Figure 5. Normative model for tilt estimation in natural scenes.

(A) The model observer estimate is the minimum mean squared error (MMSE) tilt estimate ${\displaystyle {\hat{\tau }}_{MMSE}}$ given three image cue measurements. Optimal estimates are approximated from 600 million data points (90 stereo-images) in the natural scene database: image cue values are computed directly from the photographic images and groundtruth tilts are computed directly from the distance data. (B) MMSE estimates for ~260,000 (64³) unique image cue triplets are stored in an ‘estimate cube.’ (C) Model observer estimates for the 3600 unique natural stimuli used in the experiment. For each stimulus used in the experiment, the image cues are computed, and the MMSE estimate is looked up in the ‘estimate cube.’ Excluding the 3600 experimental stimuli from the 600 million stimuli that determined the estimate cube has no impact on predictions. The optimal estimates within the estimate cube change smoothly with the image cue values; hence, a relatively small number of samples can explore the structure of the full 3D space and provide representative performance measures (see Discussion). (D) Proportion variance explained ($R^2$) by the normative model for the summary statistics (estimate counts, means, and variances; Figure 2D–F) and the conditional distributions (Figures 3 and 4). All $R^2$ values are highly significant (${\displaystyle p<{10}^{-6}}$).

Figure 5—figure supplement 1. Unsigned tilt estimates: human observers and normative model.

Each panel shows the unsigned tilt estimate of a human or model observer plotted against the groundtruth tilt of every stimulus presented in the experiment.