Concavities, negative parts, and the perception that shapes are complete

Abstract

Abstract

Abstract:

Abstract  When we perceive the shape of an object, we can often make many other inferences about the object, derived from its shape. For example, when we look at a bitten apple, we perceive not only the local curvatures across the surface, but also that the shape of the bitten region was caused by forcefully removing a piece from the original shape (excision), leading to a salient concavity or negative part in the object. However, excision is not the only possible cause of concavities or negative parts in objects—for example, we do not perceive the spaces between the fingers of a hand to have been excised. Thus, in order to infer excision, it is not sufficient to identify concavities in a shape; some additional geometrical conditions must also be satisfied. Here, we studied the geometrical conditions under which subjects perceived objects as been bitten, as opposed to complete shapes. We created 2-D shapes by intersecting pairs of irregular hexagons and discarding the regions of overlap. Subjects rated the extent to which the resulting shapes appeared to be bitten or whole on a 10-point scale. We find that subjects were significantly above chance at identifying whether shapes were bitten or whole. Despite large intersubject differences in overall performance, subjects were surprisingly consistent in their judgments of shapes that had been bitten. We measured the extent to which various geometrical features predict subjects' judgments and find that the impression that an object is bitten is strongly correlated with the relative depth of the negative part. Finally, we discuss the relationship between excision and other perceptual organization processes such modal and amodal completion, and the inference of other attributes of objects, such as the material properties.

Introduction

When we look at objects in our environment, we can make many different kinds of judgments about them. One of the most important features of an object—which we rely on for successful interaction—is shape. Most research on 3-D shape perception has focused primarily on how the brain estimates local 3-D shape properties such as position in depth (Stevens & Brookes, 1987; Bülthoff & Mallor, 1988), local surface orientation (Koenderink, van Doorn, & Kappers, 1992; Johnston & Passmore, 1994), or curvature (Rogers & Cagenello, 1989; Johnston & Passmore, 1994). This is an important line of research because of the computational challenges that the brain must overcome to reconstruct 3-D depths or surface orientations from the retinal images. However, it seems intuitively unlikely that the estimation and representation of shape in the human visual system ends with a description of surface relief. Suppose—as a thought experiment—that we had a complete model of how the brain computes maps of positions, surface orientations and other local features. Would we say that we know everything there is to know about the perception of shape? Is the computation of such maps identical to perceiving a shape, or is there something more to shape perception than estimating local surface properties? 

From a phenomenological point of view, the experience of perceiving shape seems to involve more than merely detecting different positions in space or other local surface properties. Early Gestalt psychologists identified additional experiences associated with interpreting shape as secondary and tertiary or physiognomic qualities (Koffka, 1935). Tertiary properties refer to object properties—or, more precisely, experiences of the Gestalt of objects—that we are inevitably aware of during our perception of objects, but that are not part of the physical (sensory) stimulus itself. The character of a room or the gracefulness of a statue are examples of such physiognomic qualities. A twist applied to a piece of wax leaves some kind of trace in the shape of the object, but it is not part of the object itself. However, we cannot stop ourselves from experiencing the wax as being transformed—the impression that it has been twisted is an integral part of the subjective Gestalt. 

Consider, for another example the shape in Figure 1. Most people would agree that they see not only an abstract shape with particular curvatures and angles, but also that this shape is composed of two distinct parts—a small circle and a larger ellipse. In other words the visual system's description of shape does not consist solely of a representation of the object's local geometrical properties. The visual system also decomposes the shape into parts, and to some extent makes inferences about the meanings or origins of those parts. Although the object is unfamiliar, this decomposition of shape into smaller parts can in turn help when trying to interact with it, for example by suggesting where the observer should hold it in order to pick it up, or break it into pieces. Furthermore—and central to the current work—this decomposition could also help us to make inferences about the forces or causes that might have been applied to the object in order to give it its current shape. 

Figure 1

 

Example of a composite object. The arrows indicate the most likely locations for part boundaries.

The inference of causality is a key feature of human perception. For example, Michotte (1963) created many demonstrations in which interactions between simplified moving and stationary figures yields a compelling impression of causal interactions. For example, in the launching effect, a moving circle halts when it comes into contact with a stationary circle, which at the same moment abruptly starts moving. This configuration leads to the impression that the first ball is responsible for the second ball's motion—an impression that does not occur when the temporal relationships between the motions is altered (e.g., by delaying the movement of the second ball). For a review of many similar effects in which forces and causes are inferred from motion, see Scholl and Tremoulet (2000). However, this kind of causal perception is related to moving objects and explicitly seen interactions between them. We, on the other hand, are interested in the perception of forces and interactions between objects that happened in the past and are not directly visible in the moment of observation. Considering Figure 1 again, one might argue that it is relatively unlikely that the indentations are due to biological growth of the upper part out of the lower one. Rather, the shape looks more like the two parts were somehow fused or stuck together. The fact that we can make judgments such as these suggests that perceptual organization is an important aspect of shape perception, and plays a key role in making additional inferences about objects, based on their shapes. In this article we study a specific case of perceptual organization, in which the shape is segmented into different regions or parts based on the forces or processes responsible for those parts. 

An example of this specific kind of segmentation is shown in Figure 2. Most people readily perceive that the apple has been bitten, and can distinguish the region of the bite from the rest of the surface. There are several potential cues that support this inference, such as the differences in texture and color between the “bite” and the rest of the apple. Familiarity with apples may also play a role, as apples usually do not exhibit large indentations of this kind unless they have been bitten. However, even disregarding these cues, there are grounds for believing that we can also make this kind of judgment based solely on the shape. For, example, the region is also easily distinguishable even in the crude line drawing in which color obviously plays no role. The fact that we can infer the cause of the bite should not be taken for granted. The apple is a single, bounded piece of matter, and in a physical sense, all points on the surface belong equally to the apple. The surface is continuous and the bite cannot be identified simply by detecting a concavity, because not all concavities are caused by the forceful removal of some portion of the object. The visual system must distinguish between bites and other indentations, such as the one on the top of the apple, where the stem emerges. We call the impression that portions of an object appear to be missing because they have been removed by force perceptual excision (see also Pentland, 1986a, 1986b, for a similar line of reasoning). This can be contrasted with perceptual occlusion, where portions of an object appear to be missing from the image due to being hidden from view by other surfaces. We explore the relationship between excision and occlusion in more depth in the discussion section. 

Figure 2

 

A photo and a line drawing of a bitten apple. In both cases, the bite is easily distinguishable from the rest of the apple. Black lines circumscribe indentations within the apple's surface.

In this article we asked subjects to judge whether objects have been changed by forces or not. Specifically, we are interested in the contribution of the geometrical properties of such indentations to the inferences about their causal origin. 

It is interesting to relate perceptual excision to previous work on the perceptual organization of shape, such as research on the decomposition of shape into parts (Blum & Nagel, 1978; Hoffman & Richards, 1984; Xu & Singh, 2002; Feldman & Singh, 2005; Cohen & Singh, 2007). Here the prevalent question is which rules or principles the visual system uses to parse objects into smaller subunits since the number of possible decompositions is practically infinite. One of the most prominent hypotheses in this context is the negative minima rule by Hoffman and Richards (1984). According to this rule shapes or surfaces are decomposed into subunits “at negative minima, along lines of curvature, of the principal curvatures (Hoffman, 2000, p. 89).” As noted by Hoffman and Richards, this rule mainly comes from the idea of two objects interpenetrating. For example, when a candle is pushed into a birthday cake, this results in a tangent discontinuity in the composite shape (see Figure 3). 

Figure 3

 

Genesis of tangent discontinuities when two objects intersect (after Hoffman & Richards, 1984).

