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Research Article  |   May 2008
Local determinants of contour interpolation
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Journal of Vision May 2008, Vol.8, 3. doi:10.1167/8.7.3
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      Marianne Maertens, Robert Shapley; Local determinants of contour interpolation. Journal of Vision 2008;8(7):3. doi: 10.1167/8.7.3.

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Abstract

Objects in our visual environment are perceived as integral wholes even when their retinal images are incomplete. We ask whether the perceptual precision of subjective interpolation between isolated image parts depends on the overall proportion of visible image information or rather on its geometrical arrangement. We used Varin-type subjective shapes that provide less physical stimulus information than Kanizsa-type figures because partially occluded solid inducers are replaced by partially occluded concentric arcs. We tested whether perceptual precision varies as a function of contour support, or alternatively, depends on the number of, and the distance between, line endings within the inducers. We measured performance in a probe localization task, where a small target is presented at different distances around a subjective boundary. Sensitivity, captured by the just noticeable position difference between in- and outside probes, crucially depended on the geometric arrangement of line ends in the Varin figures. This is objective evidence that the apparent subjective contour strength does not primarily depend on contour support but is determined by the number and the separation between inducers' line endings. The results suggest that neuronal mechanisms sensitive to highly localized 2D features are crucial for determining the perceived shape of visual objects.

Introduction
Visual objects in everyday life are arranged in a three-dimensional environment. As a consequence of this depth ordering, the 2D retinal projections of natural scenes are cluttered. Object parts often are obscured due to mutual overlap. Hence, the shape of objects has to be inferred on the basis of a fragmentary retinal image. To recover object properties from images, observers must decide whether two distant image parts belong to the same surface or not, and they need to know what are the crucial image cues for occlusion that trigger interpolation. Observers seem to handle these constant processes of boundary interpolation and segmentation with no apparent effort. Despite the ease of boundary completion, it is still a challenge to understand the principles of this important perceptual process. 
One factor that has been believed to be crucial for the perceptual strength of interpolated contours is the Support Ratio, e.g., the ratio between the luminance-defined contour portions and the total contour length. The influence of the Support ratio has been studied extensively in Kanizsa figures, both by means of rating methods (e.g. Shipley & Kellman, 1992) as well as by a performance-based measure (e.g. Ringach & Shapley, 1996). In subjective figures of the Kanizsa type, a number of inducing discs with a particular opening are geometrically aligned such that an occluding surface is perceived on top of solid circles (Kanizsa, 1955). With the Kanizsa stimulus, both rating-based and performance-based approaches revealed that subjective boundary completion is scale invariant, that is, appearance and performance are the same when the support ratio is kept constant (Ringach & Shapley, 1996; Shipley & Kellman, 1992). However, there are cases in which a minor increase in support ratio, e.g., by adding a thin line perpendicular to the subjective contour, leads to a disproportionate increase in apparent contour strength, as demonstrated in Figure 1. And there are other cases in which changes in the arrangement of discontinuities that do not alter the support ratio influence the perceived strength of a subjective contour (Gillam & Chan, 2002). 
Figure 1
 
Kanizsa triangle without and with additional line inducers. A triangular Kanizsa figure is observed which results from perceiving the shape as partly occluding the three underlying black discs. Due to a fairly small support ratio, the triangular surface is not very compelling in the figure on the left. It becomes strikingly more vivid in the figure on the right, which differs from the left one only by three additional line ends.
Figure 1
 
Kanizsa triangle without and with additional line inducers. A triangular Kanizsa figure is observed which results from perceiving the shape as partly occluding the three underlying black discs. Due to a fairly small support ratio, the triangular surface is not very compelling in the figure on the left. It becomes strikingly more vivid in the figure on the right, which differs from the left one only by three additional line ends.
What does the phenomenology in Figure 1 imply with respect to the principles governing contour completion? Lesher and Mingolla (1993) addressed that question using Varin-figures (Varin, 1971). In Varin figures (Figure 2), solid inducers are replaced by a number of concentric arcs. There is a large reduction in support ratio in Varin figures compared with Kanizsa figures. Yet certain Varin shapes look as vivid as Kanizsa shapes (Figure 2). Lesher and Mingolla (1993) systematically varied line density and line width within Varin inducers and found an inverted u-shaped relation between line density or line width and perceived subjective contour clarity. They concluded that there was a range within which the support ratio seemed to be crucial for perceived clarity, but other factors were playing a role as well. 
Figure 2
 
Stimuli used in the present experiments. From left to right: Kanizsa shape, four-line Varin shape, two-line Varin shape with arcs closely spaced, two-line Varin shape with arcs widely spaced. Support ratios from left to right are 0.5, 0.1, 0.05, and 0.05.
Figure 2
 
