Extremal Egdes: A powerful cue to Depth & Figure-Ground Organization

with Stephen E. Palmer (UC Berkeley)

Which side is closer? Left/Right?

Figure-1A: Two adjacent globally flat surfaces

Which side is closer? Left/Right?

Figure-1B: Two adjacent curved surfaces

The task is of decide “which side (left/right) is closer?” is very hard in Fig.1A but not in Fig 1B even though the 3D surface convexities are similar on the two sides for both the images. The goal of this project is to find the factor that strongly biases one side to appear closer in Fig.1B (These images were made using POVRAY)

Figure-Ground Organization 
When opaque objects at different environmental distances are optically projected onto a 2D surface so that their projections share an image contour, the laws of optics dictate that the shared image contour belongs to the object that is closer to the point of observation. Because such optical projections lose the spatial dimension of depth, however, the visual system must determine which side of the 2D projection “owns” the shared contour using features of the image regions to determine which side is closer.

The goal of this project is to identify and empirically study of a novel cue to figure-ground organization, the extremal edges.


Extremal edges (EEs) are defined as the set of surface points whose sight lines are tangent to the surface. EEs are viewpoint-specific horizons of self-occlusion on smoothly curved convex surfaces.

Ecological Constraints

An ecological analysis of viewpoint constraints suggests that an EE surfaces are likely to be closer to the observer than non-EE surfaces on the other side.

Four experiments examined whether EEs and 3-D surface convexity operate as strong cues to depth and figure-ground perception.

Experiment 1 used simple luminance profiles (e.g., the positive half of a sinusoid) to simulate shading gradients in simple bipartite displays. The results showed that observers are very likely to perceive both convex surfaces and EEs as closer and figural, but EEs are more potent than surface convexity alone.

The prediction that EEs should appear closer based on general viewpoint argument depends on the scene geometry only and not the optical information for EE. Therefore, the bias toward perceiveing EE as closer should hold regardless of the channel used to specify the surface property. Experiment 2 showed similar effects when EEs were rendered via texture gradients of checkerboard surfaces that contain neither shading nor occlusion cues.

Experiment 3 used shading gradients in ray-traced images of surfaces of revolution to study the effects of EEs versus other individual figure-ground cues (region size, 2-D edge convexity, surroundedness, and familiarity). The results show that EEs dominate all of these factors.

Experiment 4 used ray-traced images of EE shading patterns on simpler convex surfaces ("pillows") to control for thr dark and light horizontal striations present in the stimuli in Experiment 3.EE effects still dominated the other cues studied (2-D edge convexity, size, and their combination).

The data clearly demonstrate that extremal edges are among the most powerful cues to depth across a contour and to figure-ground organization.