**The shape of the illusory surface in stereoscopic Kanizsa figures is determined by the interpolation of depth from the luminance edges of adjacent inducing elements. Despite ambiguity in the position of illusory boundaries, observers reliably perceive a coherent three-dimensional (3-D) surface. However, this ambiguity may contribute additional uncertainty to the depth percept beyond what is expected from measurement noise alone. We evaluated the intrinsic ambiguity of illusory boundaries by using a cue-combination paradigm to measure the reliability of depth percepts elicited by stereoscopic illusory surfaces. We assessed the accuracy and precision of depth percepts using 3-D Kanizsa figures relative to luminance-defined surfaces. The location of the surface peak was defined by illusory boundaries, luminance-defined edges, or both. Accuracy and precision were assessed using a depth-discrimination paradigm. A maximum likelihood linear cue combination model was used to evaluate the relative contribution of illusory and luminance-defined signals to the perceived depth of the combined surface. Our analysis showed that the standard deviation of depth estimates was consistent with an optimal cue combination model, but the points of subjective equality indicated that observers consistently underweighted the contribution of illusory boundaries. This systematic underweighting may reflect a combination rule that attributes additional intrinsic ambiguity to the location of the illusory boundary. Although previous studies show that illusory and luminance-defined contours share many perceptual similarities, our model suggests that ambiguity plays a larger role in the perceptual representation of illusory contours than of luminance-defined contours.**

*Q*with variance

_{i}*Q*with variance

_{l}^{2}), each with a diameter of 0.8° (1.1 cm) and a 1.7° (2.2 cm) separation between the centers of adjacent inducers. In stimulus conditions without illusory contours, the inducers faced outward (see Figure 4, Stimuli: Surface conditions). Here, the inducers were shifted diagonally by 1.2° (1.5 cm), so they abutted the corners of the occluding surface. The inducers were static with a 90° circular segment removed from the outermost edge and had zero disparity relative to the reference plane. The curvature of the surface edge was defined using a surface template that represented a half cycle of a sinusoid. The peak amplitude was calculated from the disparity at the peak and the observer's IOD using the conventional formula (see Howard & Rogers, 2012, pp. 152–154). All stimuli were rendered using the same curved surface template. Figure 3 illustrates the viewing geometry for Experiment 1.

^{2}) luminance-defined surface with rotated inducing elements. The surface was filled with black to create high-contrast edges, which are optimal for good stereoscopic acuity (McKee, 1983). All stimulus conditions had the same curvature along the surface edge, but in the illusory condition, the surface peak was camouflaged with respect to the background. Thus, the largest luminance-defined disparity in the illusory condition was the relative disparity at the tip of the inducing elements, and the largest disparity signal in the remaining surface conditions was at the surface peak.

^{2}) with an array of high-contrast (65.6 cd/m

^{2}) outlined circles (radius of 0.21°) above and below the central (5.2° × 10.5°) region. The arrangement of this pattern of circles was randomized on each trial, so they provided no consistent position cue but provided a strong fusion lock and reference plane. The stimulus and fusion field were presented at a standing uncrossed disparity of 0.42°. A circular disparity probe (22.6 cm/m

^{2}) with a diameter of 0.25° was presented 2.1° to the left of the center of the screen. In preliminary testing, we assessed multiple lateral offsets (1.0°, 1.5°, 2.0°, and 2.5°) and determined that at displacements of 2.0° or greater there was no reliable influence of the probe on the interpolation of the surface.

*n*= 6). The analysis was performed using R statistical software and bootstrapped 95% confidence intervals (CIs) were calculated using Monte Carlo simulation methods run 1,000 times for each data set (Wichmann & Hill, 2001a; Wichmann & Hill, 2001b).