Stimuli used in the present experiments. From left to right: Kanizsa shape, four-line Varin shape, two-line Varin shape with arcs closely spaced, two-line Varin shape with arcs widely spaced. Support ratios from left to right are 0.5, 0.1, 0.05, and 0.05.
The aim of the experiments reported here was to measure objectively the relative contributions of support ratio on the one hand and local 2D cues, such as numerosity and spacing of line ends, on the other, to the apparent clarity of subjective contours. Although subjective measures address participants' phenomenology rather directly, they also have several drawbacks. Observers do not always have veridical access to their percepts or an ability to report them. Second, participants might differ from one another in the criterion above which they report a contour as clear, and ratings may be distorted by the adoption of criteria that are unrelated to the subjective aspects of the stimulus (for example, see Gurnsey, Iordanova, & Grinberg, 1999). Finally, as Lesher (1995) pointed out, there is a more specific concern regarding contour clarity judgements. “Clarity” is ambiguous; it might refer to recognizability or strength or instead to sharpness in terms of spatial spread. 
Objective measures have been devised in order to circumvent at least some of the problems of assessing perception of illusory figures. Ringach and Shapley (1996) asked subjects to classify Kanizsa figures according to their shape instead of relying on subjective reports. Another paradigm that has been introduced to assess more specifically the precision of subjective contour interpolation is probe localization (Guttman & Kellman, 2004; Stanley & Rubin, 2003). Here, a probe dot is presented at different distances relative to the putative path of the subjective boundary and participants are asked whether the probe appeared inside or outside the figure. Probe localization performance has been shown to be less precise in the absence of crisp contours (Stanley & Rubin, 2003) or when performance could rely only on local cues (Guttman & Kellman, 2004). Another objective technique is response classification (Gold, Murray, Bennett, & Sekuler, 2000). 
We used probe localization as a performance-based measure in order to quantify how precisely subjective boundaries are interpolated in Varin and Kanizsa figures. We quantified participants' sensitivity by the just noticeable difference (JND) in position between inside and outside probes. The major question addressed here was to what extent the precise interpolation of subjective boundaries is determined by the Support Ratio or alternatively by the geometrical arrangement of line terminations. We used curvilinear subjective shapes (Ringach & Shapley, 1996) in order to introduce some variability regarding the exact path of the subjective contour in a given trial and also to discourage participants from applying a straight-line interpolation strategy (Gerbino & Fantoni, 2006). 
General methods
Stimuli and design
Stimulus presentation and response recording were controlled using the Psychtoolbox (Brainard, 1997; Pelli, 1997) under Matlab. The monitor was 280 mm (1280 pixels) wide and 210 mm (1024 pixels) high, and the vertical refresh rate was 60 Hz. Subjects viewed the stimuli binocularly from a distance of 60 cm. Accordingly, one pixel corresponded to 0.021° or 1.25 arcmin of visual angle. Observers' heads were stabilized with a chin and forehead rest in order to minimize head movements. Experiments were performed in a dimly lit, sound-attenuated chamber. 
We presented observers with displays of Varin-type illusory figures (Varin, 1971, Figure 2). We created shapes with convex or concave sides. An inducer opening of 96° caused the perception of a convex interpolated shape, and an opening of 84° produced a concave shape. In all experiments, the inducer radius was 1.5°, and the center-to-center distance between inducers was 9°. This arrangement produced an illusory contour length of 6° visual angle. Black inducing arcs (0.05 cd/m2) were presented on a light background (20 cd/m2). Depending on the experiment, Varin inducers were composed of either two or four concentric arcs with different spacings and different line widths. 
We measured participants' performance by means of a probe localization task (Guttman & Kellman, 2004; Stanley & Rubin, 2003), where a small target dot (5 × 5′ visual angle 0.05 cd/m2) is presented at different distances relative to the subjective boundary. A hypothetical boundary path was calculated using four points: The centers of the inducers served as the outermost points, and the endpoints of their oblique openings were the innermost points. A quadratic polynomial fit (y = p1 * x2 + p2 * x + p3) was used to derive the point of maximum curvature. The probe was then presented at four distances (3.75′, 11.25′, 17.5′, and 25′ visual angle corresponding to 3, 9, 14, and 20 pixels) and to both sides (inside(−) and outside(+)) of the calculated point of maximum curvature. The probe only appeared around the vertical side edges of the shapes, and it appeared equally often on either side. Participants had to indicate the position of the target relative to the subjective boundary with a corresponding left or right button press. Instructions were as follows: “Imagine you connect the legs of the inducers by the best fitting path, was the probe left or right relative to it?” “Left” and “right” were used instead of “inside” and “outside” responses in order to keep the stimulus response mapping identical on both sides of the display. 
A trial started with a fixation period of 500 ms. Subsequently, the stimulus and the target probe were presented simultaneously for 200 ms, followed, after a blank period of 100 ms, by a mask presented for another 200 ms. The mask was composed of four checkered pinwheel elements centered at the inducer locations. Subjects were encouraged to take as much time for their response as they needed in order to minimize errors due to premature motor responses. During the response period, a fixation cross was shown, which remained present also for a variable intertrial interval (ITI range = 0.1000 in multiples of monitor refresh, equal distribution of ITIs). 
In all three experiments, participants attended two identical sessions within a 1-week period, the first of which served as training. Results from the first session were not included in the analysis. The details about number of trials were different in the different experiments, and so they are described below for each experiment separately. 
We calculated the frequency of trials in which the probe was perceived as lying outside the subjective shape as a function of its position for individual observers and fitted a two-parameter, logistic psychometric function to these data (Kuss, Jäkel, & Wichmann, 2005). In order to determine the precision of participants' boundary percepts, we assessed how accurately they could localize the probe relative to their perceived boundary. Precision of boundary interpolation was measured as the number of units in probe distance needed in order to discriminate the two possible probe positions (inside vs. outside). This difference threshold is also known as the just noticeable difference (JND), and it is calculated as the distance between F(x) = 0.25 and F(x) = 0.75 in units of stimulus intensity (here stimulus intensity equals spatial distance) divided by two. We always estimated the psychometric functions for individual observers, determined the JND, and then averaged JNDs across subjects. 
Experiment 1: Validation of probe localization, comparison of Kanizsa and Varin shapes and appearance measure
We needed to measure the power of the probe localization paradigm to reveal performance differences that specifically result from perceptual interpolation. To this end, we presented a control condition in which pairs of inducers are rotated outward by 90° such that the impression of a figure is broken down (Figure 3, outward pointing condition) and then compared probe localization performance on this control stimulus with performance with stimuli that were supposed to generate subjective surfaces and contours. One should note that our outward pointing condition constitutes a rather conservative control because under steady viewing conditions, it still contains weak subjective contours even in the absence of a subjective shape. Therefore, any performance differences that are observed between the properly aligned and the control condition should be a consequence of the presence of a subjective surface in the former but not the latter condition. 
Figure 3
 
Stimuli and individual results. The left panel depicts the stimuli used in this experiment. Kanizsa stimuli are shown in the left and four-line Varin stimuli in the right column. The upper row contains geometrically aligned and the lower row outward pointing inducers. The right panel shows logistic psychometric functions describing the relationship between probe position in pixels (x-axis) and probability of perceiving the probe as being located outside the subjective shape (y-axis). Dots depict the empirical data (the average over 20 data points), and lines depict the psychometric fits.
Figure 3
 
