Extremal edges versus other principles of figure-ground organization

Abstract

Identifying the visual cues that determine relative depth across an image contour (i.e., figure-ground organization) is a central problem of vision science. In this paper, we compare flat cues to figure-ground organization with the recently discovered cue of extremal edges (EEs), which arise when opaque convex surfaces smoothly curve to partly occlude themselves. The present results show that EEs are very powerful pictorial cues to relative depth across an edge, almost entirely dominating the well-known figure-ground cues of relative size, convexity, shape familiarity, and surroundedness. These results demonstrate that natural shading and texture gradients in an image provide important information about figure-ground organization that has largely been overlooked in the past 75 years of research on this topic.

Introduction

When opaque surfaces at different environmental distances share a contour in their 2D projected image, the laws of optics dictate that the shared contour “belongs” to the surface that is closer to the observer. Because the spatial dimension of depth is lost, however, the visual system must determine which side of the edge is closer (and thus “owns” the edge) and which side is farther (and thus extends behind the edge) based on ecologically relevant image features (Figure 1A). This problem is classically called “figure-ground organization” (FGO) but has more recently been called “edge assignment,” “border ownership,” and various other combinations of those terms (e.g., Nakayama, He, & Shimojo, 1995). 

Figure 1

 

Figure-ground organization. (A) When opaque surfaces at different environmental distances are optically projected onto a 2D surface so that their projections share an image contour, the problem of FGO is to determine which of the two adjacent sides is closer and “owns” the shared border. Flat 2D displays traditionally used to study FGO (B and D) miss the rich information present in natural images in the form of shading, highlight, and texture gradients (C and E). Extremal edges predict the correct side as being closer and figural in the natural images (C and E), whereas traditional cues of 2D edge convexity (B) and lower region (D) do not.

Rubin (1921/1958) first identified the problem of FGO and isolated several factors that influence it: All else being equal, regions that are smaller in size, surrounded by another region, higher in contrast, and horizontal or vertical in orientation tend to be seen as figural and closer than the region sharing the edges (Rubin, 1921/1958). Since then, many additional cues to FGO have been discovered, including symmetry (Bahnsen, 1928), parallelism (Metzger, 1953/2006), convexity (Kanizsa & Gerbino, 1976), familiarity (Peterson & Gibson, 1991, 1994), lower region (Vecera, Vogel, & Woodman, 2002), wider base (Hulleman & Humphreys, 2004), and edge-region grouping (Palmer & Brooks, 2008). Classical depth cues, such as binocular disparity and occlusion (T-junctions), can also function as FG cues in interpreting a scene in depth. Recent results demonstrate that even metric cues, such as binocular disparity, combine with seemingly ordinal figural depth cues to produce an integrated metric impression of depth across an edge (Bertamini, Martinovic & Wuerger, 2008; Burge, Fowlkes, & Banks, 2010; Burge, Peterson, & Palmer, 2005; but for an alternative interpretation, see Gillam, Anderson, & Rizvi, 2009). 

All of the previously mentioned FG cues have been studied using images without any of the rich shading and texture information present in natural scenes, however (see Figures 1B–1E). In this article, we show that a great deal of information about FGO can be derived from shading and highlight information around an edge by comparing the cue of “extremal edges” (Palmer & Ghose, 2008; for computational analyses of this cue, see Huggins & Zucker, 2001a, 2001b) with several previously known figural/depth cues. An extremal edge (EE) occurs when an opaque convex surface smoothly curves to partly occlude itself from the current viewpoint. Visual information about the existence of such an EE is generally carried by luminance and/or texture gradients near the self-occluding edge (see Huggins & Zucker, 2001a, 2001b; Palmer & Ghose, 2008). These gradients are relevant to the problem of FGO because the edge of such a curved self-occluding surface almost always belongs to that surface rather than the surface on the opposite side of the edge. EEs have been used previously to produce figural biases, but we know of no prior systematic psychophysical studies of the effects of EEs on FGO in the presence of other known configural cues. Shepard (1990, p. 72) used shading EEs in an ambiguous figure-ground drawing titled “Beckoning Balusters,” Huggins and Zucker (2001a, 2001b, Figure 12c) used it to disambiguate figure-ground in an ambiguous Kaniza figure, and Diane Beck used them to bias the figural side of figure-ground displays in an unpublished study in our laboratory. Also, von der Heydt and Pierson (2006) discussed border ownership in asymmetrical luminance profiles in the watercolor illusion (Pinna, Brelstaff, & Spillmann, 2001), but they did not explicitly discuss EEs as being relevant to their results. Palmer and Ghose (2008) provided the first empirical evidence of the efficacy of EEs in figural depth perception, but they were careful to eliminate all other known configural cues from their displays. 

Zucker and colleagues have called EEs “folds,” following the nomenclature used in topology (e.g., Huggins, Chen, Belhumeur, & Zucker, 2001; Huggins & Zucker, 2001a, 2001b). This label seems to apply more naturally to situations in which gradient patterns define a depth edge between different regions of the same surface, such as folds of cloth in curtains or a full skirt. Palmer and Ghose (2008) used the “extremal edge” terminology previously employed by Barrow and Tennenbaum (1978) in discussing figure-ground issues because it applies more clearly to cases in which the edge in question marks the bounding contour of the object on one side. Nevertheless, what we call an EE is the same as what Zucker and colleagues call a fold, and we acknowledge that not all EEs correspond to object boundaries. 

It is often difficult to judge relative depth across a shared edge from local information when the two adjacent regions are flat (Figure 2A), but it is generally much easier to do so when one side of the shared edge constitutes an EE (left side of Figure 2B). Technically, an EE is defined by the set of points on a smooth convex surface where the lines of sight from a given viewpoint are tangent to the surface (Figure 2). We will refer to an “extremal edge” somewhat ambiguously as referring either to the set of 3D environmental surface points or to the 2D projection of those surface points, relying on context to disambiguate them. We also talk about an EE as including the local region along the edge that provides the gradient in luminance and/or texture needed to perceive it as an EE. 

Figure 2

 

An ecological analysis of EEs based on general viewpoint constraints. (A) It is nearly impossible to judge the relative depth at the shared edge for identical flat surfaces. (B) It quite easy to judge relative depth for curved surfaces, however, if one surface projects an EE along the shared border, even if the two surfaces are identical except for a 90° rotation. Shading gradients alone (C) can be used to produce the image of a curved surface adjacent to a flat (non-EE) surface. Logically, the EE side could be farther (D), closer (E), or touching (F) the non-EE side, as indicated in the plan views. Lightly shaded regions in parts E–F indicate the range of viewpoints from which the scene projects an EE image qualitatively like C. If the non-EE surface is closer (D), moving rightward causes the flat surface to occlude the EE and moving leftward causes a gap between these two surfaces, so the qualitative structure of the image is preserved only along the single line of sight. If the EE surface is closer (E), moving in either direction will preserve the qualitative image structure. If they touch, moving in one direction (here, rightward) reduces to case D and moving in the other direction (here, leftward) reduces to case E. The viewing area consistent with EE images (C) is thus greatest if the EE side is closer (E), implying that perception should favor this outcome.

