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Research Article  |   April 2008
Classification of apparent motion percepts based on temporal factors
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Journal of Vision April 2008, Vol.8, 31. doi:10.1167/8.4.31
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      Vebjørn Ekroll, Franz Faul, Jürgen Golz; Classification of apparent motion percepts based on temporal factors. Journal of Vision 2008;8(4):31. doi: 10.1167/8.4.31.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

As pointed out by M. Wertheimer (1912), a number of qualitatively different motion impressions, such as “optimal motion,” “part motion,” and “pure phi,” may be evoked by manipulating the temporal parameters of two-element apparent motion sequences. We investigated how the transitions between the different percepts depend on temporal variables over a large range of interstimulus intervals and stimulus onset asynchronies. On the basis of these data, we present a hierarchical classification scheme describing the critical temporal conditions for alternative percepts. A particularly interesting finding is that the distinction between pure phi, on the one hand, and part and optimal motion, on the other, depends on the temporal duty cycle of the stimulus elements. It is suggested that this temporal variable may be used as a cue to resolve occlusion-related ambiguities in classical motion stimuli.

Introduction
As is well known, two stationary stimuli at different positions may evoke the impression of a single moving object if they are flashed on and off in an appropriate temporal sequence (Exner, 1875). By manipulating the temporal variables of such a simple apparent motion stimulus, it is not only possible to evoke a percept of motion instead of static flicker or succession but also to evoke a number of qualitatively different motion impressions (Neuhaus, 1930; Wertheimer, 1912; Zeeman & Roelofs, 1953, see supplementary demonstration): The impression of a single moving object is often referred to as optimal or beta motion. In optimal motion, two static stimulus elements presented sequentially are perceived as a single moving object. In part motion, the two stimulus elements are also perceived to move, but instead of merging into a single perceived object that traverses the entire distance between the two stimulus elements, they are perceived as two distinct objects, either of which moves only part of the distance toward the other. A third, rather enigmatic percept is often referred to as pure phi or objectless motion. In this case, motion is experienced even though both stimulus elements are perceived to be stationary. 
Based on Wertheimer's (1912) work, these qualitatively different percepts are often thought of as different “motion stages” associated with different time intervals. By continuously increasing the alternation rate of the two stimulus elements from a very low value, it is generally thought that an orderly sequence of qualitatively distinct percepts can be observed, such as mere succession without motion, optimal motion, part motion, pure phi, and mere flicker (Palmer, 1999). To Wertheimer (1912), who maintained that motion was a primary sensory quality in its own right (as opposed to the product of a perceptual inference based on the identification of two stimulus elements being visible at different places at different times), the percepts of part motion and pure phi were of profound theoretical significance (Neff, 1936; Sekuler, 1996). Neither of these percepts involves perceptual identification of the two stimulus elements, but motion is perceived nonetheless. 
Despite the central theoretical importance attributed to Wertheimer's (1912) original observations of pure phi, it has received relatively little attention in experimental investigations of apparent motion. One reason for this may be that researchers have found it difficult to produce stimulus conditions yielding stable and convincing percepts of pure phi. Indeed, Steinman, Pizlo, and Pizlo (2000, see also Petersik & McDill, 1981; Petersik, Schellinger, & Geiger, 2003) have presented evidence suggesting that while stable percepts of pure phi can easily be produced by multiple-element apparent motion sequences, they are—for some unknown reason—much less easily observable in the two-element displays typically used in classical studies of apparent motion (see their demonstrations at http://www2.psych.purdue.edu/Magniphi/). 
