, because of the expanded y axis, the lines from the fitted model can be seen. This shows the main effect of environment on eye visual angle (EVA): relative to Real targets, EVA was smaller for AR targets, and
smaller for VR targets. The contrast coefficients (Fig 7) give these distances. x marks a_part = .79, the intercept for participant 11.
Fig 10.
Predicting the vergence stability hypothesis (H3): reducing the maximal model to the fitted model.
See the caption for Fig 6.
Fig 11.
The vergence stability hypothesis (H3), tested with Eq 3:
The model is illustrated by the lines. Because there are no main effects or interactions with focal switching depth, the lines are horizontal, indicating that when observers fixate a target, the EVA will be stable, regardless of the focal switching depth distance or vergence direction of the eye movement. x marks the intercept of each participant. Also see Fig 12.
Fig 12.
A zoomed-in view of participant 11 from Fig 11.
Here, because of the expanded y axis, the position of the lines from the fitted model can be seen. The lines represent the levels of environment: Real, AR, and VR. Note that the y-position of the lines vary according to the interaction between end depth and environment. The contrast coefficients (Fig 13) give the distance between the grand mean for participant 11 () and each line of the fitted model.
Fig 13.
Analysis of Eq 3: the fitted model testing vergence stability (Hypothesis H3).
The fitted model is visualized in Figs 11 and 12. See the caption for Fig 7.
Table 2.
Correlation between subjective depth (D) and normalized EVA (degrees).
Fig 14.
The natural log ratio of Real target depth over XR target depth, plotted as points against the eye movement end depth, environment (AR, VR), and measure (EVA, Subjective Depth).
Unlike the previous graphs, which plot means with standard error bars, here all 156 data points are plotted. Participants are different colored points connected by lines. Real target depths are considered veridical, and therefore, for this ratio positive values overestimate veridical depth, while negative values underestimate it. The fitted linear model from Eq 4 is also shown: as gray lines surrounded by
confidence intervals. According to this model, subjective depth judgments were underestimated (see the contrasts section of Fig 16), consistent with prior findings. However, EVA measurements were much closer to unity, showing a small and nonsignificant degree of overestimation.
Fig 15.
Predicting the natural log ratio of XR target depth over real target depth (Hypothesis H4): reducing the maximal model to the fitted model.
See the caption for Fig 6.
Fig 16.
Analysis of Eq 4: the fitted model testing whether subjective depth judgments would be underestimated, and that EVA would provide a more veridical depth estimate than subjective depth judgments (Hypothesis H4).
The fitted model is visualized in Fig 14. See the caption for Fig 7.