Palmer and Ghose (2008) recently demonstrated that the presence of an EE along the shared contour is a powerful cue that the region containing the EE is closer to the observer than the opposite side. They used both shading and texture gradients to render the EE along a straight border between equal-sized regions. The ecological argument for the claim that EEs should be perceived as figural and closer is based on a general viewpoint analysis showing that, if one side of a shared edge is extremal, that side is more likely to be closer to the observer than the opposite side (see Figure 2C–2F). 

Consider the possible depth interpretations of the 2D image shown in Figure 2C. Logically, the flat surface is in front of the curved surface, the curved surface is in front of the flat surface, or they touch along the shared edge. If the flat, non-EE region is closer (Figure 2D), moving the viewpoint laterally will change the image qualitatively, either by occluding the EE or by producing a gap, because the alignment of the EE and the edge of the flat surface is highly accidental to that particular viewpoint. If the convex EE surface is closer (Figure 2E); however, changing the viewpoint laterally does not change the qualitative structure of the image: An EE remains visible on one side and no gap arises within a relative broad range of viewpoints, whose scope depends on the geometry of the scene. If the two surfaces touch (Figure 2F), moving in one direction changes the image qualitatively, whereas moving in the other does not. If the human visual system is sensitive to such ecologically relevant probabilities, as theorists as diverse as Helmholtz (1867/1925), Gibson (1950), and Rock (1983) have proposed, perception should be biased toward seeing the EE side as closer. 

Using shading gradients to render surface convexity and EEs in one experiment and texture gradients in another, Palmer and Ghose (2008) created simple displays in which one side was a projection of a convex 3D surface with a straight EE along the shared edge and the other side was either a flat surface or a convex 3D surface with no EE along the shared edge. The results for both shading and texture gradients clearly showed that EEs are powerful cues to depth and FGO across a contour, consistent with the ecological analysis of general viewpoint sketched in Figure 2. 

For clarity, we note that this claim is different than those made in classical shape-from-shading or shape-from-texture analyses, which specify the relative distance to different points on a single surface (e.g., Horn, 1975; Malik & Rosenholtz, 1994). These previous analyses were based on the optical structure of projected images quite independently of general viewpoint considerations. We note, however, that the same considerations that apply to depth at EEs also apply to depth in a single, complex, curved surface in which smooth self-occlusion occurs, such as in a fold of cloth. Zucker and colleagues have investigated the image-based determinants of such cases, referring to them as “folds” on the EE side and “cuts” on the non-EE side (e.g., Huggins et al., 2001; Huggins & Zucker, 2001a, 2001b). Finally, we note that other researchers have previously employed generic viewpoint arguments to explain other aspects of visual perception, such as stereoscopic depth (e.g., Koenderink & van Doorn, 1976), pictorial depth (e.g., Nakayama & Shimojo, 1992), shape from shading (e.g., Freeman, 1994), and figure-ground assignment (e.g., Palmer, 1999, pp. 283–284). 

Although informal, qualitative demonstrations have shown that EEs can influence depth perception in conflict with other figural factors (e.g., Huggins & Zucker, 2001a, 2001b; Palmer & Ghose, 2008; Shepard, 1990), no well-controlled, psychophysical study has compared EEs with other figure-ground cues. The present experiments link EEs to the classical literature on FGO and show that EEs almost completely dominate four powerful, previously known figural factors: smaller size, 2D edge convexity, familiarity, and surroundedness. 

Materials and methods

Experiment 1: Extremal edges vs. size and convexity in matte and specular surfaces

Cues to FGO have traditionally been formulated as ceteris paribus rules: the given factor has the stated effect if all other factors are eliminated or neutralized (Palmer, 1999). The joint effect of two or more cues can only be predicted from such rules when they are all consistent in predicting the same bias. If they conflict, it is unclear how the percept is determined. An important goal of the first experiment was to study how the cues of smaller size and 2D edge convexity combine with EEs in determining a single, unified perception of figural depth. It is of particular interest because EEs seem to dominate traditional figural cues completely, at least when they are correctly perceived as EEs (rather than as cast shadows: see below). Previous research on cue combination in grouping of artificially generated dot lattices showed additive effect for various cues (e.g., Kubovy & van den Berg, 2008). Recent research in computer vision has examined the nature of cue integration for FGO in natural images within local apertures (Fowlkes, Martin, & Malik, 2007). They found that size, lower region, and convexity combined linearly in a logistic regression model. For optimal aperture sizes and their sample of images, small size had the best predictive value for perceived figure, followed by lower region and convexity. 

A secondary goal was to determine whether specular highlights on semi-gloss surfaces would strengthen the perception of EEs as closer and figural (e.g., Figures 1A and 1E). We anticipated that highlights might be more effective because the shading gradient of an EE can be perceptually mistaken as a shadow cast by an adjacent surface. This perceptual ambiguity is generally not present with highlights because they are luminance increments rather than decrements and thus incompatible with cast shadows. 

The displays used by Palmer and Ghose (2008) eliminated all known figural factors other than the presence of the EE by using equal sized regions with a straight edge between them. These displays were ideal for establishing that EEs actually produce figural/depth effects because all other factors are eliminated or neutralized, but they do not address issues concerning their strength relative to other factors or how they combine with those other factors. In the present displays, the 3D curvature of the EE surface was convex but relatively flat except near the edge, at which point it curved smoothly to occlude its far side (Figure 3). This gave rise to “pillow-like” objects for the matte surfaces, which were defined solely by shading gradients, and “pill-like” objects for the specular surfaces, which were defined by both shading and highlight gradients. Flat conditions were also included to measure the effects of smaller size and 2D edge convexity in the absence of an EE. 

Figure 3

 

Sample displays and results for the flat, shading, and highlight conditions in Experiment 1. (A) Flat displays with no shading (light version): For the comparison side of the two-region displays (here, on the left), size decreases from left to right and edge convexity increases from bottom to top. The figural status of the left side should therefore increase from lower left to upper right based on the influence of relative size and edge convexity cues to figural status. The ordinate of the graph below the given display type represents the percentages of trials in which the comparison side (here, the left side) was chosen as figural. The error bars correspond to the standard errors of the means. The abscissa represents the relative size of the comparison (here the left) side. The green, red, and blue curves represent the results for convex, straight, and concave 2D edge contours, respectively. (B) Matte-shading condition (light version): The comparison sides of the samples are the three-dimensionally convex EE surfaces rendered by ray-traced superellipsoids with matte surface properties. (C) Specular-highlight condition (dark version): The comparison sides of the samples shown are convex surfaces rendered by ray-tracing superellipsoids with glossy surface properties that produce specular highlights. The shared edge is an EE for the region on the comparison side of each display (here on the left) in the shading-only and shading-plus-highlight displays but not in the flat displays.

