Perception, 2014, volume 43, pages 1033 – 1048
doi:10.1068/p7629
The shape of a hole is perceived as the shape of its interior
Rolf Nelson1, Jason E Reiss1, Xue Gong1, Sherri Conklin2, Laura Parker1, Stephen E Palmer3 1
Department of Psychology, Wheaton College, 26 E Main St, Norton, MA 02766, USA; Department of Philosophy, University of California, Santa Barbara, CA 93106, USA; 3 Department of Psychology, University of California, Berkeley, CA 94720, USA; e‑mail: [email protected] Received 16 October 2013, in revised form 16 August 2014, published online 8 October 2014 2
Abstract. In typical figure–ground displays the figure has shape and is perceived as being in front, whereas the ground is shapeless and recedes to the back. The recent literature on the visual perception of holes has questioned the nature of this coupling between shape and depth both theoretically and empirically. In this paper we provide a theoretical framework that clarifies the underlying issues and we report new evidence supporting the view that the shape of a hole is perceived as the shape of its interior region. Palmer, Davis, Nelson, and Rock (2008 Perception, 37, 1569–1586) showed that the shape of the interior region of a hole is remembered as such, even though the surface visible through it is perceived as farther in depth. The present paper extends this evidence to perceiving holes. Participants performed a speeded shape-matching task in which they compared a surrounded interior region (of either a hole or an object) or its exterior complement with one of several shapes. The results indicate that holes are perceived as shaped in the same way as their material counterparts. We conclude that the shape of a hole is encoded as the shape of its interior region, even though that region contains no surface material. These results can be reconciled with recent experiments that have provided evidence that holes are perceived differently from their material counterparts. Keywords: perceptual organization, holes, figure–ground perception, shape
1 Introduction The distinction between figure and ground in visual perception is central to understanding how depth edges in two-dimensional (2‑D) images are interpreted in terms of the layout of corresponding surfaces in the three-dimensional (3‑D) environment. Figure 1 shows a classic example: two image regions sharing a single contour (figure 1a) that tend to be perceived spontaneously as two different surfaces at different distances from the observer. As indicated, the image is perceptually ambiguous(1) along the contour because it can be perceived either as a white surface with a sharp, pointy edge in front of a farther black surface (figure 1b) or as a black surface with a smooth, bumpy edge in front of a farther white surface (figure 1c). This ambiguity (and its resolution, ideally toward the true state of affairs) lies at the heart of the figure–ground phenomenon and demonstrates its importance for understanding real-world perception. The figure–ground phenomenon was initially described and investigated by gestalt psychologists early in the 20th century. Most significantly, Rubin (1915/1958) initially identified a set of important stimulus factors that influence which region is perceived as figure, including surroundedness, smaller size, higher contrast with the background, and symmetry.
(1)
This is not to say that the two interpretations are equally likely to occur perceptually, of course. All of the cues to figural status (but primarily convexity in figure 1a) operate in such situations to bias perception toward one or the other interpretation.
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Modern researchers have identified many additional factors, including convexity (Kanizsa & Gerbino, 1976), lower region (Vecera, Vogel, & Woodman, 2002), familiarity (Peterson & Gibson, 1994), extremal edges (Palmer & Ghose, 2008), and edge-region grouping (Palmer & Brooks, 2008). Nevertheless, the exact definition and perceptual consequences of the figure–ground phenomenon have never been entirely clear.
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Figure 1. A classic demonstration of an ambiguous figure–ground phenomenon in a bipartite display. (a) The two-dimensional image. (b) Its three-dimensional (3‑D) interpretation in which the white surface is closer and shaped by the edge. (c) Its 3‑D interpretation in which the black surface is closer and shaped by the edge.
2 Theoretical considerations We argue that the confusion about whether the shape of a hole is perceived as the shape of its interior or exterior regions has arisen largely because, in its usual manifestation (eg figure 1), the figure–ground phenomenon confounds three potentially separable distinctions: a closer–farther distinction in terms of relative depth across the shared contour, a material–immaterial distinction with respect to surface presence in the closer depth plane, and an inside–outside distinction with respect to determining the shape defined by the contour. [Previous arguments related to aspects of the present position can be found in Davis (1985),(2) Casati and Varzi (1995), Subirana-Vilanova and Richards (1996), Palmer (1999), Peterson (2003), Palmer et al. (2008), and Nelson, Thierman, & Palmer (2009)]. (2)
To the best of our knowledge, Davis and Rock should be credited with first addressing theoretical and empirical issues concerning the relation between figure–ground phenomena and hole perception through Davis’s doctoral thesis at Rutgers in 1985. Relevant results of this thesis, in addition to a further series of experiments [initially cited in Palmer (1999, page 286) as Rock, Palmer, and Hume (in preparation)] were published only much later as Palmer, Davis, Nelson, and Rock (2008), but Davis’s dissertation preceded all of the modern work on the subject.
