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Research Article  |   June 2009
Background motion and the perception of shape defined by illusory contours
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Journal of Vision June 2009, Vol.9, 5. doi:10.1167/9.6.5
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      Wang O. Li, Sieu K. Khuu, Anthony Hayes; Background motion and the perception of shape defined by illusory contours. Journal of Vision 2009;9(6):5. doi: 10.1167/9.6.5.

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A Kanizsa triangle is usually generated by placing three circular tokens, with deleted wedges, at the apexes of an equilateral triangle. If the token angles do not each subtend 60°, a Kanizsa triangle may still be evident, but with illusory contours that appear to be curvilinear. We investigated whether this curved-contour distortion in shape can be nulled by radial background image motion utilizing a two-alternative forced-choice procedure. We report that a test Kanizsa figure with concave illusory contours appeared to form a perfectly regular equilateral triangle when it was superimposed on a globally expanding pattern. Conversely, a test Kanizsa triangle with convex illusory contours was perceived as regular when it was superimposed on a globally contracting pattern. This distortion effect was most distinct for fast dot speeds and was greater for contracting motion. Additionally, the effect was observed regardless of a polarity difference between background dots and tokens. However, shape distortion was not evident when the Kanizsa figure was defined by “real,” luminance-defined, contours. Our findings support the conclusion that background image motion plays an important role in the perception of shape, especially when there is an “insufficiency” in the position information that specifies the shape's contours.

Introduction
Where an object in the visual world is localized is dependent not only on its spatial position on the retina, but on a number of other visual attributes, such as image motion, which has frequently been shown to affect the apparent spatial position of objects (e.g., De Valois & De Valois, 1991; Ramachandran & Anstis, 1990). In a now classic experiment, De Valois and De Valois reported that the apparent position of a Gabor pattern (a Gaussian-windowed sinusoid) that contains grating motion has its apparent position shifted in the direction of motion. The extent of the apparent displacement is small—approximately 2–10 min arc for a Gabor pattern with a Gaussian envelope space constant of 0.13°—and this effect is dependent on both stimulus eccentricity and speed of grating motion. Since the work of De Valois and De Valois, a number of studies have sought to characterize the stimulus conditions under which ‘internal’ motion results in an apparent shift in position. For example, motion has also been shown to affect the following: the apparent position of objects located in depth (see Tsui, Khuu, & Hayes, 2007a); the binding of contour elements (Hayes, 2000); and the perception of global shape (Rainville & Wilson, 2004, 2005). Recently, Tsui, Khuu, and Hayes ( 2007b) have argued that this perceived spatial displacement is a result of object-centroid shift. 
It is important to note that the studies referred to above have used luminance-defined stimuli for which there is a corresponding retinal representation, such spatial positions can in principle be derived from their coordinates on the retinal image. Furthermore, apparent displacement of position induced by background or internal motion occurs only for objects, such as Gabor patterns, that do not have sharp edges. Motion of a pattern generated by a “box-car” windowed sinusoid, for example, does not result in apparent displacement of the pattern as it does with a Gaussian-windowed sinusoid (a Gabor pattern—see De Valois & De Valois, 1991). This distinction is likely because there is greater certainty regarding the boundary of a sharp-edged object as compared to an object with soft edges. However, perception of objects is possible without this requisite, as evidenced by illusions of form such as Varin and Ehrenstein figures (Ehrenstein, 1987). A classic example is subjective, or illusory, contours that have no retinal representation, in which the perception of distinct lines or edges is derived via perceptual heuristics, or inferences, about a particular stimulus configuration. 
Perhaps the best-known figures formed by illusory contours are Kanizsa figures. Kanizsa figures, for example, Kanizsa triangles (Kanizsa, 1976, 1979), are, for most observers, vivid examples of illusory contours that appear as edges linking wedge-shaped tokens that form the corners of an equilateral triangle (or, indeed, other geometric shapes). A regular triangle is perceived in Figure 1a when three circles, each with a 60° wedge-shaped section removed, are placed such that radii forming wedges are in alignment. Illusory contours appear to connect the radii of each wedge section to produce a perceptually closed stimulus that, in some instances, not only has defined edges but appears brighter or darker than its background. It is thought that this visual modal completion arises because the visual system, given no other cues, interprets the stimulus as a completed triangle with its corners occluding the tokens (see Rock & Anson, 1979). 
Figure 1
 
Kanizsa triangles. (a) A typical Kanizsa triangle placed on a gray background. (b) A Kanizsa triangle with its circular missing-wedge segments subtending an angle slightly greater than 60°, giving rise to illusory convex curvilinear contours, and with the appearance of being a “fat” Kanizsa triangle. (c) A Kanizsa triangle is perceived to be a “thin” triangle with concave edges when its tokens' wedge-shaped segments subtend less than 60°. (d) A stimulus used in the present study—a Kanizsa triangle was superimposed on 200 light-increment dots, which underwent either expanding or contracting motion. The Kanizsa triangle as a whole slowly, rigidly, rotated.
Figure 1
 