Stimuli and individual results. The left panel depicts the stimuli used in this experiment. Kanizsa stimuli are shown in the left and four-line Varin stimuli in the right column. The upper row contains geometrically aligned and the lower row outward pointing inducers. The right panel shows logistic psychometric functions describing the relationship between probe position in pixels (x-axis) and probability of perceiving the probe as being located outside the subjective shape (y-axis). Dots depict the empirical data (the average over 20 data points), and lines depict the psychometric fits.
In Experiment 1, the inducers for the Varin figures were all composed of four concentric arcs as in Figure 2, second from the left. The line width of each arc was 4 pixels. 
In order to provide a standard of comparison for the Varin figures, we also assessed probe localization performance in corresponding Kanizsa figures. Kanizsa and Varin stimuli were equivalent in the overall size of the display and in the separation between the centers of inducers, but the former were composed of solid, inked-in pacmen. The support ratio of the Kanizsa figure was 0.33 (2 × 72/432 pixel), whereas that of the Varin was 0.07 (2 × 4 × 4 pixel/432 pixel). 
Finally, we wanted also to address the phenomenological aspects of participants' percepts. We asked them to execute a second response subsequent to their probe localization judgment. They had to indicate with a two-alternative button press of the left hand whether a figure had been presented or not. Participants were instructed to focus on probe localization and to perform figure detection only as well as they could without sacrificing localization accuracy. 
One experimental session consisted of five experimental blocks of 128 trials each preceded by a practice block of 32 trials. In the practice block, we presented two repetitions for each inducer type and alignment combination and only used the two most extreme probe distances for each direction (±14, 20). In the practice block only, auditory feedback was given after each “correct” response. Due to the rapid stimulus presentation, participants may get discouraged when they have the impression they are performing at chance. Therefore, they underwent this supervised training in the practice blocks with only the easier stimulus conditions in order to counteract those notions. One should note however, that there is no absolute “correct” in the sense that we do not know in advance how participants interpolate. Hence, as we were interested in participants' unbiased percepts, no feedback was provided in the main experimental blocks. One experimental block of trials took about 5 min and participants were encouraged to take breaks of at least one min duration between blocks. Trial order was randomized across experimental conditions, and each combination of alignment (2) × inducer type(2) × probe distance (8) appeared equally often and 20 times in total per session. As mentioned above, each participant participated in two experimental sessions, a training session and the main session, and data for analysis were taken only from the main session. 
Thirteen subjects took part in this experiment; eleven of them were female. All of them were university students with a mean age of 23 years (SD = 2.5) with normal or corrected-to-normal vision. Five subjects received course credits for their participation while the other eight were paid. 
Results
Detection
We calculated hit and false alarm rates (i.e., the proportion of correct detections in the aligned and incorrect detections in the outward pointing trials, respectively). For the Kanizsa figure, the hit rate was 82% and the false alarm rate was 13%, and for the Varin figure the corresponding numbers were 84% and 20%. In both conditions performance was significantly different from chance level (p = 0 in both χ2 tests with df = 1). That is, for the stimuli we used, subjects could “see” the illusory figure and discriminate it from the outward pointing condition. 
Probe localization
We estimated separate psychometric functions describing the relationship between probe distance and probability of “outside” response for each combination of alignment and inducer type in individual observers. Example psychometric functions for one observer are shown in Figure 3
JNDs were derived from the individual psychometric functions as described in the General methods section and averaged across observers (Figure 4). In order to test for the statistical significance of the observed effects, a two-way repeated measures ANOVA with the factors alignment and inducer type was performed on the mean JNDs. There was a main effect of alignment (F(1,12) = 8.44, p = .01). The just noticeable difference to perceive dots as either inside or outside the subjective figure was much smaller for aligned than for outward pointing stimuli for both Kanizsa and Varin figures. The effect of alignment was a reduction in JND by 5 pixels for the Kanizsa figure and a reduction by 8 pixels for the Varin figure. Probe localization accuracy for the Varin shape as measured by the JNDs was about 83% of that of the Kanizsa shape. 
Figure 4
 
Group results of Experiment 1. The graph depicts mean JNDs in units of pixels averaged across 13 observers as a function of inducer type (x-axis) and inducer alignment (differently colored bars). Error bars indicate standard errors of the mean.
Figure 4
 
Group results of Experiment 1. The graph depicts mean JNDs in units of pixels averaged across 13 observers as a function of inducer type (x-axis) and inducer alignment (differently colored bars). Error bars indicate standard errors of the mean.
In order to test for a possible speed accuracy trade-off, we performed a corresponding repeated measures ANOVA with the factors alignment and inducer type for response times. We observed a main effect for alignment (F(1,12) = 8.1, p = .01), that is, aligned stimuli evoked faster responses as well as enabling more precise probe localization (Figure 5). 
There was a bias in the psychometric functions. As mentioned above, we presented concave and convex subjective contour curvatures in random order. Observers tended to localize the probe more frequently as being “outside” the subjective shape, and that effect was more pronounced for convex than for concave shapes. Therefore, the point of subjective equality, when observers responded “outside” on 50% of the trials, was biased toward inside the shapes. However, we have too few observations per combination of curvature, alignment, and pacmen type (n = 10) to explore the issue here but rather note that the bias needs further study. There was no effect of curvature on the slope of the psychometric functions. 
Figure 5
 
Mean response latencies in Experiment 1. The bars show the mean response times as a function of inducer type (x-axis) and inducer orientation (panel variable). Error bars indicate standard errors of the mean.
Figure 5
 
