Three Gradients and the Perception of Flat and Curved Surfaces

SUMMARY Researchers of visual perception have long been interested in surfaces. Most psychologists have been interested in the perceived slant of a surface and in the gradients that purportedly specify it. Slant is the angle between the line of sight and the tangent to the planar surface at any point, also called the surface normal. Gradients are the sources of information that grade, or change, with visual angle as one looks from one's feet upward to the horizon. The present article explores three gradients—perspective, compression, and density— and the phenomenal impression of flat and curved surfaces. The perspective gradient is measured at right angles to the axis of tilt at any point in the optic array; that is, when looking down a hallway at the tiles of a floor receding in the distance, perspective is measured by the x-axis width of each tile projected on the image plane orthogonal to the line of sight. The compression gradient is the ratio of y/x axis measures on the projected plane. The density gradient is measured by the number of tiles per unit solid visual angle. For flat surfaces and many others, perspective and compression gradients decrease with distance, and the density gradient increases. We discuss the manner in which these gradients change for various types of surfaces. Each gradient is founded on a different assumption about textures on the surfaces around us. In Experiment 1, viewers assessed the three-dimensional character of projections of flat and curved surfaces receding in the distance. They made pairwise judgments of preference and of dissimilarity among eight stimuli in each of four sets. The presence of each gradient was manipulated orthogonally such that each stimulus had zero, one, two, or three gradients appropriate for either a flat surface or a curved surface. Judgments were made for surfaces with both regularly shaped and irregularly shaped textures scattered on them. All viewer assessments were then scaled in one dimension. Multiple correlation and regression on the scale values revealed that greater than 98% of the variance in scale values was accounted for by the gradients. For the flat surfaces a mean of 65% of the variance was accounted for by the perspective gradient, 28% by the density gradient, and 6% by the compression gradient. For curved surfaces, on the other hand, a mean of 96% of the variance was accounted for by the compression gradient, and less than 2% by either the perspective gradient or the density gradient. There were no differences between results for surfaces with regularly shaped and irregularly shaped textures, demonstrating remarkable tolerance of the visual system for statistical variation. The differential results for the flat and curved surfaces suggest independent channels of information that are available in the optic array to observers for their use at different times and in different situations. We argue that perspective information seems to be most important for flatness judgments because that information is a component of an invariant available to'viewers about flat surfaces. We also argue that compression is important for curvature judgments because it reveals potential nonmonotonicities in change of slant, the angle between line of sight and a line orthogonal to the local surface plane. In Experiment 2 we show that when the height of a curved surface is diminished enough to create a nearly monotonic compression function, viewers cannot distinguish such a surface from one that is flat. Finally, we suggest two things with regard to the existing literature on surface perception. First, although psychologists have been very interested in the perception of slant for a flat surface, we argue that slant is a largely irrelevant variable for the perception of flatness. That is, slant information is functionally related to compression gradients, and compression