The results of the present experiments offer a clarification of this issue. They suggest that two-element and multi-element displays are not fundamentally different with respect to their ability to produce pure phi: We report data showing that stable pure phi may also be produced by two-element apparent motion sequences, provided that certain temporal conditions are met. The most important one is that the temporal duty cycle of the stimulus elements (i.e., the fraction of the animation cycle during which a stimulus element is present) must be fairly large (larger than about 0.6). This explains why pure phi has been seldom reported: In the majority of studies of apparent motion, zero or positive interstimulus intervals have been used, i.e., the two stimulus elements were presented in strict temporal succession. Provided that the on-times of the two stimulus elements are equal, which has also been the case in most studies, this means that the temporal duty cycle never exceeded a value of 0.5. 
While the primary aim of our experimental work was to determine the temporal conditions favoring the occurrence of pure phi, we were also interested in gaining a better understanding of how the different motion and non-motion stages relate to each other. In our experiments, we therefore recorded the tendency to perceive flicker, pure phi, part motion, optimal motion, and succession over a large range of stimulus onset asynchronies and interstimulus intervals. On the basis of these data, we develop a categorization scheme that captures the dependence of the different percepts on the relevant temporal variables in a fairly simple manner. We also consider possible theoretical interpretations of our results. 
Phenomenology and terminology
The phenomenon of pure phi is known by many names. Saucer (1953, 1954) seems to have rediscovered the phenomenon independently of Wertheimer (1912) and introduced the term omega motion, which was also used by Tyler (1973). Zeeman and Roelofs (1953) used the term afterimage motion, while Petersik and McDill (1981) spoke of kinetic optical occlusion. In the remainder of this paper, we shall use the descriptive term shadow motion introduced by Allport (1968). Generally, when the phenomenon is observed, the stimulus elements are perceived as stationary while something of blurry appearance is perceived to move in front of them. The term “shadow” is appropriate with respect to the blurriness, but it should be noted that the “shadow” is not necessarily dark. Typically, it is similar in color to the background, sometimes with a slight tinge of the color complementary to that of the stimulus elements (which is of course suggestive of Zeeman and Roelof's notion of “afterimage motion”). 
A further phenomenological aspect that warrants mention is that when shadow motion is perceived, the static stimulus elements are typically perceived to persist behind the moving “shadow” in the amodal sense of Michotte, Thinés, and Crabbé (1991). In this respect, as well as many others, the phenomenon is similar to the occlusion-related modifications of apparent motion reported by Sigman and Rock (1974, see also Rock, 1983). 
Experiments
Stimuli and methods
The stimuli, which were presented on a computer monitor and viewed in a dark chamber from a distance of approx. 85 cm, consisted of two black squares A and B, each with a diameter of 15 mm (≈ 1 deg. vis. angle). Their center-to-center distance was 35 mm, and they were presented in the middle of a gray (≈ 29 cd/m 2) square background (15 × 15 cm). The separation of the stimulus elements in the present experiment is within the range of values used in similar studies (Tyler, 1973; Zeeman & Roelofs, 1953) and well outside of the working range attributed to short-range motion mechanisms (Braddick, 1974). The remaining parts of the monitor were black (< 0.5 cd/m2). A black fixation cross consisting of a horizontal and a vertical line (length 9 mm, width 0.3 mm) was always present at a central location 18 mm above a virtual horizontal line connecting the centers of the black squares (thus the fixation cross was outside the motion path). 
The temporal parameters of the stroboscopic sequences used are defined in Table 1 and illustrated in the space–time diagrams of Figure 1. In the figure, the stimulus onset asynchrony (SOA) is fixed at an arbitrary value. Due to the symmetry of the sequence, the cycle time P is 2 × SOA. From left to right, sequences with increasing interstimulus intervals are shown, ranging from the extreme negative (ISI = −SOA) to the extreme positive (ISI = SOA). We have ISI = SOA − D. Thus, whenever the stimulus duration D exceeds the SOA, the ISI is negative, and the two stimulus elements are simultaneously presented for a time interval corresponding to the absolute value of the ISI. 
Table 1
 