Mean response latencies in Experiment 1. The bars show the mean response times as a function of inducer type (x-axis) and inducer orientation (panel variable). Error bars indicate standard errors of the mean.
Discussion
Probe localization was more precise for aligned shapes than for the outward pointing condition, indicating that the presence of a subjective figure compared to two half-figures (control condition) increased the precision of probe localization. This suggests that probe localization is an appropriate tool to measure subjective boundary completion based on performance. The figure detection results indicate that participants' percepts depend on the alignment of the stimuli in the way similar to probe localization because a subjective shape was perceived in the aligned but not in the outward-pointing condition. We also learned from this experiment that Varin inducers work almost as well as Kanizsa inducers with respect to their potential to induce a subjective surface. The present results already provide an initial answer to our main experimental question regarding the importance of overall contour support. While the Varin figure's support ratio was only 21% of that of the Kanizsa figure, the sensitivity to the Varin figure's boundary was about 83% that of the Kanizsa figure. However, this advantage in probe localization performance for the Kanizsa over the Varin stimulus is already evident in the control condition. Varin inducers are different from the Kanizsa inducers because even within the inducers there is no actual contour. The benefit for Kanizsa shapes in the outward pointing condition could be based—in part—on this real contour, and the same might be true for the aligned condition. Also the improvement of probe localization for aligned over misaligned was greater for the Varin than for the Kanizsa figure (8 vs. 5 pixels). This may indicate that the evidence for illusory surface formation is stronger for the Varin 4-line figure than for the corresponding Kanizsa pattern. The differential sensitivity to shapes versus non-shapes was not at all due to a speed–accuracy trade-off, but the results on localization-precision pattern were rather confirmed by the reaction times. 
Experiment 2: Support ratio vs. line density
In order to test whether the perceptual strength of interpolated contours is determined by the luminance-defined contour support (Shipley & Kellman, 1992) as compared to the number of discontinuities (Lesher & Mingolla, 1993), we varied line density and the line width independently in Varin figures resulting in conditions with equal contour support but different line density and vice versa. Inducers were composed of two or four lines that were either four or eight pixels wide. In that way, we created two conditions of equal contour support (2 lines × 8 pixels and 4 lines × 4 pixels) but different line density. If contour support were the major determinant of contour clarity, performance would be identical for these conditions. 
The global stimulus parameters as well as the sequence of events in one trial were identical to those described in the General methods section. One experimental session consisted of five experimental blocks of 128 trials each preceded by a practice block of 64 trials. In the practice block, we presented four repetitions for each line number and line width combination. During practice, we only presented the two most extreme probe distances for each direction (±14.20 pixels), and we played auditory feedback after each “correct” response (see Experiment 1). Trial order was randomized across conditions and each combination of number of lines (2) × line width (2) × probe distance (8) appeared 20 times in total, resulting in 480 trials per session. As before, there were two sessions, the second of which was the main session used for the data analysis. Again, we randomized subjective contour curvature within each condition but averaged across curvature in the analyses. 
Sixteen new participants took part in this experiment. All of them were female university students with normal or corrected-to-normal vision (mean age of 22 years, SD = 1.4) in the Department of Psychology of Otto-von-Guericke University Magdeburg, and they received course credits for their participation. 
Results
A two-by-two analysis of variance calculated for the JNDs revealed a significant main effect for the number of line ends (F(1,15) = 80.07, p < .001) and also a significant main effect for line width (F(1,15) = 7.41, p < .015; Figure 6). These effects reflect that participants' judgments were more precise with four than with two line ends, and with eight than with four pixels. More importantly, however, to the current question, there was a significant difference between the two conditions of identical support ratio (t(15) = 4.83, p < .001). Four-line stimuli were better than two-line stimuli, even when 4-line and 2-line stimuli had identical support ratio by a doubling of the line widths of the 2-line stimuli (Figure 6). 
Figure 6
 
Group results in Experiment 2. The bar graph shows the mean JNDs in units of pixels averaged across 16 observers as a function of the number of inducing line ends (x-axis) and line width (panel variable). Error bars indicate standard errors of the mean. The bracket signifies conditions of equal support ratio. The support ratios of the stimuli were (left to right) 0.035, 0.07, 0.07, and 0.14.
Figure 6
 
Group results in Experiment 2. The bar graph shows the mean JNDs in units of pixels averaged across 16 observers as a function of the number of inducing line ends (x-axis) and line width (panel variable). Error bars indicate standard errors of the mean. The bracket signifies conditions of equal support ratio. The support ratios of the stimuli were (left to right) 0.035, 0.07, 0.07, and 0.14.
We were also interested in the relative contributions of number of line ends vs. line width to the performance difference between conditions of maximally different support ratio, e.g., two lines and four pixels vs. four lines and eight pixels. In order to compare the relative effects of line ends vs. support ratio, we calculated to what extent their main effects accounted for the difference between the most extreme conditions. That is, how much of the difference between the 2-line Varin shape with 4 pixel line width and the 4-line Varin shape with 8 pixel line width is attributable to differences in line width (4 pixels vs. 8 pixels averaged across number of arcs) or instead to differences in the number of arcs (2 vs. 4 averaged across line width). We found that 71% of the difference between the extreme support ratio conditions was attributable to the effect of number of line ends, whereas only 29% was accounted for by the line ends' width or in other words support ratio. 
There was a main effect of the number of lines on reaction times (F(1,15) = 9.32, p < .008) as responses to four-line stimuli (923 ms) were faster than those to two-line stimuli (970). Also, there was a reaction-time difference between configurations of identical support ratio (t(15) = −2.50, p = .02): Responses to four-line stimuli with 4 pixel line width (926 ms) were faster than those to two line stimuli with 8 pixel line width (973 ms). 
Discussion
This experiment shows that the major determinant of precise perception of a subjective boundary's path was not the support ratio but rather the number of discontinuities delineating the putative path of the subjective contour. The lion's share of the performance difference between conditions of maximally different support ratios is attributable to their differing in the number of discontinuities and not in the amount of ink being used (support ratio). Moreover, conditions of identical support ratio, but different numbers of discontinuities, resulted in very different performance levels. This is direct evidence showing that factors which are clearly distinct from support ratio can determine the perceptual strength of a subjective boundary. Our results are in line with those of Lesher and Mingolla (1993) who also found that illusory contour strength in Varin figures did not increase monotonically with contour support but instead appeared to be maximal at an intermediate value of support ratio. 
Experiment 3: Line density vs. line spacing
The third experiment addressed the question whether, in addition to the number of line ends, the spatial distribution of line ends within inducers plays a role in subjective boundary interpolation. Visual inspection of Figure 2 suggests that it is not only the number of discontinuities that determines perceived boundary clarity, but also the spacing of these 2D features. The boundaries in the two-line Varin shape with widely spaced arcs appear to be less sharp than those in the Varin shape with neighboring arcs. We addressed this question using Varin inducers with equal line width (6 pixel) but a different number and distribution of arcs (Figure 2). In the two-line Varin shapes that were used in this experiment, arcs could be either closely or widely spaced. In addition, we included the corresponding outward pointing inducers as a control condition. The sequence of events within one trial was identical to that in the other experiments. 
Again, participants attended two experimental sessions that consisted of five experimental blocks of 96 trials and one practice block of 48 trials. We only analyzed data from the second experimental session. In the practice block, participants were shown two repetitions of each alignment and line type combination for the two most extreme probe distances (±14, 20), and auditory feedback was given in each trial. In the experimental blocks, each combination of alignment (2) × inducer type (3) was shown 10 times per probe distance, and trial order was randomized across all conditions. Subjective contour curvature was randomly varied within all factor combination but data were averaged across curvature. 
Fifteen new participants (7 male) with a mean age of 26 years (SD = 5.5) and with normal or corrected-to-normal vision took part in this experiment. All participants were university students at Department of Psychology Otto-von-Guericke University, Magdeburg and received course credits for their participation. 
Results
A two-by-three repeated-measures ANOVA calculated on the mean JNDs yielded a main effect for alignment (F(1,14) = 11.25, p = .004) and a main effect for inducer type (F(2,28) = 8.31, p = .001). It is obvious from the graph in Figure 7 that the main effect for alignment reflects the superiority in probe localization for aligned over outward pointing stimuli. Post hoc paired t tests were performed in order to test which of the three levels of the factor inducer type were significantly different from each other. They revealed that sensitivity was higher for four-line than for two-line Varin shapes with widely spaced arcs (t(14) = −3.41, p = .004) and also for two-line Varin shapes with closely spaced arcs than for two-line Varin shapes with widely spaced arcs (t(14) = −3.29, p = .005). There was no statistical difference in performance between the four-line Varin and the two-line Varin shape with close arc spacing (p = .25). Even though the interaction between alignment and inducer type was not significant, the effect of alignment (that is, perceptually completed surfaces) differed between inducers. Proper alignment reduced the JND by 6 pixels for the four-line Varin inducers, by 5 pixels for the closely spaced two-line Varin inducers, but only by 3 pixels for the widely spaced two-line Varin inducers. As can be inferred from the error bars in Figure 7, the 3 pixel difference in the two-line wide space condition seems to be in the noise, whereas the differences between aligned and outward pointing stimuli in the other two conditions do not seem to be attributable to noise. 
Figure 7
 
