Magnitude of perceived depth of multiple stereo transparent surfaces

Abstract

According to the geometric relational expression of binocular stereopsis, for a given viewing distance the magnitude of the perceived depth of objects would be the same, as long as the disparity magnitudes were the same. However, we found that this is not necessarily the case for random-dot stereograms that depict parallel, overlapping, transparent stereoscopic surfaces (POTS). The data from five experiments indicated that (1) the magnitude of perceived depth between the two outer surfaces of a three- or a four-POTS configuration can be smaller than that for an identical pair of stereo surfaces of a two-POTS configuration for the range of disparities that we used (5.2–19.4 arcmin); (2) this phenomenon can be observed irrespective of the total dot density of a POTS configuration, at least for the range that we used (1.1–3.3 dots/deg²); and (3) the magnitude of perceived depth between the two outer surfaces of a POTS configuration can be reduced as the total number of stereo surfaces is increased, up to four surfaces. We explained these results in terms of a higher-order process or processes, with an output representing perceived depth magnitude, which is weakened when the number of its surfaces is increased.

Appendix

A: Predicted ratios of magnitude of perceived depth for Experiments 2 and 5

To calculate the difference in perceived depth between the two-POTS and three-POTS configurations in Experiment 2 and that between the three-POTS and four-POTS configurations in Experiment 5, we assumed that the magnitude of perceived depth (d′) is expressed by the following equation:

$$ \mathrm{d}\mathrm{^{\prime }}\fallingdotseq Kn\delta, $$

(A1)

where K is constant for each POTS, n denotes the number of stereoscopic surfaces, and δ is the disparity. In Experiment 2, the perceived depth obtained with the two-POTS configuration with disparity (δ ₂) was the same as that for the three-POTS configuration with disparity (δ ₃). From Eq. A1, we obtain

$$ {K}_2{\delta}_2={K}_3{\delta}_3, $$

(A2)

where δ ₂ is the matched disparity of the two-POTS configuration and δ ₃ is the corresponding disparity of the three-POTS configuration. From Eq. A2, we calculate K ₂ over K ₃ (K ₂ /K ₃), which is the predicted ratio of the perceived depth for the three-POTS configuration to that for the two-POTS configuration when their two outer surfaces have the same disparity. Similarly, we calculated K ₃ over K ₄ (K ₃ /K ₄), which is the predicted ratio of the perceived depth for the four-POTS configuration to that for the three-POTS configuration when their two outermost surfaces have the same disparity.

B: Details of the cross-correlation analysis

We simulated the psychophysical experiments described in this article to examine whether the cross-correlation analysis can explain the depth reduction phenomenon. The procedure used to calculate the cross-correlation values was the same as that used in Filippini and Banks (2009). After calculating the values, subpixel interpolation of the correlation function was carried out in order to obtain decimal values of disparity. The subpixel disparity was computed by fitting parabolas to three cross-correlation values around each of the local maxima in the correlation function (Shimizu & Okutomi, 2002). We regarded a maximum (mode) of the parabolas as the perceived location of a stereo surface, as in Stevenson et al. (1991), and estimated the perceived disparity of two outermost surfaces in a multi-POTS configuration.

In the simulation, we generated two-, three-, and four-POTS configurations that had five disparities (6.2, 9.4, 12.5, 15.6, and 18.7 arcmin), which were similar to those used in Experiments 1–5. The total dot density was 2.2 dots/deg², which was within the range of dot densities used in Experiments 1–5. The dots were distributed equally on each stereo surface, and there were eight trials for each of the five disparities and the three POTS configurations. Thus, 120 trials in total were presented for each dot-density condition.

We averaged the simulated disparities between the two outermost surfaces over eight trials, which was calculated from local modes in the function for each POTS configuration, and examined whether or not the depth reduction phenomenon occurred. Table 1 shows the ratios of the depth magnitudes, which were calculated from the averaged simulated disparities, between each two POTS configurations (i.e., the ratio of the depth magnitude for a three-POTS configuration to that for a two-POTS configuration, the ratio of the depth magnitude for a three-POTS configuration to that for a two-POTS configuration, and the ratio of the depth magnitude for a three-POTS configuration to that for a four-POTS configuration). We judged that the depth reduction phenomenon occurred when the ratio was less than unity. As can be seen from Table 1, the depth reduction phenomenon can be seen for smaller disparities (6.2, 9.4, and 12.5 arcmin), but not for the larger disparities (15.6 and 18.7 arcmin). Note, however, that the degree of depth reduction in the simulation was much smaller than that observed in the psychophysical experiments described in this article. Thus, the results of the simulated experiment using the cross-correlation methods are not necessarily consistent with those obtained in the psychophysical experiments.

Table 1 Cross-correlation simulation results: Ratio of the magnitude of predicted depth of the two-POTS configuration to that of the three-POTS configuration (3P/2P), ratio of the magnitude of predicted depth of the two-POTS configuration to that of the four-POTS configuration (4P/2P), and ratio of the magnitude of predicted depth of the three-POTS configuration to that of the four-POTS configuration (4P/3P)