Definition of symbols used to describe the temporal parameters of our stimuli (see Figure 1).
Table 1
 
Definition of symbols used to describe the temporal parameters of our stimuli (see Figure 1).
Symbol Meaning Relations
D The identical duration of the two stimulus elements A and B.
SOA Stimulus onset asynchrony. In all of our stimuli the SOA was symmetric (i.e., SOA AB = SOA BA).
P Duration of a single animation cycle. P = 2 SOA
ISI Interstimulus interval. ISI = SOA − D
LII Local interstimulus interval, i.e., the time interval between offset and the next onset of a given stimulus element (A or B). In all of our stimuli the LII was symmetric (i.e., LII A = LII B). LII = PD = SOA + ISI
δ Temporal duty cycle of either stimulus element (i.e., the fraction of the animation cycle during which the stimulus element is visible). δ = D / P = D / ( D + LII)
Figure 1
 
Space–time diagrams of 5 different apparent motion sequences with a fixed stimulus onset asynchrony (SOA) and interstimulus intervals (ISI) in the range from −SOA to SOA. The middle space–time diagram represents a stimulus with a zero ISI, i.e., stimulus A is turned off as stimulus B is turned on and vice versa. Consequently, the temporal duty cycle at either of the stimulus locations equals 0.5, i.e., the local stimulus is on half of the cycle time. Toward the right, the ISI increases. The rightmost space–time diagram shows the purely notional, limiting case ISI = SOA, where the stimuli would always be absent. Toward the left, the ISI decreases below zero. The leftmost space–time diagram shows the limiting case ISI = −SOA where the stimuli would never be turned off. As is evident by comparing the upper and the lower scale, varying the ISI from −SOA to SOA is equivalent to varying the local temporal duty cycle of the stimulus elements from 1 to 0. D is the duration of the stimulus, the local interstimulus interval (LII) is the time interval for which either stimulus element is blanked.
Figure 1
 
Space–time diagrams of 5 different apparent motion sequences with a fixed stimulus onset asynchrony (SOA) and interstimulus intervals (ISI) in the range from −SOA to SOA. The middle space–time diagram represents a stimulus with a zero ISI, i.e., stimulus A is turned off as stimulus B is turned on and vice versa. Consequently, the temporal duty cycle at either of the stimulus locations equals 0.5, i.e., the local stimulus is on half of the cycle time. Toward the right, the ISI increases. The rightmost space–time diagram shows the purely notional, limiting case ISI = SOA, where the stimuli would always be absent. Toward the left, the ISI decreases below zero. The leftmost space–time diagram shows the limiting case ISI = −SOA where the stimuli would never be turned off. As is evident by comparing the upper and the lower scale, varying the ISI from −SOA to SOA is equivalent to varying the local temporal duty cycle of the stimulus elements from 1 to 0. D is the duration of the stimulus, the local interstimulus interval (LII) is the time interval for which either stimulus element is blanked.
The monitor was running with a frame rate of 85 Hz, so all time intervals used were integer multiples of the frame duration (1000/85 ≈ 11.8 ms). The SOA was varied in 19 steps in the range from 47 ms (4 frames) to 941 ms (80 frames): The interval from 4 to 20 frames was sampled in steps of 2 frames, the interval from 24 to 40 in steps of 4 frames, and the interval from 48 to 80 in steps of 8 frames. At an SOA of n frames the longest and briefest possible interstimulus intervals are ( n − 1) and −( n − 1) frames, respectively. The former corresponds to the case when the stimuli are flashed on for just one frame during each cycle, the latter to the case when they are flashed off for just one frame. We sampled the range of possible ISIs from the interval [−( n − 1),( n − 1)] in steps of 2 (or less) at the briefest SOA levels (SOAs of up to 20 frames), in steps of 4 (or less) at the intermediate SOA levels (SOAs from 24 to 40 frames), and in steps of 8 (or less) at the longest SOA levels (48 to 80 frames). These combinations of SOA and ISI yielded at basic set of 269 different stimuli. 
The subjects reported on the presence of 5 different percepts in separate sessions:
  1.  