When applied to Figure 1 the minima rule would parse the object into two distinct parts at the locations indicated by the arrows, and this is consistent with most people's subjective impression of the shape. 

However, this does not mean that the rule can only be applied to objects whose parts have been put together by force. It also works to some extent with objects whose parts are the result of molding or biological growth. For example, a hand—whether the result of biological growth or modeling by a sculptor—would be parsed, according to the minima rule, into a palm and five fingers. Support for the plausibility of the negative minima rule comes not only from everyday experience but also from a series of experiments. For example, Cohen and Singh (2007) showed that subjects were better at identifying line segments when the segments were bounded by negative minima then when they were bounded by positive maxima or inflections (i.e., when the segment was a natural part of the object, rather than crossing part boundaries). This not only suggests that negative minima play an important role in the perceptual decomposition of shape, but also suggests that segmentation occurs automatically. 

However, as mentioned earlier, in this article, we focus on another aspect of perceptual organization—indentations—which are sometimes called negative parts. In contrast to a positive part, a negative part is a region, such as a keyhole, which does not belong to an object per se but is surrounded or partly surrounded by one (Hoffman & Richards, 1984), and nevertheless has some clearly discernible shape. In other words, it belongs in some sense to the ground, because it contains no matter. However, in contrast to other ground regions, negative parts are perceived as having a shape. These shapes can either be the consequence of natural growth of positive parts or the result of an external force changing the previous shape. Examples for the former are the indentations at the top and bottom of the apple or the gaps between the fingers of a hand. The latter case in turn corresponds to the bite in the apple or water washing out a bay in an island. We suggest that geometrical properties of a shape's negative parts can be an important source of information for the visual system to make inferences about the forces that might have given an object its current (i.e., observed) shape. 

A number of other authors have also argued for generative models in shape perception. For example, the idea of generative models is latent in much of Whitman Richard's work (Hoffman & Richards, 1984; Richards, 1988). Similarly, Feldman and Singh's (2006) work on skeletal representations of shape is also based on the idea that different shapes—and different portions of a single shape—are brought about by different statistical processes. In that context, the statistical processes responsible for negative and positive parts are clearly different, and this could play a role in inferring excision. In Pentland's (1986a, 1986b) model, shapes are represented by a Boolean combination of prototype shapes that are modified or transformed (e.g., by twisting, tapering, or bending). This tacitly assumes that different processes are responsible for different parts of a shape (e.g., excised parts versus positive parts), since, e.g., excision results from the subtraction of two prototype shapes. 

However, while others have considered the potential significance of concavities for shape representation, most previous work does not consider the different kinds of indentations that, on a perceptual and behavioral level, are treated differently (and have distinct causal histories). For example, consider the two shapes in Figure 4. Why does the brain treat the indentations in the two shapes differently? Although the two objects have similar silhouettes, most people would agree that the negative part in the cookie (Figure 4, left)—the bite—is extrinsic, in the sense that it seems to have been caused by some kind of external force, which has changed the original or complete shape of the cookie. By contrast, for the croissant, the situation is quite different. Here we accept the negative part as an intrinsic or intended part of the croissant's shape. 

Figure 4

 

A cookie (left) and a croissant (right) exhibiting a very similar shape. However, for the cookie the negative part is perceived as being a bite, whereas in the case of the croissant, it is not.

In the case of the cookie, the processes responsible for the original shape and those for the bite are different in nature, whereas for the croissant they are the same—the negative and positive parts are the result of one single process applied to a piece of dough. Thus, one criterion for judging whether a negative part is an intrinsic part of an object's shape (as opposed to a bite or dent, which is perceived as extrinsic) could be whether the process responsible for the negative part is the same in form as for the positive parts of the object. This is not only the case for external forces but also for internal (biological) forces as can be seen by the indentation at the top of the apple (Figure 2) that is perceived as an intrinsic negative part of the apple. In contrast, a bite or a dent is perceived as extrinsic when the force responsible for its formation is different from the forces forming the original object. As mentioned earlier, there are several cues that might aid the visual system in distinguishing between these different situations. However it is not clear to what extent purely geometrical properties of the negative parts play a role. 

For our experiments, we sought to identify geometrical properties that are important for judging a shape to be bitten as opposed to whole. Note that the decision about a shape's state as being bitten or whole is a categorical one. Nevertheless, we can express this binary decision in a continuous manner by asking subjects about their certainty or confidence in the judgment of a shape as being bitten. In order to study the geometrical factors that influence the subjects' ratings, our stimuli had to meet several criteria. First, they should be new to our subjects in order to minimize any familiarity effects. Second, we decided they should not be curved, since bites almost always result in tangent discontinuities (see Figure 5). This could potentially make the task too easy as subjects could simply detect the presence of the local tangent discontinuities, rather than basing their judgment on a holistic impression of whether the object is bitten or complete. Third, whole objects should not have any concavities. This was necessary to ensure that the correct response was well defined since discriminating between a randomly shaped concavity that is caused by another object and a naturally occurring concavity in a random object is underconstrained. Based on these criteria, we developed a highly restricted generative shape model—which we call hexagon world—in which we create objects by intersecting irregular 2-D hexagons. 

Figure 5

 

When two curved objects intersect, they generically produce tangent discontinuities at the intersection. This provides a strong cue that the cause of the resulting negative part is removal of material by an external force (excision).

In the first experiment, we investigated whether subjects can perform the task at all. In other words, we ask whether subjects can discriminate between bitten and whole shapes without knowing how these shapes have been created. In order to test this, we presented subjects with a large number of different objects, half of which had had portions removed by intersecting with another object. We asked subjects to rate the extent to which the shapes appeared to be complete as opposed to having had some portion removed. We were further interested in identifying shape properties that could explain the ratings of subjects. Are there properties of the shape—especially those of the negative parts—that predict the subjects' ratings? 

Experiment 1

Method

Subjects

Ten subjects participated in the experiment (five female, five male; age range: 29−74 years; mean age: 47.5 years). Subjects were either asked directly to participate in the experiment or responded to an email advertisement. All participants reported having normal or corrected to normal vision, and were unaware of the purpose of the experiment. Student subjects were rewarded with course credits; all other subjects participated without reward. 

Stimuli

Stimuli were produced using Matlab 2007. The stimulus set consisted of complete and bitten objects, which were derived from 360 different convex irregular hexagons. Each complete shape was created by finding a set of six random points forming a convex 2-D hexagon within a rectangle of 1600 × 900 pixels. In order to ensure stimuli were at least a minimum size, the distance between any two vertices, and the distance between any vertex and the hexagon's centroid, was at least 100 pixels. 

To create the bitten objects, a portion of each shape was deleted by intersecting it with another randomly placed hexagon, and removing the region of overlap (Figure 6A). Occasionally, this procedure cleaves one of the initial shapes into two pieces; we rejected these cases, keeping only those cases that created exactly two final objects. Note also, that the two resulting objects do not necessarily both contain concavities—we included these convex bitten or bitten off shapes in the experiment to test whether subjects perceived them as bitten or complete. These criteria led to a wide variety of different shapes, with the number of vertices ranging from three to 13. In total the stimulus set consisted of 360 polygons, with 180 being whole and 180 bitten (30 being convex bitten off) objects. Examples stimuli in Figure 6B are shown as black shapes on white background for practical reasons only. During the experiment, stimuli were presented as a white silhouette on a uniform black background at random orientations on each trial. 

Figure 6

 

(A) Generation of a bitten stimulus. Two irregular convex hexagons (left) were randomly intersected and the overlapping region (red) was removed (middle). Note that the resulting two shapes (right) do not necessarily both contain a negative part. (B) Example stimuli, arranged in ascending order from upper left to lower right according to the average rating of subjects. Notice that during the experiment all shapes where white silhouettes on black background.