Methods

Participants

Eight students at the University of California, Berkeley, volunteered to take part in the experiment for partial course credit in an undergraduate psychology course. All participants had normal or corrected-to-normal vision, were naïve to the purpose and nature of the experiment, and gave informed consent in accord with the policies of the University of California, Berkeley, Committee for the Protection of Human Subjects, which approved the experimental protocol. 

Apparatus

Displays were generated and presented on a 15-inch LCD screen of a Sony VAIO notebook computer (screen size = 20.5 × 33 cm) with 1280 × 800 pixel resolution. The sequencing and presentation of the displays were controlled by a program written in the MATLAB programming language (Mathworks Ltd.) using routines from Psychophysics Toolbox (Brainard, 1997). The size of the displays was 6.5 × 6.5 cm, which was about 6.5° × 6.5° at a viewing distance of 57 cm in an otherwise dark testing booth. The screen was oriented perpendicular to the line of sight, and the observer's head was stabilized using a chin-rest. 

Design and displays

All displays were two-region images in which one of the sides (the “standard” side) was a flat homogeneous gray surface without any EE gradient whose 2D shape was the complement of the manipulated “comparison” side (see Figure 3). The target side was either flat (no EE), contained an EE defined by shading, or contained an EE defined by highlights and shading. In the flat conditions, both sides of the displays were different shades of gray and were devoid of any shading gradients. The purpose of these conditions was to measure the relative strengths of the flat cues (size and/or edge convexity) on their own and also how they combined with each other (Figure 3A). For other conditions, either shading gradients (Figure 3B) or shading-plus-highlight gradients (Figure 3C) were added to the target side of the displays to render 3D curvature and EEs. 

The complete set of 108 displays was generated by a five-way factorial within-subject design. The first factor was the nature of gradient information: absent (flat, with no shading or highlights), shading (matte surfaces), or shading-plus-highlights (specular surfaces). The second factor was the shape of the edge with respect to the manipulated comparison side (convex, straight, or concave), which also dictated the shape of the homogeneous standard side (concave, straight, and convex, respectively). The third factor was the relative size of the comparison side (smaller, equal, or larger than the flat side), which also determined the relative size of the flat standard side (larger, equal, or smaller, respectively). The fourth factor was the spatial position of the manipulated comparison side (left or right), and the fifth was the luminance level of the manipulated comparison surface (a lighter gray than the standard side, as in Figures 3A and 3B, or a darker gray, as in Figure 3C). 

Two different reflectance characteristics were used to render the EE surfaces: one was strongly Lambertian—i.e., a matte surface of homogeneous isotropic reflectance (Figure 3B)—and the other had higher specularity, resulting in a surface with a semi-gloss sheen that introduced highlights (Figure 3C). The specular highlights made the curvature of the EE surface more unambiguously evident perceptually, and we therefore expected it to produce stronger figural/depth effects at EEs that the shading gradients in the matte surfaces. 

POVRAY (an open-source ray-tracing program) was used to construct the convex EE surfaces. The surfaces of different edge shapes were generated by varying the two parameters (n ₁, n ₂) of superellipsoids. The equation for a superellipsoid is: 

$f(x,y,z)\equiv {[{\left|\frac{x}{r_x}\right|}^{2/n_2}+{\left|\frac{y}{r_y}\right|}^{2/n_2}]}^{n_2/n_1}+{\left|\frac{z}{r_z}\right|}^{2/n_1}$

(1)

(Bourke, 1990). The shape of the shared 2D edge of the surface had the following parameters: (n ₁ = 5.0, n ₂ = 0.25) for concave edges, (n ₁ = 0.25, n ₂ = 0.25) for straight edges, and (n ₁ = 1.0, n ₂ = 0.5) for convex edges. The shared contour was an EE on the shaded side for the matte and specular displays. Adobe Photoshop was used to make 5.0 × 5.0 cm images by putting these convex surfaces next to flat, homogeneous, 50% gray regions. A 6.5 × 6.5 cm square annulus of random noise surrounded this image to differentiate it from the black screen on which the displays were presented during the experimental trials. The bigger, equal, and smaller sized displays occupied 75%, 50%, and 25%, respectively, of the display. 

Different surface properties of the comparison sides were created by varying the diffuse and specular terms in the POVRAY shading model. The light and dark flat displays were made in Adobe Photoshop by stripping the convex surfaces of all shading gradients and filling them with solid grays of 85% and 35% of the maximum luminance, respectively. There were three presentations of each of the 108 displays defined by the design given above. 

Procedure

The nature of figure-ground organization was explained in terms of perceived depth to the participants (“the ‘figure’ appears to be closer to you than the ‘ground’”), who were then told that their task would be to choose the side in each display that appeared closer and figural by pressing one of two keys. The natural left/right response mapping was explained, and they were then given 10 practice trials. The experimental trials were presented in 3 blocks, during which participants were allowed to take a break after any response, if they wished. 

Each trial began with a 6.5 × 6.5 cm random noise field containing a large fixation cross in the center of a black screen. The observer started the presentation of the next display with a key press when he/she was ready. The display was presented for a maximum of 2 sec and was then replaced with a black screen with a random noise field containing a large, centered question mark, which prompted participants to respond by pressing a key corresponding to the region that appeared closer and figural to them in the preceding display. This slide stayed on until the participant responded, although they could respond before the question mark appeared, if they were immediately sure of their perception. As soon as the response was made, a black screen and random noise field containing the large fixation cross indicated the start of the next trial. 

Results and discussion

The data were coded as the percentage of trials on which participants chose the comparison side as closer/figural for the three factors of size, edge convexity, and gradient condition (see Figures 3A, 3B, and 3C). There were no significant main effects of left–right reversals or the surface color (dark/light) (p > 0.08 in all cases). Main effects were present for EE conditions (F(2, 14) = 75.0, p < 0.001), edge convexity (F(2, 14) = 11.7, p < 0.01), and size (F(2, 14) = 8.3, p < 0.01). Although size and edge convexity did not interact reliably (F(4, 28) = 0.53, p > 0.70), EE gradient conditions interacted with both edge convexity (F(4, 28) = 11.9, p < 0.001) and size (F(4, 28) = 4.5, p < 0.01). Because the choice probabilities were so close to 1.0 in several conditions, we performed another ANOVA on the same data following an arcsine transformation (Keppel & Wickens, 2004), but the results were essentially the same as for the raw data. 

To study the nature of cue combination, we performed a logistic regression analysis based on the following logistic function: 

$P(\mathrm{f}\mathrm{i}\mathrm{g}\mathrm{u}\mathrm{r}\mathrm{e}|c\left(p\right))=\frac{1}{1+e^{-{\beta }^{T}c\left(p\right)}}$

(2)

For a given bipartite display, this function takes the linear combination of cues present on the left side of the image (c(p)) after applying a sigmoidal nonlinearity. The output is “0” if the likelihood of the left side being figural is less than 0.5 and “1” if it is greater than or equal to 0.5. The model parameters (β) were fit using the collected data to maximize the log likelihood function. The correct classification rate of each cue and their combination was used as a measure of the predictive power of each cue and their combination. 