    In the shadow motion task, they were asked to respond positively if they had the impression of “something shadowlike” moving back and forth in front of two stationary black squares. They were told that both motion of the “shadow” and stationarity of the two black squares were prerequisites for a positive judgment. They were also instructed to respond negatively if the black squares appeared to materialize and disappear as opposed to being uncovered and covered by the moving “shadowlike” object.
  2.  
    In the optimal motion task, the target percept was that of a single black square moving back and forth. The subjects were told to respond negatively when they had the impression of more than a single black square moving.
  3.  
    In the partial motion task, the subjects were asked to respond positively if they had the impression of two black squares, both moving or making small jumps.
  4.  
    In the flicker task, the subjects were to respond positively if the two black squares were perceived to flicker, but there was no sense of motion back and forth.
  5.  
    In the disappearance–appearance task, the target percept was that of two black squares at different positions simply appearing and disappearing, without any kind of motion.
Every stimulus was presented in a repeating sequence for a total of N full cycles, each of duration P = 2 × SOA with N chosen such that the duration N × P of the trial was as close to 5 s as possible with integer values of P. Actual durations of the stimulus presentation thus ranged between 4.5 and 5.6 s. The subjects were asked to direct their gaze at the fixation cross before initiating a trial by pressing the return button of a keyboard and to maintain fixation until the trial was finished. They were told that their experience of the target percept (one of the abovementioned ones) may be unstable, and that it may appear later during the stimulus presentation even though it was not present immediately. They were instructed to look for the target percept and indicate how easily it occurred to them on a scale from 0 to 5. A zero rating was to be given if they never experienced the target percept during the trial, and a rating of 5 if they experienced it clearly during the entire trial. They were told that they could use the proportion of the trial duration during which they experienced the target percept as a guide for assigning intermediate scores. Immediately after the trial, the subjects could scroll through the numbers 0 to 5, presented at the bottom of the screen, using the up and the down keys. By pressing return, the chosen rating was recorded, and a new trial was started immediately. The subjects were also told that the target percept may occur very seldom or very often, and that they were not to worry if they found this to be the case. 
The basic set of 269 different stimuli (see above) was used in all of the 3 motion tasks (shadow, optimal, and part). The subset thereof corresponding to SOAs of up to 20 frames (235 ms) was used in the flicker task, and the subset with SOAs larger than that was used in the disappearance–appearance task. The stimuli were presented in random order, with separate sessions for the different tasks, and every measurement was repeated 3 times for each observer. Single sessions typically lasted 1 to 2 hours, during which the subjects could make smaller breaks at their own discretion. In addition to one of the authors (VE), two student research assistants and one paid subject participated in the experiment. 
Results
Although there were appreciable differences between the subjects which we shall discuss later, the mean data pooled across all subjects, which are shown in Figure 2, can be regarded as fairly typical. The data from each SOA level are plotted separately against ISI (lower horizontal scale). The horizontal axis of all plots ranges from ISI = −SOA (stimulus elements always present) to ISI = SOA (stimulus elements never present). Thus, the left end of each plot corresponds to a local temporal duty cycle δ (upper horizontal scale) of one, while the right end corresponds to a local temporal duty cycle of zero (this relationship can be appreciated in Figure 1). The mean ratings of the different target percepts are stacked vertically in the same vertical order as the legend. 
Figure 2
 