Procedure

Stimuli were presented on a Sony Vaio Z-3 Notebook at a resolution of 1600 × 900 pixels. Subjects were seated about 80 cm away from the monitor. On each trial, the observer was shown a stimulus in the center of the screen for an unlimited amount of time. The task was to indicate with a cursor on a 10-point scale the extent to which the presented shape appeared to them as being bitten (i.e., part of the shape was caused by removing a portion of an original shape) or not (i.e., the presented shape was a complete intact object). By pressing the space bar observers confirmed their response and the next shape was presented. Low values (one to five) indicated a low probability that a shape had been bitten, and therefore a high probability that it was whole, whereas high values (six to 10) indicated a high probability that it had been bitten, and therefore a low probability that it was whole. Stimuli were presented in random order and at random orientations in six blocks of 60 trials each with breaks in between. Before the experiment, observers were informed about the anonymous treatment of their data as well as their right to abort whenever they wanted without any effect on their reward. After reading the instructions subjects were given 36 practice trials in order to familiarize them with the procedure and stimuli. Importantly, subjects were not informed about the exact procedure used to generate the shapes; they had to identify for themselves what counted as a whole or bitten shape. 

Results

To test whether the observers were able to correctly distinguish shapes that had been bitten (i.e., created by removing a portion of the shape) from those that were whole, we can transform the 10-point ratings into a binary scale by treating values 1–5 as a judgment that the shape was whole and values 6–10 as judgments that the shape was bitten. This was then compared to the actual class of the shapes (Figure 8A). The mean proportion of correct answers across subjects was 0.73 (SEM = 0.04). Two-sided binomial tests for the individual subjects confirmed that all but one observer performed significantly above chance level (50% correct answers), indicated by the asterisks above each bar in Figure 7A. 

Figure 7

 

(A) Mean proportion of correct answers for all 10 subjects. The black line indicates chance performance and the purple line the mean performance across subjects. Two different strategies to achieve near perfect performance by identifying concave hexagons as bitten objects (green line) or simply counting the number of vertices (red line) are displayed in addition. Error bars indicate SEM. Stars indicate the level of significance: no star, p > = 0.05; one star, p < 0.05; two stars, p < 0.01. (B) Interindividual differences in judging bitten versus whole shapes. Each blue point indicates proportion correct for a single subject. Subjects are more consistent at identifying bitten shapes than whole ones. Thus, differences in performance between subjects were due primarily to the ability to correctly identify whole shapes. The green and red dot indicate two different strategies subjects could have adopted to achieve near perfect performance by identifying concave hexagons as bitten objects (green dot) or simply counting the number of vertices (red dot) are displayed in addition.

Figure 8

 

Relative frequency within which the different values of the 10-point scale were used.

This shows that without knowing the method by which the shapes were created, they could spontaneously distinguish those shapes that had been subject to the biting transformation. However, they could also have achieved near perfect performance simply by identifying concave hexagons as bitten (green line in Figure 7A), or by counting the number of vertices (red line in Figure 7A). Evidently they did not notice this. 

We also found that there were substantial interindividual differences in subjects' overall performance (see Figure 7B). Some subjects showed almost no difference in their performance at identifying bitten or whole shapes, whereas others were better at judging one class than the other. In particular, the two worst performing subjects were significantly below chance at identifying whole shapes (p < 0.0207, as indicated by a two-sided binomial test on those subjects data for whole shapes), whereas all subjects were above chance performance at identifying bitten objects (p < 0.0001, shown by a two-sided binomial test for the individual subjects). This shows that observers rarely mistook bitten shapes for whole ones but sometimes thought whole shapes were actually bitten. Put another way, if the task of identifying bitten is thought of as the process of detecting a negative part, observers' errors were more likely to be false positives than misses, which is surprising because the whole shapes never contained concavities, whereas the bitten ones sometimes did not. This provides further evidence that subjects did not simply interpret concave objects as bitten and convex ones as whole. 

Figure 8 shows the relative frequency with which the observers used the different categories of the 10-point scale to rate the stimuli. Extreme values were used more often than central ones resulting in a V-shaped function. Thus, in contrast to the tendency to the middle often observed with rating scales, this suggests that observers tended to be confident in their judgments about the extent to which a shape appears to be bitten or not. 

Together, these findings suggest that subjects are relatively consistent in their judgments that given objects are bitten. Nevertheless, the deeper question remains: Which geometrical properties of the objects cause them to appear bitten as opposed to whole? In order to investigate this, we measured a number of geometrical properties of the stimuli and sought correlations between the values of these features and the ratings reported by subjects. It is important to note that our goal at this stage is not to model exactly which image measurements the visual system makes, but rather to identify a small number of geometrical properties of the stimuli themselves that correlate with subjects' judgments, and to rule out properties that correlate poorly. Before presenting the results, we first briefly describe each of the four properties we tested. These are schematized in Figure 9. 

Figure 9

 

Four different geometrical properties of negative parts. (A) The relative area of a negative part, (B) the mean of interior angles, (C) the relative aperture width, and (D) the relative depth of a negative part.

1.  
   Relative area of the negative part (Figure 9A): This property is derived by relating the area of the negative part to the area of the total shape's convex hull. In other words, it is the ratio of the area of the negative part to the area of the shape's convex hull. This is intended to capture the intuition that small concavities lead to only weak impressions of excision, while larger concavities lead to a stronger impression that the object has been bitten. The measure is zero for all whole objects and positive for 83.3 % of the bitten objects in our stimulus set—those that are not convex. It progressively increases the larger the negative part is, but does not take into account the shape of the object or its negative part, only the relative size. Note that the negative part defined this way is not always equal in size to the portion that was deleted to create the shape.
2.  
   Mean of interior angles (Figure 9B): Sum of the shape's interior angles divided by the number of vertices. This is always  

    

   where n is the number of vertices. This property is intended to capture the intuition that as more of the intruding shape enters into the bitten object, the number of vertices in the negative part—and therefore the shape as a whole—increases. We reason that because the shapes are polygonal, the vertices play a special role in indicating that more of the shape is intruded. Thus, this measure is a discrete (rather than continuous) measure of the degree of penetration of one object by another, reflecting the special informational status of vertices (Attneave, 1954; Feldman & Singh, 2005). The measure has a constant value of 120° for whole objects, but is lower for convex bitten objects, and higher for concave bitten objects. Thus, when treated as a monotonic measure of excision, this property actually rates some bitten as more whole than the whole objects.
3.  
   Relative aperture width (Figure 9C): Distance in pixels between the two points on the shape's outline where the bite has been taken out in relation to the perimeter of the shape's convex hull. The intuition behind this measure is that larger apertures in relation to the perimeter of the shape's convex hull may be treated as stronger evidence of excision. The value is zero for all whole objects and positive for all nonconvex bitten objects.
4.  
   Relative depth (Figure 9D): Depth of a bite in relation to its width. Hoffman and Singh (1997) identify this as an important factor in part salience (see also Kim & Feldman, 2009, who also specifically consider part salience in the context of negative parts). In order to derive the depth and width of a bite the smallest rectangle enclosing the negative part was calculated. If one side of the rectangle and the line connecting the two intersection points (i.e., points on the outline where the bite enters the shape) overlap, the length of this side is taken as the negative part's width. The depth of the negative part is given by the length of the rectangle's adjacent edge. If the two lines do not overlap, the width of the negative part corresponds to the length of the side of the rectangle that forms the smallest angle with an extension of the intersection line. The perpendicular side (i.e., the one with the second smallest angle) represents the negative part's depth. In case of the two angles being equal (45°) the width is represented by the shorter one of the rectangle's side (the longer side represents the depth). This measurement is intended to capture a simple intuition about the shape of negative parts, namely that deeper concavities are probably judged to be stronger evidence of excision than shallow bites of the same total area. The measurement is zero for all whole shapes and positive for all nonconvex bitten shapes.

