PSYCHOLOGICAL SCIENCE Research Report CONTOUR COMPLETION AND RELATIVE DEPTH: Petter’s Rule and Support Ratio Manish Singh, 1 Donald D. Hoffman, 1 and Marc K. Albert 2 1 Department of Cognitive Sciences, University of California, Irvine, and 2 Department of Psychology, Harvard University Abstract—The ability to see complete objects despite occlusion is critical to humans’ visual success. Human vision can amodally com- plete visual objects that are partially occluded, and modally com- plete visual objects that occlude other objects. Previous experiments showed that the perceived strength of a completed contour depends on its support ratio: the ratio of the length of the physically speci- fied contour to the total length of the contour. Other experiments showed that human vision prefers to make modal completions as short as possible, an effect known as Petter’s rule. The experiment reported here examined the relationship between Petter’s rule and support ratio, showing that both affect modal completion in figures of homogeneous color, but that when they compete Petter’s rule dominates. Finally, our results confirm that Petter’s rule is an effect of relative gap lengths and not of relative size. Occlusion is a ubiquitous feature of the visual world. Most objects people see are partly hidden behind other objects. Despite this, people see them not as isolated fragments, but as complete, unitary objects. In Figure 1a, for example, the viewer sees a single cat behind a window pane, not four different cat pieces. And in Figure 1b, the unity of the partially occluded object is apparent even though the object is unfa- miliar. This ability to see unitary objects despite occlusion is fortunate, because otherwise fragments and pieces would appear and disappear as people moved around, and humans’ visual world would resemble the “blooming, buzzing, confusion” that William James described. Indeed, this ability is not unique to humans. All visual animals face the occlusion problem, and recent studies show that many readily solve the problem, including chicks (Lea, Slater, & Ryan, 1996; Regolin & Vallortigara, 1995), hens (Forkman & Vallortigara, 1998), and mice (Kanizsa, Renzi, Conte, Compostela, & Guerani, 1993). The ability to complete partially occluded objects has been termed amodal completion (Michotte, Thines, & Crabbe, 1964/1991). The term amodal indicates that the completion has no sensory characteris- tics, such as a perceived brightness gradient. In Figure 1, for example, although one is aware of the unity of the objects behind the window panes, one does not see a brightness gradient, or contour, in the occluded regions. In modal completion (Michotte et al., 1964/1991), which occurs when an object partially occludes a surface of the same color, one does see a brightness gradient along the completed contours. In Figure 2, for example, not only is one aware of completed triangular shapes that occlude the disks, but one also sees a clear difference in brightness between the inside and outside of the completed contours—even though there are, in fact, no gradients in any image property in those regions: In Figure 2a, the inside of the completed triangle looks whiter than the surrounding white; and in Figure 2b, the inside of the com- pleted triangle looks blacker than the surrounding black. Such com- pleted figures, and their associated contours, have been termed “anom- alous,” “illusory,” or “subjective” (see, e.g., Petry & Meyer, 1987; Spillmann & Dresp, 1995). SUPPORT RATIO What determines whether human vision will interpolate between two given edges to construct a completed contour? One factor is that the two edges must be appropriately aligned. In Figure 3, for example, each of the “Pacman” shapes in Figure 2 is slightly rotated, and, as a result, one no longer sees the illusory figures that are so striking in Figure 2. Kellman and Shipley (1991) proposed a precise criterion for edge alignment, termed relatability. According to this criterion, the extensions of the two edges must meet, and their exterior angle of intersection must be acute (see Kellman & Shipley, 1991, pp. 174–176). Although relatability is an all-or-none criterion, it can be extended to a graded measure of edge alignment (Singh & Hoffman, 1999) that can better track the gradedness in psychophysical data. Furthermore, Shipley and Kellman (1992) have proposed that the strength of an interpolated contour depends on its support ratio, that is, the ratio of the length of the physically specified contour (i.e., the contour specified by a luminance gradient) to the total length of the contour (physically specified plus interpolated). In a series of experi- ments, Shipley and Kellman (1992) demonstrated that the support ratio predicts perceived strengths of illusory figures. Figure 4a demon- strates the role of the support ratio. In each display, the length of the physically specified contour is 2r, where r is the radius of the black disks in the display; the total length of a contour is l, the length of the side of the square in the display. Hence, the support ratio is 2r/l. As predicted, the strength of the illusory square increases with r, and decreases with l (Shipley & Kellman, 1992). An important property of support ratio is that it is scale invariant: It does not change if a display is uniformly blown up, or shrunk, so that any unit-formation processes that are based on support ratio will not change unit assignments as the viewer moves toward or away from a scene. By contrast, the length of an interpolated contour is not scale invariant: If the display shrinks to half its size, so does the length of the interpolated contour (see Fig. 4b). Hence, any theory of interpola- tion strength based on absolute length of the interpolated contour would predict that this strength would vary with the viewer’s distance to the display. In their experiments, however, Shipley and Kellman (1992) found that the perceived strength of interpolation is largely independent of scale (but see Dumais & Bradley, 1976). You can see this by comparing Figure 4a with 4b. Hence, scale invariance is a desirable property of the support ratio. PETTER’S RULE Consider the display in Figure 5a, which consists of a single, irreg- ular shape of homogeneous color. One sees this display not as a single VOL. 10, NO. 5, SEPTEMBER 1999 Copyright © 1999 American Psychological Society 423 Address correspondence to Manish Singh, Department of Brain and Cog- nitive Sciences, Perceptual Science Group, NE20-451, Massachusetts Institute of Technology, Cambridge, MA 02139; e-mail: manish@psyche.mit.edu.