Stacked mean ratings pooled across all subjects. The stacking order corresponds to the vertical order of the legend. Each panel shows the mean ratings plotted against the interstimulus interval (ISI) for a fixed level of stimulus onset asynchrony (SOA). Each data point is the mean of 12 single observations (3 repetitions × 4 subjects). The lower horizontal axis ranges from ISI = −SOA to ISI = SOA in all plots. This range corresponds to local temporal duty cycles δ between 1 and 0 (upper horizontal scale, increasing from right to left). Note that data for the flicker task were only collected for SOAs less or equal to 235 ms, while data for the disappearance–appearance task were only collected for the longer SOA levels. The solid line is the model fit to the shadow motion data.
Figure 2
 
Stacked mean ratings pooled across all subjects. The stacking order corresponds to the vertical order of the legend. Each panel shows the mean ratings plotted against the interstimulus interval (ISI) for a fixed level of stimulus onset asynchrony (SOA). Each data point is the mean of 12 single observations (3 repetitions × 4 subjects). The lower horizontal axis ranges from ISI = −SOA to ISI = SOA in all plots. This range corresponds to local temporal duty cycles δ between 1 and 0 (upper horizontal scale, increasing from right to left). Note that data for the flicker task were only collected for SOAs less or equal to 235 ms, while data for the disappearance–appearance task were only collected for the longer SOA levels. The solid line is the model fit to the shadow motion data.
If one assumes that the 5 target percepts are exhaustive and mutually exclusive, one would expect the top of the stacked data to be constant at a value of 5 (maximal possible rating). Three systematic deviations from this ideal expectation are apparent: First, there is an increasing sub-additivity toward higher positive ISIs at the higher SOA levels. Secondly, there is a rather sharp sub-additive “trough” somewhat below ISI = 0 evident in the SOA levels from 212 to 282 ms. Third, there is a slight super-additivity for positive interstimulus intervals at the SOA levels from 212 ms downward. We shall discuss possible reasons for these deviations from additivity later. At present, we note that the former sub-additivity is evident in three of the four individual data sets (all except author VE). The latter sub-additivity (the “trough”) is discernible in all data sets. The super-additivities at the lowest SOA levels were however mainly contributed by single subjects. 
The following general trends are evident in Figure 2: Flicker is restricted to the lower SOA levels, where it dominates over all other percepts to a degree which is approximately independent of ISI. Shadow motion occurs predominantly at negative ISIs (i.e., temporal duty cycles > 0.5) and optimal motion predominantly at positive ones (temporal duty cycles < 0.5). In the intermediate ISI range, between shadow motion and optimal motion, either part motion (lower SOA levels) or appearance–disappearance (higher SOA levels) or both take over. Flicker decreases very swiftly with SOA, generally even swifter than suggested by the mean data in Figure 2. For three of the four subjects, there is essentially no flicker above the three lowest SOA levels. Thus, the fact that data on flicker were collected only for SOAs up to 235 ms is likely to be unproblematic. Slightly unfortunate, though, in retrospect, is the fact that data on disappearance and appearance were only collected for SOAs above that. The prevalence of this percept is strongest at the long SOA levels and decreases steadily toward lower SOAs, but it is not entirely abolished at the lowest SOA levels used for that task (282 ms). Thus, by extrapolation, one may surmise that slight tendencies to perceive disappearance and appearance exist at the lower SOA levels where we failed to collect data. It should be noted that the failure to collect data on this percept below the SOA level of 282 ms cannot completely explain the abovementioned trough-shaped sub-additivity since it is also apparent at and above this level. 
Figure 3 shows our model fit of the data in Figure 2. Before we consider data and modelling in more detail, we give a brief overview of the general logic of our analysis. Generally, we assumed that the transition from one percept to the other depends on one or more parameters of the stimulus. Figure 4 shows the relation between relevant parameters and the stimulus percepts suggested by our data analysis. For instance, whether a stimulus gives rise to a percept of flicker or not depends only on whether the stimulus onset asynchrony s is smaller than a certain critical value μ s or not. The occurrence of other percepts depends on a combination of several similar constraints. For shadow motion to occur, for instance, three constraints must be met simultaneously: As is suggested in Figure 4, SOA must be greater than the critical value μ s, the temporal duty cycle δ must be greater than the critical value μ δ, and the local interstimulus interval LII must be less than a critical value μ l. Whether a stimulus is classified as this or that category given the parameter value x is assumed to be a matter of probabilities, and we use cumulative Gaussians f( x, μ x, σ x) to represent the probability of a certain categorization given the parameter value x, where the mean μ x may be taken to be the critical value of the variable and the standard deviation σ x to be a measure of the uncertainty. The complementary cumulative Gaussian,  
f c ( x , μ x , σ x ) : = 1 f ( x , μ x , σ x ) ,
(1)
then represents the probability of the complementary categorization. If the occurrence of a percept depends on two stimulus parameters x and y, then the probability of its occurrence is given by the product of the corresponding Gaussians f and g. In the following, we will reserve the letters f, g, h, u, and v for cumulative Gaussians associated with specific stimulus parameters. For brevity, we sometimes write just f( x) instead of f( x, μ x, σ x). The subjects' ratings on the scale from 0 to 5 will be treated as estimates of probabilities. The necessary scaling with the factor 5 will of course always be performed after multiplication of the relevant cumulative Gaussians. We now analyze the data from the different perceptual tasks in more detail and begin with the non-motion percepts. 
Figure 3
 
Model fit to the data in Figure 2.
Figure 3
 
Model fit to the data in Figure 2.
Figure 4
 
Overview of the hierarchical categorization of the input into different percepts as suggested by our data analysis. The colors correspond to those used in Figures 2 and 3. The letters in the brackets denote the cumulative Gaussians used in our model.
Figure 4
 
Overview of the hierarchical categorization of the input into different percepts as suggested by our data analysis. The colors correspond to those used in Figures 2 and 3. The letters in the brackets denote the cumulative Gaussians used in our model.
Non-motion percepts
Figure 5 shows the individual mean ratings from the flicker task. The most obvious systematic effect is the rapid decrease in flicker with increasing SOA. The systematic decrease with ISI is particular to a single subject, but the distinct peak around ISI = 0 at the SOA level of 94 ms is common to all subjects. Ignoring these peculiarities, the data curves are essentially flat, so the data may be reasonably well summarized in terms of SOA only. Figure 6 shows the individual means from the flicker task pooled across all ISI levels and plotted against SOA. The lines represent best-fitting complementary cumulative Gaussian ( f c in Equation 1, with x = SOA) according to a least squares criterion. The means and standard deviations of the best-fitting complementary cumulative Gaussian are listed in Table 2, based on the individual data sets and the pooled data set. The means μ s of the best-fitting individual functions ranged between 77 and 86 ms for three of the observers and was 122 ms for observer JH. The latter value may possibly be regarded as an outlier because it was the data of this subject that most clearly deviated from constancy across ISI. 
Table 2
 
Means μ s and standard deviations σ s of the complementary cumulative Gaussian f c fitted to the flicker data (see Figure 6) for each subject individually and for the pooled data.
Table 2
 
