Testing optimal Gaussian cue combination models with possibly correlated depth cues

Estimates of depth from different depth cues are often modeled as Gaussian random variables that are typically assumed to be uncorrelated. Whether the cue estimates are correlated or not, the optimal cue combination rule is a weighted average of cue estimates whose weights can be computed directly from the variances of the cue estimates and their correlations. We test whether human cue combination performance is consistent with the optimal Gaussian rule with possibly non-zero correlation. Stimuli were slanted planes defined by linear perspective (a grid of lines) and texture gradient (diamond-shaped texture elements). The observers's task was to adjust the slant of the plane to 75 degrees. Feedback was provided after each setting and the single observer trained extensively until her setting variances in all conditions stabilized. We chose a HIGH and LOW variance version of each cue type and measured setting variability in the four single-cue conditions (LOW, HIGH for each cue) and in the four possible combined-cue conditions (LOW-LOW, LOW-HIGH, etc) in randomized order. We first fit the uncorrelated Gaussian model to the data by maximum likelihood estimation of its parameters. The results reproduced the observer's setting variances in all conditions to within 9% on average. However, there was an evident pattern in the deviations between fit and data: the observer's variances in the combined cue conditions were higher than would be expected given the variances in the single cue conditions (nested hypothesis test, p < 0.01). When we refit the model with correlation as a free parameter, the deviations between model and data were 4% on average (the largest deviation was 10%), but the same pattern of deviations was present (nested hypothesis test, p < 0.02). Our results indicate that, while the observer's performance was close to that predicted by the Gaussian models, the observer did not combine information from cues as efficiently as either model would predict.