Kanizsa triangles. (a) A typical Kanizsa triangle placed on a gray background. (b) A Kanizsa triangle with its circular missing-wedge segments subtending an angle slightly greater than 60°, giving rise to illusory convex curvilinear contours, and with the appearance of being a “fat” Kanizsa triangle. (c) A Kanizsa triangle is perceived to be a “thin” triangle with concave edges when its tokens' wedge-shaped segments subtend less than 60°. (d) A stimulus used in the present study—a Kanizsa triangle was superimposed on 200 light-increment dots, which underwent either expanding or contracting motion. The Kanizsa triangle as a whole slowly, rigidly, rotated.
The cortical origin of illusory contour perception is, likely, early in the visual pathway given that visual attributes that are commonly processed in the early stages of visual processing, such as luminance contrast, also influence illusory contours. Dresp and Bonnet (1991) showed that the detection threshold of a spot of light varied at different positions on Kanizsa figures with thresholds higher when the stimulus was near the illusory contours of the figures. However, thresholds remained normal at the position of the illusory contours. Similarly, the detection thresholds of Gabor patches imaged on Kanizsa figures are only affected when the Gabors are aligned collinearly to illusory contours (Danilova & Kojo, 2001). A number of studies have implicated area V2 of the primate visual cortex as being important in the processing of illusory contours (Ffytche & Zeki, 1996; Ritzl et al., 2003; Seghier et al., 2000). For example, both single cell recording studies (e.g., Peterhans & von der Heydt, 1989) and neuroimaging studies (Ffytche & Zeki, 1996; Ritzl et al., 2003) have reported selective activation of area V2 in response to illusory contours. 
While there is a substantial literature that reports the interaction of image motion and the spatial position of objects generated from “real” elements, the extent to which, and the conditions in which, image motion affects the position of illusory contours has received little attention. Published studies have, however, reported that the perception of illusory contours shares similar characteristics to perception of real contours, especially in the perception of line orientation. For example, Westheimer (1990) compared the magnitude of the tilt illusion, and orientation discrimination thresholds, of real contours with several types of illusory contours induced by offset lines, Kanizsa tokens, and moving dots. Westheimer reported that orientation discrimination performance was superior for real contours and that both illusory and real contours produced measurable tilt effects, but a larger effect was noted for illusory contours. These findings have led Westheimer to argue that illusory contours provide a weaker perceptual representation of orientation, thus resulting in poorer discrimination leading to a larger tilt effect. Support for this argument comes from Guttman and Kellman ( 2004) who investigated the ability to discriminate the location of dots relative to contours defined by real lines or illusory lines, or interpolated from discontinuous edges. Guttman and Kellman showed that localization acuity was best for real contours, while localization acuity for both illusory and interpolated contours was comparatively lower. Importantly, the studies by Westheimer and Guttman and Kellman imply that the main difference between illusory and real contours is that illusory contours provide a comparatively degraded spatial signal leading to comparatively poorer perceptual judgements. 
Recent evidence suggests that illusory and real (luminance-defined) contours share a common site of processing in the visual cortex. Ritzl et al. (2003) compared the neural activation of real (luminance-defined) contour figures, Kanizsa figures, and figures that consisted of rotated inducers; i.e., same tokens as for Kanizsa figures, but the inducers were not orientated to form illusory figures. They reported that figures with contours, either real or illusory, activate extrastriate cortex earlier than figures without contours. Additionally, Seghier et al. (2000) reported that activation in V1 was particularly prominent when illusory figures were perceived. Figures that consisted of the same inducers, but where the inducers did not induce illusory contours, resulted in less V1 activation. Importantly, given that illusory and real contours share common cortical representation, it is likely that there will be similarities in their processing. As noted above, image motion affects the position of real objects especially those that have “soft” boundaries without sharp edges but not to objects with “hard” boundaries. Given that illusory contours provide a weaker spatial signal than real contours (Guttman & Kellman, 2004; Westheimer, 1990), they are susceptible to motion-induced displacement in an analogous fashion to soft-edged objects. 
It is important to note that published investigations on motion-induced spatial distortions have usually required observers to judge the apparent position of an object containing motion with reference to flankers (e.g., De Valois & De Valois, 1991). The generation of vivid illusory contours is limited to specific configurations of tokens that serve to the induce illusory contours and, if a “reference to flanker” technique is used, the tokens may interfere with vernier-acuity judgments since they themselves signal the position of the illusory contour; i.e., observer judgments may be based on the position of the tokens, rather than on the position of the illusory lines that the tokens induce. To avoid this potential confound, we used a Kanizsa figure and examined whether image motion can distort its apparent shape. This methodology offers the advantage of quantifying the effect of image motion on the position of illusory contours without requiring direct report of relative or absolute spatial position; rather, the effect is reflected as a distortion of shape. 
A property of Kanizsa figures is that when the angle subtending each of the three wedge sections is not exactly 60°, the illusion is not abolished, but the illusory contours appear curvilinear and the resulting illusory triangles have concave or convex boundaries (Ringach & Shapley, 1996; Spehar, 2000). As shown in Figure 1b, when the angle of each wedge section is slightly greater than 60°, illusory contours appear convex in shape. Conversely, when the wedge angles are slightly less than 60°, illusory contours that are concave result (see Figure 1c). These two versions of the triangle appear to an observer as “fat” and “thin” Kanizsa triangles, respectively. 
In the present study, we examine whether apparent shape produced by changing the angle of the wedges of a Kanizsa triangle can be perceptually nulled by radially expanding or radially contracting background motion. If image motion acts to distort the spatial perception of illusory contours in the motion direction, a convex “fat” Kanizsa figure will appear to be of regular equilateral form when viewed in the presence of dots undergoing contracting motion, and vice versa for a concave “thin” Kanizsa triangle when viewed in the presence of dots undergoing expanding motion. Hayes (2000) demonstrated that the illusory positions of Gabor patterns containing moving sinusoids, rather than their veridical positions, govern the binding of contour elements. Similarly, motion may induce a shift in apparent position of illusory contours, and in the case of “fat” and “thin” Kanizsa figures may act to correct their shape. In Experiments 1 and 2, we examine this possibility by superimposing a Kanizsa triangle on a radial global-dot motion stimulus. In Experiment 3, we investigate the role of contrast polarity in the interaction of motion and contour spatial position. In Experiment 4, we examine whether motion-induced distortion of shape is evident for “real” luminance-defined contours. 
