Interaction of disparity size and depth structure on perceived numerosity in a three-dimensional space

Abstract

The number of elements in two stereo-surfaces parallelly overlapped in depth is overestimated compared to that in a single flat surface, even when both have the same number of elements. Using stereoscopic pairs of elements, we evaluated two hypotheses on the overestimation: one that a higher-order process, forming a background surface, increases the number of perceived elements, and the other that the number of elements potentially occluded by the elements on a front surface is taken accounted for. The data from four experiments showed that (a) when binocular disparity between (or among) stereoscopic elements was small, the overestimation occurred for the stimuli we used-a two-surface-overlapping stimulus, where the likelihood for the process to operate was manipulated by changing the averaged luminance of each surface, a volumetric stimulus, where the likelihood for the background surface to be formed would decrease, and a two-non-overlapping-surface stimulus, where the surfaces in depth were not overlapped-, and (b) when binocular disparity was large, the overestimation occurred for the two-surfaces-overlapping stimulus, when the averaged luminance of the two surfaces were the same, and for the volumetric stimulus, but diminished for the surface-overlapping stimulus, when the averaged luminance differed between the surfaces and for the surfaces-non-overlapping stimulus. These results cannot be explained either hypothesis only. We explain the results by postulating that the sensory system processing disparities of elements interferes with that estimating the number of elements, resulting in an overestimation of the elements in a stereo-stimulus, and the disparity range within which the interference occurs may depend on the stimulus depth structure.

Fig 1. Schematic illustrations of 3-D stimuli included in the experiments.

(a) a two -POTS stimulus, (b) a volumetric stimulus, and (c) a stepwise stimulus (up-back/down-front, up-front/down-back, right-front/left-back, or right-back/left-front stimulus) (from left to right). Illustrations were drawn from the side view except for those of right-front/left-back and left-front/right-back stimuli, which were drawn from the top view. Gray lines illustrate the monitor planes, and the solid black rectangular boxes represent the position of the elements. See the text for a detailed description.

Fig 2. Schematic explanation of how to calculate the PSE and JND.

The percentage (%) (y-axis) of responses that the number of 2-D stimulus elements was judged to be higher than that of 3-D stimulus elements was plotted against the number of 2-D stimulus elements (x-axis). The x-axis value yielding a 50% response in the fitted psychometric function represents the PSE of the 2-D stimulus. Half of the difference between the x-axis values yielding a 25% and a 75% response in the function represents the JND. For instance, when the number of right-front/left-back stimulus elements was fixed at 300, the number of 2-D stimulus elements varied at five levels from 196–404. In this case, PSE and JND of the observer14 were determined to be 316.1 and 25.2, respectively. The red vertical arrow shows the point producing a 50% response in the fitted function. The left and right blue vertical arrows show the points producing a 25% and a 75% response, respectively, in the fitted function.

Fig 3. Schematic illustrations of three different two-POTS stimuli used in Experiment 1.

(a) black-white, (b) front-black/back-white and (c) front-white/back-black two-POTS stimuli (from left to right). Illustrations were drawn from the top view. While, for descriptive purpose, black and white elements were depicted as if they were placed alternatively for the black-white stimulus, their locations were randomly determined in the experiment.

Fig 4. Results from Experiment 1.

Mean biases of PSE for the three types of two-POTS stimuli. The vertical and horizontal axes represent the mean biases of PSE and binocular disparity, respectively. Error bars indicate 95% confidence interval.

Fig 5. Results from Experiment 2.

Mean biases of PSE for volumetric stimuli. The vertical and horizontal axes represent the mean biases of PSE and binocular disparity, respectively. Error bars indicate 95% confidence interval.

Fig 6. The number of observers as a function of PSE bias.

The number of observers as a function of PSE bias for the smaller-disparity condition (a) and the larger-disparity condition (b). The vertical and horizontal axes represent the number of observers and the mean biases of PSE, respectively.

Fig 7. Results from Experiment 3.

Mean biases of PSE for each of the five different 3-D standard stimuli. The vertical and horizontal axes represent the mean biases of PSE and binocular disparity, respectively. Error bars indicate 95% confidence interval.

Fig 8. Schematic front view of stimulus used in Experiment 4.

The width and the height of the 2-D stimulus were the same as and half of the 3-D stimulus, respectively.

Fig 9. Results from Experiment 4.

Mean biases of PSE for the four estimation conditions (see text for details). The vertical and horizontal axes represent the mean biases of PSE and binocular disparity, respectively. Error bars indicate the 95% confidence interval.