The cues we used to predict the results were (a) region size (+1 for smaller, 0 for equal, −1 for larger), (b) region convexity (+1 for convex, 0 for straight, −1 for concave), (c) EE shading (+1 for shading-only gradient, 0 otherwise), and (d) EE highlights (+1 for highlight gradient, 0 otherwise). The β values therefore estimate the relative weights of each cue in predicting perception of the closer figure. This model is generally consistent with a Bayesian formulation of cue integration (e.g., Ernst & Banks, 2002). 

The overall correct classification rate of the logistic regression model for the average data (Figure 3) was 88.5% and included all four cues as reliable predictors. The correlation between the predicted and observed values was +0.988. The β values were 0.35 for size, 1.27 for convexity, 2.52 for EE shading, and 3.76 for EE highlights. The same model was fit to the data for subjects individually, and their β values were analyzed by difference-score t-tests to determine the relative strength of the cue weights. The β estimates for EE highlights were reliably higher than those for EE shading (t(7) = 2.78, p < 0.05), and those for EE shading were reliably higher than for either size (t(7) = 10.43, p < 0.001) or convexity (t(7) = 2.90, p < 0.05), which differed only marginally from each other (t(7) = 2.05, p < 0.08). 

The EE shading-only displays (Figure 3B) were perceptually ambiguous in a way that may account for the fact that EEs were less dominant in this condition than in the highlights condition: The shading pattern along the shared edge can be interpreted either as an EE or a cast shadow. One perception is to see the gradient as the EE of a convex surface on the comparison side (the left side in Figure 3) in front of a flat surface on the homogenous standard side (the right side in Figure 3). The other is to see it as a shadow cast by a closer flat surface on the standard side (the right side in Figure 3) onto a farther flat surface on the comparison side (the left side in Figure 3). 

The ambiguity of the shading-only gradient can most readily be observed in the bottom row of Figure 3B with a concave 2D edge convexity on the shaded side (left side in Figure 3). This alternative interpretation arose most often when the EE side was concave because the convex edge of the flat side produced a strong enough bias toward seeing it as closer that the cast-shadow perception was triggered. Either highlights (Figure 3C) or texture gradients effectively eliminate the cast-shadow interpretation, however. In the EE highlights condition (Figure 3C), for example, the influence of both edge convexity and size were largely overpowered by the EE cues. It therefore appears that unambiguous EEs as rendered by both shading and highlight gradients are so powerful as cues to depth and FGO that they nearly dominate both classical cues. For example, when both regions were flat, the side that was simultaneously smaller and convex (top right display of Figure 3A) produced 90% closer/figural responses. When EEs were depicted on the opposite side using highlights (top right display of Figure 3C), however, the very same classical figural cues produced only 9% closer/figural responses. Caution must be exercised in claiming that EEs “dominate” other cues, however, because each cue is actually a dimension that can vary substantially in strength (e.g., different degrees of convexity and different size disparities). For Cue A to dominate Cue B categorically, every value of A must dominate every value of B, a condition that is not fully met even in the present, lopsided results because edge convexity clearly has some effect in the EE shading conditions, but it is probably because these displays were not always perceived as an EE. 

Experiment 2: Extremal edges vs. configural cues in surfaces of revolution

We conducted a second study to compare the strength of EEs with a wider range of known FG cues, including not only small size and edge convexity, but familiarity and surroundedness. In previous experiments, familiarity and surroundedness have been shown to be particularly strong cues to figural status (e.g., Peterson & Gibson, 1991, 1994; Rubin, 1921/1958). (Surroundedness is necessarily confounded with edge convexity, because the outer contour of any surrounded region must be globally more convex than the inner contour of the surrounding region.) These four factors will be referred to as “flat cues” for simplicity to contrast them with EEs. 

We performed a direct test of the perceptual strength of the flat cues relative to EEs by examining three conditions: the flat cue alone, the flat cue combined consistently with EEs (where both factors biased the same side of the edge), and the flat cue combined in opposition to EEs (where the flat cue and EEs biased opposite sides of the edge). To portray the more complex edge shapes required for familiarity and surroundedness and to generalize the findings of the previous experiment to another kind of convex 3D surface, the present displays depicted surfaces of revolution rendered by ray-tracing software (POVRAY) (see Figure 4). 

Figure 4

 

Sample displays and results for Experiment 2. The top row shows the flat displays that were used to measure the independent effect of each flat cue—Smaller Size, Convexity, Familiarity, and Surroundedness—with no EEs in the display. From the second row downward the two columns for each cue show Consistent conditions, in which the shading gradient is present on the same side biased by the flat cue, and Inconsistent conditions, in which the shading gradient is present on the opposite side. Row 2 shows the EE-visible condition, in which the shared contour is a fully visible EE for the side with a shading gradient. Row 3 shows the EE-occluded condition, in which the evidence for the EE was weakened by occluding 25 pixels along the EE where the shading gradient was steepest. Row 4 shows the flattened condition in which the EE shading gradient was completely eliminated by spreading a thin vertical slice from the center of the SOR throughout the entire region. Row 5 shows the misaligned condition in which the edge of the flat side in the flattened displays was shifted vertically so that the convexities and concavities in the edge were not properly aligned with the horizontal striations. The percentages of trials on which participants chose each side as closer and figural are indicated above the corresponding sides of the display, except for the Surrounded displays, for which the percentages of choosing the central, surrounded region are given above the left side and those of choosing the peripheral, surrounding region are given above the right side. Colored stars within stimulus displays represent significant deviations from chance for each display type (yellow: p < 0.001 and pink: p < 0.05). Stars between stimulus displays represent significant differences between the Consistent and Inconsistent displays for the condition represented in that row (green: p < 0.001, red: p < 0.05). The data in Row 2 show that EEs are powerful enough that they overcome the other cues, as indicated by the results for the Inconsistent conditions. The data in Row 3 show less influence of shading gradients on figural/depth judgments, when information about the presence of the EE is diminished. The influence was even lower for the flattened (Row 4) and misaligned (Row 5) conditions.