Experiment 1: The effect of motion on the perception of illusory shape
We investigate whether the motion of elements in the background affects the apparent shape of an illusory Kanizsa triangle. Observers were presented with a Kanizsa triangle on a background of white dots that moved in a radial direction, and they were required to compare the shape of the Kanizsa triangle with a reference stimulus. We controlled the direction of motion (expanding, contracting, and random) as well as the dot speed (2.5°/s and 6.0°/s). 
Methods
Observers
Six observers participated in Experiment 1. All observers had normal, or corrected to normal, visual acuity and were naive to the purpose of the study. 
Stimuli
Custom software written in MATLAB 7.1 and the PsychToolBox (Brainard, 1997; Pelli, 1997) was used to generate and present stimuli. Stimuli, shown in Figure 1d, were Kanizsa triangles placed within an otherwise blank gray square (set to a background luminance of 43.1 cd/m2) that subtended 8.83° × 8.83° of visual angle when viewed from a distance of 100 cm. The image sequence was generated by a Dual-Core PC (3.0G Hz) and presented on a linearized CRT monitor. To generate a Kanizsa triangle, three incomplete white circles, of diameter 1.79° and luminance 132 cd/m2, were equidistantly placed (with their origins 3.5° of visual angle from the center of the stimulus) to form three corners of an equilateral triangle. The corners of the triangle were indicated by wedge-shaped sectors removed from each of the three “token” circles. The Kanizsa triangle was then superimposed on a background consisting of 200 white anti-aliased dots (diameter 0.09° visual angle, 132 cd/m2; see Figure 1d). The resulting average dot density was 2.57 dots/deg 2. Dots moved at a particular speed either in an expanding, contracting, or random direction, depending on the experimental condition. The distribution of the dots followed a radial Gaussian with the peak midway between the center of the stimulus and the boundaries, and a standard deviation equal to one fourth of that distance (1.10° visual angle). This stimulus property reduced the uneven distribution of moving dots in the stimulus caused by rapid replotting of dots when they disappeared from the stimulus at the inner or outer boundary. 
The Kanizsa triangles, in addition to being placed on a background of moving dots, also underwent slow rigid rotation in a clockwise or anti-clockwise direction at a speed of 0.17°/s. Moving Kanizsa figures were presented at a movie frame rate of 24 Hz. The slow rotation of the Kanizsa triangle was employed to prevent observers from simply attending to the tokens, or making geometric comparisons with the screen edge, when judging the apparent shape of the stimulus. Additionally, observers were required to maintain fixation on a mark, with a size of 0.12° × 0.12° visual angle, placed in a 0.70° × 0.70° blank region at the center of the stimulus. 
Procedure
Data collection was implemented using Method of Constant Stimuli in conjunction with a temporal two-alternative forced-choice procedure. In one interval, the test stimulus, a Kanizsa triangle with all three token angles set to a value of between 56° and 64° (in 1° steps), was presented. The test stimulus was superimposed on a background with dots moving in a radial direction at a speed of either 2.5°/s or 6.0°/s. Thus, for the test stimuli there were nine possible token angles and two background-dot speeds. The other interval presented a reference stimulus, which was a similar Kanizsa figure with token angles set to 60°. Similar to the test stimulus, the reference stimulus was superimposed on a background of dots that moved at 2.5°/s, but in random directions. Each interval had a 2.0-s duration and was separated by a period of 0.5 s during which the screen was set to the background luminance. The presentation order of the reference and test intervals was randomized from trial to trial. The task of the observer was to indicate the interval containing a perceptually “fatter” triangle. Observers indicated their response by pressing one of two buttons on a keyboard. 
The above stimulus procedures were used for conditions with test stimuli in which background dots moved in an expanding direction, and for conditions in which the background dots moved in a contracting direction. A condition in which dots moved in random directions at 2.5°/s, in which there was no coherent motion, was included to provide a control measure of performance. In total, there were nine levels of test token angle, two speed conditions, two different motion directions, and a baseline random motion-direction condition. Thus, there were 45 stimulus conditions. Each condition was repeated 20 times, and the order in which conditions were presented was randomized. 
Results and discussion
The results of Experiment 1 were collated to indicate the proportion of times an observer perceived the test stimulus to be “fatter” than the reference stimulus for different dot speeds (2.5°/s and 6.0°/s) and directions (expanding, contracting, and random). Logistic functions fit to data provided an estimate of the point of subjective equivalence (PSE) for each speed-direction condition, and is indicative of the token angle of the test stimulus when the test stimulus is judged to be equal (in shape) to the reference stimulus. Figure 2 presents representative data for one observer and plots (along with modeled psychometric functions) the results for contracting, expanding, and random, motion, conditions with a dot speed of 6°/s. Logistic functions well fit the data with an average R 2 of 0.98. The horizontal dashed line indicates the subjective equivalence line, while the vertical dashed line indicates the physical token angle of the reference stimulus of 60°. As can be seen from this figure, when background dots moved in a random direction (solid line, black) the 0.5 point of the corresponding psychometric function is similar to the reference token angle. This result clearly indicates that no shape distortion is evident for the control condition. However, when the background motion was either expanding or contracting, the corresponding psychometric functions were shifted to the left or to the right of the control random-dot condition. For contracting motion, the 0.5 point of the psychometric function (red line) corresponds to a token angle that is larger than the reference stimulus token angle. This indicates that the test stimulus for this motion condition needed to be “fatter” than the reference stimulus for it to appear similar in shape. A converse pattern is seen for expanding motion. For expanding motion, the psychometric function (green line) is shifted to the left of the 60° line, indicating that the wedges of Kanizsa stimulus must be smaller than 60° to achieve a perceptual match to the reference stimulus. 
Figure 2
 