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It is incontrovertible and universally agreed that, when two adjacent image regions share a contour corresponding to a depth discontinuity, the figure is defined as the region seen as closer to the observer, whereas the ground is seen as farther. This closer–farther distinction constitutes the depth-based definition of the figure–ground phenomenon. It also seems incontrovertible and universally agreed that the figure is defined as the region seen as ‘owning’ the shared contour, whereas the ground does not. That is, the closer surface, corresponding to the figure, has a material existence up to the corresponding edge in the closer depth plane, whereas there is no material surface in that depth plane on the ground side. This material–immaterial distinction is central to the edge-based definition of the figure–ground phenomenon. It should be clear that the depth-based and edge-based definitions are referentially equivalent, since the behavior of light falling on opaque surfaces dictates that the visible edge must signal the end of the closer surface (because it has material existence in the closer depth plane) rather than the farther surface. The coupling of the depth-based and edge-based aspects of the figure– ground phenomenon is thus complete, and both are mentioned in virtually every theoretical discussion of the figure–ground phenomenon (eg Nakayama et al., 1995; Palmer, 1999; Peterson, 2003; Rock, 1983). The fact that the figure is perceived as ‘shaped’ by the contour is also frequently mentioned as an aspect of figure–ground perception. The shape claim is similar to the ownership claim—so similar, in fact, that they are usually considered together as ‘figural shape’—but the shape claim goes beyond border ownership by asserting that one perceives the shape of the figure but not that of the ground. However, several researchers have argued that the shape-based aspects of the figure–ground phenomenon, although highly correlated with its depth-based and edge-based aspects in standard cases (eg figure 1), are dissociable in the case of holes (eg Palmer, 1999, pages 285–287; Palmer et al., 2008; Peterson, 2003). In the present paper we address the relation between shape perception and figure–ground perception with the aim of resolving a controversy that has arisen between researchers who have argued that observers naturally and spontaneously perceive the interior shapes of holes (eg Casati & Varzi, 1995; Nelson & Palmer, 2001; Nelson et al., 2009; Palmer, 1999; Palmer et al., 2008; Peterson, 2003; Subirana-Vilanova & Richards, 1996), and those who have claimed that observers naturally and spontaneously perceive the exterior shapes of holes (eg Bertamini, 2006, 2010; Bertamini & Croucher, 2003; Bertamini & Helmy, 2012; Bertamini & Mosca, 2004). For example, proponents of the former ‘interior’ or ‘hole-based’ position would assert that the shape of a round hole in a rectangular surface (eg a round window in a rectangular wall) is spontaneously perceived as circular, whereas proponents of the latter ‘exterior’ or ‘surround-based’ position would claim that the shape that is spontaneously perceived is that of the surrounding surface and that the circular shape of the hole is subsequently inferred by performing something like a figure–ground reversal on the shape of that surrounding surface.(3) For the present analysis of hole shape we need provide only a few important observations about shape perception. One is that a two-sided image contour is perceptually shapeambiguous because it does not specify inside versus outside. Once it is interpreted as the (3)
Although the ‘surround-based’ or ‘exterior’ position may seem unlikely, it has been endorsed explicitly by Bertamini and Croucher (2003). In particular, they say: ““Let us take the example of a fish-shaped hole (discussed in Casati and Varzi, 1995). It is true that people can recognize this as a fish, but our claim is that to do that people must be doing one of two things; either they enforce a figure/ground reversal so that they do not see this region as a hole, or, alternatively, they rely on a slower process that compensates by means of our vast cognitive resources for the fact that perceptually that figure is not the figure of a fish (albeit it is a figure containing enough information to infer the presence of a fish). We are aware that this is a strong claim” (page 40).
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boundary of a physical surface, it becomes one-sided (an edge) with an inside (the closer surface) and an outside (the absence of that closer surface), thus marking the termination of the material extent of the closer surface. Perhaps the most intuitive and important shape feature for present purposes is local concavity/convexity of a bounded region.(4) As figure 1 illustrates, the concavities and convexities on the two sides of the same contour are always complementary, giving rise to the ambiguous perception of the closer region either as pointy (mainly concave with narrow convexities) or as rounded (mainly convex with narrow concavities) along the shared contour. The important observation here is that, although the figural side of a contour (ie the closer and material side) is usually the same as its inside (for convexity/concavity determination of shape processing), this is not always the case. In particular, the contour can be assigned to one side with respect to determining the closer/ farther surfaces and the material/immaterial sides, but to the opposite side with respect to determining the inside/outside distinction required to perceive shape as being convex versus concave. This is precisely what we believe happens in the case of holes: the inside of the contour for purposes of shape perception is not the same as the material side in the closer depth plane. The conflation between the material/immaterial distinction and the inside/ outside distinction may be what has given rise to the concept of ‘quasi-figural’ status for holes (eg Bertamini, 2006); that is, the region denoting a hole may be quasi-figural with respect to shape even if it is not figural with respect to depth and edge ownership. Figure 2a shows an example of the kind of stimuli we study in the experiments described below: a 2‑D image that contains two, nested, closed, image contours; an inner contour and an outer contour. It can be interpreted in either of two ways: as arising from two solid surfaces, the smaller of which is closer and partly occludes the larger (figure 2b), or as arising from a closer surface that contains a hole and a single, farther, solid surface behind the hole (figure 2c). The shape of the two solid surfaces (figure 2b) is straightforward, in that the shape of each surface is the shape of the interior of its single bounding edge. What, however, is the shape of the hole-bearing surface (figure 2c)? And what, if anything, is the shape of the hole itself ? It is not obvious how to extrapolate the case of a simple shape of a solid single-edged surface to one in which the surface contains a hole because of the topological difference between one versus two nested, continuous, closed edges. The material of a hole-bearing surface clearly lies between these two contours, but how can the shape of that surface be described? The most intuitively obvious way is to give a shape description of the outer contour plus a description of the shape of the hole itself, augmented by a description of the hole’s location, orientation, and size. People use this format quite naturally in everyday language when they describe figure 2c as “a square surface with a clover-shaped hole in its center” (cf Giralt & Bloom, 2000, for developmental evidence that young children name holes in the same way that they name material objects). Specifying the outside shape of the surface as ‘square’ indicates that the surface lies inside the square-shaped outer edge that defines its exterior, as for any solid surface that would accurately be described as a square. (4)
A segment of a contour is convex (ie has positive curvature) if all of the straight-line segments connecting points on that contour segment lie wholly inside the region, and it is concave (ie has negative curvature) if those line segments lie wholly outside the region. The point along the contour at which the sign of its curvature changes from convex to concave (or vice versa) is called an inflection point. These definitions are important because the determination of this crucial shape feature (convexity/concavity) necessarily implies the assignment of inside/outside with respect to the contour, a distinction that is analogous (but not equivalent) to the figure–ground distinction in the sense that it interprets a two-sided contour as one-sided, but in this case for purposes of shape description rather than border ownership or relative depth.
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Figure 2. An example of an ambiguous figure–ground phenomenon involving a hole. (a) The twodimensional image. (b) Its three-dimensional (3‑D) interpretation that consists of two solid surfaces at different depths: a closer, light-gray, clover-leaf-shaped surface and a farther, dark-gray, squareshaped surface. (c) A 3‑D interpretation of the image in (a) that consists of a closer, dark-gray, square-shaped surface that contains a clover-leaf-shaped hole [the complement of the closer surface in (b)] and a farther light-gray surface.