A surface of revolution (SOR) is a surface created by rotating a given 2D region bounded by a curve (called the generatrix) around a straight line (the axis). For example, a straight line generatrix rotated about an axis that is coplanar with and parallel to the axis gives rise to a surface of revolution that is a cylinder, and a circular generatrix rotated about one of its diameters gives rise to a sphere, which is another example of an SOR. Many SORs have the appearance of 3D vases with different profiles/outlines. Each cut through the surface perpendicular to the axis of revolution is circular and therefore convex at every tangent point. When SORs are orthographically projected to a viewpoint along a line perpendicular to the axis that passes through the surface's center, they produce an EE whose projected 2D shape is equivalent to the edge of the 2D region from which it was generated. The surfaces of revolution used for this experiment were 3D convex surfaces generated from a flat 2D region whose shape was specified to conform to known configural cues. The only exception to this general rule was the torus, which we used to study the inconsistent condition for the flat cue of surroundedness (see below). A torus is an SOR, but its generator is a circle whose axis of revolution lies outside the circle and coplanar with it. Moreover, to display the torus so that the surrounded “hole” is visible as a circle, the axis of revolution was collinear with the line of sight. 

The surfaces of the rendered objects were predominantly Lambertian and produced shading gradients along the EE consistent with self-occlusion. The first step in creating them was to construct the “flat alone” conditions: two-region displays of the standard type in which a known configural cue biased one side (i.e., the smaller, more convex, more familiar, or surrounded region) over the other. Two different shapes were used for each configural cue, except for surroundedness, where the two instances were both generated from circles that differed only in radius. To generate the “consistent” conditions, the configurally biased side was rotated around the appropriate axis to produce an SOR whose EE had the same shape as the original display on the side biased by the flat cue. We expected that the consistently combined conditions would show stronger figural effects than the flat cue alone. To generate the “inconsistent” conditions (other than the sphere and torus), the non-biased side (i.e., the complement of the biased side according to the flat cue) was rotated around the appropriate axis to produce an SOR whose EE had the same shape as the original flat display on the side opposite to that biased by the flat cue. If the EE is more powerful than the flat cue, observers will perceive the EE side as the figure in these inconsistent conditions more often. If the flat cue is more powerful than the EE cue, the reverse will occur. 

There are factors other than EEs that may be at work in these images, however. One such factor is the presence of horizontal shading striations that result from variations in surface depth orthogonal to the EE. Another is the alignment of those horizontal shading striations with the 2D edge. In an attempt to unravel some of these effects we created four types of displays that manipulated potentially relevant aspects of image structure. In the EE-visible conditions, the entire EE was present in the image and the horizontal striations were properly aligned with the edge. In the EE-occluded conditions, a small (25 pixel) portion of the EE just along the shared edge was eliminated by shifting the flat side horizontally over the outer EE. These images provide reduced evidence of the EE because the steepest portion of the shading gradient was removed from the region adjacent to the shared edge. In the flattened conditions, the EE was entirely eliminated along the shared edge by selecting a single-pixel vertical strip of the projection of the SOR along the axis of revolution and spreading it throughout the entire region, producing images that looked much like an extruded or corrugated sheet of metal. In these displays both the horizontal shading striations and their proper alignment with the shape of the 2D image contour were preserved. In the misaligned conditions, no EE was present and the shading striations were misaligned with the shape of the 2D image contour by shifting the latter vertically while keeping the shading striations in place. 

The flattened and misaligned images contain “gradient cuts” (Ghose & Palmer, in preparation; Huggins & Zucker, 2001a, 2001b), a potentially important factor that actually biases perception toward seeing the side opposite the shaded side as figure. Gradient cuts arise when the shared edge cuts through the equiluminance contours of a shading pattern instead of being roughly parallel to them (as unoccluded EEs are), thus making it appear as if the 2D flat surface is occluding the shaded surface and hence lies in front of it. We therefore expected that the misaligned displays not only would produce the lowest bias toward seeing the shaded side as closer and figural but that this bias might actually be negative in some cases (i.e., that perception in the misaligned conditions might sometimes be biased toward seeing the shaded side as the farther ground). 

Methods

Participants

Eleven students at the University of California, Berkeley, volunteered to take part in the experiment for partial course credit in an undergraduate psychology course. The subject pool for this experiment did not overlap with that for the previous experiment. All participants had normal or corrected-to-normal vision, were naïve to the purpose and nature of the experiment, and gave informed consent in accord with the policies of the University of California, Berkeley, Committee for the Protection of Human Subjects, which approved the experimental protocol. 

Apparatus

The equipment used to generate and display the images was the same as in Experiment 1. 

Design and displays

Each display contained two regions. In the flat conditions, only the flat cue was present (i.e., there was no EE), allowing the strength of each flat cue in isolation to be measured in unshaded 2D displays (Row 1 in Figure 4). These correspond to classical demonstration displays for these four factors, except for the addition of a random noise texture, which was used to counteract the fact that shaded surfaces have a slight figural bias compared to homogenous surfaces (Palmer & Ghose, 2008). 

In all other conditions, the relative strength of EEs versus flat cues to FGO was studied by comparing the results for “consistent” conditions, in which the EE-shaded side was also the side biased by the flat cue (see Figure 4 columns labeled “Consistent”) with “Inconsistent” conditions, in which the EE-shaded side was opposite to that biased by the flat cue (see Figure 4 columns labeled “Inconsistent”). Both the consistent and the inconsistent displays included the four different image conditions described above: EE-visible, EE-occluded, flattened, and misaligned (see Figure 4). 

Procedure

The participants were given instructions similar to those for Experiment 1 in terms of explaining figure-ground and depth. They were asked whether they had any questions and then were given 10 practice trials to learn the response mapping, which required them to press a key (left arrow or right arrow) corresponding to the region that appeared closer and figural to them. For the torus displays using the flat cue of surroundedness, participants were instructed to use the left arrow if the outer (surrounding) region appeared figural and the right arrow if the inner (surrounded) region appeared figural. In all other respects, the procedure was the same as in Experiment 1. 

Results and discussion

The purpose of this study was to compare the strength of EEs with a wider range of previously known FGO cues, including not only small size and edge convexity, but familiarity and surroundedness, both of which were expected to produce stronger figural biases. 

In the flat 2D displays, the percentages of trials on which the side biased by the flat cue alone was chosen as figural are shown in the top row of Figure 4 as judged by eight independent participants who saw only the flat 2D conditions. The probability of choosing the flat cue as figural was significantly greater than chance for all cues except smaller size: t(10) = 1.1, 2.3, 3.3, and 2.2, for size, convexity, familiarity, and surroundedness, p < 0.05, one-tailed. 

Figure 5 shows how the probability of seeing the shaded side as closer changed relative to the corresponding flat condition. It plots the difference between the probability of choosing the shaded side minus the probability of choosing the correspondingly shaped flat side (delta-probability). In this graph, positive values indicate that the shaded side is seen as closer more frequently than the correspondingly shaped flat region, negative values that the shaded side is seen as closer less frequently than the correspondingly shaped flat regions, and zero indicates that there is no difference due to shading. 