Representative data for one observer from Experiment 1 are plotted for contracting, expanding, and random motion conditions—dot speed of 6.0°/s. Dashed horizontal and vertical lines indicate the 0.5 proportion-correct performance level and the token angle of the reference stimulus, respectively.
Figure 2
 
Representative data for one observer from Experiment 1 are plotted for contracting, expanding, and random motion conditions—dot speed of 6.0°/s. Dashed horizontal and vertical lines indicate the 0.5 proportion-correct performance level and the token angle of the reference stimulus, respectively.
The PSEs of all observers for different motion direction conditions are plotted in Figures 3a and 3b for test stimulus dot speeds of 2.5°/s and 6.0°/s. There is some variability between observers, but overall the results confirm that the background motion affects the shape of illusory Kanizsa figures, especially at faster dot speeds. A contracting background resulted in the perception of a “thinner” Kanizsa figure, while an expanding background produced the opposite result. The average PSE for a Kanizsa figure superimposed on a contracting background moving at 2.5°/s was 60.90°, and this value was 61.28° for a dot speed of 6.0°/s. Importantly, these PSE values reflect the situation in which the test stimulus needs a larger token angle (was a “fatter” stimulus) to offset the distortion effect of motion. In the converse observation, Kanizsa figures superimposed on an expanding background were perceived to be equilateral when their token angles were 59.43° and 58.97° for dot speeds of 2.5°/s and 6.0°/s, respectively. Thus, for both dot speeds the Kanizsa figure needed to be “thinner” to counteract the distortion effect of expanding motion, with a larger effect noted for the faster speed conditions. Statistical comparison with a one-tailed t-test between the expanding and contracting conditions revealed a statistically significant difference in shape (dot speed 2.5°/s: t 5 = 2.04, p < 0.05; 6.0°/s: t 5 = 2.65, p < 0.05). All t-tests in the present article are one-tailed with an alpha value of 0.05, unless otherwise specified. When contracting and expanding motion conditions were compared to the random dot condition, a statistically significant difference was evident for the contracting motion condition at 6.0°/s ( t 5 = 3.13, p < 0.05). The difference between expansion conditions at 6.0°/s and the random dot condition is close to being significantly different ( t 5 = 1.944, p = 0.0548). 
Figure 3
 
Results of Experiment 1. Plots of the PSE for the different direction conditions for dot speeds of (a) 2.5°/s and (b) 6.0°/s of Experiment 1. Dark solid line symbols represent the mean result of all participants. Error bars indicate ±1 standard error of the mean.
Figure 3
 