Figure 10 plots subjects' ratings as a function of each of the four shape properties, and Table 1 shows the correlations of the properties with ratings, as well as the intercorrelations between the different properties. All properties significantly correlate with subject's ratings (first column). The correlation was positive for relative area of the negative part, mean of interior angles, and relative depth. This means that the higher any of the measures were, the higher the subjects tended to rate the shapes as being bitten. However, relative aperture width showed a negative correlation, meaning that the higher this measure was, the lower the subjects tended to rate the shapes as being bitten. Except for the relative aperture width, which showed no linear relationship with subjects' ratings (see Figure 10C), all other shape properties explained at least 63% of the variance of subjects' ratings (see Figures 10A, B, and D). This means that the higher the mean of interior angles of a negative part, the bigger its area or the deeper it goes into the shape compared to its width, the more likely it was to be considered a bite than an intrinsic part of the object's shape. 

Figure 10

 

Relationship between subjects' ratings and the different shape properties: (A) relative area of the negative part, (B) mean of interior angles, (C) relative aperture width, and (D) relative depth.

Table 1

 

Correlations (and R2) between subjects' ratings and the different shape properties as well as their intercorrelations between the different shape properties. all ps < 0.0001.

	Rating	Relative area	Ø interior angles	Relative aperture
Relative area	0.79 (0.63)
Ø interior angles	0.95 (0.90)	0.54 (0.29)
Relative aperture	−0.67 (0.45)	−0.37 (0.14)	−0.74 (0.54)
Relative depth	0.87 (0.76)	0.77 (0.59)	0.51 (0.26)	−0.60 (0.36)

What is also evident from Table 1 is that there are highly significant intercorrelations between the different properties, especially combinations of relative area of the negative part, mean of interior angles, and relative depth. All three are in some way related to the size and shape of the negative part, but because the different properties are not independent for this stimulus set, it is hard to tease apart their relative importance. In Experiment 2, we sought to explicitly decorrelate the different factors in order to study their explanatory power independently. 

Discussion

The results of Experiment 1 showed that observers can discriminate between bitten and whole hexagonal objects even without having explicit information about the process responsible for generating the stimuli (Figure 6A). In other words, there is something about the shape of the objects themselves that indicates whether they had been bitten or not. It is interesting to ask which geometrical factors they used to make their judgments. 

Concavities as a cue to excision

Perhaps unsurprisingly, in the debriefing after the experiment, most subjects reported that they relied on the presence of concavities to distinguish between bitten and whole shapes. Indeed, they could have achieved near perfect performance (91.67% correct) by judging concave shapes as bitten and convex shapes as whole. Furthermore, Figure 7B shows that observers rarely mistook bitten shapes for whole ones but sometimes thought whole shapes were actually bitten, i.e., that most errors made by observers were due to false positives rather than misses. This suggests that they relied heavily on identifying concave objects as bitten, with specific properties of the concavities determining how bitten the object appears. This interpretation is further supported by the high correlations between observers' ratings and different properties of these negative parts. In short, a shape is identified as bitten when a negative part is detected. 

However, the converse is not true: Convex shapes are not automatically interpreted as being whole. With this in mind, there are several reasons for believing that subjects' ratings were not simply due to detecting whether a concavity was present or not. First, their performance (73% correct on average) was substantially below what would be predicted from using this strategy. While random errors (e.g., due to lapses of attention, or inability to detect concavities) may explain a proportion of their failure to perform as well as would be predicted from this strategy, this is unlikely to be the whole story. Most concavities were clearly above threshold visibility, suggesting that subjects would be unlikely to be limited by internal noise in their ability to detect concavities. Thus, rather than simply failing to detect or attend to concavities (i.e., noise), there appears to be a deliberate decision not to use these as the only criterion for deciding whether the object is bitten or not (i.e., a bias). 

Second, the pattern of errors also suggests that subjects' performance was not simply determined by how noisy they were. If subjects were limited by noise, lapses of attention, or criterion variability, then their errors should be equally likely to be misses as false positives. This predicts that the data in Figure 7B should cluster along the diagonal, and that there should also be a negative correlation between standard error and position along the diagonal, with reliability progressively increasing with accuracy. The fact that the data are significantly more spread out along the whole than the bitten axis, suggests a systematic bias. Thus, subjects did not use concavities as their only criterion, because not all convex objects appear whole. Some other factor(s) must also contribute to their rating. 

A Bayesian model based on turning angles

The way the stimuli are constructed suggests that subjects could also have used the distribution of vertex angles in the shape to distinguish between whole and bitten objects. In the Results section, we consider predictions based on the number of vertices, which is equivalent to the mean of the distribution of internal angles. However, additional information could be obtained by considering the probabilities associated with the entire distribution of vertex angles in each shape. Here we present a simple Bayesian model that uses this entire distribution to distinguish bitten from whole objects. To anticipate, we again find that subjects' ratings were substantially less decisive than such an ideal observer. 

To develop such a model, we first need to characterize the likelihood distributions representing the probability of each vertex angle for bitten and whole objects. Because of the way the stimuli are created, we expect these two distributions to be different. Specifically, each shape can be thought of as a series of turning angles α (Feldman, 1997, 2001; Feldman & Singh, 2005, 2006) that were drawn from a particular distribution. A turning angle here is defined as the change in direction between the extensions of two consecutive edges of a polygon as one follows the shape's outline in clockwise direction. 

For the whole shapes, the distribution contains any combination of six positive angles whose sum is 360° (whole shape angle distribution). For the bitten shapes, the distribution is somewhat more complex, consisting of three components. Since bitten shapes were created by intersecting two whole shapes, their outline in part consists of a series of turning angles drawn from the distribution for whole shapes (positive values of α, first component). The series of angles in the negative part however, come from a second distribution, which is the inverse of the whole shape distribution (negative values of α, second component). Finally, two additional angles (the outer most angles of a negative part) are created at the locations where one shape intrudes into the other (although note that an overlap does not always lead to an intrusion, see Figure 6A). These angles depend on the relative orientation of the two shapes, and form the third component of the distribution of turning angles for bitten objects (bitten shape angle distribution). 

Based on these distributions, we built a simple Bayesian classifier and compared its ability to distinguish bitten from whole shapes with our subjects' performance. To obtain the posterior distribution that determines whether a given shape is whole or bitten from its distribution of turning angles,  

 

we need to know two things: (a) the distribution of turning angles within the pool of whole (whole shape angle distribution) and the pool of bitten shapes (bitten shape angle distribution) and (b) the prior probability of a shape being whole or bitten:  

 

 

Since half of our stimuli were whole and the other half bitten a shape has equal probability of being whole or bitten. In addition, we explicitly informed our subjects about these probabilities. Therefore we assume an equal prior for our subjects to see how well their data can be fit by the model:  

 

 

The whole shape angle distribution and the bitten shape angle distribution we simply measure from our stimulus set by sampling all angles of all shapes of a particular shape category and then pooling them in bins of 5° (angular degrees) between −180° and 180°. 

