Figure 1.
(a) Example stimuli from the four conditions of Experiment 1. The star indicates which object is the target to be judged. The two-letter label denotes whether the judged object and the non-judged distractor were matte (M) or shiny (S), respectively. (b–d) Proportion of objects perceived as convex, as a function of shading direction (in deg), averaged across observers. The solid lines indicate the fit of the model to the averaged data. Shaded region indicates standard error (SEM) from the mean. (e) Proportion of convex responses for the four conditions, across observers. Error bars indicate ±1SEM across observers. (f) Perceived shape in the SM (red) and MM (black) conditions, and the corresponding mutual information (green diamonds) between the highlight and the perceived shape.
Figure 2.
Geometry of specular highlights for concave stimuli.
(a) We render our stimuli from a scene geometry in which the angle θr between the viewing direction and the rim is greater than the angle θl between the viewing direction and the lighting direction required to produce the visible specular highlight, so that the specular highlight is visible. However, the apparent Lambertian shading pattern is also consistent with the geometry shown in (b), where a deeper surface causes θl to exceed θr, making the specular highlight infeasible. (c) The blue curve shows the family of (depth, illuminant slant) scene solutions consistent with the rendered image (the bas-relief ambiguity). (The depth expansion factor is the ratio of the depth to the half-width of the hemi-ellipsoidal surface.) The red curve shows how the direction to the rim of the surface changes as the surface depth increases. When the illumination slant exceeds the slant of the rim (shaded region), the light source is occluded, making the appearance of a specular highlight infeasible.
Figure 3.
Schematic of illumination configurations and evaluation of simplified models.
a) Schematic of the one- and two-light configurations. b) Bayesian Information Criterion for three simplified models and the full Bayesian model (see Table 1 and Figure 7).
Table 1.
Model parameters.
Figure 4.
Experiment 2 stimuli and data.
(a) 6 examples of the 120 stimulus configurations. For the stimuli in the left and middle columns, most observers will perceive the highlight as a specularity on a shiny object. However, the misaligned highlights in the rightmost column are more often perceived as the result of a local patch of illumination on a matte object. (b) Data averaged across observers. The yellow star indicates the polar angle of the highlight. Black circles and red stars give data for objects with and without a highlight, respectively. Solid lines show the model fit and shaded regions indicate ±1SEM.
Figure 5.
Experiment 2 highlight analyses.
Mutual information (MI) between the presence of a highlight and the reported sign of surface curvature, as a function of the shading and highlight orientation. As this analysis is only possible for observers whose perception is modulated by a highlight, we weight each observer's data by his/her MI(highlight, shape) over both experiments. Green diamonds indicate weighted average over observers, and shaded region indicates ±1SEM. The black line indicates the model fit. (a) MI(highlight, shape) as a function of shading orientation, averaged over highlight location. The highlight only has influence when the illumination is not directly overhead. (b–d) MI(highlight, shape) for three example highlight locations. Mutual information is generally highest when the highlight is consistent with the shading. (e) Variation in the probability assigned by the model to the specular interpretation of the highlight (vs. the local illuminant interpretation) as a function of the angular offset between the highlight and shading gradient direction.
Figure 6.
(a) 4 examples of the 336 stimulus configurations. In the left column, object depth is fixed (depth = ±0.75× half-width) but illuminant slant varies between the top (25°) and bottom (55°) images. In the right column, illuminant slant is fixed (65°) but object depth varies between the top (0.5) and bottom (1) images. Further stimulus examples can be found in Figure S2. (b) The effect of adding a highlight on the perception of surface curvature sign, as a function of illuminant slant, averaged across object depth. The dashed green lines in (b) and (c) give the highlight eccentricity, i.e. distance from the object's centre/object radius. (c) The highlight effect as a function of object depth, averaged across illuminant slant. The data are averaged across the four observers. Error bars indicate ±1SEM.
Figure 7.
Graphical representation of the model.
Shown are the observable variables (rectangles), generative object and illumination components (rounded rectangles) and the model's 9 free parameters (ellipses).
Table 2.
Maximum likelihood parameter estimates for the full model for individual observers.