Results of Experiment 1. Plots of the PSE for the different direction conditions for dot speeds of (a) 2.5°/s and (b) 6.0°/s of Experiment 1. Dark solid line symbols represent the mean result of all participants. Error bars indicate ±1 standard error of the mean.
To further clarify the strength of the relationship between the extent of distortion and our stimulus manipulations, Cohen's d was computed for the comparison between expansion and contraction of the two speeds (dot speed 2.5°/s: d = 1.45; 6.0°/s: d = 2.10). Both comparisons have a “large” difference according to Cohen's standard. Similar observations are found in the comparisons between the radial motion and the random motion conditions. Cohen's d with a range of 0.80 to 1.47 was noted, which suggest a systematic distortion of the radial patterns in the stimulus. 
The results reveal an additional finding—the effect of background-dot speed on the distortion of illusory shape. For the two dot-speed conditions, the shape distortion of illusory contours is larger for the faster speed condition. This observation is very much consistent with previous research that reports a speed dependency on the extent of position shift of real objects (see De Valois & De Valois, 1991; Tsui et al., 2007b). 
It should be noted that background motion may interact with the apparent spatial position of the tokens used to induce illusory contours, and this potential apparent change in position may account for a change in triangle shape. However, this effect, if present, is an unlikely explanation for our results, since, if motion displaced the apparent spatial position of tokens, it would do so equally, resulting in a percept of a larger or smaller equilateral triangle, not a fatter or thinner triangle. 
Experiment 2: The effect of image motion on apparent token angles
In Experiment 1, we proposed that the apparent change in shape and position of illusory contours is a consequence of background motion shifting their apparent position in the direction of motion. In the context of a curvilinear Kanizsa figure, this process has the effect of perceptually “correcting” the shape distortion that arises from token angles greater than or less than 60°. It can be argued that our findings are the result not of background motion acting on the spatial position of illusory contours, but the result of background motion acting directly on the angle of wedge-shaped tokens, which in turn determine the shape of illusory contours they induce. We repeated Experiment 1 with stimuli in which motion was excluded from both in and around each token. 
Methods
The stimuli were similar to those of Experiment 1 with the exception that no moving dots appeared within the wedges of the tokens. The same procedure as in Experiment 1 was repeated for all the observers. Only one dot speed—6.0°/s—was used since this speed produced the strongest distortion effect. Each condition was repeated 20 times, and the results were fit with a logistic function to provide an estimate of the PSE for the three conditions of contraction, expansion, and random motion. 
Results and discussion
The results of Experiment 2 are plotted in Figure 4 in a similar format to Figures 2 and 3. The pattern of results is similar for all observers, and they mirror those obtained in Experiment 1. In Figure 4a, representative data and corresponding logistic functions for one observer are shown, while in Figure 4b, the PSEs extracted from function fits are plotted for each motion condition. To perceive a perceptually regular equilateral triangle, observers judged the PSE token angle to be in the opposite direction to motion—the PSE is smaller for expanding motion (mean = 59.24°) and larger for contracting motion (mean = 60.86°) than the reference stimulus (mean = 60.03°). Moreover, the difference in PSEs for expanding and contracting motions is statistically significant ( t 5 = 2.68, p < 0.05). Importantly, these findings suggest that distortion effect found in Experiment 1 is not a consequence of local motion within the token wedges directly affecting its perceived angle. 
Figure 4
 
Results of Experiment 2 plotted in a similar format to Figures 2 and 3. (a) The psychometric functions of a representative observer for different motion conditions. (b) Plots of the PSE for different motion conditions for different observers. Error bars indicate ±1 standard error of the mean.
Figure 4
 
Results of Experiment 2 plotted in a similar format to Figures 2 and 3. (a) The psychometric functions of a representative observer for different motion conditions. (b) Plots of the PSE for different motion conditions for different observers. Error bars indicate ±1 standard error of the mean.
Experiment 3: The effect of luminance polarity on the distortion of shape
Illusory contours of Kanizsa figures are induced through the appropriate arrangement of tokens that serve to highlight the corners of a geometric form. A noteworthy question has been whether the perception of illusory contours is dependent on the stimulus characteristics of inducers; particularly, their luminance polarity. Published studies have reported that Kanizsa figures are perceptible even when corner tokens are of different polarities (see Matthews & Welch, 1997; Prazdny, 1983; Shapley & Gordon, 1983, 1985; Victor & Conte, 2000), though the percept is less salient (Spehar, 2000). While polarity is not a factor in the formation of illusory contours, we investigate in Experiment 3 whether the polarity of moving dots is a factor in the distortion effect noted in Experiments 1 and 2. In the previous experiments, the luminance polarities of inducers and dots were identical. We question whether reversing the polarity of dots (and thus the Kanizsa figure and background will be, potentially, processed by separate systems) produces a similar distortion effect. The observation that the induction of illusory contours is not dependent on luminance contrast implies that, regardless of the polarity of dots, the apparent position of illusory contours is distorted by motion. However, it is important to note that a luminance polarity difference usually provides a strong segmentation cue between stimulus and background, and under these conditions motion may not affect the perception of Kanizsa shape. Indeed, we have recently demonstrated that the shape perception of a motion pattern is unaffected by another complex motion pattern, only when the polarities of each motion pattern are of opposite luminance polarity (Li, Khuu, & Hayes, 2008). 
The purpose of the present experiment is to resolve whether a polarity difference between moving background dots and the Kanizsa figure will result in a distortion of shape as found in the same-polarity conditions. We repeated Experiment 1 with dots of the opposite polarity to the Kanizsa-triangle-inducing tokens with all observers. The stimulus and procedure were identical to Experiment 1, except all dots, including those in the reference interval, were black and set to a luminance of 0.06 cd/m 2. The tokens were white, set to a luminance of 132 cd/m 2. Only the dot speed of 6.0°/s was used. 
Results and discussion
The results of Experiment 3 are plotted in Figure 5, in a similar format to Figures 2 and 3. The pattern of results is similar for all observers: despite the polarity difference between the dots and the tokens, background motion distorts the apparent curvature of the illusory edge in the direction of the background motion, with a significant difference between expanding and contracting motion conditions ( t 5 = 4.09, p < 0.01). The mean PSEs for expanding and contracting motions are 59.36° and 60.64°, respectively. This finding demonstrates that illusory contours are susceptible to the effect of background motion regardless of the polarity of moving elements. Moreover, for illusory contours, luminance polarity does not segregate the Kanizsa figure and remove it from influence from a moving background. This finding reinforces to the notion that illusory contours can best be considered as polarity neutral. Mixing up the luminance polarity of tokens does not limit the perception of illusory contours, and computational models have been developed in which illusory contours are extracted by second-order mechanisms that perform full wave rectification through the combination of “on” and “off” pathways (see Wilson, 1999). 
Figure 5
 