Figure 5

 

Results for Experiment 2 relative to baseline (flat) probabilities. The histograms show how the probability of seeing the shaded side as closer changed relative to the corresponding flat (baseline) condition by plotting the difference between the probability of choosing the shaded side minus the probability of choosing the correspondingly shaped flat side. The error bars correspond to the positive standard errors of the means. Positive values indicate that the shaded side is seen as closer more frequently than the correspondingly shaped flat region, negative values indicate that the shaded side is seen as closer less frequently than the correspondingly shaped flat regions, and zero indicates that there is no difference due to shading. The large positive values for EE-visible conditions indicate that the shading pattern corresponding to EEs biased the shaded side to appear figural and closer compared to the corresponding flat side, with the effect being stronger for the inconsistent than consistent conditions. As shading information indicating the presence of an EE is systematically removed in the EE-occluded and flattened conditions, the change measures decrease correspondingly. In the misaligned condition, virtually no overall bias toward the shaded side is evident.

In the consistent trials of the EE-visible conditions (Figure 4, row 2), the presence of the EE dramatically increased the probability that the side with the flat cue was chosen as figural when compared with the flat 2D displays (Figure 4, row 1) for all four configural cues (t(10) = 12.5, 20.1, 21.1, 17.5 p < 0.001, for size, convexity, familiarity, and surroundedness, respectively, as listed in this order from here on). There are also reliable increases in the inconsistent conditions for each cue relative to the corresponding side of the flat displays (t(10) = 14.6, 60.5, 13.0, 6.2, p < 0.001) that are even larger than for the consistent conditions (t(10) = 5.8, 14.6, 8.5, 3.2, p < 0.01). Indeed, inspection of the EE-visible data in Figure 4 shows that the probability of choosing the shaded side when combined consistently with the flat cue was not significantly different from the probability of choosing it when it was combined inconsistently with the flat cue for any cue (t(10) = 0.8, −0.8, 1.1, 1.4, p > 0.05, one-tailed). Fully visible EEs along the shared contour of surfaces of revolution are thus so powerful as a cue to depth and figural status that the consistency of flat configural cues did not produce significant effects. 

In the EE-occluded conditions (Figure 4, row 3), where 25 pixels were occluded on the shaded side of the edge, there was still a measurable increment in the probability of seeing the shaded side as closer and figural for all cues in the inconsistent conditions relative to the corresponding side of the flat displays (t(10) = 8.2, 14.0, 7.4, 3.9, p < 0.05) and all cues except size in the consistent conditions (t(10) = 4.6, 4.7, 8.6, p < 0.01). The difference between the consistent and inconsistent conditions was significant for all cues (t(10) = 5.6, 4.5, 6.9, 1.9, p < 0.05 one-tailed), indicating that when the steepest part of the EE shading gradient was removed, the flat cues had a measurable influence. The effects of the shading gradients in the EE-occluded conditions were reduced relative to the EE-visible conditions in 7 of 8 conditions (all except convexity; see Figure 4, rows 2 vs. 3). This reduction was greatest for the size-consistent condition, probably because occluding 25 pixels from the small SOR eliminated a larger percentage of the shading gradient than in the other conditions. All of these effects indicate that partly occluding the shading gradient along an EE reduces, but seldom eliminates, the strong bias toward seeing the shaded side as closer and figural. 

The flattened conditions eliminated the EE gradients parallel to the edge entirely but maintained the horizontal striations that indicated surface curvature orthogonal to the curvature of the EE. As expected, the influence of shading was reduced in the flattened conditions (Figure 4, row 4) relative to the EE-visible conditions (Figure 4, row 2) in all 8 conditions, being significantly different for all inconsistent conditions (t(10) = 3.6, 3.9, 3.4, 3.7, p < 0.01), and for all consistent conditions except familiarity (t(10) = 4.8, 3.2, 3.1, p < 0.01). These effects may not be due entirely to eliminating EE gradients, however, because gradient cuts (Ghose & Palmer, in preparation; Huggins & Zucker, 2001a, 2001b) are present along the shared edge in the flattened conditions. The flat surfaces may therefore tend to have been seen as closer because the horizontal shading gradients ended abruptly at the edge, as though occluded by the flat region on the other side. 

The purpose of the misaligned condition was to examine the importance of the alignment between the structure of the flat 2D edge and the dark horizontal striations. Here the misalignment of the shared edge with the characteristic shading striations of an SOR makes the evidence of occlusion of the shading stripes by the adjacent region even stronger than in the flattened conditions. As expected, the bias toward seeing the shaded side as figural in the misligned conditions was less than in the flattened conditions in all six relevant cases (misalignment is meaningless for surroundedness because no shading information is present in the flattened condition). These results suggest that the alignment of the 2D edge convexities and concavities with the horizontal shading striations are important for the shaded side to appear figural. Subsequent studies exploring such alignment effects have confirmed their importance in perceiving figural depth (Ghose & Palmer, 2007, 2008, in preparation). 

We modeled the delta-probability results shown in Figure 5 using linear regression through the origin with 9 binary predictor variables. We chose a zero-intercept model because the probability change from the flat (baseline) condition should be zero if there is no effect of any factor other than the flat cue, and the effects of the flat cues themselves were eliminated in computing the delta-probability measure. Any significant deviation from zero thus indicates the figural bias of the corresponding shading gradient condition (EE-visible, EE-occluded, flattened, or misaligned), the consistency of flat and shading information, or the type of flat cue present in the display (size, convexity, familiarity, and surroundedness). A no-intercept linear regression analysis showed that six predictors accounted for 92% of the variance in the 30 data points: EE-visible accounted for 44% of the variance (F(1, 29) = 22.5, p < 0.001), EE-occluded for an additional 24% (F(1, 28) = 20.2, p < 0.001), consistency for an additional 7% (F(1, 27) = 7.0, p < 0.05), flattened for a further 13% (F(1, 26) = 27.6, p < 0.001), misaligned for 1% more (F(1, 25) = 3.0, p > 0.05), and Convexity for a final 3% (F(1, 24) = 11.6, p < 0.05). The only flat cue that accounted for a significant percentage of variance in the delta-probability data was convexity. It seems that the data obtained for the flat displays with convexity somehow underestimated the strength of convexity as a figural cue. The coefficients of the regression analysis were as follows: EE-visible: 0.55 (t(24) = 12.4, p < 0.001), EE-occluded: 0.43 (t(24) = 9.8, p < 0.001), consistency: −0.28 (t(24) = −7.2, p < 0.001), flattened: 0.25 (t(24) = 5.7, p < 0.001), misaligned: 0.05 (t(24) = .97, p > 0.05), convexity: 0.15 (t(24) = 3.4 p < 0.01). The regression model reconfirmed that the presence of a shading gradient corresponding to an EE changed the figural bias of the corresponding side of a bipartite display and that the influence of the shading gradient declines as shading information corresponding to an EE is removed from the displays. 