Group results in Experiment 3. The bar graph shows the mean JNDs in units of pixels averaged across 15 observers as a function of inducer type (x-axis) and inducer alignment (panel variable). Error bars indicate standard errors of the mean.
Figure 7
 
Group results in Experiment 3. The bar graph shows the mean JNDs in units of pixels averaged across 15 observers as a function of inducer type (x-axis) and inducer alignment (panel variable). Error bars indicate standard errors of the mean.
Again, reaction times paralleled the accuracy pattern since there was a significant main effect for alignment (F(1,14) = 15.30, p = .002) with aligned stimuli (996 ms) eliciting faster responses than outward pointing stimuli (1078 ms). However, no significant timing difference was observed between responses to different Varin stimuli in this experiment. 
Discussion
The results from this experiment show that the spacing of 2D features, or in other words the stimulus geometry, is important for the precision of our percepts of subjective boundaries. Probe localization precision differed between two conditions that had an equal number of discontinuities but a different spacing. On the other hand, performance was comparable for stimuli with an unequal number of discontinuities but identical spacing. These findings indicate that the precision of illusory contour perception depends neither exclusively on support ratio nor on the number of 2D features but rather on a number of factors including the geometrical arrangement of the features that signal occlusion. 
General discussion
In summary, our results show that subjective boundary interpolation, which was objectively measured in the current experiments by means of probe localization accuracy, does not depend exclusively on support ratio. 
Performance can be very different for stimuli with identical support ratios: (i) In Experiment 2, performance was more precise for Varin figures composed of four-line inducers than for Varin figures composed of two-line inducers, even though their respective line widths were adjusted such as to yield identical support ratios. (ii) In Experiment 3, we observed pronounced differences in probe localization accuracy between the two types of two-line Varin figures, which had identical support ratios, but differed in arc spacing. 
Very different support ratios can result in similar performance: (i) Experiment 1 compared probe localization accuracy between four-line Varin and Kanizsa figures. The Varin figure's support ratio was reduced by about 80% relative to the Kanizsa figure, but the sensitivity to the Varin figure's boundary was reduced by only 17%. (ii) In Experiment 3, the four-line Varin shape and the two-line Varin shape with closely spaced arcs differed in support ratio by a factor of two. However, the JND for probe localization for the two-line Varin figure amounted to 88% of that for the four-line Varin figure. Hence, a subjective boundary will appear crisp not because of the overall contour support, but rather because of a crucial number and distribution of discontinuities that delineate the putative contour path. 
Relation to previous work
The present results, obtained with an objective technique, confirm the earlier findings of Lesher and Mingolla (1993), who reported that there is no simple monotonic relationship between the amount of ink supporting a bounding contour and its apparent strength or crispness. Our current findings go further: They reveal that not only the number of discontinuities determines the strength of an occlusion signal, but also their geometrical arrangement. We showed that the same two line ends evoked very different subjective contour percepts depending on their exact spacing. One could argue that the two-line inducers—besides differences in spacing—also differed in their salience because of the relatively small inner arc in the widely spaced inducer. However, as demonstrated by the configuration in Figure 8a, it is unlikely that lower salience accounts for the degraded crispness of the subjective contours in this condition because even with an increased salience of the inducing arcs only weak subjective contours are perceived. 
Figure 8
 
Line spacing demonstrations. (a) Increasing inducers' salience by increasing the length of the inner arc (upper stimulus) does not counteract the perceptual weakening of the subjective contours in the two line (lower stimulus) compared to the four-line Varin shape (b, upper left). (b) Upper row—orderly arranged concentric arcs in Varin inducers with four (left) and three (right) rings. Lower row—disordered Varin-type inducers with four (left) and three (right) rings. In Varin stimuli, the disordered arrangement of line terminations does not seem to improve the perceptual strength of the occluded surface or the interpolated boundaries.
Figure 8
 