By dividing the frequency of every bin by the summed frequency of all bins, we obtain the likelihood function of turning angles (see Figure 11A) given a whole (green)  

 

or a bitten (blue) shape  

 

 

Figure 11

 

(A) Likelihood functions p(α | whole) in green and p(α | bitten) in blue. Bin size = 5°. (B) The posterior p(whole | α)—probability that a given shape is whole given a specific turning angle α. The posterior p(bitten | α) (not shown) is simply the reciprocal of this function since the probability of the two alternatives sums to one. Bin size = 5°. (C) Relative frequency of likelihood ratios within our set of whole (green) and bitten (blue) stimuli. The red dashed line indicates the Bayesian model's classification boundary and corresponding classification accuracy. Note the different scaling for the ranges < 1 and > 1. Bin size = 0.2 (below one) and ∼ 12.6 (above one).

Note that the likelihood distributions are substantially different from one another. Now that we have the likelihood functions and the prior probability, the posterior (see Figure 11B) is obtained using Bayes' Theorem (the formula for Display Formula is equivalent to Equation 7):  

 

 

This is the marginal distribution for the probability of whole versus bitten for each individual angle in the shape. Note that the posterior distribution for bitten objects is simply the reciprocal of the distribution for whole objects. Note that there is a large range of angles, which, if present, uniquely determine that the shape must be bitten. Thus, already at this stage (considering only individual turning angles), we expect the classifier to be generally rather good at distinguishing bitten from whole. 

We are interested in classifying based on the entire set of turning angles in a given shape, not just on a single angle, as in Equation 7. Therefore we must take the product Π of the likelihoods of all individual angles in each shape:  

 

 

In the same way we can also calculate the probability that a shape is bitten taking into account the entire ensemble of turning angles it contains:  

 

 

The relative probability that a given shape is bitten versus whole is determined from the likelihood ratio:  

 

 

A likelihood ratio of one means a given series of turning angles was as likely to be drawn from the whole shape angle distribution as from the bitten shape angle distribution. Values greater than one indicate the shape is more likely to be whole whereas values lower than one indicate the shape is more likely to be bitten. 

Figure 11C illustrates the distribution of likelihood ratios for whole (green) and bitten (blue) shapes for our stimuli. Note that we have stretched the axis between zero and one to aid visualization of the distribution for bitten objects. The complete distribution is bimodal, with clearly distinct modes for bitten and whole shapes, indicating that it should be easy to distinguish the two. Note also, that the bitten distribution has many shapes with a likelihood ratio of zero. This corresponds to those shapes that contain one or more negative turning angles, which never occur for the whole shapes. 

The accuracy of the classifier is given by the proportion of correct classifications given the optimal decision boundary at likelihood ratio of one. This leads to an accuracy of 98.3%. Thus, a Bayesian model based on the distribution of turning angles in a each shape therefore performs as well as one would by counting the number of vertices and clearly outperforms our human observers (∼73% correct). There are several possible explanations of the suboptimal performance. The model assumes noiseless estimates of the distribution of turning angles, whereas human angle perception is limited by noise. However, to match human performance would require positing an unrealistically high level of noise. Another possibility is that subjects do not base their classification on turning angles, although this is inconsistent with the high correlation found between subjects' ratings and the mean of interior angles. Another possibility is that subjects have a different prior on the relative frequency of whole and bitten shapes, or incorrect likelihood models for the distribution of turning angles. The model assumes complete knowledge of the distribution of turning angles within the distinct shape categories. Subjects in contrast did not have this prior knowledge and would have had to rely solely on assumptions about the distributions or estimates acquired during the 20 training trials. 

Distinguishing between cues

Because the relative area, mean of the interior angles, and relative depth all correlate with subjects' ratings (and with one another), we suggest that the subjects rely on some intuitive concept of the ‘extent' of the concavity to assign their ratings. Since these shape properties were highly intercorrelated, the results of Experiment 1 cannot clearly distinguish which factors are most important. The two factors that explain the most variance in subjects' settings were mean of interior angles and relative depth. Therefore to separate more clearly the relative contribution of these two factors, we conducted a second experiment in which we decorrelated the mean of interior angles and relative depth. 

Experiment 2

Methods

Experiment 1 showed that of the shape properties we tested, the mean of interior angles and the relative depth of a negative part explained most of the variance in our observers' ratings. However, since the investigated shape properties in Experiment 1 were highly intercorrelated, we could not make definite conclusions about the independent contributions of these variables to observers' ratings. In order to overcome this limitation and tease apart the contributions of the different cues, we conducted a second experiment with a set of stimuli for which, by design, the correlation was kept near zero. 

Subjects

A new group of three male and seven female undergraduate psychology students from the University of Giessen participated in exchange for course credits. Ages ranged from 20 to 38 years (M = 24.5 years). Subjects were recruited by their response to an advertisement sent by mail through the university's internal mail distributor. All participants reported to have normal or corrected to normal vision. 

Stimuli

The stimuli were again created in Matlab following the same procedure as in Experiment 1 except for the following differences (see Figure 12A). This time, in contrast to the first experiment, the intersection procedure was not done randomly but in such a way as to ensure that the mean of interior angles and the relative depth of a negative part were not correlated. 

Figure 12

 

(A) Generation of a bitten stimulus in Experiment 2. Two irregular convex hexagons (left) were intersected at their centroid. One hexagon (here gray) was shifted in either one of the four cardinal or their intermediate directions. The overlapping part (red) was cut out of the stationary hexagon (black) when certain criteria were met (see text for further details). (B) Example stimuli from the decorrelated stimulus set. Shapes with identical mean of interior angles exhibit negative parts with different relative depths (columns) and vice versa (rows).

Specifically, we intersected two different convex, irregular hexagons at their centroids. Then we shifted one of the hexagons successively in one of eight possible directions in steps of 10 pixels, while holding the other hexagon fixed at its position. This was done until all vertices of the shifted hexagon lay outside the border of the fixed one. Directions could be either horizontal (left-right), vertical (up-down), or a combination of the two (diagonal). The direction was determined once for a hexagon pair at the beginning and kept constant during the whole procedure. A region of overlap was cut out of the fixed hexagon when the following criteria were met: (a) At least one of the shifted hexagon's vertices had to lie outside and one inside the borders of the fixed hexagon, and (2) when cutting out the region of overlap, the remaining shape of the fixed hexagon should only consist of one single coherent shape (rather than multiple independent shapes). This procedure was repeated with thousands of hexagon pairs producing hundreds of thousands of different bitten shapes. From this large pool of potential stimuli, a subset was selected in which the relative depth of the bite and the mean of the interior angles were almost completely uncorrelated. We call this the decorrelated stimulus set (see Figure 12B for an example). 

It has to be pointed out that if random sampling is used, this approach does not make it possible to create stimuli covering the entire range of both means of interior angles and relative depth at the same time. It is highly unlikely, even with hundreds of thousands of stimuli, to produce a stimulus with a high mean of interior angles (e.g., 155°) while at the same time having a low relative depth of the negative part or vice versa. In other words, the shift-and-intersect procedure produces a substantial correlation between the two variables if samples are drawn uniformly from the entire range. 

Therefore, in selecting the decorrelated stimulus set from the large pool of potential stimuli, we constrained the range of the mean of interior angles to 90°, 108°, 120°, 128.57°, 135°, and 140° and the range of the log relative depth of the negative part to six intervals equal in size between −5.3 and zero. This way, we ensured that different shapes with the same mean of interior angles covered the whole range of relative depth of the negative part and vice versa. We randomly sampled 10 different stimuli from each of the 36 possible variable ranges (i.e., 6 Mean of Interior Angles × 6 Intervals of Relative Depth), resulting in a decorrelated stimulus set of 360 different shapes. 

Procedure

The apparatus and experimental procedure were identical to Experiment 1. 

