As summarized above, previous experimental work has shown that observers are able to judge object size in inanimate dynamic systems governed by gravity. The experiments reported here provide the first empirical evidence that those findings can be extended to the domain of animate motion as well. The human visual system uses the physically determined relation between spatial and temporal scales to obtain the size of a moving animal in the absence of other cues.
In both experiments conducted to test the spatio-temporal scale hypothesis, we found the predicted effect of stride frequency on perceived size.
Nevertheless, when investigating the individual size estimations in terms of the parameters of the proposed model, substantial interindividual differences became evident. These differences were more pronounced in Experiment 1 than in Experiment 2. The results obtained in the modified setting show that observers retrieved the motion-mediated size information more efficiently. The data show less intersubject variability and larger values for k1 when compared to Experiment 1, in which we had attempted to provide a method for transforming observers’ size impression into a corresponding response while maintaining a constant retinal size of the stimulus.
In the two experiments reported here, we presented to the observers a single scaling relation between time and space with the requirement to yield judgment of spatial scale based on temporal variations. One might argue that observers simply assign numbers to the temporal variations without really detecting these variations as information about scale. However, if this were the case, one would expect observers to assign the direction of the mapping between time and space arbitrarily. Only one of 32 observers showed a reversed correlation between perceived size and stride frequency. Moreover, we found a quadratic relation rather than a simple linear one, which reflects the physical properties of the temporal spatial relation. Simply assigning numbers to temporal variations would probably lead to a linear relation instead of a quadratic one.
Altogether, seven observers in Experiment 1 and five observers in the optimized setting in Experiment 2 neglected the temporal-spatial scaling relation by showing a random pattern in their results. A reason for this pattern of results might be the methodological approach. We used a method similar to
Pittenger (1985), in which participants were given only timing as information about spatial scale in pendulum motion. Pittenger’s results were similar to the current results in that they were noisy with strong individual differences. In a related study concerning pendulum motion (
Pittenger, 1990), the observers were given precise information about spatial scale, but the timing of the event was manipulated to be either consistent or inconsistent with the pendulum law. Rather than having to readjust the correct timing, observers had to judge only its correctness. Observers performed with high accuracy on this task. According to Pittenger’s results, observers seem to be more sensitive to violation of the temporal-spatial scaling relation than to transforming temporal information about spatial parameters into size judgments. A similar effect may have also played a role in our setup.
Given constant stride length, a higher stride frequency goes along with a higher locomotion speed. One might be concerned about this confoundation of stride frequency and locomotion speed, arguing that the current results could depend on simple translational speed rather than on the details of the gait itself. In a previous study (
Jokisch, Midfort & Troje, 2001), we used point-light displays of biological motion of dog animations, having subtracted the translational motion component. Consequently, the position of the point-light animal remained constant in the center of the screen. Varying the stride frequency, we found a significant effect on perceived size. Therefore, we are confident that the crucial source conveying size information in the experiments we are reporting here is the stride frequency itself.
Nevertheless, we cannot entirely exclude that translational speed may contribute to the size judgment. In a natural display stride frequency, locomotion speed and stride length cannot be unconfounded. However, we did not want to make any issue about the details of the perceptual cues used to derive size from biological motion. Instead, we wanted to test whether the human visual system is able to employ the relation between temporal and spatial scales, which is physically defined through gravitational acceleration.
Human observers seem to be able to employ the general inverse quadratic relation between size and stride frequency to derive information about size from temporal parameters. In addition to this qualitative result, the measurements taken in Experiment 2 can also be used to make quantitative comparisons between the absolute size indicated by the observers and the size of real animals that walk with the respective stride frequencies. The relation between size and stride frequency of walking animals is expressed by the factor
c1 in
Equation 3. Summarizing the results of Experiment 2, we compute
c1 as the median of the 11 observers that did respond in a consistent manner. The resulting value amounts to 435 cm s
−2.
Unfortunately, the only set of data that we are aware of which can be used to derive the spatio-temporal relation factor from natural locomotion patterns is the one reported by
Pennycuick (1975), who compared stride frequencies and shoulder heights of 14 African quadruped mammal species for different gait patterns. The smallest animal in this study (Thomson’s gazelle) had a shoulder height of 60 cm; the largest one (elephant) had a shoulder height of 310 cm. From Pennycuick’s Figure 13, we calculated
c1 to amount to 410 cm s
−2 for cantering animals. This value is very close to the one obtained from our data.
The close matching between the empirical data for cantering animals (
Pennicuick, 1975) and the data obtained in our experiments seems to imply that the human visual system not only takes into consideration the general inverse quadratic relation between stride frequency and size but also takes advantage from implicit knowledge about the particular observed gait pattern. We want to note, however, that the good quantitative fit between Pennicuick’s and our data may well be accidental. There are a number of factors that introduce uncertainty into the absolute value of the spatio-temporal scaling factor
c1 as derived from our experiments. For instance, the perceived height of the reference objects in the scenery may deviate from their “real” height. The posts were intended to have a height of 1 m and the cactuses a height of 2 m. Those numbers were given to the observers in their introduction to the experiment. However, the reference objects may still have been perceived to be larger or smaller, changing the reference frame used to indicate the dog’s size. Another critical point is the determination of the constant
c2 in
Equation 5. In the second subtask of Experiment 2, we tried to measure the perceived size as given by cues that are independent from stride frequency. We did that by asking the observers to estimate the size of a static stick-figure display. However, this procedure may not be sufficient to accurately derive the desired information. It is still possible that a moving dog does provide cues about its size, which are not available in the static display but which are still not depending on the stride frequency. A last factor that adds uncertainty is the fact that living animals, even if they try to minimize energy consumption during locomotion, are still different from inanimate dynamic systems. In a swinging pendulum or a bouncing ball, the relation between temporal and spatial parameters is exactly defined by gravity, because no other forces affect these motions. In contrast, in dynamic animate systems, muscular forces controlled by intentional behavior play an important role. They are not used only to simply compensate for damping effects in the articulated pendulum system of the body; they can also be used to significantly alter the motion pattern to cover a wider range of stride frequencies within a given gait pattern.