*F*(3, 15) = 123.05,

*p*< 0.001,

*η*

^{2}= 0.95. The differences in the PSE between stimulus conditions were examined using pairwise

*t*tests with Benjamini and Hochberg's (1995) correction for false discovery rate. For all observers, the estimated disparity of the low- and high-contrast peaks was similar, and estimates of the illusory and combined surface peaks were shifted downward. Pairwise

*t*tests on the means confirmed that the difference between the illusory (PSE

_{I}= 0.087°,

*SE*= ±0.002), low-contrast (PSE

_{LC}= 0.123°,

*SE*= ±0.001), and combined (PSE

_{C}= 0.107°,

*SE*= ±0.002) surface conditions were all significant (

*p*< 0.01). There was no significant difference in perceived depth between the low- and high-contrast (PSE

_{HC}= 0.126°,

*SE*= ±0.001) luminance-defined surfaces (

*p*= 0.17). Importantly, the PSEs obtained using the high- and low-contrast surfaces closely matched the disparity at the peak of the surface template for all observers. This confirms that observers could accurately localize the peak of these surfaces and that the disparity probe did not introduce a bias.

^{1}On average, PSEs were lower when the surface was illusory than in the low-contrast condition (

*p*< 0.001). This indicates that the trajectory of the interpolated surface was shallower for these illusory surfaces than in the surface template used to create the stimulus. In the combined condition, the peak was consistently localized as lying between the illusory and low-contrast surface peaks (

*p*< 0.001 and

*p*= 0.001, respectively). Thus, when the surface was defined by luminance edges that occluded the inducing elements, the presence of the inducers consistently

*reduced*the perceived depth at the peak of the surface. This occurred even though, when presented on its own, the luminance-defined signal was consistently matched to a larger disparity.

*SE*= ±0.002) was approximately the same as the disparity at the tip of the inducing edge. In this respect, the interpolation seen here is consistent with studies of surface structure (Anderson, 2003) and disparity interpolation (Mitchison & McKee, 1987) that show, when a disparity signal is interpolated across an ambiguous region, the interpolated disparity tends to be equivalent to the disparity of the nearest unambiguous element. This pattern of results is consistent with observers simply matching the disparity of the inducer tips rather than relying on the perceived depth of the interpolated surface. However, if this were the case, then the tips of the inducers should appear to bend toward the observer in depth. Instead, inspection of the stimuli reveals that the disparity signal at the inducer tip appears to be assigned to the illusory surface while the inducer itself appears almost fronto-parallel (Figure 4). To determine if this impression was shared by observers, in a follow-up study we evaluated the perceived depth of the inducer tips and illusory surface boundary. If observers explicitly relied on the depth of the inducer tips in the preceding experiment, then we would predict that the perceived offset in depth (depth magnitude) of these two regions would be the same; furthermore, they should show the same dependence on disparity. Here we asked observers to estimate the perceived depth in the region of the tip of one of the inducing elements. On half the trials, they were told to indicate the depth of the high-contrast inducer tip, and on the remaining trials, they were asked to base their judgments on the perceived depth of the adjacent (illusory) surface at that location. The trial type was randomized and indicated by displaying the word “tip” or “surface” prior to the stimulus presentation. Observers had unlimited time to make their responses and indicated depth magnitude using a custom-built (and previously validated) haptic sensor strip (see Deas & Wilcox, 2014; Hartle & Wilcox, 2016). We tested four observers using three inducer tip disparities (0°, 0.04°, and 0.09°).

*t*test confirmed the increase from zero to the first test disparity for inducer depth estimates is not significant (

*p*= 0.05) even with the variability at zero disparity excluded. Overall, these results confirm that observers can and do report the depth at the peak of the illusory surface separately from the depth at the inducer edge. These results are also consistent with the well-known boundary ownership phenomenon reported in figure–ground literature (Anderson, 2003; Anderson, Singh, & Fleming, 2002), whereby the depth from disparity can disambiguate boundary ownership along the contour by assigning the depth to one side of the contour or the other but not both (Anderson & Julesz, 1995).

*Q*, luminance-defined contours

_{i}*Q*, and the combined cue condition

_{l}*Q*. The goal of Experiment 2 was to estimate the variance associated with these perceived depth estimates. The variance associated with the combined MLE estimate is

_{c}*SD*of the fitted cumulative Gaussian function divided by

*F*(3, 15) = 18.22,

*p*< 0.001,

*η*

^{2}= 0.53. Thus, as predicted, precision varied across the four surface conditions. Pairwise

*t*tests showed that despite the absence of luminance-defined boundaries or features in the illusory surface, mean estimates were equally precise in the illusory (JND