Results

The results show a much higher contribution of log relative depth to the variance in subjects' ratings (R2 = 0.59; Figure 13A) than the contribution of mean of interior angles (R2 = 0.10; Figure 13B) when the two properties were decorrelated. This suggests that the correlation between the mean of interior angles and observers' ratings was mainly due to the intercorrelation of negative parts' relative depth and the mean of interior angles. Note, however, that the proportion of the variance accounted for by the log relative depth is also lower than in Experiment 1. Thus, although the subjects clearly rely more on the part salience (Hoffman & Singh, 1997) of concavities, other unknown factors also play a substantial role. 

Figure 13

 

Relationship between subjects' ratings and (A) log relative depth and (B) mean of interior angles.

General discussion

Our experiments suggest subjects can discriminate unfamiliar bitten and whole shapes even without explicit knowledge about the generative process responsible for the stimuli. Even though subjects had never seen the shapes before and therefore had no experience with or knowledge about them, they correctly identified them as bitten or whole in 73% (Experiment 1) and 72% (Experiment 2) of cases. This suggests that subjects had a clear understanding of the task and could distinguish shape features that are intrinsic to the object from those that are caused by external forces. 

Interestingly, with these stimuli, they could have achieved near perfect performance simply by judging shapes exhibiting a concavity as bitten and those without a concavity as whole. The fact that they did not perform perfectly suggests that in general identifying an unknown shape as bitten or whole does not depend solely on the detection of a concavity. This makes sense given that in the real world, not all concavities are caused by extrinsic forces. Even when a shape is unfamiliar, observers treat some concavities as extrinsic shape features and others as intrinsic to the shape, similar to the cookie-croissant example in Figure 4. 

By using our hexagon world stimuli, we tried to minimize the role of other cues such as familiarity, color, or texture so that we could investigate the contribution of purely geometrical properties of our shapes to the ratings of the subjects. We focused on four different shape properties, which were intended to measure the size and shape of the negative part in different ways. The aim was to test the intuition that larger or more salient negative parts lead to a stronger impression that the object is bitten. Specifically, we tested the predictive power of: (a) the relative area of the negative part; (b) the mean of interior angles; (c) the relative width of the aperture; and (d) the relative depth of the negative part. 

In Experiment 1 we found that three of the four factors were substantially positively correlated with subjects' ratings. In other words, subjects on average rated a shape as being more likely to have been bitten when the negative part was relatively large or deep, or the mean of the interior angles was higher (see Table 1). Alternatively, having identified that a concavity is due to excision, rather than intrinsic to the shape, the size and depth of the concavity alters the strength or confidence of the interpretations, with larger, more salient concavities yielding stronger ratings. 

Since the shape properties we tested in Experiment 1 were themselves highly intercorrelated we performed a second experiment in order to separate the effects of the two factors that accounted for the largest proportion of the variance in Experiment 1 (mean of interior angles and relative depth). To do this, we explicitly decorrelated the two factors by selectively sampling shapes from a large pool of randomly generated stimuli. In Experiment 2 we found that relative depth was still correlated with subject's rating whereas mean of interior angles was not anymore. When mean of interior angles and relative depth are not correlated only relative depth predicts subjects' ratings. Thus, it seems that of the geometrical properties we tested, the size and salience of negative parts are the most important factors in determining whether excision has occurred. Again, either large, deep concavities appear more likely to be caused by some extrinsic process removing a portion of the object, or, having identified that a concavity is a bite, these factors strengthen the confidence in the interpretation. 

It is important to clarify that we do not claim that the brain computes exactly these quantities in order to work out the mechanical history of a shape. There are potentially many other ways that the size or depth of negative parts could be represented, and indeed, there are almost certainly other conditions that affect the interpretation of shapes, some of which we discuss in greater detail below. As this was a correlational study, we cannot make strong claims about the precise parameterization that the brain uses, or rule out other factors. Our goal, instead, was to describe a few shape properties that correlate with perceptual excision, to provide initial constraints in developing a theory of how the visual system works out the meaning or causes shape features. In the following discussion, we consider how discriminating bitten from whole shapes relates to other perceptual organization processes. 

By using stimuli that we created from irregular convex hexagons, we were able to isolate the contribution of purely geometrical properties of negative parts to subjects' perception of excision for a limited class of objects. However, the current results cannot readily be generalized to shapes that were created by intersecting other classes of shape. Of particular importance is the fact that intersecting smoothly curved shapes generically leads to salient tangent discontinuities in the boundary that provide an additional potential cue that excision has occurred (Figure 5). We focused on polygonal objects to avoid this cue, which could dominate judgments and make the task too easy. An alternative approach would be use shapes that are piecewise smooth, so that they contain salient tangent discontinuities whether they have been bitten or not. Future studies are required to determine how subjects distinguish which corners are due to bites and which are naturally occurring features of the object itself. 

Further observations about excision

So far we have focused on the interpretation of our empirical findings. In the following sections, we make a number of more general theoretical and phenomenological observations about excision and how it relates to other perceptual effects. Readers who are interested primarily in our experimental results can skip directly to the Conclusions section. 

Connection between excision and visual completion

Perceptual excision—as we have posed it—is also clearly related to visual completion (Kanizsa, 1979; Kellman & Shipley, 1991; Van Lier, van der Helm, & Leeuwenberg, 1995; Nakayama & Shimojo, 1992; Tse, 1999). Visual completion refers to the phenomenon in which portions of a shape that are not present in the retinal image are nevertheless perceived by an observer (see black shape in Figures 14A and B). This absence from the retinal image is usually due to another object obscuring parts of the object either by occlusion or by camouflage. By contrast, in excision, a portion of the object is missing from the image because it has been forcibly removed from the object itself. Despite some obvious differences, which we consider below, it is interesting to speculate whether the conditions that induce visual completion might be related to the conditions in which a negative part is perceived to be due to excision. It is unclear to what extent we can infer the complete original shape of an excised object; nevertheless, to the extent to which we can, it seems plausible that the interpolation may be similar to what occurs in amodal completion. 

Figure 14

 

Is there a relationship between perceptual excision and visual completion? (A) A portion of the black square is hidden behind an occluder, leading to amodal completion. (B) The occluder is camouflaged against the background, leading to modal completion of the occluder in addition to the amodal completion of the black square. (C) The same missing portion as in (A) and (B) but in this case, it has been physically removed (bitten out). Notice that in all cases the black shape is visually completed to form a square despite different causes. The arrows indicate the direction of border ownership assignment (arrow points from figure to ground).

One important difference between traditional completion and excision is the extent to which the cause of the missing portion is visible in the image. In occlusion, the cause (i.e., the occluder) is visible, either directly, or due to modal completion if the occluder is itself camouflaged (see Figure 14B). For modal completion of the occluder to occur, at least some parts of the camouflaged occluder must be visible (e.g., the portions that overlap the inducers in Figure 14B). By contrast, in excision, the cause of the missing portion can be completely absent from the image. It is interesting to speculate whether one of the preconditions for perceiving excision (rather than occlusion) might be the presence of local cues indicating that a portion of the shape is missing, but the absence of evidence for an occluder. In the absence of evidence for occlusion, the visual system is left with an alternative interpretation, namely that the portions of the shape that do not appear in the image really are missing from the shape. Put differently, when figures that look like inducers appear in the image—but don't match up so that completion can occur—the visual system may infer that a different cause is responsible for the missing portions of the shape(s). 

