Identifying and understanding the visual cues that determine relative depth across image contours (i.e., figure-ground organization) are central problems of vision science. In this monograph we report the figural cue of extremal edges (EEs), which arise when an opaque convex surface smoothly curves to occlude part of itself. We also compare classical cues to figure-ground organization with the recently discovered cue of EE. Our results show that EEs are surprisingly powerful pictorial cues to relative depth across a contour, almost entirely dominating the well-known figure-ground cues of relative size, convexity, surroundedness, and shape familiarity. These results demonstrate that natural shading and texture gradients in an image provide important additional information about figure-ground organization that has been overlooked in the past 75 years in research on figure-ground organization.