Results of Experiment 3 plotted in a similar format to Figures 2 and 3. (a) The psychometric function of a representative observer. (b) Summary of results plotted as perceived regularities in terms of angular subtense of token wedge. Error bars indicate ±1 standard error of the mean.
Figure 5
 
Results of Experiment 3 plotted in a similar format to Figures 2 and 3. (a) The psychometric function of a representative observer. (b) Summary of results plotted as perceived regularities in terms of angular subtense of token wedge. Error bars indicate ±1 standard error of the mean.
Experiment 4: The effect of image motion on the shape of “real” triangles
Several studies that have examined illusory contours have concluded that, due to their subjective nature, there is less certainty regarding their spatial position, which leads to a reduction in sensitivity (leading to poorer discrimination) as compared to “real,” luminance-defined contours (see Guttman & Kellman, 2004; Westheimer, 1990; Westheimer & Li, 1996, 1997). This observation about illusory and real contours may be the basis of our effect and necessarily implies that the amount of shape distortion previously noted is less, or negligible, in the case of “real” figures. Experiment 4 examined this possibility by repeating the experiments with a triangle defined by contrast edges. 
Methods
In constructing “real” triangles, it is important to note that, given the subjective nature of Kanizsa figures, there is no “real” objective counterpart, especially for conditions in which the figure is defined by curvilinear contours. To our knowledge, no studies have directly attempted to systematically define the shape of curvilinear Kanizsa figures with changing token angle. However, a good perceptual approximation of the curvilinear shape of Kanizsa figures can be achieved through cosine modulation of the border of a triangle according to the following equations:  
T ( r 0 , ϕ 0 ) = r 0 cos ( π / 3 ) / cos ( π / 3 ϕ 0 ) ,
(1)
 
M ( r 0 , ϕ 0 ) = A ( r 0 T ( r 0 , ϕ 0 ) ) ,
(2)
 
T ( r 0 , ϕ 0 ) = T ( r 0 , ϕ 0 ) + M ( r 0 , ϕ 0 ) ,
(3)
where T( r 0, ϕ 0) is the polar coordinate of any point of a regular equilateral triangle with respect to its centroid; r 0 denotes the distance between its centroid to its vertex; while ϕ 0 denotes the angle deviating from this line. The modulation applied, M( r 0, ϕ 0), is defined relative to a circle sharing its center with the centroid of the triangle and having a radius equal to r 0. A represents the magnitude of the modulation. T′( r 0, ϕ 0) denotes the modulated coordinate. A fully modulated triangle with A equal to 1 will result in the polar coordinate T′( r 0, ϕ 0) of a full circle with a radius equal to r 0. The parameter r 0 is kept constant at 2.61° and is similar to the distance of the center of Kanizsa tokens from the center of the stimulus in the previous experiments. Nine modulations, A = −0.08, −0.06, −0.04, −0.02, 0.0, 0.02, 0.04, 0.06, 0.08, were tested. Figures 6a and 6b show the maximally modulated stimuli tested in the present experiment with A equal to −0.08 and 0.08. Naturally, 0.0 represents a “regular” triangle, while negative and positive values correspond to “thinner” and “fatter” figures. 
Figure 6
 
Luminance-defined “real” triangle used in Experiment 4. (a) A “real” triangle with a modulation A = −0.08 simulating a “thin” Kanizsa figure. (b) A “real” triangle with a modulation A = 0.08 simulating a “fat” Kanizsa figure.
Figure 6
 
Luminance-defined “real” triangle used in Experiment 4. (a) A “real” triangle with a modulation A = −0.08 simulating a “thin” Kanizsa figure. (b) A “real” triangle with a modulation A = 0.08 simulating a “fat” Kanizsa figure.
Similar to Experiment 1, Method of Constant Stimuli in conjunction with a temporal two-alternative forced-choice procedure was used. In one interval, the reference stimulus was presented: a luminance-defined filled triangle (50.1 cd/m 2) against a gray surface (43.1 cd/m 2) with a background of white dots (132 cd/m 2) moving in random directions. In the other interval, the test stimulus was presented with one of the aforementioned nine shape modulations. For both reference and test stimuli, background dots moved at a speed of 6.0°/s. The task of the observer was to indicate the interval containing the “fatter” figure. The above procedure was repeated for dots moving in expanding, contracting, and random directions. Each modulation and dot-direction condition was repeated 20 times. 
Results and discussion
The results were collated to indicate the number of times the test stimulus was judged to be “fatter” for each motion direction condition. A logistic function was fit to the data to reveal the PSE—the magnitude of modulation according to Equations 13 (A) of the test stimulus resulting in a triangular shape judged equal to the reference stimulus; these values are plotted in Figure 7. Representative data are shown in Figure 7a, while Figure 7b illustrates the PSEs extracted from psychometric functions of the observers. Three noteworthy patterns in the data are found. First, the results of all observers were very similar. Second, in comparison to previous findings, the standard error of the mean for the different direction conditions is much reduced. This result indicates that observers are very accurate in perceiving the shape of a luminance-defined equilateral triangle. Third, and most importantly, the distortion effect noted in previous experiments was eliminated for most observers (observer KWY registered a very small distortion effect) with a luminance-defined triangle ( t 5 = 0.97, p > 0.05), though not all had PSE modulations that corresponded perfectly to an equilateral triangle (A = 0). These results demonstrate that a significant distortion effect is not found when the triangle is luminance defined and support the argument that spatial-position uncertainty with illusory contours, as reported by Westheimer (1990) and Westheimer and Li (1996, 1997), underlies our findings with Kanizsa triangle background-motion-induced apparent distortion. 
Figure 7
 