Specifying the shape of the hole as ‘clover-shaped’ indicates that, even though the material surface lies outside the inner edge, the shape description is that of the interior region, which is composed primarily of four broad, rounded convexities separated by narrow concavities. Speakers are thus describing the shape of the empty space of the hole in the same way as they would its hole-filling material complement, as if the hole were a ‘subtraction’ from the otherwise solid surrounding surface that effectively changes a portion of a material surface into an immaterial region of that surface (cf Palmer et al., 2008). We will call this the interior shape of the hole. In this scheme the shape representation of a hole-bearing surface includes, as an integral part, a representation of the shape of the hole itself, very much like the corresponding case of two surfaces (figure 2b), except that in the two-surface case, the inner contour signals a closer surface whose material lies inside that contour. It is far less clear how else the global shape of a hole-bearing surface can be represented in its entirety. It is not difficult to understand how the shape of a portion of such a holebearing surface might be represented, however, including concave portions along its inner edge. Try covering half of figure 2c with your hand and attend to the shape of the visible portion. It is now quite easy to see that the materially defined inner edge (which borders the hole) is broadly concave with a few very sharp points, quite unlike the convex and distinctly rounded clover shape seen as that of the hole itself. Moreover, it is hard to understand how that shape description can be extrapolated to encompass the entire shape
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of the hole-bearing surface without employing some description of the interior shape of the hole. The apparent difficulty of constructing such a representation does not mean that the visual system cannot do so, however, as it may just reflect a lack of imagination.(5) We will refer to the representation of the shape of a hole-bearing surface that does not require a description of the interior shape of the hole—whatever that representation might be—as the exterior shape of the hole. Finally, it is important to clarify the depth status of a hole: does it lie in the closer plane of the physical surface and edge, in the farther plane of the ground, or perhaps in a plane that is even closer to the viewer than the physical surface and edge? Bertamini (2006) states the ground-plane view quite explicitly: “In summary, I define holes as background regions that are surrounded by a foreground figure” (page 884). Others disagree, identifying the hole explicitly with the foreground surface (eg Nelson & Palmer, 2001; Nelson et al., 2009; Palmer, 1999; Palmer et al., 2008; Peterson, 2003). Perhaps the most obvious problem with identifying a hole as a background region is that it is incompatible with common sense and everyday language. People quite naturally say: “The background surface is visible through the hole”, but this statement clearly conflicts with any claim that the hole actually is some portion of that background surface, because it simply does not make sense for the background to be visible through itself. Another difficulty with identifying a hole as a background region is that the ground region visible through the hole is a purely accidental feature of that surface that changes with viewing positions. The fact that the perceived shape of the hole is invariant over such changes in viewpoint suggests that the hole is properly identified as an immaterial part of the hole-bearing foreground surface, within which it is a nonaccidental feature of that surface. For these reasons, we conceive the hole as a distinct part of the foreground surface— albeit a missing part—whose absence enables the viewer to see a portion of the background surface through it. 3 Empirical evidence To date, the best evidence that the shape of a hole is perceived as that of its interior region has come from experiments studying memory for the shapes of (immaterial) holes. In a long-term memory paradigm, Palmer et al. (2008) found that the interior shapes of holes are remembered just as well as those of their hole-filling material complements.(6) Their initial experiment used classic bipartite images without holes (like that shown in figure 1) and instructed participants to perceive one particular region as figure and the other as ground. Observers remembered the shape of the perceived figural region, but not that of the visible portion of the ground, for which performance was no different than for unpresented distractor foils. In a subsequent experiment the same instructions were given for surrounded regions that could be perceived as either material objects against a background or as immaterial holes in a surrounding surface. Here, participants remembered the interior shapes of the holes just as well as the shapes of correspondingly shaped material objects, and they remembered both of those much better than the (accidental) shapes of ground regions, which were no different than for distractor foils. The authors extended this result to unambiguous 3‑D stimuli, in which no instructions were required to induce the perception of depth and holes. These results (5)
There are some special cases in which it does seem relatively easy to construct a shape description of a hole-bearing surface without explicitly describing the shape of the hole. One is a picture-frame-like object—that is, an outer edge with nearby parallel inner edges—in which the surface can be viewed as an outline of that shape whose material happens to be so thick that it has distinct outer and inner edges. But the general case of a hole within a surface is not easily handled in this way because the shapes of the inner and outer contours may be different. (6) This is not the case, however, when the shape of the hole is given by accidentally aligned contours from different objects (see Nelson et al., 2009).
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clearly demonstrated that the interior shapes of holes are typically remembered just as well as their solid, hole-filling, object complements. Palmer et al. (2008) interpreted this pattern as evidence that the shapes of both material objects and immaterial holes are perceived in terms of their interior regions, despite the opposite-sided assignment of relative depth and edge ownership in the case of holes. Although performance in these memory tasks has provided evidence that people remember the shape of the interior rather than the exterior region of a hole, other researchers have reported evidence to the contrary in perceptual tasks. For example, Bertamini and Croucher (2003) measured the time observers required to verify the same features of two image regions when they were perceived as holes versus when they were perceived as solid objects. When the image regions like those shown in figure 3 were perceived as solid objects, observers were faster at discriminating whether the left-side or right-side vertex was lower when the object had a convex shape (an asymmetrical hexagonal ‘barrel’) than when the object had a concave shape (an asymmetrical hexagonal ‘hourglass’). They referred to this as a ‘convexity advantage’. However, when the same 2‑D shapes were perceived as holes, the speed of the same left–right discrimination of the lower vertex reversed, at least when the shapes were equated for difficulty: responses were faster for the concave (hourglass-shaped) hole than for the convex (barrel-shaped) hole. The authors inferred that, because there were no differences in the shape of the contours between objects and holes, observers must have performed the task on the exterior shape of the holes, consistent with the convexity advantage they found for solid shapes. In a similar set of experiments, Bertamini (2010) found that detection of reflection is faster in an image region perceived as an object than in one perceived as a hole. Bertamini and Helmy (2012) required participants to identify the shape of an interior region that was surrounded by an exterior region which was either congruent or incongruent in shape (eg a circle surrounded by a circle vs a circle surrounded by a square). Conflicting shapes influenced performance more when the interior region depicted a hole rather than an object. Such results suggest that there is sometimes a distinct difference in the processing of a region when it is perceived as a hole versus when it is perceived as an object.
Response time (log)/ms
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Figure 3. Adapted from experiment 3 of Bertamini and Croucher (2003). Barrel and hourglass shapes were presented as objects or holes, and observers were asked to determine whether the lower vertex was on the left or the right side. When seen as objects, responses to the convex barrel were faster than for the concave hourglass; whereas when seen as holes, responses to the concave hourglass were faster than for the convex barrel.