Means μ s and standard deviations σ s of the complementary cumulative Gaussian f c fitted to the flicker data (see Figure 6) for each subject individually and for the pooled data.
Subject μ s (ms) σ s (ms)
VE 83 11
EG 86 8
JH 122 45
WM 77 15
Pooled 88 19
Figure 5
 
Individual mean ratings in the flicker task.
Figure 5
 
Individual mean ratings in the flicker task.
Figure 6
 
Individual mean ratings in the flicker task, pooled across all ISI levels and plotted against SOA. Best-fitting complementary cumulative Gaussians f c, scaled with a factor of 5 (maximal possible rating) are also shown.
Figure 6
 
Individual mean ratings in the flicker task, pooled across all ISI levels and plotted against SOA. Best-fitting complementary cumulative Gaussians f c, scaled with a factor of 5 (maximal possible rating) are also shown.
As a natural consequence of the predominance of flicker at low SOAs, essentially no motion is reported at the two smallest SOA levels in Figure 2, be it shadow, part or optimal motion. A clear picture of how these three motion impressions are related to each other can thus only be gained for SOA levels greater than that. Referring back to Figure 4, the first categorization from the left (flicker vs no flicker) is modeled by the complementary cumulative Gaussian f c (and its complementary f). 
The data from the second non-motion task, namely the disappearance–appearance task, are shown in red in the plots of Figure 2 (remember that data on this percept were only collected for SOA levels of 282 ms or greater). It is particularly evident at the longest SOA levels that disappearance and appearance is perceived when the interstimulus interval is longer than those leading to shadow motion and shorter than those leading to optimal motion. The transition point between disappearance–appearance and optimal motion appears to be fairly constant across SOA levels and located somewhat below an ISI of zero, i.e., somewhat above a temporal duty cycle of 0.5, but the transition point between disappearance–appearance and shadow motion is not constant across SOA levels. Instead, it drifts toward more negative ISIs (or, equivalently, toward temporal duty cycles larger than 0.5) as SOA increases. The critical variable for this transition thus appears to be neither ISI nor temporal duty cycle, but the sum of SOA and ISI. This sum, which we shall refer to as the local interstimulus interval (LII), is simply the time interval between stimulus element disappearance and reappearance at the same location, as is illustrated in Figure 1. In order to make this more readily apparent, the data from the appearance–disappearance task are replotted as a function of LII in Figure 7. The thick gray line fitted to the data is the product of two cumulative Gaussians g and h, which are functions of LII and δ, respectively, multiplied by the maximal possible rating (5). As is evident in the plots, g is identical across the SOA levels. The second cumulative Gaussian h refers to the local temporal duty cycle δ, which is related to LII by the equation δ = 1 − LII / (2 SOA). The scale of temporal duty cycles corresponding to the scale of LIIs is shown at the top of each plot. With reference to this upper scale, it can be seen that h is also identical across the plots. Thus, the curve fitted to the data in all panels of Figure 7 corresponds to a single function of δ and LII. Although this function has just four free parameters ( μ and σ for g and h), a rather good fit to the entire data set was obtained, as is evident in Figure 7. The parameters of the fit are given in Table 3 for the data pooled across all subjects and for each of the subjects individually. It can be seen that there are some considerable differences between observers, but the data from three of four observers yield estimates of μ l of about 200 ms and estimates of μ δ of about 0.6. The basic finding is thus that the tendency to perceive appearance and disappearance depends on two conditions: The local interstimulus interval must exceed a certain critical value, and so must the local temporal duty cycle. Referring back to Figure 4, the Gaussian h( δ) models the categorization between “stimulus motion” and “no stimulus motion” while g(LII) models the categorization between shadow motion and the disappearance–appearance percepts. Following the scheme in Figure 4, the Gaussian f should also have been used to model the data, i.e., we should have used the function f × g × h instead of just g × h. However, at the high SOA levels used in the present task, f can be expected to be effectively equal to 1 based on the flicker data, so that it is irrelevant whether it is multiplied in or not. 
Table 3
 
Parameters of the product function g × h yielding the best fit to the data from the disappearance–appearance task. Parameters are given for the individual data as well as the pooled data (see Figure 7).
Table 3
 