Results of Experiment 4 plotted in a similar format to Figures 2 and 3. (a) The psychometric function of a representative observer. (b) Summary of results plotted as perceived regularities in terms of modulations applied to illusory contours. Error bars indicate ±1 standard error of the mean.
Figure 7
 
Results of Experiment 4 plotted in a similar format to Figures 2 and 3. (a) The psychometric function of a representative observer. (b) Summary of results plotted as perceived regularities in terms of modulations applied to illusory contours. Error bars indicate ±1 standard error of the mean.
General discussion
Using a Kanizsa equilateral triangle with variable token wedge angles, we demonstrated that the apparent inwards or outwards curved shape of the illusory contours of the triangle can be systematically nulled by radial background motion. Our conclusion is that background motion distorts spatial perception of illusory contours, and our results complement a body of literature that shows that motion affects the spatial position of objects under many stimulus arrangements, though particularly where the object has boundaries that are not composed of sharp luminance edges (see, e.g., De Valois & De Valois, 1991; Ramachandran & Anstis, 1990). 
In accounting for the Kanizsa illusion, a major issue has been to provide an explanation for the phenomenon of curvilinear contours when the angle of the wedges does not equal 60° (Ringach & Shapley, 1996; Spillman & Dresp, 1995). Lesher (1995) proposed a “completion filter” model, related to the idea of an association field proposed by Field, Hayes, and Hess ( 1993), as a low-level determinant of illusory contours. Considering the tokens as contour elements, tokens activate low-level visual processors as if there existed real contours and they induce a percept of a contour. Lesher argued that the strength of contour completion depends on how much the line ends of Kanizsa tokens stimulate these low-level determinants. Tokens with a curvilinear alignment give a perhaps suboptimal response according to their relative alignment offset and generate a sense of curvilinear contour. 
Hayes ( 2000) showed that motion within contour fragments can shift in their apparent relative spatial position and, under appropriate conditions, give rise to a coherent percept of a contour where contour fragments are not retinally aligned. The distortion effect noted in the present experiments cannot be explained with reference to apparent shift of position of contour elements since there is no contour, only terminators. A possible account for the distortion is that the radial motion may have affected the spatial registries of the processes responsible for the illusory contours. If an illusory contour activates visual neurons in a similar manner to a real contour, but with no physical signal, other nearby physical signals may be misperceived as contributing to the activation. The spatiotemporal activation by radial motion may be considered as part of the activation due to the illusory contours and is therefore integrated with it, and thus the perceived position of the illusory contours is shifted. At positions near the Kanizsa tokens, strong spatial signals are induced, which makes them less susceptible to a motion-induced apparent spatial shift than at the middle of the contour. 
Experiment 4 illustrates the consequence of having luminance-defined spatial information. The apparent shape of a luminance-defined triangle is not influenced by the motion of background dots. This is likely a consequence of the difference in the spatial positional signal provided by “real” and “illusory” contours (see Guttman & Kellman, 2004; Westheimer, 1990; Westheimer & Li, 1996, 1997). Since low-contrast edges are used in the experiment, it is possible to conclude that the distortion found is exclusive to illusory contours. Spatial position signals are affected much more by virtue of the contours being illusory than by virtue of the edges being of low contrast. Despite this degraded spatial property of illusory contours, published studies have found evidence showing that real and illusory contours share neural mechanism in early visual processing (Ritzl et al., 2003; Seghier et al., 2000). We argue that motion plays a crucial part in the metric of shape perception but only becomes apparent when there is an “insufficiency” in the spatial positional signal. Indeed, a motion-induced distortion effect is more profound when the stimuli are viewed in the periphery where spatial acuity is coarse (De Valois & De Valois, 1991). 
Fu, Shen, Gao, and Dan (2004), using a drifting sinusoidal grating, recorded from single cells in the primary visual cortex of the cat in response to a motion-induced, position-shift stimulus. Their results indicate that the stimulus causes receptive-field displacement. Where may be the brain locus for motion-induced position shifts of illusory contours? Physiological studies, including single-cell recording (e.g., Peterhans & von der Heydt, 1989) and neuroimaging (Ffytche & Zeki, 1996; Ritzl et al., 2003), have shown a selective response of V2 neurons to illusory contours. It is possible that, similar to the observation made by Fu et al. (2004), image motion functions to shift the receptive field of V2 neurons responsible for the perception of illusory contours. 
Kanizsa figures are still perceived when tokens of differing polarity with the background are used (see Dresp & Fischer, 2001; Matthews & Welch, 1997). The results of Experiment 3, where opposite luminance-polarity background motion is shown to be as effective as same-polarity motion, is likely related to this phenomenon. A possible explanation for insensitivity to luminance polarity is that illusory contours may activate visual processors of different luminance polarity, and the perception of illusory contours involves a cortical area that is insensitive to polarity information. While this study cannot directly comment on the cortical locus of this effect, as discussed above, published research indicates that neurons in area V2 are important in the perception of illusory contours, and that also such neurons implement signal rectification, which effectively renders them sensitive to both luminance increment and luminance decrement information (see, e.g., Baker, 1999; Wilson, 1999; Zhan & Baker, 2006). This property of V2 neurons provides an explanation as to why dots of opposite polarity to the tokens produce a distortion effect: areas responsive to illusory contours are sensitive to both. 
Acknowledgments
This work was supported by a Hong Kong Shue Yan University Research and Staff Development Committee Grant Award to W. O. Li; a UNSW Science Faculty Research Grant and Early Career Researcher Award to S. Khuu; and an E.T.S. Walton Award from Science Foundation Ireland to A. Hayes. 
Commercial relationships: none. 
Corresponding author: Wang O. Li. 
Email: woli@hksyu.edu. 
Address: LG215, Main Building, Hong Kong Shue Yan University, Wai Tsui Crescent, Braemer Hill, North Point, Hong Kong. 
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Figure 1
 