Another important difference between occlusion and excision is the assignment of border ownership in the missing portion of the shape. Since borders are perceived as being owned by the figure, not the ground (Palmer, 1999; Bertamini, 2006), the interpretation of a shape as being figural or being part of the ground co-varies with the assignment of border ownership (Kim & Feldman, 2009). When occlusion is inferred, the contours of the concavity are assigned to the occluder, not to the occluded object. The fact that these contours do not belong to the occluded object indicates that there really is no concavity in the shape itself. Put differently, there is a reversal in the sign of border ownership at the transition from the true boundary of the occluded object to the concavity (which belongs to the occluder). Indeed, the assignment of the contours to the front layer—and their removal from the representation of the occluded shape—may be a key stage in the amodal completion of the occluded object (Grossberg & Mingolla, 1985a, 1985b). By contrast, when excision is inferred, the boundaries within the concavity strictly ought to belong to the object itself. This means there should be no reversal in the sign of border ownership at the transition into the concavity in the excised shape. Thus, again, the presence of cues indicating that a portion is missing from the shape, but the absence of a reversal of figure-ground assignment may be a key process in perceptual excision. Surprisingly, however, Kim and Feldman (2009) found evidence that—on a local level—border ownership also tends to reverse in negative parts. They measured border ownership using a small local probe (a perturbation of the contour) and found that when a concavity is seen as a negative part, subjects tend to assign border ownership to the hole rather than to the object. This tendency was clearly related to the saliency (e.g., convexity, see Bertamini & Wagemans, 2012, for a review) or figurehood of the concavity, with small openings and wide interiors exhibiting the strongest effect. Put simply, the more figural a negative part, the higher the probability to interpret it as being due to occlusion (and the less likely it is to have been excised). We however, find that the perception of excision increases with part saliency. Although at first glance, this seems to be inconsistent, we argue in favor of an interpretation of Kim and Feldman's results as being a different parameterization of the same measure. One possibility, e.g., is that border ownership assignment is scale dependent. In other words, subjects assign border ownership in one direction when considering the object as a whole (as measured in our task), whereas local assignment (as measured in Kim and Feldman's task) can be in the opposite direction. Another possibility is that border ownership is not a unitary phenomenon, so that a border can be owned by one side at one level of perception, but to the other side at another level. For example, when a cookie cutter removes a gingerbread man from some dough, at a low level, figural status is assigned to the man-shaped hole, and perceptual judgments about the shape of the hole treat it as if the hole owned the boundary. At the same time, it is clearly the remaining dough—and not the hole—which is made of matter, and at a high level we understand that the hole is a hole and not an object: For example, nobody in their right mind would try to pick up the man-shaped hole, indicating that at another level, the border ownership is correctly assigned to the dough. Interestingly, sculptors also exploit this ambiguity in their experimentations with negative space. 

There may also be some important differences in the way the incomplete shape is completed, depending on whether occlusion or excision is inferred. In Figure 14, if asked to judge the shape of the black objects, most people would probably report seeing a square in all three cases, suggesting that perceptual organization principles underlying amodal completion in all three cases may be similar (Kellman & Shipley, 1991; although see also Anderson, Singh, & Fleming, 2002; Singh, 2004; Anderson, 2007). However, this is not always the case, as can be seen in Figures 15A–C. In Figure 15A, for example, the shape that is partly occluded by the gray figure can't be completed to form a square as otherwise additional parts of the square should be visible (Figure 15B). However, this is more ambiguous in Figure 15C. Here, although the shape of the black region is identical to Figure 15A, one might validly complete the bitten shape to form a square, rather than a truncated square as in Figure 15A. 

Figure 15

 

Visual completion following occlusion can be different from visual completion following excision even when the inducing element is the same. With occlusion, the black shape in (A) cannot be completed as a square because a corner is missing, as shown in (B). However with excision, as in (C), the black region can be completed to form a square, even though the black inducer is identical in shape to (A).

This may be in part due to a fundamental difference between the physics of occlusion and excision in the real world. When an object is occluded, only those parts of the object that are hidden by the occluder disappear from the image. If the occluder spans the object, it can lead to spatially disconnected image regions, which belong to the same object and which must be grouped together visually (the black body and the corner of the square in Figure 14B). By contrast, with excision, a bite that spans the object can physically disconnect the corner from the rest of the object, causing it to fall off. In other words, the effects of excision are not limited to the size of the object (or process) causing the excision. Thus, excision generally does not lead to spatially disconnected image regions that must be grouped together; residual portions that survive excision may nevertheless be completely absent from the image. As depicted in Figure 16 (top row), this means that the shape shown in Figure 15C could initially have been a square—the missing corner is not inconsistent with this interpretation. Thus, completing the shape to a square is a valid interpretation in the case of excision, even though it is not in the case of occlusion. 

Figure 16

 

Multiple explanations of an excised shape. An initial shape (A), which could be either a square or a truncated square, is excised by the gray shape (B), leaving the remaining portions of the shape (C). The disconnection of the corner from the rest of the object in (D) leads to the final shape (E). The fact that two different initial shapes subject to the same excision result in the same final shape, means that inferring the shape of the object before excision is ambiguous.

Figure 16 also shows that excision with the same gray object causes both a truncated square and a complete square to end up as the same shape. Because multiple different initial shapes can lead to the same final shape, the interpretation of the final shape is necessarily ambiguous. It is interesting to ask why the excised shape in Figures 15C and 16E is generally seen as having originally been a square rather than a truncated square. Several types of explanation come to mind, such as Prägnanz (a square is more symmetrical than a truncated square), familiarity (a square is more common than a truncated square), or the fact that a truncated square may itself be the result of a prior excision process, which removed the corner from a square—thus the original shape may have been a square anyway. 

The ambiguity outlined in Figure 16 is not the only ambiguity involved in interpreting excision. The form and process of the excision itself is also highly ambiguous. So far we have considered excision as a cookie cutter process in which one shape stamps a hole in another shape. If this process is inferred, then the shape of cookie cutter is itself ambiguous. Only those portions of the excising shape that were in contact with the remaining portions of the excised shape are specified—the rest of the excising object is completely unknown. This is another reason why the black figure in Figure 15C is completed to a square: Depending on the inferred shape of the excising object, the corner may simply have been excised along with the rest of the excised portion, rather than having dropped away after excision. Future work is required to investigate not only how the visual system infers the original shape of the excised object but also the shape of the excising object—presumably regularity and symmetry constraints—also affect this latter inference. 

Furthermore, the process causing excision is itself highly ambiguous. Many possible processes, such as smashing, biting, or carving, could be responsible for the absence of part of the object. The fact that multiple possible processes can yield potentially identical final shapes means that it is not possible to uniquely identify the process given only the final shape. Nevertheless, many processes leave specific signatures such as scratches in the excised shape. It is possible that there may be some cues in the specific details of the local geometry that can increase the relative probability of some explanations rather than others. 

An interesting example is a process of intrusion in which a shape carves out material from the excised object as it progresses into its body, as shown in Figure 17. For short intrusion distances, this leads to an excised shape that is very similar to cookie cutting, as shown in Figures 17D and E. However, longer intrusions (e.g., Figure 17F) can lead to a distinctive shape profile, due to the tracing out of the features of the intruding shape. It is interesting to ask how different processes impose specific regularities on the shape of the excised region. In principle the visual system might be able to use such regularities to distinguish specific excision processes, although how exactly such regularities could be extracted from the contour requires further investigation. 

Figure 17

 

Excision due to intrusion of one object into the other. As the intruding shape burrows into the black square (A)–(C), material is ejected from the square. (D) The result of removal by stamping with a cookie cutter shape. (E) A very similar shape results from intrusion of the same shape in the plane of the square, if the intrusion is not deep. (F) However, when the intrusion is deeper, the effects of intrusion are more clearly visible as parallel streaks in the contour.

Taken together, it is clear that there are both similarities and important differences in visual completion between occluded and excised objects, even when the concavities in the inducing image regions are identical. 