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Bertamini (2006) has claimed that these perceptual differences provide strong evidence that the contour of a hole is, in fact, owned by the material exterior region and that holes therefore do not have figural or ‘quasi-figural’ status. (Because the relative depth of surfaces at a hole is not in dispute by anyone, we take quasi-figural status to mean that a hole is perceived to have the shape of its immaterial interior rather than its material exterior.) In order to demonstrate such a claim for holes, he describes three conditions that need to be met: (1) that the stimuli are perceived as holes on a given task involving shape perception; (2) that performance should be similar for holes and for material objects; and (3) that, on the same task, holes should behave differently from other background regions. He further states that the results reported by Palmer et al. (2008) provide the best evidence that the perceived shape of a hole is the shape of its interior, but criticizes them because they are based on people’s performance on shape memory tasks, which might not reflect differences in perceptual processing The experiments reported below test Bertamini’s three requirements in a perceptual task. To the extent that the results support all three predictions, they would provide evidence that holes are perceived as having the shapes of the interiors of the hole contours, even though the material surfaces that own them physically lie outside those contours. We used a perceptual shape-matching task, similar to that employed by Baylis and Driver (1995) in that it requires a speeded response to match contour shapes. We asked participants to match object and hole stimuli in terms of either the interior shape (the ‘inner region’) or the exterior shape (the ‘outer region’) along one side. Because solid objects clearly own their surrounding contour, we expected that matching the shape of an object’s inner region would be faster than matching the shape of its outer region. If our proposal that observers perceive the interior shape of holes is correct, then matching the shape of a hole should be similar to matching the shape of an object: faster for matching its interior region than for matching its exterior region. However, if the shape of a hole is available only indirectly—being derived from the shape of its attached, closer, material surface—then participants should find it faster to match the holes in the exterior shape condition than in the interior shape condition. Such a finding would indicate that the shapes of holes and those of their hole-filling materially complementary solid objects are fundamentally different and that a hole’s interior shape is perceived only indirectly from prior perception of its exterior shape, as claimed by Bertamini and Croucher (2003). 4 Experiment 1 4.1 Method 4.1.1 Participants. Sixteen undergraduate students from Wheaton College (Norton, MA) were recruited through fliers posted on campus. Compensation was a US $5 gift certificate from the campus bookstore. All participants reported normal or corrected-to-normal vision and were unaware of the purpose of the experiment. There were five males and eleven females, whose ages ranged from 18 to 22 years old (M = 20.1 years). All experiments were carried out under approval of the Wheaton College IRB and in accordance with the World Medical Association Helsinki Declaration as revised in October 2008. 4.1.2 Apparatus. Stimulus images were presented on a 17″ Dell LCD monitor positioned approximately 60 cm in front of participants. Presentation, timing, and response collection were controlled by SuperLab experiment software and a Cedrus RB‑834 response pad (Cedrus Corporation, 2006) running on a Dell OptiPlex computer. The experiment was conducted in an isolated, diffusely lighted room.
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4.1.3 Stimuli and procedure. Figure 4 illustrates sample trial stimuli. Object and hole targets were blue inner regions, defined by a complex contour on the right side and straight contours on the remaining three sides (subtending approximately 2.4 × 3.6 deg of visual angle), which were positioned at the center of a red square (each side approximately 5.4 deg), which itself was surrounded by a blue square (each side approximately 8.9 deg). Shading was used to disambiguate each of the same 36 unique shapes as hole and object versions. Previous evidence has demonstrated that such shading information is highly effective in differentiating relative surface depth (Nelson & Palmer, 2001). The response options presented below the target stimulus were grayscale versions taken from the same set as the target shapes, one of which always matched the target stimulus for the given set of instructional conditions. Target type
hole
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Figure 4. Illustration of the four trial types in experiment 1. Participants were presented with an object or hole target stimulus and were required to indicate which of four response options matched the shape of (a) the object’s inner region, (b) the hole’s inner region, (c) the object’s outer region (ie the complement of the object’s shape), or (d) the hole’s outer region. The correct answer is shown in the top-left quadrant of each quartet of options.
Each trial began with the target stimulus appearing alone at the top of the display for 800 ms before the four matching shapes were added underneath. This combined target-plusmatching display remained visible until a response was made. Participants were instructed to make their responses by pressing one of four buttons on the button box whose positions were spatially isomorphic to those of the four response options on the computer screen. The instructions emphasized response speed without sacrificing accuracy. Correct answers initiated the next trial, whereas incorrect responses prompted a text box reading “Incorrect answer! Please wait to try again.” This message was followed by a 3 s delay and the repetition of that trial. The location of the matching shape was randomized across trials such that each button served as the correct response an equal number of times for each participant.