Line spacing demonstrations. (a) Increasing inducers' salience by increasing the length of the inner arc (upper stimulus) does not counteract the perceptual weakening of the subjective contours in the two line (lower stimulus) compared to the four-line Varin shape (b, upper left). (b) Upper row—orderly arranged concentric arcs in Varin inducers with four (left) and three (right) rings. Lower row—disordered Varin-type inducers with four (left) and three (right) rings. In Varin stimuli, the disordered arrangement of line terminations does not seem to improve the perceptual strength of the occluded surface or the interpolated boundaries.
Using a pattern (“New York Titanic,” Gillam, 1997) composed of a set of horizontally aligned line terminations, Gillam and Chan (2002) have already shown that the perceived strength of subjective contours can differ between stimuli with identical support ratios. They found that spatially periodic compared to disorganized discontinuities resulted in weaker subjective contours. Gillam and Chan interpreted their results to mean that illusory contours are enhanced when perceptual grouping of inducers is reduced by disorder. While the configuration used by Gillam and Chan (2002) elicits the perception of occlusion, it does not involve the interpolation of a surface analogous to that in Varin or Kanizsa shapes. It is conceivable that different principles apply to isolated occluding contours as opposed to occluding surfaces as in Varin and Kanizsa figures. As illustrated in the demonstration depicted in Figure 8b, it seems as if disordered inducers do not outperform inducers in a more orderly arrangement regarding the persuasiveness of subjective contours in Varin stimuli. Indeed what seems more important in the Varin figures is the number and/or spacing of discontinuities, as we have shown in the Results section. As is evident in the demonstration in Figure 8b, with disordered discontinuities, it is more likely that the removal of one individual discontinuity will introduce a gap between neighboring end points that goes beyond a critical spacing and hence will weaken the evidence for a common occlusion process (Figure 8b). Since Gillam and Chan (2002) did not vary the spacing of line terminations, it is an open question how terminator spacing would influence the perceptual strength of subjective contours in their patterns. 
Computational and neural mechanisms
There are two different aspects of the current results that relate to different computational mechanisms that may be involved in the process of subjective contour formation. One is the overall alignment effect: Probe localization performance in the aligned condition always outperformed that in the outward pointing condition. We attribute this performance benefit to the availability of a surface representation which is generated only in response to properly aligned inducers and which might act to strengthen the interpolated boundary. In addition to the effect of global geometrical alignment, we found that local image constraints substantially affected the perceived strength of the interpolated contour. In particular, in order for the stimuli to generate perceptually compelling subjective contours, there was a requirement for a critical distribution or spacing of local image discontinuities. 
We suggest the following hypothetical scenario for the neural mechanisms of subjective contour completion in order to account for the two effects: global alignment and local feature-spacing. The line ends within each inducer elicit responses in end-stopped cells in primary visual cortex V1 (Heitger, Rosenthaler, von der Heydt, Peterhans, & Kübler, 1992). These cells generate an occlusion signal which is transferred to several extrastriate areas (Mendola, Dale, Fischl, Liu, & Tootell, 1999), specifically to neurons in a cortical area termed the lateral occipital complex (LOC; Malach et al., 1995). LOC neurons have been shown to be sensitive to the presence of salient regions, e.g., to contiguous parts of an image that belong to the same surface (Stanley & Rubin, 2003). Salient regions or surfaces are inferred from the proper geometrical alignment of local cues signaling occlusion; thus, the responses of LOC neurons to salient regions are important because they allow us to differentiate between aligned and outward pointing stimuli. We assume that proportional to their response, LOC neurons also send stronger feedback signals to early visual areas V2 and V1 for aligned than for outward pointing stimuli (Stanley & Rubin, 2003), and in turn the subjective contour signal in early visual cortex including V1 (Grosof, Shapley, & Hawken, 1993; Peterhans & von der Heydt, 1989) will be amplified for aligned stimulus configurations (Maertens & Pollman, 2005). Probe localization is likely to depend on signals in retinotopically organized visual areas because there the dot and the subjective contour evoke spatially discriminable neural responses. To account for the differences in interpolation accuracy between the two types of two-line Varin figures, there are at least two explanations. It is possible that the more closely spaced lines already evoke stronger responses in V1 because they might stimulate end-stopped neurons at a close to optimal sampling rate. Then these stronger responses are simply inherited by LOC and fed back to V1. Alternatively, the difference may arise first in responses of neurons in LOC. LOC neurons may learn to respond better to closely spaced Varin inducers because the narrow spacing has proven (by experience) to be a more reliable signal for occlusion than the wider one. Of course the hypothesized circuit is overly simplified and presumably involves a number of intermediate neural populations as well. But our major purpose here was to provide a sketch of cortical machinery that can account for both the effect of global geometrical alignment (LOC) and the effect of local geometry of line terminations (V1). 
The perception of occlusion with Kanizsa figures is more problematical. Two types of subjective contours have been distinguished because of their presumably different underlying completion mechanisms (Lesher, 1995). Edge-induced subjective contours as in the Kanizsa figure originate collinear to the supporting edge, whereas line-end-induced subjective contours as in abutting gratings or Ehrenstein-type figures emerge roughly perpendicular to the supporting edges. It has been pointed out that there is no sharp boundary between both types, since an edge-type inducer has two corners, and a line-end becomes an edge-type inducer with increasing line width. However, if the Kanizsa edge-type inducer were producing its effect only because of its terminating corners, subjective boundaries should be comparably strong in the Kanizsa and in the two-line Varin figure with widely spaced arcs. The two can be regarded as discontinuity equivalents (Lesher & Mingolla, 1993; Figure 9). On the contrary, phenomenologically the two-line, widely spaced Varin figure generates much weaker subjective contours than the Kanizsa figure. Furthermore, we found that probe localization performance in the widely spaced, two-line Varin figure was considerably less precise than in the Kanizsa figure that is its discontinuity equivalent (compare Figure 4 with Figure 7). This finding is in agreement with Lesher and Mingolla (1993) who reported that the mean clarity ratings for stimuli composed of thick lines always were higher than that of their discontinuity equivalents. 
The complementary portions of the Kanizsa inducers, that is, the contours that form the collinear flanks, are also not sufficient by themselves to induce the percept of a subjective shape (Figure 9). This has been shown quantitatively by Stanley and Rubin (2003) who found that inducers with rounded corners result in less precise probe localization performance compared with otherwise identical proper inducers. However, two line ends can be sufficient to generate almost as strong subjective contours as their four-line counterparts as long as they are close enough to each other in space. As has been pointed out by Rubin (2001), it is reasonable to assume that local cues should play an important role as cues to occlusion because they do not require global image computations. The observation that the local arrangements of two line ends had a big effect on the subjective interpolation result underlines the importance of local cues, even though we do not understand the exact mechanism yet. So in fact, one question raised by the current results is, how is completion accomplished in the Kanizsa shape given that there seems to be a critical spacing for local discontinuities to trigger completion, and the parameters of the Kanizsa shapes used here did not meet those criteria? The possible cooperation of the local discontinuities and the contours of the Kanizsa inducers should be investigated as a possible answer to this open question. 
Figure 9
 
Kanizsa square and two of its putative but insufficient cues for boundary completion. A compelling subjective square is evident only in the left (Kanizsa), but not the middle (two-line Varin) and the right (corner outline) figure. Neither a discontinuity (middle) nor a contour support (right) equivalent of the Kanizsa shape is sufficient to generate as strong a subjective square by itself.
Figure 9
 