Distinguishing excision from intrinsic concavities

Even when a concavity is clearly not due to occlusion, but to the true border of the shape, there is still the problem of establishing whether the concavity is intrinsic to the shape (as in the croissant in Figure 4), or extrinsic, and therefore due to excision (as in the bitten cookie in Figure 4). It is interesting to ask which factors influence this distinction. One important factor may be related to amodal completion. Consider the shapes in Figure 18. For some of the shapes (e.g., Figure 18A), the impression of excision is relatively strong. For others (e.g., Figures 18B and E), the impression that the concavity is intrinsic is strong. Finally, for the remaining figures, the impression maybe somewhat ambiguous. One important difference between the figures that appear bitten and those whose concavities appear to be intrinsic is the extent to which the neighborhood of the concavity lends itself to amodal completion. For example in Figure 18A, there are two clear tangent discontinuities in the border flanking the concavity, and the two edges on either side of the concavity are aligned (relatable) and therefore readily enable amodal completion of the shape (Kellman & Shipley, 1991; Rubin, 2001). By contrast, the concavity in Figure 18B is flanked by continuous curves, rather than tangent discontinuities. Although there exist points on either side of the concavity where the two tangents are aligned, the local context of the curves does not readily support amodal completion between these points. Thus, one important difference between intrinsic and extrinsic concavities may be the extent to which amodal completion is supported by the immediate context of the concavity. If the concavity could be removed from the shape by amodal completion, then it is plausible that it was added to the shape by some external force. Thus, when amodal completion is strongly supported (but occlusion is ruled out), then the concavity may be due a bite. By contrast, when amodal completion is not strongly supported, then the concavity may be an intrinsic feature of the shape, produced by some process other than excision of matter from the object. 

Figure 18

 

Five shapes that contain concavities. Figure (A) appears to be a square from which a portion has been bitten. However, in (B) the support for amodal completion is low, resulting in the perception of the concavity is intrinsic to the shape. In (C) and (D) the interpretation of the negative part is more ambiguous. (E) A shape exhibiting tangent discontinuities but without strong cues for amodal completion. Most people see the concavity as intrinsic rather than due to excision.

We suggest, however, that the perceived causes of different portions of shapes may not be determined solely by the local geometry in the neighborhood of the concavity. For example, in Figure 18D, the concavity is flanked by tangent discontinuities and relatable edges, but the concavities are somewhat ambiguous, and can be readily seen as intrinsic to the shape. This may be partly due to the symmetry between the two concavities, suggesting a nonaccidental relationship between the process generating the concavity and the rest of the object (see below). It may also be due to the internal structure of the concavities—which appears related to the processes that have shaped the rest of the object—or familiarity with the object, which looks like a chess piece. Clearly, these higher-order factors also play a role in determining whether a shape is seen as bitten or complete. 

In essence, perceiving excision involves working out whether the concavity and the rest of the shape share a common origin, or whether they are due to distinct physical processes. Symmetries and genericity (Binford, 1981; Richards, Feldman, & Jepson, 1992; Freeman, 1994; Feldman, 1996; Albert & Hoffman, 2000; Albert, 2001) can play an important role in this inference (see Figure 19). When portions of the shape are related to one another by non-transverse (i.e., nongeneric) geometrical relationships—through symmetries such as being parallel, or bisecting one another—then there are probabilistic grounds for inferring that these relationships are not an accident. In other words, symmetries relating the concavity to the rest of the shape provide evidence that the concavity is intrinsic. By contrast, when the concavity is randomly located within the shape, and has transverse (i.e., generic) relationships with the other features of the object, there is less evidence that the concavity was created systematically, which should make the concavity appear more like a bite. This is demonstrated in Figure 19. From left to right, the symmetries relating the concavity to the rest of the shape are progressively removed, leading to a subtly stronger impression that the concavity was caused by some accident. 

Figure 19

 

Effects of symmetry on the interpretation of concavities. Regularities in the shape are progressively reduced from left to right, leading to an increase in the perception that the concavity is due to excision. In (A) the concavity bisects both the horizontal and vertical axes of the object, and is perpendicular to the vertical edge. In (B) the horizontal symmetry is removed, in (C) the vertical symmetry is also removed, and in (D) the orthogonality is also removed.

Relationship to material perception

The final, observed shape of an object is the product of both the forces and processes applied from the outside, and the intrinsic, dispositional properties of the object itself, which determine how it responds to external forces. If an object is struck by a hammer, the result will be very different depending on whether it is made of thin glass, thin metal, or solid wood. As mentioned in the Introduction, in addition to indicating that excision has occurred, in many cases the local geometrical properties of a negative part can also contribute to judgments about the material properties and internal structure of the object. For example, biting something with a material like meringue results in a different local geometry in the concavity than biting something soft like a doughnut (Figures 20A and B). Another example is shown in Figure 20C. Based on the diagonal ridges and furrows in the surface, most observers readily infer that the object has been twisted, and furthermore, most people would probably agree that the shape is probably made of a soft, malleable material (rather than metal, for example), based on the way it has responded to the twisting process. These judgments are, of course, heavily based on knowledge about materials and their characteristic behavior in response to different forces. Interestingly, Vrins, de Wit, and van Lier (2009) have shown that knowledge of material properties affects judgments of amodal completion, suggesting that there could be similar effects in the context of perceiving excision. At the same time, Gerbino and Zabai (2003) showed that object knowledge (e.g., material and its behaviour in response to forces) does not always trump geometrical factors in determining the perceptual interpretation of a scene. They presented subjects with 3-D stimuli in which an elongated object appears to partially intersect the edge of a block, forming a joint. The region of intersection—which must be amodally completed—could theoretically belong to either of the objects. In one example, a banana has a right-angled chunk cut out of it, so that when placed on the edge of the brick it appears to penetrate it. Despite the fact bananas are soft, and thus it is more likely that the brick penetrates the banana than vice versa (based on object knowledge) subjects were more likely to amodally complete the banana than the brick. 

Figure 20

 

Three different materials, which exhibit different geometrical properties depending on the applied forces. From the resulting geometrical properties it might be possible to infer properties of the material.

However, the local geometrical structure of objects presumably also provides many important cues that can aid the estimation of material properties. The processes by which the visual system infers material properties from 3-D shape is an important research topic, which has received little attention to date. 

Conclusions

We studied whether subjects can determine if objects have been bitten or are complete, based solely on their shape. We found that subjects were both accurate and consistent in their judgments of whether specific shapes were the result of excision. We suggest that they based their ratings on a number of geometrical properties related to the relative size and salience of the concavities in the shapes, and identified a number of other factors, including symmetry and the extent to which the remaining shape supports amodal completion, which may also play a role more generally. 

This study forms part of an emerging interest in the experimental study of the relationship between perceptual organization and inferences about the causal history or origins of objects (e.g., Savova, Jäkel, & Tenenbaum, 2009; Wilder, Feldman, & Singh, 2011). More generally, the findings suggest that shape may not be represented in purely geometrical terms (concavities, slants, curvatures, etc). In addition to geometrical properties, the functional meanings or causes of different shape features are also represented in some way by the human visual system. 

Acknowledgments

The authors wish to thank Peter Vangorp, Steven Cholewiak, and Manish Singh for invaluable comments throughout the research project. Part of this research was funded by a grant from the BMBF-NSF Joint Program in Computational Neuroscience (FKZ: 01GQ1111—“Towards a Neural Theory of 3D Shape Perception”). 

Commercial relationships: none. 

Corresponding author: Patrick Spröte. 

Email: [e-mail]. 

Address: Department of General PsychologyJustus-Liebig-Universität Giessen, Giessen, Germany.