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Trials were divided into four blocks based on the type of target stimulus (object vs hole) and the type of matching response (match the inner region vs match the outer region). Instructions were provided at the beginning of each block regarding both what type of target stimulus would be presented and what participants were asked to match. Specifically, participants were told to select the response shape that “matches the one shown” (object target + match inner region block; see figure 4a), “fits into a slot’”(hole target + match inner region block; see figure 4b), “fits into the one shown” (object target + match outer region block, see figure 4c), or “is the complement of the shape of the hole” (hole target + match outer region block; see figure 4d). To ensure that participants understood the task, they were shown what each block instruction meant using a demonstration on the computer monitor. For instance, the two outer region tasks were demonstrated by showing how the correct answer complemented the right side of the target shape in a lock-and-key fashion to form a complete square. After the instructions were given, participants were prompted to ask the experimenter any questions they might have. They were also given the opportunity for a break before beginning the experimental trials. Within each block, all target shapes were presented twice in a random order, resulting in 72 trials per block. Thus, there were a total of 288 ‘core’ trials per participant, not including any trial repetitions due to an incorrect response. Every participant completed all four blocks, and block order was determined by a between-subjects Latin squares design. Reaction times were based on the first 288 trials with correct responses (disregarding errors), and accuracy measures were based on the first 288 trials of each stimulus type as defined by the experimental design (disregarding retaken trials). 4.2 Results and discussion Accuracy and reaction time data (see figure 5) for the 288 core trials were analyzed using separate within-subjects analyses of variance (ANOVAs) that each contained two factors, target type (object vs hole) and matching response type (inner region vs outer region). The analysis of reaction times, which included only correct trials and excluded any trials in which the time exceeded 3 SDs of the population-correct trial mean (1.96% of correct trials removed), revealed a significant main effect of matching response type (F1, 15 = 39.54, p
doi:10.1068/p7629
The shape of a hole is perceived as the shape of its interior
Rolf Nelson1, Jason E Reiss1, Xue Gong1, Sherri Conklin2, Laura Parker1, Stephen E Palmer3 1
Department of Psychology, Wheaton College, 26 E Main St, Norton, MA 02766, USA; Department of Philosophy, University of California, Santa Barbara, CA 93106, USA; 3 Department of Psychology, University of California, Berkeley, CA 94720, USA; e‑mail: [email protected] Received 16 October 2013, in revised form 16 August 2014, published online 8 October 2014 2
Abstract. In typical figure–ground displays the figure has shape and is perceived as being in front, whereas the ground is shapeless and recedes to the back. The recent literature on the visual perception of holes has questioned the nature of this coupling between shape and depth both theoretically and empirically. In this paper we provide a theoretical framework that clarifies the underlying issues and we report new evidence supporting the view that the shape of a hole is perceived as the shape of its interior region. Palmer, Davis, Nelson, and Rock (2008 Perception, 37, 1569–1586) showed that the shape of the interior region of a hole is remembered as such, even though the surface visible through it is perceived as farther in depth. The present paper extends this evidence to perceiving holes. Participants performed a speeded shape-matching task in which they compared a surrounded interior region (of either a hole or an object) or its exterior complement with one of several shapes. The results indicate that holes are perceived as shaped in the same way as their material counterparts. We conclude that the shape of a hole is encoded as the shape of its interior region, even though that region contains no surface material. These results can be reconciled with recent experiments that have provided evidence that holes are perceived differently from their material counterparts. Keywords: perceptual organization, holes, figure–ground perception, shape
1 Introduction The distinction between figure and ground in visual perception is central to understanding how depth edges in two-dimensional (2‑D) images are interpreted in terms of the layout of corresponding surfaces in the three-dimensional (3‑D) environment. Figure 1 shows a classic example: two image regions sharing a single contour (figure 1a) that tend to be perceived spontaneously as two different surfaces at different distances from the observer. As indicated, the image is perceptually ambiguous(1) along the contour because it can be perceived either as a white surface with a sharp, pointy edge in front of a farther black surface (figure 1b) or as a black surface with a smooth, bumpy edge in front of a farther white surface (figure 1c). This ambiguity (and its resolution, ideally toward the true state of affairs) lies at the heart of the figure–ground phenomenon and demonstrates its importance for understanding real-world perception. The figure–ground phenomenon was initially described and investigated by gestalt psychologists early in the 20th century. Most significantly, Rubin (1915/1958) initially identified a set of important stimulus factors that influence which region is perceived as figure, including surroundedness, smaller size, higher contrast with the background, and symmetry.
(1)
This is not to say that the two interpretations are equally likely to occur perceptually, of course. All of the cues to figural status (but primarily convexity in figure 1a) operate in such situations to bias perception toward one or the other interpretation.
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Modern researchers have identified many additional factors, including convexity (Kanizsa & Gerbino, 1976), lower region (Vecera, Vogel, & Woodman, 2002), familiarity (Peterson & Gibson, 1994), extremal edges (Palmer & Ghose, 2008), and edge-region grouping (Palmer & Brooks, 2008). Nevertheless, the exact definition and perceptual consequences of the figure–ground phenomenon have never been entirely clear.
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Figure 1. A classic demonstration of an ambiguous figure–ground phenomenon in a bipartite display. (a) The two-dimensional image. (b) Its three-dimensional (3‑D) interpretation in which the white surface is closer and shaped by the edge. (c) Its 3‑D interpretation in which the black surface is closer and shaped by the edge.
2 Theoretical considerations We argue that the confusion about whether the shape of a hole is perceived as the shape of its interior or exterior regions has arisen largely because, in its usual manifestation (eg figure 1), the figure–ground phenomenon confounds three potentially separable distinctions: a closer–farther distinction in terms of relative depth across the shared contour, a material–immaterial distinction with respect to surface presence in the closer depth plane, and an inside–outside distinction with respect to determining the shape defined by the contour. [Previous arguments related to aspects of the present position can be found in Davis (1985),(2) Casati and Varzi (1995), Subirana-Vilanova and Richards (1996), Palmer (1999), Peterson (2003), Palmer et al. (2008), and Nelson, Thierman, & Palmer (2009)]. (2)
To the best of our knowledge, Davis and Rock should be credited with first addressing theoretical and empirical issues concerning the relation between figure–ground phenomena and hole perception through Davis’s doctoral thesis at Rutgers in 1985. Relevant results of this thesis, in addition to a further series of experiments [initially cited in Palmer (1999, page 286) as Rock, Palmer, and Hume (in preparation)] were published only much later as Palmer, Davis, Nelson, and Rock (2008), but Davis’s dissertation preceded all of the modern work on the subject.