Experiment 3: Effects of degree of occlusion and compression

Experiment 2 showed that the figural bias along an EE is diminished when evidence of it is reduced by occluding 25 pixels from the steepest part of the shading gradient on the convex, curved side. Our decision to occlude 25 pixels was based simply on our introspections about the effects of this much occlusion, which was sufficient to reduce the figural bias of EEs, while leaving enough of the gradient intact to signal the presence of curvature. Several problems arise in interpreting the results of this occlusion condition, however, due to other factors that were not fully controlled. First, occluding part of the shading gradient also eliminated some of the area on the occluded side, making it smaller. The difference in relative region size in the occluded condition is potentially relevant because smaller size is a well-known cue to figural status (Rubin, 1921/1958). Second, the remaining gradient covered a smaller area as well as containing less of the gradient. This is relevant because the reduction in figural responses we measured might have been due simply to the smaller area covered by the shading gradient rather than eliminating part of the gradient itself. Third, the surfaces of revolution studied in Experiment 2 produced complex shading gradients that included horizontal shading striations arising from variations in surface depth orthogonal to the EE gradient (see Figure 4 row 4 and row 5). These striations may well have influenced the figural bias of the shaded side in ways that are independent of simple occlusion. 

The present experiment examines the effect of occluding part of an EE shading gradient more thoroughly by eliminating the confounds mentioned above and by measuring the figural bias quantitatively as a function of the amount of occlusion, which is progressively increased from 0 pixels (0%) to 70 pixels (58%) of a shading gradient that spanned a total of 120 pixels. To eliminate relative region size as a factor, we equated the sizes of the two sides of the display. This was accomplished by filling in as many pixels on the far (non-shared) side of the gradient as were removed on the central (shared) side (Figures 6C and 6D). To ensure that any effects are due specifically to eliminating the given portion of the EE gradient, rather than simply reducing the area covered by a gradient, we compared the effects of gradient occlusion with the effects of gradient compression. Compression equated the reduction in the area of the compressed gradient with that of the corresponding partly-occluded gradient—from 0% to 58% of the area of the full, 120-pixel shading gradient—without eliminating any of the gradient itself. We expected that the figural bias of the shaded side would be significantly reduced by progressive occlusion of the EE shading gradient (see Figures 6D and 7D), but would not be reduced by corresponding compression of the gradient, because there is essentially full evidence of the EE along the shared edge in the compression conditions (Figures 6C and 7C). Finally, to eliminate possible effects of the striations in Experiment 2, we used simple quarter-cylinder gradients in the present study that produce a simple shading gradient along a straight central edge with no orthogonal striations. 

Figure 6

 

Sample displays and results for the shading-only conditions in Experiment 3. (A) Sample display with a 10% gray and white textured checkerboard on one side and a 0-pixel occlusion/compression version of a quarter-cylinder gradient rendered by shading only on the other side of the shared edge. (B) The ordinate of the graph represents the percentages of trials in which the gradient side (here, the left side) was chosen as figural. The error bars correspond to the standard errors of the means. The abscissa represents the amount of reduction in terms of the number of pixels eliminated from the gradient side (here, on the left). The data with square symbols represent the results of gradient compression and the data with circular symbols represents the results of gradient occlusion. The blue curve and the equation show the best fitting polynomial to model the effect of amount of occlusion on figural responses. (C) Compression condition: The spatial extent of the gradient was decreased without removing any part of it, so that the evidence for an EE along the shared edge on the shaded side was intact. The gradient side is compressed by 10 to 70 pixels and the farther side was filled by a homogeneous gray strip. (D) Occlusion condition: Corresponding numbers of pixels were removed from the steepest part of the gradient along the shared edge, thereby reducing evidence for the presence of an EE. The size of the gradient side was equated for all sizes of occlusion by adding a homogeneous gray strip along the far side of the display.

Figure 7

 

Sample displays and results for the shading + texture conditions in Experiment 3. (A) Sample display with a flat gray-and-white textured checkerboard on one side and a 0-pixel occlusion/compression version of a quarter-cylinder gradient rendered by shading gradient with a correlated texture gradient on the other side of the shared edge. (B) The ordinate of the graph represents the percentages of trials in which the gradient side (here, the left side) was chosen as figural. The error bars correspond to the standard errors of the means. The abscissa represents the amount of reduction in terms of the number of pixels eliminated from the gradient side (here the left). The data with square symbols represent the results for the compression conditions and the data with circular symbols represent the results for the occlusion conditions. The blue curve and the equation show the best fitting polynomial to model the effect of occlusion size on figural bias. (C) Examples of the compression condition gradients. (D) Examples of the occlusion condition gradients. (Note: The full displays were square, as illustrated in part A, and thus twice as tall as they are shown in parts C and D. Also, compressions and occlusions of 20, 40, and 60 pixels are not shown in this figure due to space limitations.)

In Experiments 1 and 2, the displays with EEs were rendered exclusively with shading gradients. As we argued in the discussion of Experiment 1, however, depth information from EE shading gradients can be ambiguous because the same luminance change can also be perceptually consistent with a shadow cast by an adjacent flat surface that is closer to the observer (see Figure 3B). Therefore, we included a second gradient condition (shading + texture) in which we added an unambiguous texture gradient to the shading gradient to specify that the two correlated gradients were due to the presence of an EE. We therefore expected that the figural bias toward the shaded side in the shading + texture conditions would be greater than for the corresponding shading-only conditions. 

Methods

Participants

Fourteen students at the University of California, Los Angeles, volunteered to take part in the experiment for partial course credit in an undergraduate psychology course. All participants had normal or corrected-to-normal vision, were naïve to the purpose and nature of the experiment, and gave informed consent in accord with the policies of the University of California, Los Angeles, Committee for the Protection of Human Subjects, which approved the experimental protocol. 

Apparatus

The equipment used to generate and display the images was the same as in Experiment 1. 

Design and displays

The complete set of 64 displays was generated by a four-way factorial within-subject design: gradient cue type (shading-only or shading + texture), gradient reduction type (occlusion or compression), reduction amount (from 0 to 70 pixels occluded or compressed), and spatial position of the gradient side (left or right). The complete set of displays with the gradient side on the left side is shown in Figures 6C and 6D for the shading-only cue and Figures 7C and 7D for shading + texture cue. 

Each display consisted of one flat side that contained a checkerboard texture and one shaded side that contained either a simple shading gradient (the shading-only conditions) or the same shading gradient with an additional, correlated texture gradient. The shaded and flat sides were rendered separately in POVRAY and combined in Adobe Photoshop. The full shading gradient was consistent with the image of a quarter cylinder (Figure 6A). Two different conditions were used to render the curvature of the surface on the shaded side: shading-only (Figure 6A) and a combination of shading and texture (Figure 7A). The surfaces of the rendered objects were predominantly Lambertian and produced shading gradients along the EE consistent with self-occlusion. The texture gradients consisted of a checkerboard of alternating dark-gray (10% of maximum luminance) and white (100% of maximum luminance) squares. To ensure that size cues did not favor the EE region, the square checks on the flat regions were the same size as the largest checks on the shaded region and thus substantially larger than the smaller checks along the shared border on the shaded side. To minimize occlusion cues to depth, the texture elements were aligned along the central contour so that the edges of the individual texture elements of the flat region fell along the straight line of the contour such that there was no evidence of partial occlusion of texture elements on either side. 

