Stereo-transparency is an intriguing, but not well-understood, phenomenon. In the present experiment, we simultaneously manipulated the number of overlaid planes, density of elements, and depth separation between the planes in random dot stereograms to evaluate the constraints on stereoscopic transparency. We used a novel task involving identification of patterned planes among the planes constituting the stimulus. Our data show that observers are capable of segregating up to six simultaneous overlaid surfaces. Increases in element density or number of planes have a detrimental effect on the transparency percept. The effect of increasing the inter-plane disparity is strongly influenced by other stimulus parameters. This latter result can explain a difference in the literature concerning the role of inter-plane disparity in perception of stereo-transparency. We argue that the effects of stimuli parameters on the transparency percept can be accounted for not only by inhibitory interactions, as has been suggested, but also by the inherent properties of disparity detectors.

- Glass-transparency—light passes through objects made of clear transparent materials such as glass.
- Translucency—translucent materials allow light to pass through them only diffusely and cannot be clearly seen through. Examples of such materials are frosted glass and certain types of cloth.
- Pseudo-transparency—light passes through gaps in non-transparent lacy objects such as wire fences or tree branches.

^{2}, and the Michelson contrast was 99%. Each stereogram, when fused, depicted several overlaid planes of dots. The first plane (closest to the observer) was presented at fixation, and the rest were presented with uncrossed disparities with respect to fixation. Each adjacent pair of planes was separated in depth by disparity

*d*(see Figure 2). Antialiasing was used to allow subpixel positioning.

*n*planes the number of Glass planes was in the range [0,

*n*].

^{2}. When new planes were added to an RDS with a certain density, the overall density was kept constant. The dots were simply redistributed across the planes.

*both*the correct number of planes and the correct number of Glass planes. It was necessary to combine the two tasks since the number of Glass planes task served as a check for the coherency of the perceived pseudo-transparent planes. Separate analysis of the two judgments confirmed the main findings. Logistic regression analysis showed that subject, number of planes, disparity, density, combination of density and disparity, and combination of planes and disparity influenced performance significantly.

*χ*

^{2}= 1603.66;

*p*< .000;

*df*= 4; Exp(B) for indicator categorical contrasts with the performance with two-plane stimuli as reference were 3 planes = .253, 4 planes = .055, 5 planes = .016, 6 planes = .007). However, determining an upper limit on the number of overlaid planes that can be perceived simultaneously requires careful analysis. Recall that the experimental task consisted of two parts: stating the number of planes (NP task) and the number of Glass planes (NGP task) in the stimulus. Both responses had to be correct for a trial to be scored as successful. To determine the upper limit on performance, we first had to define chance level for this task. There were five possible answers in the NP task, thus the corresponding chance level was 1/5 or 20%. However, even when the observers were incorrect, they were close to the true number of planes. For example, when viewing two or three planes, they were very unlikely to respond five or six. This is evident in Figure 6, which shows the distributions of the observers' responses to the NP task, for stimuli with different numbers of planes. The maxima of the distributions are always located at the correct answer. Moreover, the distributions have substantial values only in immediate proximity of the maxima (two neighboring values). Based on these observations, we defined the neighborhood of any given number of planes

*n*as the set {

*n*− 1,

*n, n*+ 1} (for marginal cases the neighborhood is defined as {

*n, n*+ 1,

*n*+ 2} and {

*n*− 2,

*n*− 1,

*n*}, respectively). It is apparent from Figure 6 that vast majority of the answers (100–84%) for each type of stimuli with

*n*planes, falls within the respective neighborhood of

*n*as defined above. Consequently, the number of possible answers in the NP task is effectively reduced from five to three, the size of the neighborhood. Thus, we set the chance level of performance for the NP task to 1/3 or 33%.

*pn*since the observers knew that the number of Glass planes fell into the interval [0,

*pn*]. Since the smallest

*pn*present in the stimuli was two, the smallest number of possible answers for the NGP task was three. This implies that the chance level of performance for the NGP task was always equal to or smaller than 1/3 or 33%. In order to judge performance conservatively on the NGP task, the highest of the levels of chance performance has to be taken. Recognizing that the NGP task was subordinate to the NP task, we set the chance level of performance on the overall experimental task to 33% and the criterion for successful performance to 66%, which is halfway between chance and perfect performance. According to this criterion, as Figure 5 shows, the observers were able to perceive up to three planes under almost all conditions. Under optimal conditions, including low density and high disparity, observers were able to perceive up to six overlaid planes.

*χ*

^{2}= 250.9;

*p*< .000;

*df*= 1; Exp(B) = .907). In conditions for which the effect of density cannot be observed, the overall performance was either near perfect (ceiling effect) or almost at chance levels (floor effect).

*χ*

^{2}= 32.37;

*p*< .000;

*df*= 3). The smallest inter-plane disparity, of 1.9′, corresponded to the poorest performance for all stimuli (see Figure 5). Initial increases in inter-plane disparity resulted in improved performance.

*χ*

^{2}= 44.215;

*p*< .000;

*df*= 3). Specifically, peak disparity was smaller for dense stimuli than for sparse stimuli (for stimuli with the smallest density, the peak disparity presumably falls outside the range of disparities tested here). Moreover, the curvature of the plots of percent correct with respect to disparity increased and thus became more peaked with increasing density. This was reflected in a significant negatively signed interaction effect between the quadratic terms of the polynomial contrast of disparity and density (Wald

*χ*

^{2}= 24.318;

*p*< .000;

*df*= 1; Exp(B) = .942). Peak disparity was also tied to the number of planes (Wald

*χ*

^{2}= 78.717;

*p*< .000;

*df*= 12). As shown in Figure 5, for all subjects the peak disparity phenomenon was not evident for two- and three-plane stimuli. Moreover, for subjects IT and YS, the peak disparity for densities 15.6 dots/deg

^{2}and 10.6 dots/deg

^{2}was first apparent in stimuli with four planes while the peak disparity for 6.2 dots/deg

^{2}density was first apparent in five-plane stimuli. Quadratic terms of the polynomial contrasts of disparity confirmed that the increase in the number of planes increased the curvature of the performance curve with respect to disparity (B values for 4 planes = −.722, 5 planes = −1.410, 6 planes = −1.531).

*a*

_{s}and

*a*

_{d}are the areas of the RDS and a single dot respectively,

*p*is the number of planes, and

*n*is the number of elements in an RDS given a certain density. Figure 8 shows that for a wide range of disparity gradients performance clearly decreased with increasing number of planes, irrespective of the disparity gradient. Therefore, violation of the disparity gradient limit cannot explain the loss of segregation ability that occurred as the number of superimposed planes increased.

^{2}(we assumed that the dot size in their study was 1 pixel) and a maximum disparity of 112′, while Wallace and Mamassian used stimuli with a fixed density of 8.9 dots/deg

^{2}and a maximum disparity of 63′.

^{2}for one observer and 50 dots/deg

^{2}for another (these data were estimated from the graphs on page 2919). As suggested by Gepshtein and Cooperman's data, Wallace and Mamassian's (2004) stimuli were well within the upper performance limit imposed by density. However, Akerstrom and Todd's (1988) stimuli likely exceeded, this critical density so their results show the expected drop in performance with high disparities.