Do Explicit Estimates of Angular Declination Become Ungrounded in the Presence of a Ground Plane?

Abstract

In a series of seven experiments (total N = 220), it is shown that explicit angular declination judgments are influenced by the presence of a ground plane in the background. This is of theoretical importance because it bears on the interpretation of the relationship between angular declination and perceived distance on a ground plane. Explicit estimates of ground distance are consistent with a simple 1.5 gain in the underlying perceived angular declination function. The experiments show that, in general, functions of estimates of perceived angular declination have a slope of 1.5, but that an additional intercept can often be observed as a result of incorporating changes in ground distance into reports of changes in angular declination. By varying the background context, a variety of functions were observed that are consistent with this contamination hypothesis.

Keywords: angular declination; distance perception; magnitude estimation.

Figure 1.

Overestimation of angular declination, relative to straight ahead, by 1.5 predicts underestimation of egocentric distance by about 0.7. If a ball viewed with an angular declination of 12 ° appears to be along an 18 ° line of sight, it should appear to be at a distance of about 5 m (i.e., about 3 eye-heights), much closer than its true 7.5 m distance.

Figure 2.

Experiment 1A. The upper panel depicts the set of 11 angular locations (6 ° to 36 °) of both suspended balls (shown here, for clarity, in blue; all 1.9 m from the observer) and their corresponding ground locations and larger ball sizes (shown as green) that project the same horizontal angular extents and directions when embedded in the ground. The three square images across the middle represent details (originally projected at 64 × 64 cm) from individual trials in which a single ball is shown either embedded in the ground (left) like the schematically green balls, or suspended 1.9 m from the observer, like the schematically blue balls, either in the absence of a ground surface (center) or in the presence of a visible ground surface (right). A full screen (256 × 144 cm) view of the latter trial type is shown at lower left. All displays used binocular disparity tuned to the measured interpupillary distance of the observer to specify distance and aligned the virtual ground with the floor of the laboratory in which the observers were seated. The laboratory is shown, with room lights on for the photo, in the lower right panel.

Figure 3.

Experiment 1B. The lower right panel shows the physical viewing set-up from a headrest in front of a large back-projection screen while in use. The other three images show all ball positions simultaneously (only one ball was visible during each actual trial) in environments in which the balls were shown either embedded in the ground (upper left), as if embedded in the ground in the absence of a ground plane (upper right), or suspended along a virtual arc 3.9 m from the observer in the absence of a ground plane (lower left). All displays were stereoscopic.

Figure 4.

Results of Experiments 1A (left) and 1B (right) as a function of visual direction and environment. The predicted angular gain of 1.5 is shown as a dashed line. Standard errors are shown for each mean.

Figure 5.

Combined estimates of first environments tested in Experiments 1A and 1B as a function of visual direction and presence or absence of a visible ground plane. The predicted angular gain of 1.5 is shown as a dashed line. Standard errors are shown for each mean.

Figure 6.

The upper figures show the six equiangular (∼5 °) ball locations used in Experiment 2A and the six equidistant (2.5 m) ball locations used in Experiment 2B. The lower image shows the field location where the studies were conducted. Ball elevation relative to standing location, horizontal ground distance, and eye-height were used to compute the exact angular declination for each target relative to each observer.

Figure 7.

Results of Experiments 2A and 2B. Angular declination estimates (left panel) as a function of actual angular declination are shown with standard errors. The solid line shows the fit to the data from visible ground plane conditions in virtual environments from Experiments 1A and 1B (see Figure 5 earlier). Distance estimates (right panel) as a function of ground distance are shown with standard error bars.

Figure 8.

Environments used in Experiment 3. In the experimental condition (left panel), the upper 25 ° of the ground plane is occluded by a large tree trunk. Target balls appear in front of either the trunk or the near grass. In the control condition (right panel), the trunk is moved 1 m off to one side, so the balls appear in front of the ground plane on all trials.

Figure 9.

Results of Experiment 3. Mean declination estimates (with standard error bars) as a function of actual declination and ground plane occlusion beyond the target area.

Figure 10.

Details of the close reference pole (left) and far reference pole (right) conditions of Experiment 4. The target ball was presented on the ground at various distances; the farthest target (6 ° declination) is shown in each image.

Figure 11.

Results of Experiment 4. Angular declination judgments as a function of reference-pole position (near or far) and actual declination (left panel). The right panel plots the same data with a logarithmic abscissa to show that the estimates approximated a linear function of log declination.

Figure 12.

Results of Experiment 5. Estimates of angular declination to balls viewed on a ground plane from a window at an eye-height of 4 m (left panel) are replotted (right panel) with a logarithmic abscissa for comparison with the results of Experiment 4. Standard errors of the mean are shown.