Procedure

The participants were given instructions similar to those for Experiment 1 in explaining the task of choosing the side that appeared closer and figural. They were asked whether they had any questions and then were given 10 practice trials to learn the response mapping, which required them to press a key (left arrow or right arrow) corresponding to the region that appeared closer and figural to them. In all respects, the procedure was the same as in Experiment 1. 

Results and discussion

The primary purpose of this experiment was to study how the strength of the figural bias arising from a simple EE would diminish with progressive occlusion of the steepest portion of its gradient. Because occluding such gradients reduces their overall area as well as eliminating a portion of the gradient, we included a compression condition in which displays were compressed by the same number of pixels as their occluded counterparts, leaving the gradient itself intact. 

The percentage of trials on which the shaded side was chosen as closer and figural is plotted in Figure 7B as a function of number of pixels occluded or compressed for the shading-only and shading + texture gradients. Main effects were present for gradient cue type, with shading + texture producing stronger figural biases than shading-only (F(1, 13) = 8.0, p < 0.05), gradient reduction type, with compressed gradients showing stronger figural biases than occluded gradients (F(1, 13) = 33.9, p < 0.001), and amount of reduction, with smaller reductions producing larger figural biases than larger reductions (F(7, 91) = 7.2, p < 0.001). Although gradient cue type and gradient reduction type did not interact reliably (F(1, 13) = 3.6, p > 0.07), amount of reduction did interact with gradient cue type (F(7, 91) = 5.3, p < 0.001) and with gradient reduction type (F(7, 91) = 8.6, p < 0.001). There was also a small, but significant, three-way interaction (F(7, 91) = 3.1, p < 0.05). 

For the compressed conditions (Figures 6C and 7C), there were no significant effects other than a marginal increment in the figural bias for the shading + texture gradient over the shading-only gradient (F(1, 13) = 4.48, p = 0.05). As predicted, however, there were no significant differences in the magnitude of the figural bias as a function of the amount of compression for either the shading-only condition (F(7, 91) = 2.0, p > 0.07) or for the shading + texture condition (F(7, 91) = 1.3, p > 0.27). Compression presumably had no systematic effect because it did not eliminate any of the EE gradient; it merely increases its slope. (More extreme amounts of compression would eventually have an effect, of course, when the gradient itself can no longer be detected.) For the occluded conditions (Figures 6D and 7D), however, the magnitude of the figural bias decreases significantly for both the shading-only condition (F(7, 91) = 12.5, p < 0.001) and the shading + texture condition (F(7, 91) = 3.0, p < 0.01). The significant three-way interaction among the type of reduction (occlusion vs. compression), gradient type (shading-only vs. shading + texture), and amount of reduction (in pixels) arises because the amount of reduction for the occlusion conditions are greater for the shading-only gradients than for the shading + texture gradients. In both cases, the occlusion curves contained a linear and a quadratic component. For the shading-only conditions, these two components together accounted for 94% of the variance, and for the shading + texture conditions they accounted for 93% of the variance. It is worth noting that the “neutral” or “chance” level of response for the shading-only condition is probably lower than 50% because the flat side contained a high-contrast checkerboard texture, which is more likely to be seen as figural than an untextured gray region with a flat gradient. This means that we cannot claim that the figural bias is eliminated when more than half of the gradient is occluded because chance performance would probably be lower than 50%. 

The data from Experiment 3 support several conclusions. First, when the entire gradient of an EE is present, as it was in all of the compressed conditions, no changes in the magnitude of the figural bias were evident as a function of the amount of compression. This fact suggests that the steepness of the gradient per se did not influence the perception of relative depth across the adjacent regional contour. (It is possible that there was an effect of gradient slope that was exactly offset by the reduction in the area covered by the gradient, but this seems unlikely.) Second, when the steepest part of the gradient was eliminated, as in the occlusion conditions, its figural bias decreased monotonically with the amount of gradient occluded. Third, adding the monocular depth cue of a correlated texture gradient to a shading gradient increased the net figural bias, presumably by diminishing alternative interpretations of the shading gradient as resulting from a shadow cast by the adjacent surface. And fourth, the effects of occluding the gradient are more pronounced for the shading-only condition than for the shading + texture condition This interaction may also be due to the texture gradient disambiguating the interpretation of the shading gradient as being a shadow cast by a closer flat region. 

Discussion

The results of the experiments reported here clearly show that EEs are very powerful cues to figural status and depth across an edge, capable of dramatically reversing the figural bias for all four of the flat cues we studied when they conflicted. It even overcame the figural bias of a perfect circle, which was not only smaller and completely surrounded, but familiar and maximally convex, appearing to be seen as ground viewed through a donut hole on 85% of the trials (Figure 4, right column). The results for the EE-visible, EE-occluded, flattened, and misaligned conditions further indicate that these depth and figural effects for surfaces of revolution are complex and involve several different factors, including the strength of the evidence for the EE itself on the gradient side of the shared edge (as indicated by the reduced effects in the EE-occluded and flattened condition) and the strength of the evidence for gradient cuts on the non-EE side of the shared edge (as indicated by the further reductions in the misaligned conditions). These additional factors require further study but do not detract from the main conclusion of the present experiments: Extremal edges are a more potent cue to figural status and perceived depth across an edge than any flat figure-ground cue we have studied. Experiment 3 further showed that the figural bias of an EE is systematically diminished with progressive occlusion of the steepest part of the shading gradient. However, corresponding compression of the gradient, which leaves evidence of an EE intact, does not reduce its figural bias. Addition of correlated texture cues to the shading gradient increases the figural bias of the gradient side with an EE, probably by reducing alternative interpretations of the shading gradient, e.g., as a shadow cast by the adjacent side. 

Together with previous results demonstrating the efficacy of shading and texture gradients in producing the perception of extremal edges (Palmer & Ghose, 2008), we conclude that shading, highlights, and texture gradients are crucial factors in understanding how the visual system determines which side of an edge is closer to the viewer. The present experiments have barely scratched the surface of this topic, however. Future research is needed to investigate important further issues, such as the influence of additional context on FGO near an EE. We have conducted experiments on factors such as the radius of curvature of the EE surface and the presence of additional gradients farther from the EE (e.g., quarter cylinders vs. half cylinders vs. multiple half cylinders), which we will report in a subsequent publication. 

Acknowledgments

We thank the members of PalmerLab for their support and encouragement in all phases of the research reported in this article. Its publication was supported by NSF grant BCS-0745820 and a gift from Google to S. Palmer. We also thank two anonymous reviewers for their comments and criticisms. 

Commercial relationships: none. 

Corresponding author: Stephen E. Palmer. 

Email: [e-mail]. 

Address: Department of Psychology, University of California, Berkeley, Berkeley, CA 94720-1650, USA.