_{I}= 0.014°,

*SE*= ±0.002°) and high-contrast (JND

_{HC}= 0.013°,

*SE*= ±0.001°) surface (

*p*= 0.63) conditions. Importantly, the average precision was poorer in the low-contrast surface condition (JND

_{LC}= 0.025°,

*SE*= ±0.003°) than in any other condition (

*p*= 0.01); thus, the perceived depth of the low-contrast luminance-defined surface was

*less reliable*than the depth of an illusory surface with the same dimensions. The reliability of the mean estimates in the illusory and combined (JND

_{C}= 0.014°,

*SE*= ±0.002°) conditions was similar (

*p*= 0.73), suggesting that the presence of the inducers improved the precision of depth estimates, bringing them to a level similar to that of a salient, high-contrast, luminance-defined surface.

*σ*) of the estimates are the JNDs measured in the 2IFC task in Experiment 2. The linear model in Equation 1 was used to predict PSEs and JNDs in the combined condition (PSE

_{c}and

*σ*, respectively) for each observer who had full data sets from both Experiments 1 and 2 (

_{c}*n*= 5). To assess whether the observed PSE

_{c}and

*σ*estimates were consistent with MLE model predictions, we compared the 95% CI of the observed and predicted PSE

_{c}_{c}and

*σ*for each observer (Figure 8). The results revealed that the

_{c}*σ*of observed depth estimates were consistently within the 95% CI of the MLE predictions for all five observers. However, comparison of the 95% CI for the PSE

_{c}_{c}showed that the observed PSEs were much higher than the predicted PSEs for all observers.

*σ*are consistent with the MLE model predictions, the observed PSEs show a systematic bias; depth from the illusory surface is consistently underweighted, resulting in a combined estimate that is larger than MLE predictions. Observers underweight the depth estimate from illusory boundaries despite the considerable influence of the illusory boundary on the location of the surface peak in Experiment 1. Given that the results of Experiment 2 showed that the depth from illusory boundaries was more reliable than the depth from low-contrast luminance-defined edges, the MLE model predicts that when estimating the depth of the combined surface observers should assign even more weight to the illusory boundaries.

_{c}*I*(

*v*) of images that depict partly camouflaged surfaces. We assume that the images are ambiguous: An image

*I*(

*v*) may depict many partly camouflaged surfaces that have the same visible components but different camouflaged (i.e., invisible) components.

*I*(

*V*) and judges the depth of a point of interest in the camouflaged region. Here

*V*is a random variable whose value is the parameter

*ν*that picks out the randomly chosen image. The depth of the point of interest is also a random variable

*D*. To model the ambiguous information that each image

*I*(

*v*) provides about

*D*, we assume that

*D*is conditionally distributed as

*ϕ*(

*x*,

*μ*,

*σ*) is the normal probability density function, and

*σ*is a parameter that quantifies the depth ambiguity of the images. Thus, given an image

_{D}*I*(

*v*), the depth of the point of interest follows a normal distribution with mean

*σ*. Equation 3 implies that the image parameter

_{D}*v*is not arbitrary: We have parameterized the family of images

*I*(

*v*) such that the mean depth of the point of interest over all partly camouflaged surfaces depicted by the image

*I*(

*v*) is equal to

*v*. That is, the images are parameterized by the mean depths that they depict at the camouflaged point of interest.

*I*(

*V*) and computes an unbiased but noisy depth cue

*X*for the point of interest, given by

*N*is a normally distributed random variable with mean zero and standard deviation

_{X}*σ*. We assume that

_{X}*V*and

*N*are independent. The observer uses the depth cue

_{X}*X*to estimate the depth

*D*of the point of interest. In the Appendix, we show that, under the model outlined here, the probability distribution of the depth cue

*X*given true depth

*D*=

*d*is closely approximated by

*d*that maximizes this expression, so

*x*, and the uncertainty associated with this estimate is

*X*to estimate the depth

*D*of the point of interest. The ambiguity does not affect the observer's maximum likelihood depth estimate or their JND in the single-cue condition. This means that in a single-cue depth-discrimination task such as our Experiment 2, we can use the slope of the psychometric function to estimate the depth cue noise parameter

*σ*, and this slope is not affected by the ambiguity parameter

_{X}*σ*.