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It is incontrovertible and universally agreed that, when two adjacent image regions share a contour corresponding to a depth discontinuity, the figure is defined as the region seen as closer to the observer, whereas the ground is seen as farther. This closer–farther distinction constitutes the depth-based definition of the figure–ground phenomenon. It also seems incontrovertible and universally agreed that the figure is defined as the region seen as ‘owning’ the shared contour, whereas the ground does not. That is, the closer surface, corresponding to the figure, has a material existence up to the corresponding edge in the closer depth plane, whereas there is no material surface in that depth plane on the ground side. This material–immaterial distinction is central to the edge-based definition of the figure–ground phenomenon. It should be clear that the depth-based and edge-based definitions are referentially equivalent, since the behavior of light falling on opaque surfaces dictates that the visible edge must signal the end of the closer surface (because it has material existence in the closer depth plane) rather than the farther surface. The coupling of the depth-based and edge-based aspects of the figure– ground phenomenon is thus complete, and both are mentioned in virtually every theoretical discussion of the figure–ground phenomenon (eg Nakayama et al., 1995; Palmer, 1999; Peterson, 2003; Rock, 1983). The fact that the figure is perceived as ‘shaped’ by the contour is also frequently mentioned as an aspect of figure–ground perception. The shape claim is similar to the ownership claim—so similar, in fact, that they are usually considered together as ‘figural shape’—but the shape claim goes beyond border ownership by asserting that one perceives the shape of the figure but not that of the ground. However, several researchers have argued that the shape-based aspects of the figure–ground phenomenon, although highly correlated with its depth-based and edge-based aspects in standard cases (eg figure 1), are dissociable in the case of holes (eg Palmer, 1999, pages 285–287; Palmer et al., 2008; Peterson, 2003). In the present paper we address the relation between shape perception and figure–ground perception with the aim of resolving a controversy that has arisen between researchers who have argued that observers naturally and spontaneously perceive the interior shapes of holes (eg Casati & Varzi, 1995; Nelson & Palmer, 2001; Nelson et al., 2009; Palmer, 1999; Palmer et al., 2008; Peterson, 2003; Subirana-Vilanova & Richards, 1996), and those who have claimed that observers naturally and spontaneously perceive the exterior shapes of holes (eg Bertamini, 2006, 2010; Bertamini & Croucher, 2003; Bertamini & Helmy, 2012; Bertamini & Mosca, 2004). For example, proponents of the former ‘interior’ or ‘hole-based’ position would assert that the shape of a round hole in a rectangular surface (eg a round window in a rectangular wall) is spontaneously perceived as circular, whereas proponents of the latter ‘exterior’ or ‘surround-based’ position would claim that the shape that is spontaneously perceived is that of the surrounding surface and that the circular shape of the hole is subsequently inferred by performing something like a figure–ground reversal on the shape of that surrounding surface.(3) For the present analysis of hole shape we need provide only a few important observations about shape perception. One is that a two-sided image contour is perceptually shapeambiguous because it does not specify inside versus outside. Once it is interpreted as the (3)
Although the ‘surround-based’ or ‘exterior’ position may seem unlikely, it has been endorsed explicitly by Bertamini and Croucher (2003). In particular, they say: ““Let us take the example of a fish-shaped hole (discussed in Casati and Varzi, 1995). It is true that people can recognize this as a fish, but our claim is that to do that people must be doing one of two things; either they enforce a figure/ground reversal so that they do not see this region as a hole, or, alternatively, they rely on a slower process that compensates by means of our vast cognitive resources for the fact that perceptually that figure is not the figure of a fish (albeit it is a figure containing enough information to infer the presence of a fish). We are aware that this is a strong claim” (page 40).
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boundary of a physical surface, it becomes one-sided (an edge) with an inside (the closer surface) and an outside (the absence of that closer surface), thus marking the termination of the material extent of the closer surface. Perhaps the most intuitive and important shape feature for present purposes is local concavity/convexity of a bounded region.(4) As figure 1 illustrates, the concavities and convexities on the two sides of the same contour are always complementary, giving rise to the ambiguous perception of the closer region either as pointy (mainly concave with narrow convexities) or as rounded (mainly convex with narrow concavities) along the shared contour. The important observation here is that, although the figural side of a contour (ie the closer and material side) is usually the same as its inside (for convexity/concavity determination of shape processing), this is not always the case. In particular, the contour can be assigned to one side with respect to determining the closer/ farther surfaces and the material/immaterial sides, but to the opposite side with respect to determining the inside/outside distinction required to perceive shape as being convex versus concave. This is precisely what we believe happens in the case of holes: the inside of the contour for purposes of shape perception is not the same as the material side in the closer depth plane. The conflation between the material/immaterial distinction and the inside/ outside distinction may be what has given rise to the concept of ‘quasi-figural’ status for holes (eg Bertamini, 2006); that is, the region denoting a hole may be quasi-figural with respect to shape even if it is not figural with respect to depth and edge ownership. Figure 2a shows an example of the kind of stimuli we study in the experiments described below: a 2‑D image that contains two, nested, closed, image contours; an inner contour and an outer contour. It can be interpreted in either of two ways: as arising from two solid surfaces, the smaller of which is closer and partly occludes the larger (figure 2b), or as arising from a closer surface that contains a hole and a single, farther, solid surface behind the hole (figure 2c). The shape of the two solid surfaces (figure 2b) is straightforward, in that the shape of each surface is the shape of the interior of its single bounding edge. What, however, is the shape of the hole-bearing surface (figure 2c)? And what, if anything, is the shape of the hole itself ? It is not obvious how to extrapolate the case of a simple shape of a solid single-edged surface to one in which the surface contains a hole because of the topological difference between one versus two nested, continuous, closed edges. The material of a hole-bearing surface clearly lies between these two contours, but how can the shape of that surface be described? The most intuitively obvious way is to give a shape description of the outer contour plus a description of the shape of the hole itself, augmented by a description of the hole’s location, orientation, and size. People use this format quite naturally in everyday language when they describe figure 2c as “a square surface with a clover-shaped hole in its center” (cf Giralt & Bloom, 2000, for developmental evidence that young children name holes in the same way that they name material objects). Specifying the outside shape of the surface as ‘square’ indicates that the surface lies inside the square-shaped outer edge that defines its exterior, as for any solid surface that would accurately be described as a square. (4)
A segment of a contour is convex (ie has positive curvature) if all of the straight-line segments connecting points on that contour segment lie wholly inside the region, and it is concave (ie has negative curvature) if those line segments lie wholly outside the region. The point along the contour at which the sign of its curvature changes from convex to concave (or vice versa) is called an inflection point. These definitions are important because the determination of this crucial shape feature (convexity/concavity) necessarily implies the assignment of inside/outside with respect to the contour, a distinction that is analogous (but not equivalent) to the figure–ground distinction in the sense that it interprets a two-sided contour as one-sided, but in this case for purposes of shape description rather than border ownership or relative depth.
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(a)
(b)
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Figure 2. An example of an ambiguous figure–ground phenomenon involving a hole. (a) The twodimensional image. (b) Its three-dimensional (3‑D) interpretation that consists of two solid surfaces at different depths: a closer, light-gray, clover-leaf-shaped surface and a farther, dark-gray, squareshaped surface. (c) A 3‑D interpretation of the image in (a) that consists of a closer, dark-gray, square-shaped surface that contains a clover-leaf-shaped hole [the complement of the closer surface in (b)] and a farther light-gray surface.