Parameters of the product function g × h yielding the best fit to the data from the disappearance–appearance task. Parameters are given for the individual data as well as the pooled data (see Figure 7).
Subject μ l (ms) σ l (ms) μ δ σ δ
VE 301 157 0.54 0.004
EG −7 284 0.59 0.044
JH 201 74 0.52 0.008
WM 182 81 0.62 0.058
Pooled 195 146 0.57 0.050
Figure 7
 
Pooled data for the disappearance–appearance task. Each panel shows the data for one SOA level plotted against the local interstimulus interval LII (lower horizontal axis). The local temporal duty cycle δ corresponding to the values of the local interstimulus interval LII is given on the upper horizontal axis. The thick gray line is the product g × h of two cumulative Gaussians g and h, the former in terms of LII (dashed blue lines) and the latter in terms of temporal duty cycle δ (solid red lines). Note that multiplication of the cumulative Gaussians is performed before scaling to the maximal possible rating. Each data point is based on 12 observations (4 subjects × 3 repetitions). Error bars represent one SEM in each direction.
Figure 7
 
Pooled data for the disappearance–appearance task. Each panel shows the data for one SOA level plotted against the local interstimulus interval LII (lower horizontal axis). The local temporal duty cycle δ corresponding to the values of the local interstimulus interval LII is given on the upper horizontal axis. The thick gray line is the product g × h of two cumulative Gaussians g and h, the former in terms of LII (dashed blue lines) and the latter in terms of temporal duty cycle δ (solid red lines). Note that multiplication of the cumulative Gaussians is performed before scaling to the maximal possible rating. Each data point is based on 12 observations (4 subjects × 3 repetitions). Error bars represent one SEM in each direction.
Shadow motion
Having considered the conditions for non-motion percepts, we now turn to the data from the shadow motion task. For clarity, they are replotted in Figure 8. As before, the lower horizontal axis (ISI values) ranges from −SOA to SOA in all plots, which corresponds to a range of local temporal duty cycles from 1 to 0 (upper horizontal axis). If we restrict attention to an intermediate range of SOA levels (say, from 118 to 212 ms), the data curves of all plots look rather similar. Therefore, within this limited range of SOA values, the tendency to perceive shadow motion appears to be a fixed function of the local temporal duty cycle δ. More specifically, the relation between δ and the tendency to perceive shadow motion is very nearly a cumulative Gaussian in terms of δ (note that δ increases from right to left, so that the cumulative Gaussian decreases from left to right in the plots). At briefer and longer SOA levels, however, systematic deviations from this simple scheme are readily apparent. The data curves flatten at the lower SOA levels (in the SOA range below about 118 ms) and shift leftward at the longer SOA levels (in the range above about 212 ms). 
Figure 8
 
Mean ratings from the shadow motion task, pooled across subjects. Each data point corresponds to 12 measurements (4 subjects × 3 repetitions). Error bars represent one SEM in each direction.
Figure 8
 
Mean ratings from the shadow motion task, pooled across subjects. Each data point corresponds to 12 measurements (4 subjects × 3 repetitions). Error bars represent one SEM in each direction.
The flattening of the data curves is presumably due to the increasing dominance of the flicker percept at the lowest SOA levels. The green horizontal lines in the plots represent an estimate of how much room there is for other percepts when the tendency to perceive flicker is subtracted. Specifically, these lines were computed as the cumulative Gaussian f complementary to the function f c fitted to the flicker data (parameters from Table 2). These lines differ significantly from the maximal rating (5) only at the very briefest SOA levels (47, 71, and 94 ms), where their reduced height correspond rather well to the vertical flattening of the data curves. 
The leftward shift of the data curves at the highest SOA levels is presumably related to a kind of competition with the disappearance–appearance percept (see Figure 2). The function g c plotted in Figure 8 is the complementary of the cumulative Gaussian g used in the fit of the product g × h to the disappearance–appearance data (parameters from Table 3). This curve describes the data well at the longest SOA levels, but at SOA levels below about 329 ms the data fall systematically below it. Also, the specific form of this deviation is rather similar across the SOA levels in which it is discernible. Thus, in order to describe the shadow motion data it seems necessary to take further factors into account. A good fit of the data can be obtained by including the temporal duty cycle δ of the stimuli as a third relevant factor. We did this by using the function h( δ), which was already used above in the fit of the disappearance–appearance data. Since δ is 1 at the left end and 0 at the right end of each plot, the shape of this function is the same in all plots. As is evident in Figure 8, the product f × g c × h of these three functions yields an excellent fit to the data (note that multiplication of the cumulative Gaussians is performed before multiplication with 5 to accommodate the range of possible ratings). 
Fitting the product of these three cumulative Gaussians to the data involves estimating 6 parameters (mean μ and standard deviation σ for each Gaussian). The parameters of f and g c were estimated in two different ways, yielding similar results. On the one hand, they were estimated based on the flicker and disappearance–appearance data, respectively, as described above, and on the other hand they were estimated based on the shadow motion data. The parameter estimates obtained by fitting f × g c × h to the shadow motion data are given in Table 4. The function f × g c × h drawn in Figure 8 uses the parameters for f estimated from the flicker data, the parameters of g estimated from the disappearance–appearance data, and the parameters of h estimated from the shadow data. Virtually the same curve (not shown) was obtained by replacing the parameter estimates from the other tasks by the corresponding ones based on the shadow data. 
Table 4
 