_{D}*X*is

*σ*for each observer using his or her observed PSE for the combined surface (Figure 9). The ambiguity parameter

_{D}*σ*for observers BH, LD, LW, MC, and MJ were 0.012, 0.030, 0.014, 0.048, and 0.022, respectively. The estimated intrinsic ambiguity due to the ambiguity of the illusory contour is on average 1.8 times the observed standard deviation of illusory depth estimates. This estimate seems reasonable given it is within an order of magnitude of the observed standard deviation of the perceived depth of the combined surface for all observers. Thus, very little depth ambiguity is necessary to account for our findings. In addition, comparison of the observed standard deviations in Figure 8 to the predicted standard deviations in Figure 9 shows that the inclusion of intrinsic ambiguity in the model yields predicted standard deviations that are within the 95% CI of the observed standard deviations for four of the five observers. Given that the inclusion of intrinsic ambiguity produces a combined estimate that is consistent with our observed estimates, our results are consistent with the explanation that observers attribute additional uncertainty to the position of illusory contours and consequently underweight their contribution to perceived depth.

_{D}*ρ*, then the optimal choice of weight for a single cue is

*r*

_{1}>

*r*

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*r*

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*r*

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^{1}To evaluate whether the contrast polarity (black inducers on a gray background) was important to the pattern of results, in a followup study a subset of observers (

*n*= 3) compared the perceived peak of a high-contrast white surface to the black surface used in Experiment 1. There was no significant difference in the PSE of the peaks.

*X*=

*x*, is the value of

*d*that maximizes

*P*(

*X*=

*x*|

*D*=

*d*). To find this conditional probability, we start by partitioning over the possible values of the stimulus image parameter

*V*:

*V*, the true depth

*D*of the camouflaged point gives no additional information about the observer's depth cue

*X*(i.e.,

*X*is conditionally independent of

*D*, given

*V*):

*P*(

*V*=

*v*)/

*P*(

*D*=

*d*) with weights given by a normal probability density function whose value is negligible outside the range

*μ*′ ± 3

*σ*′. Within this small range, if the priors on depicted depth

*V*and actual depth

*D*are weak, then the ratio

*P*(

*V*=

*v*)/

*P*(

*D*=

*d*) is approximately constant, and we use this approximation here. Letting this constant be

*x*, and the uncertainty associated with this estimate is

*X*to judge the depth of the point of interest, and we can recover

*σ*from the slope of the observer's psychometric function.

_{X}*X*and

*Y*, are available, the maximum likelihood depth estimate depends on both cues. Here we assume that

*X*is unbiased but ambiguous in the manner described by the model outlined here and that

*Y*is unbiased and unambiguous as in classic cue-combination models. We also assume that

*X*and

*Y*are conditionally independent given the true depth

*D*. The maximum likelihood depth estimate is the value of

*d*that maximizes

*d*=

*μ*″, so the maximum likelihood depth estimate is a weighted sum of depth cues

*X*and

*Y*with weights determined by the reliabilities

*r*and

_{X}*r*, much as in classic cue-combination models. However, here the ambiguous depth cue

_{Y}*X*has reliability

*Y*can mean both the inverse of the cue's variance,

*r*used to construct the cue's weight

_{X}*r*/(

_{X}*r*+

_{X}*r*) in an optimal weighted sum of depth cues because these values are equal. In our revised model, these values are not necessarily equal. In this case, it may be more descriptive to refer to the inverse variance

_{Y}*v*is not arbitrary in that the images

*I*(

*v*) are parameterized by the mean depth

*v*that they depict at the point of interest. This is a genuine restriction on the set of stimulus images because in general there could be distinct images

*I*

_{1}and

*I*

_{2}that depict distinct sets of partly camouflaged surfaces that have the same mean depth

*v*at the point of interest whereas in a one-dimensional family of images

*I*(

*v*) there can only be one image for each value of

*v*. However, this restriction is valid in many depth judgment tasks, including the experiments we report here, in which the stimulus images are a one-dimensional family

*I*(

*v*) of stereoscopic Kanizsa squares, each of which can be interpreted as depicting a range of partly camouflaged surfaces with a unique mean depth

*v*at the point midway along a vertical illusory contour.