Specifying the shape of the hole as ‘clover-shaped’ indicates that, even though the material surface lies outside the inner edge, the shape description is that of the interior region, which is composed primarily of four broad, rounded convexities separated by narrow concavities. Speakers are thus describing the shape of the empty space of the hole in the same way as they would its hole-filling material complement, as if the hole were a ‘subtraction’ from the otherwise solid surrounding surface that effectively changes a portion of a material surface into an immaterial region of that surface (cf Palmer et al., 2008). We will call this the interior shape of the hole. In this scheme the shape representation of a hole-bearing surface includes, as an integral part, a representation of the shape of the hole itself, very much like the corresponding case of two surfaces (figure 2b), except that in the two-surface case, the inner contour signals a closer surface whose material lies inside that contour. It is far less clear how else the global shape of a hole-bearing surface can be represented in its entirety. It is not difficult to understand how the shape of a portion of such a holebearing surface might be represented, however, including concave portions along its inner edge. Try covering half of figure 2c with your hand and attend to the shape of the visible portion. It is now quite easy to see that the materially defined inner edge (which borders the hole) is broadly concave with a few very sharp points, quite unlike the convex and distinctly rounded clover shape seen as that of the hole itself. Moreover, it is hard to understand how that shape description can be extrapolated to encompass the entire shape
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of the hole-bearing surface without employing some description of the interior shape of the hole. The apparent difficulty of constructing such a representation does not mean that the visual system cannot do so, however, as it may just reflect a lack of imagination.(5) We will refer to the representation of the shape of a hole-bearing surface that does not require a description of the interior shape of the hole—whatever that representation might be—as the exterior shape of the hole. Finally, it is important to clarify the depth status of a hole: does it lie in the closer plane of the physical surface and edge, in the farther plane of the ground, or perhaps in a plane that is even closer to the viewer than the physical surface and edge? Bertamini (2006) states the ground-plane view quite explicitly: “In summary, I define holes as background regions that are surrounded by a foreground figure” (page 884). Others disagree, identifying the hole explicitly with the foreground surface (eg Nelson & Palmer, 2001; Nelson et al., 2009; Palmer, 1999; Palmer et al., 2008; Peterson, 2003). Perhaps the most obvious problem with identifying a hole as a background region is that it is incompatible with common sense and everyday language. People quite naturally say: “The background surface is visible through the hole”, but this statement clearly conflicts with any claim that the hole actually is some portion of that background surface, because it simply does not make sense for the background to be visible through itself. Another difficulty with identifying a hole as a background region is that the ground region visible through the hole is a purely accidental feature of that surface that changes with viewing positions. The fact that the perceived shape of the hole is invariant over such changes in viewpoint suggests that the hole is properly identified as an immaterial part of the hole-bearing foreground surface, within which it is a nonaccidental feature of that surface. For these reasons, we conceive the hole as a distinct part of the foreground surface— albeit a missing part—whose absence enables the viewer to see a portion of the background surface through it. 3 Empirical evidence To date, the best evidence that the shape of a hole is perceived as that of its interior region has come from experiments studying memory for the shapes of (immaterial) holes. In a long-term memory paradigm, Palmer et al. (2008) found that the interior shapes of holes are remembered just as well as those of their hole-filling material complements.(6) Their initial experiment used classic bipartite images without holes (like that shown in figure 1) and instructed participants to perceive one particular region as figure and the other as ground. Observers remembered the shape of the perceived figural region, but not that of the visible portion of the ground, for which performance was no different than for unpresented distractor foils. In a subsequent experiment the same instructions were given for surrounded regions that could be perceived as either material objects against a background or as immaterial holes in a surrounding surface. Here, participants remembered the interior shapes of the holes just as well as the shapes of correspondingly shaped material objects, and they remembered both of those much better than the (accidental) shapes of ground regions, which were no different than for distractor foils. The authors extended this result to unambiguous 3‑D stimuli, in which no instructions were required to induce the perception of depth and holes. These results (5)
There are some special cases in which it does seem relatively easy to construct a shape description of a hole-bearing surface without explicitly describing the shape of the hole. One is a picture-frame-like object—that is, an outer edge with nearby parallel inner edges—in which the surface can be viewed as an outline of that shape whose material happens to be so thick that it has distinct outer and inner edges. But the general case of a hole within a surface is not easily handled in this way because the shapes of the inner and outer contours may be different. (6) This is not the case, however, when the shape of the hole is given by accidentally aligned contours from different objects (see Nelson et al., 2009).
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clearly demonstrated that the interior shapes of holes are typically remembered just as well as their solid, hole-filling, object complements. Palmer et al. (2008) interpreted this pattern as evidence that the shapes of both material objects and immaterial holes are perceived in terms of their interior regions, despite the opposite-sided assignment of relative depth and edge ownership in the case of holes. Although performance in these memory tasks has provided evidence that people remember the shape of the interior rather than the exterior region of a hole, other researchers have reported evidence to the contrary in perceptual tasks. For example, Bertamini and Croucher (2003) measured the time observers required to verify the same features of two image regions when they were perceived as holes versus when they were perceived as solid objects. When the image regions like those shown in figure 3 were perceived as solid objects, observers were faster at discriminating whether the left-side or right-side vertex was lower when the object had a convex shape (an asymmetrical hexagonal ‘barrel’) than when the object had a concave shape (an asymmetrical hexagonal ‘hourglass’). They referred to this as a ‘convexity advantage’. However, when the same 2‑D shapes were perceived as holes, the speed of the same left–right discrimination of the lower vertex reversed, at least when the shapes were equated for difficulty: responses were faster for the concave (hourglass-shaped) hole than for the convex (barrel-shaped) hole. The authors inferred that, because there were no differences in the shape of the contours between objects and holes, observers must have performed the task on the exterior shape of the holes, consistent with the convexity advantage they found for solid shapes. In a similar set of experiments, Bertamini (2010) found that detection of reflection is faster in an image region perceived as an object than in one perceived as a hole. Bertamini and Helmy (2012) required participants to identify the shape of an interior region that was surrounded by an exterior region which was either congruent or incongruent in shape (eg a circle surrounded by a circle vs a circle surrounded by a square). Conflicting shapes influenced performance more when the interior region depicted a hole rather than an object. Such results suggest that there is sometimes a distinct difference in the processing of a region when it is perceived as a hole versus when it is perceived as an object.
Response time (log)/ms
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Figure 3. Adapted from experiment 3 of Bertamini and Croucher (2003). Barrel and hourglass shapes were presented as objects or holes, and observers were asked to determine whether the lower vertex was on the left or the right side. When seen as objects, responses to the convex barrel were faster than for the concave hourglass; whereas when seen as holes, responses to the concave hourglass were faster than for the convex barrel.