Kanizsa square and two of its putative but insufficient cues for boundary completion. A compelling subjective square is evident only in the left (Kanizsa), but not the middle (two-line Varin) and the right (corner outline) figure. Neither a discontinuity (middle) nor a contour support (right) equivalent of the Kanizsa shape is sufficient to generate as strong a subjective square by itself.
Signals for occlusion
It has been pointed out before that whenever a number of line ends terminates along the same curve in space, there is a high probability that the alignment of their end points has been caused by occlusion (Heitger et al., 1992). Heitger et al. (1992) linked this particular type of occlusion situation to the functional properties of end-stopped cells and hypothesized that they “have evolved not primarily for the detection of certain object features but for a contour mechanism where they serve to resolve situations of occlusion” (p. 979). Heitger and coworkers developed a model which, including operators analogous to simple, complex, and end-stopped cells, is capable to identify 2D features (“K-points”) in real-world images that were generated by occlusion. The authors propose a strategy by which every abrupt termination of an edge or a line is being treated as equally effective evidence for occlusion. Our results show that as a signal for occlusion, not only the number of terminators or discontinuities is important but also their geometrical arrangement. That means that even though individual terminators might be interchangeable, and in that sense “equally effective,” the joint effectiveness of multiple line ends might involve interactions and hence depend on spatial arrangement. 
The present results speak against the idea that the perceptual strength of a filled-in contour or surface increases with the mere amount of physical contour support (e.g., increasing number of line ends). Instead of simply accumulating physical evidence for occlusion, the brain rather seems to rely on factors such as the number and arrangement of discontinuities as reliable evidence for occlusion. 
Acknowledgments
We wish to thank S. Pollmann and M. Hanke for helpful comments regarding the design of the experiments. We are also grateful to N. Rubin for discussions about the implications of our results. This research was supported by grants to M. Maertens by the German Academic Exchange Service and the Humboldt Foundation and by NIH grant EY 01472 to Robert Shapley. 
Commercial relationships: none. 
Corresponding author: Marianne Maertens. 
Email: marianne.maertens@tu-berlin.de. 
Address: Technical University Berlin, FG Modelling of Cognitive Processes, Sekr. FR 6-4, Franklinstr. 28/29, 10587 Berlin. 
References
Brainard, D. H. (1997). The Psychophysics Toolbox. Spatial Vision, 10, 433–436. [PubMed] [CrossRef] [PubMed]
Gerbino, W. Fantoni, C. (2006). Visual interpolation is not scale invariant. Vision Research, 46, 3142–3159. [PubMed] [CrossRef] [PubMed]
Gillam, B. Pelli, D. Torres, A. M. (1997). New York Titanic. Thresholds: Limits of perception. New York: Arts Magazine.
Gillam, B. Chan, W. M. (2002). Grouping has a negative effect on both subjective contours and perceived occlusion at T-junctions. Psychological Science, 13, 279–283. [PubMed] [CrossRef] [PubMed]
Gold, J. M. Murray, R. F. Bennett, P. J. Sekuler, A. B. (2000). Deriving behavioural receptive fields for visually completed contours. Current Biology, 10, 663–666. [PubMed] [Article] [CrossRef] [PubMed]
Grosof, D. H. Shapley, R. M. Hawken, M. J. (1993). Macaque V1 neurons can signal ‘illusory’ contours. Nature, 365, 550–552. [PubMed] [CrossRef] [PubMed]
Gurnsey, R. Iordanova, M. Grinberg, D. (1999). Detection and discrimination of subjective contours defined by offset gratings. Perception & Psychophysics, 61, 1256–1268. [PubMed] [CrossRef] [PubMed]
Guttman, S. E. Kellman, P. J. (2004). Contour interpolation revealed by a dot localization paradigm. Vision Research, 44, 1799–1815. [PubMed] [CrossRef] [PubMed]
Heitger, F. Rosenthaler, L. von der Heydt, R. Peterhans, E. Kübler, O. (1992). Simulation of neural contour mechanisms: From simple to end-stopped cells. Vision Research, 32, 963–981. [PubMed] [CrossRef] [PubMed]
Kanizsa, G. (1955). Margini quasi-percettivi in campi con stimulazione omogenea. Rivista di Psicologia, 49, 7–30.
Kuss, M. Jäkel, F. Wichmann, F. A. (2005). Bayesian inference for psychometric functions. Journal of Vision, 5, (5):8, 478’492. http://journalofvisionorg/5/5/8/. doi:101167/558. [PubMed] [Article] [CrossRef]
Lesher, G. W. (1995). Illusory contours: Toward a neurally based perceptual theory. Psychonomic Bulletin & Review, 2, 279’321. [CrossRef] [PubMed]
Lesher, G. W. Mingolla, E. (1993). The role of edges and line-ends in illusory contour formation. Vision Research, 33, 2253–2270. [PubMed] [CrossRef] [PubMed]
Maertens, M. Pollmann, S. (2005). fMRI reveals a common neural substrate of illusory and real contours in V1 after perceptual learning. Journal of Cognitive Neuroscience, 17, 1553–1564. [PubMed] [CrossRef] [PubMed]
Malach, R. Reppas, J. B. Benson, R. R. Kwong, K. K. Jiang, H. Kennedy, W. A. (1995). Object-related activity revealed by functional magnetic resonance imaging in human occipital cortex. Proceedings of the National Academy of Sciences of the United, 92, 8135–8139. [PubMed] [Article] [CrossRef]
Mendola, J. D. Dale, A. M. Fischl, B. Liu, A. K. Tootell, R. B. (1999). The representation of illusory and real contours in human cortical visual areas revealed by functional magnetic resonance imaging. Journal of Neuroscience, 19, 8560–8572. [PubMed] [Article] [PubMed]
Pelli, D. G. (1997). The VideoToolbox software for visual psychophysics: Transforming numbers into movies. Spatial Vision, 10, 437–442. [PubMed] [CrossRef] [PubMed]
Peterhans, E. von der Heydt, R. (1989). Mechanisms of contour perception in monkey visual cortex. II. Contours bridging gaps. Journal of Neuroscience, 9, 1749–1763. [PubMed] [Article] [PubMed]
Ringach, D. L. Shapley, R. (1996). Spatial and temporal properties of illusory contours and amodal boundary completion. Vision Research, 36, 3037–3050. [PubMed] [CrossRef] [PubMed]
Rubin, N. (2001). The role of junctions in surface completion and contour matching. Perception, 30, 339–366. [PubMed] [CrossRef] [PubMed]
Shipley, T. F. Kellman, P. J. (1992). Strength of visual interpolation depends on the ratio of physically specified to total edge length. Perception & Psychophysics, 52, 97–106. [PubMed] [CrossRef] [PubMed]
Stanley, D. A. Rubin, N. (2003). fMRI activation in response to illusory contours and salient regions in the human lateral occipital complex. Neuron, 37, 323–331. [PubMed] [Article] [CrossRef] [PubMed]
Varin, D. (1971). Fenomeni di contrasto e diffusione cromatica nell' organizzazione spaziale del campo percettivo. Rivista di Psicologia, 65, 101–128.
Figure 1
 