Kanizsa triangles. (a) A typical Kanizsa triangle placed on a gray background. (b) A Kanizsa triangle with its circular missing-wedge segments subtending an angle slightly greater than 60°, giving rise to illusory convex curvilinear contours, and with the appearance of being a “fat” Kanizsa triangle. (c) A Kanizsa triangle is perceived to be a “thin” triangle with concave edges when its tokens' wedge-shaped segments subtend less than 60°. (d) A stimulus used in the present study—a Kanizsa triangle was superimposed on 200 light-increment dots, which underwent either expanding or contracting motion. The Kanizsa triangle as a whole slowly, rigidly, rotated.
Figure 1
 
Kanizsa triangles. (a) A typical Kanizsa triangle placed on a gray background. (b) A Kanizsa triangle with its circular missing-wedge segments subtending an angle slightly greater than 60°, giving rise to illusory convex curvilinear contours, and with the appearance of being a “fat” Kanizsa triangle. (c) A Kanizsa triangle is perceived to be a “thin” triangle with concave edges when its tokens' wedge-shaped segments subtend less than 60°. (d) A stimulus used in the present study—a Kanizsa triangle was superimposed on 200 light-increment dots, which underwent either expanding or contracting motion. The Kanizsa triangle as a whole slowly, rigidly, rotated.
Figure 2
 
Representative data for one observer from Experiment 1 are plotted for contracting, expanding, and random motion conditions—dot speed of 6.0°/s. Dashed horizontal and vertical lines indicate the 0.5 proportion-correct performance level and the token angle of the reference stimulus, respectively.
Figure 2
 
Representative data for one observer from Experiment 1 are plotted for contracting, expanding, and random motion conditions—dot speed of 6.0°/s. Dashed horizontal and vertical lines indicate the 0.5 proportion-correct performance level and the token angle of the reference stimulus, respectively.
Figure 3
 
Results of Experiment 1. Plots of the PSE for the different direction conditions for dot speeds of (a) 2.5°/s and (b) 6.0°/s of Experiment 1. Dark solid line symbols represent the mean result of all participants. Error bars indicate ±1 standard error of the mean.
Figure 3
 
Results of Experiment 1. Plots of the PSE for the different direction conditions for dot speeds of (a) 2.5°/s and (b) 6.0°/s of Experiment 1. Dark solid line symbols represent the mean result of all participants. Error bars indicate ±1 standard error of the mean.
Figure 4
 
Results of Experiment 2 plotted in a similar format to Figures 2 and 3. (a) The psychometric functions of a representative observer for different motion conditions. (b) Plots of the PSE for different motion conditions for different observers. Error bars indicate ±1 standard error of the mean.
Figure 4
 
Results of Experiment 2 plotted in a similar format to Figures 2 and 3. (a) The psychometric functions of a representative observer for different motion conditions. (b) Plots of the PSE for different motion conditions for different observers. Error bars indicate ±1 standard error of the mean.
Figure 5
 
Results of Experiment 3 plotted in a similar format to Figures 2 and 3. (a) The psychometric function of a representative observer. (b) Summary of results plotted as perceived regularities in terms of angular subtense of token wedge. Error bars indicate ±1 standard error of the mean.
Figure 5
 
Results of Experiment 3 plotted in a similar format to Figures 2 and 3. (a) The psychometric function of a representative observer. (b) Summary of results plotted as perceived regularities in terms of angular subtense of token wedge. Error bars indicate ±1 standard error of the mean.
Figure 6
 
Luminance-defined “real” triangle used in Experiment 4. (a) A “real” triangle with a modulation A = −0.08 simulating a “thin” Kanizsa figure. (b) A “real” triangle with a modulation A = 0.08 simulating a “fat” Kanizsa figure.
Figure 6
 
Luminance-defined “real” triangle used in Experiment 4. (a) A “real” triangle with a modulation A = −0.08 simulating a “thin” Kanizsa figure. (b) A “real” triangle with a modulation A = 0.08 simulating a “fat” Kanizsa figure.
Figure 7
 
Results of Experiment 4 plotted in a similar format to Figures 2 and 3. (a) The psychometric function of a representative observer. (b) Summary of results plotted as perceived regularities in terms of modulations applied to illusory contours. Error bars indicate ±1 standard error of the mean.
Figure 7
 
Results of Experiment 4 plotted in a similar format to Figures 2 and 3. (a) The psychometric function of a representative observer. (b) Summary of results plotted as perceived regularities in terms of modulations applied to illusory contours. Error bars indicate ±1 standard error of the mean.
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