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Bertamini (2006) has claimed that these perceptual differences provide strong evidence that the contour of a hole is, in fact, owned by the material exterior region and that holes therefore do not have figural or ‘quasi-figural’ status. (Because the relative depth of surfaces at a hole is not in dispute by anyone, we take quasi-figural status to mean that a hole is perceived to have the shape of its immaterial interior rather than its material exterior.) In order to demonstrate such a claim for holes, he describes three conditions that need to be met: (1) that the stimuli are perceived as holes on a given task involving shape perception; (2) that performance should be similar for holes and for material objects; and (3) that, on the same task, holes should behave differently from other background regions. He further states that the results reported by Palmer et al. (2008) provide the best evidence that the perceived shape of a hole is the shape of its interior, but criticizes them because they are based on people’s performance on shape memory tasks, which might not reflect differences in perceptual processing The experiments reported below test Bertamini’s three requirements in a perceptual task. To the extent that the results support all three predictions, they would provide evidence that holes are perceived as having the shapes of the interiors of the hole contours, even though the material surfaces that own them physically lie outside those contours. We used a perceptual shape-matching task, similar to that employed by Baylis and Driver (1995) in that it requires a speeded response to match contour shapes. We asked participants to match object and hole stimuli in terms of either the interior shape (the ‘inner region’) or the exterior shape (the ‘outer region’) along one side. Because solid objects clearly own their surrounding contour, we expected that matching the shape of an object’s inner region would be faster than matching the shape of its outer region. If our proposal that observers perceive the interior shape of holes is correct, then matching the shape of a hole should be similar to matching the shape of an object: faster for matching its interior region than for matching its exterior region. However, if the shape of a hole is available only indirectly—being derived from the shape of its attached, closer, material surface—then participants should find it faster to match the holes in the exterior shape condition than in the interior shape condition. Such a finding would indicate that the shapes of holes and those of their hole-filling materially complementary solid objects are fundamentally different and that a hole’s interior shape is perceived only indirectly from prior perception of its exterior shape, as claimed by Bertamini and Croucher (2003). 4 Experiment 1 4.1 Method 4.1.1 Participants. Sixteen undergraduate students from Wheaton College (Norton, MA) were recruited through fliers posted on campus. Compensation was a US $5 gift certificate from the campus bookstore. All participants reported normal or corrected-to-normal vision and were unaware of the purpose of the experiment. There were five males and eleven females, whose ages ranged from 18 to 22 years old (M = 20.1 years). All experiments were carried out under approval of the Wheaton College IRB and in accordance with the World Medical Association Helsinki Declaration as revised in October 2008. 4.1.2 Apparatus. Stimulus images were presented on a 17″ Dell LCD monitor positioned approximately 60 cm in front of participants. Presentation, timing, and response collection were controlled by SuperLab experiment software and a Cedrus RB‑834 response pad (Cedrus Corporation, 2006) running on a Dell OptiPlex computer. The experiment was conducted in an isolated, diffusely lighted room.
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4.1.3 Stimuli and procedure. Figure 4 illustrates sample trial stimuli. Object and hole targets were blue inner regions, defined by a complex contour on the right side and straight contours on the remaining three sides (subtending approximately 2.4 × 3.6 deg of visual angle), which were positioned at the center of a red square (each side approximately 5.4 deg), which itself was surrounded by a blue square (each side approximately 8.9 deg). Shading was used to disambiguate each of the same 36 unique shapes as hole and object versions. Previous evidence has demonstrated that such shading information is highly effective in differentiating relative surface depth (Nelson & Palmer, 2001). The response options presented below the target stimulus were grayscale versions taken from the same set as the target shapes, one of which always matched the target stimulus for the given set of instructional conditions. Target type
hole
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Figure 4. Illustration of the four trial types in experiment 1. Participants were presented with an object or hole target stimulus and were required to indicate which of four response options matched the shape of (a) the object’s inner region, (b) the hole’s inner region, (c) the object’s outer region (ie the complement of the object’s shape), or (d) the hole’s outer region. The correct answer is shown in the top-left quadrant of each quartet of options.
Each trial began with the target stimulus appearing alone at the top of the display for 800 ms before the four matching shapes were added underneath. This combined target-plusmatching display remained visible until a response was made. Participants were instructed to make their responses by pressing one of four buttons on the button box whose positions were spatially isomorphic to those of the four response options on the computer screen. The instructions emphasized response speed without sacrificing accuracy. Correct answers initiated the next trial, whereas incorrect responses prompted a text box reading “Incorrect answer! Please wait to try again.” This message was followed by a 3 s delay and the repetition of that trial. The location of the matching shape was randomized across trials such that each button served as the correct response an equal number of times for each participant.
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Trials were divided into four blocks based on the type of target stimulus (object vs hole) and the type of matching response (match the inner region vs match the outer region). Instructions were provided at the beginning of each block regarding both what type of target stimulus would be presented and what participants were asked to match. Specifically, participants were told to select the response shape that “matches the one shown” (object target + match inner region block; see figure 4a), “fits into a slot’”(hole target + match inner region block; see figure 4b), “fits into the one shown” (object target + match outer region block, see figure 4c), or “is the complement of the shape of the hole” (hole target + match outer region block; see figure 4d). To ensure that participants understood the task, they were shown what each block instruction meant using a demonstration on the computer monitor. For instance, the two outer region tasks were demonstrated by showing how the correct answer complemented the right side of the target shape in a lock-and-key fashion to form a complete square. After the instructions were given, participants were prompted to ask the experimenter any questions they might have. They were also given the opportunity for a break before beginning the experimental trials. Within each block, all target shapes were presented twice in a random order, resulting in 72 trials per block. Thus, there were a total of 288 ‘core’ trials per participant, not including any trial repetitions due to an incorrect response. Every participant completed all four blocks, and block order was determined by a between-subjects Latin squares design. Reaction times were based on the first 288 trials with correct responses (disregarding errors), and accuracy measures were based on the first 288 trials of each stimulus type as defined by the experimental design (disregarding retaken trials). 4.2 Results and discussion Accuracy and reaction time data (see figure 5) for the 288 core trials were analyzed using separate within-subjects analyses of variance (ANOVAs) that each contained two factors, target type (object vs hole) and matching response type (inner region vs outer region). The analysis of reaction times, which included only correct trials and excluded any trials in which the time exceeded 3 SDs of the population-correct trial mean (1.96% of correct trials removed), revealed a significant main effect of matching response type (F1, 15 = 39.54, p