Kanizsa triangle without and with additional line inducers. A triangular Kanizsa figure is observed which results from perceiving the shape as partly occluding the three underlying black discs. Due to a fairly small support ratio, the triangular surface is not very compelling in the figure on the left. It becomes strikingly more vivid in the figure on the right, which differs from the left one only by three additional line ends.
Figure 1
 
Kanizsa triangle without and with additional line inducers. A triangular Kanizsa figure is observed which results from perceiving the shape as partly occluding the three underlying black discs. Due to a fairly small support ratio, the triangular surface is not very compelling in the figure on the left. It becomes strikingly more vivid in the figure on the right, which differs from the left one only by three additional line ends.
Figure 2
 
Stimuli used in the present experiments. From left to right: Kanizsa shape, four-line Varin shape, two-line Varin shape with arcs closely spaced, two-line Varin shape with arcs widely spaced. Support ratios from left to right are 0.5, 0.1, 0.05, and 0.05.
Figure 2
 
Stimuli used in the present experiments. From left to right: Kanizsa shape, four-line Varin shape, two-line Varin shape with arcs closely spaced, two-line Varin shape with arcs widely spaced. Support ratios from left to right are 0.5, 0.1, 0.05, and 0.05.
Figure 3
 
Stimuli and individual results. The left panel depicts the stimuli used in this experiment. Kanizsa stimuli are shown in the left and four-line Varin stimuli in the right column. The upper row contains geometrically aligned and the lower row outward pointing inducers. The right panel shows logistic psychometric functions describing the relationship between probe position in pixels (x-axis) and probability of perceiving the probe as being located outside the subjective shape (y-axis). Dots depict the empirical data (the average over 20 data points), and lines depict the psychometric fits.
Figure 3
 
Stimuli and individual results. The left panel depicts the stimuli used in this experiment. Kanizsa stimuli are shown in the left and four-line Varin stimuli in the right column. The upper row contains geometrically aligned and the lower row outward pointing inducers. The right panel shows logistic psychometric functions describing the relationship between probe position in pixels (x-axis) and probability of perceiving the probe as being located outside the subjective shape (y-axis). Dots depict the empirical data (the average over 20 data points), and lines depict the psychometric fits.
Figure 4
 
Group results of Experiment 1. The graph depicts mean JNDs in units of pixels averaged across 13 observers as a function of inducer type (x-axis) and inducer alignment (differently colored bars). Error bars indicate standard errors of the mean.
Figure 4
 
Group results of Experiment 1. The graph depicts mean JNDs in units of pixels averaged across 13 observers as a function of inducer type (x-axis) and inducer alignment (differently colored bars). Error bars indicate standard errors of the mean.
Figure 5
 
Mean response latencies in Experiment 1. The bars show the mean response times as a function of inducer type (x-axis) and inducer orientation (panel variable). Error bars indicate standard errors of the mean.
Figure 5
 
Mean response latencies in Experiment 1. The bars show the mean response times as a function of inducer type (x-axis) and inducer orientation (panel variable). Error bars indicate standard errors of the mean.
Figure 6
 
Group results in Experiment 2. The bar graph shows the mean JNDs in units of pixels averaged across 16 observers as a function of the number of inducing line ends (x-axis) and line width (panel variable). Error bars indicate standard errors of the mean. The bracket signifies conditions of equal support ratio. The support ratios of the stimuli were (left to right) 0.035, 0.07, 0.07, and 0.14.
Figure 6
 
Group results in Experiment 2. The bar graph shows the mean JNDs in units of pixels averaged across 16 observers as a function of the number of inducing line ends (x-axis) and line width (panel variable). Error bars indicate standard errors of the mean. The bracket signifies conditions of equal support ratio. The support ratios of the stimuli were (left to right) 0.035, 0.07, 0.07, and 0.14.
Figure 7
 
Group results in Experiment 3. The bar graph shows the mean JNDs in units of pixels averaged across 15 observers as a function of inducer type (x-axis) and inducer alignment (panel variable). Error bars indicate standard errors of the mean.
Figure 7
 
Group results in Experiment 3. The bar graph shows the mean JNDs in units of pixels averaged across 15 observers as a function of inducer type (x-axis) and inducer alignment (panel variable). Error bars indicate standard errors of the mean.
Figure 8
 
Line spacing demonstrations. (a) Increasing inducers' salience by increasing the length of the inner arc (upper stimulus) does not counteract the perceptual weakening of the subjective contours in the two line (lower stimulus) compared to the four-line Varin shape (b, upper left). (b) Upper row—orderly arranged concentric arcs in Varin inducers with four (left) and three (right) rings. Lower row—disordered Varin-type inducers with four (left) and three (right) rings. In Varin stimuli, the disordered arrangement of line terminations does not seem to improve the perceptual strength of the occluded surface or the interpolated boundaries.
Figure 8
 
Line spacing demonstrations. (a) Increasing inducers' salience by increasing the length of the inner arc (upper stimulus) does not counteract the perceptual weakening of the subjective contours in the two line (lower stimulus) compared to the four-line Varin shape (b, upper left). (b) Upper row—orderly arranged concentric arcs in Varin inducers with four (left) and three (right) rings. Lower row—disordered Varin-type inducers with four (left) and three (right) rings. In Varin stimuli, the disordered arrangement of line terminations does not seem to improve the perceptual strength of the occluded surface or the interpolated boundaries.
Figure 9
 
Kanizsa square and two of its putative but insufficient cues for boundary completion. A compelling subjective square is evident only in the left (Kanizsa), but not the middle (two-line Varin) and the right (corner outline) figure. Neither a discontinuity (middle) nor a contour support (right) equivalent of the Kanizsa shape is sufficient to generate as strong a subjective square by itself.
Figure 9
 
Kanizsa square and two of its putative but insufficient cues for boundary completion. A compelling subjective square is evident only in the left (Kanizsa), but not the middle (two-line Varin) and the right (corner outline) figure. Neither a discontinuity (middle) nor a contour support (right) equivalent of the Kanizsa shape is sufficient to generate as strong a subjective square by itself.
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