Parameters of the product function f × g c × h yielding the best fit to the shadow motion data.
Table 4
 
Parameters of the product function f × g c × h yielding the best fit to the shadow motion data.
Subject μ s (ms) σ s (ms) μ l (ms) σ l (ms) μ δ σ δ
VE 85 6 165 61 0.63 0.00
EG 78 19 22 80 0.63 0.05
JH 94 0 347 159 0.58 0.08
WM 71 0 122 27 0.47 0.21
Pooled 81 11 154 142 0.59 0.07
In the left panel of Figure 9, the parameter estimates of μ s based on the flicker data are plotted against those based on the shadow motion data. It can be seen that the data from subjects EG, WM, and VE yield similar estimates (about 80 ms) in both tasks (note the restricted scale) while the data from subject JH yield higher estimates, particularly in the flicker task (as already mentioned). In the middle panel of Figure 9, the parameter estimates μ l based on the appearance–disappearance task are plotted against those based on the shadow motion task. The low estimates of μ l resulting from the data of subject EG are clearly not particular to the task. The right panel of Figure 9 shows the estimates of μ δ, which are fairly consistent across subjects. For all subjects except WM, the estimates are above the value of 0.5 that corresponds to a zero interstimulus interval. 
Figure 9
 
Left: Estimates of μ s based on the shadow motion task plotted against the corresponding ones based on the flicker task. Middle: Estimates of μ l based on the shadow motion task plotted against the corresponding ones based on the disappearance–appearance task. Right: Individual estimates of μ δ based on the shadow motion task.
Figure 9
 
Left: Estimates of μ s based on the shadow motion task plotted against the corresponding ones based on the flicker task. Middle: Estimates of μ l based on the shadow motion task plotted against the corresponding ones based on the disappearance–appearance task. Right: Individual estimates of μ δ based on the shadow motion task.
Referring back to the overview in Figure 4, we see that the shadow motion and the disappearance–appearance percept may be grouped together in the category that we have called “no stimulus motion” because their occurrence seem to depend on two common conditions: SOA must be larger than the critical value μ s, and δ must be larger than the critical value μ δ. The distinction between shadow motion and the disappearance–appearance percept seems to depend on LII: If this parameter exceeds a critical value μ l, disappearance–appearance is perceived, if it is less, shadow motion is perceived. 
Stimulus motion
The data on part motion and optimal motion were fairly difficult to analyze in isolation, particularly because the distinction between these two percepts was subject to substantial individual differences. We begin our analysis of these data by first considering the sum of the part motion and optimal motion ratings because this made it easier to discern general patterns in the findings. A certain theoretical rationale for conflating the data in this way may also be given by noting that part motion and optimal motion together constitute a natural complement of shadow motion within the set of all motion percepts: In contrast to the shadow motion percept, in which the stimulus elements are perceived to be stationary, both part motion and optimal motion involve impressions of the stimulus elements moving. For that reason, we shall use stimulus motion as a generic term subsuming both percepts (see the overview in Figure 4). 
The mean of the stimulus motion data pooled across observers is shown in Figure 10. The “raw” stimulus motion data were obtained by adding the part motion ratings and the optimal motion ratings trial-wise (the stimuli were presented in the same sequence across tasks). The thick gray line may be thought of as a prediction based on the flicker data and the shadow motion data. It is the product of the Gaussians f and h c. The parameters of the former are those estimated from the flicker data ( Table 2), and the parameters of the latter are those estimated from the shadow motion data ( Table 4). 
Figure 10
 
Mean pooled stimulus motion ratings. The gray line is a “prediction” based on the flicker data and the shadow motion data (see text). The red line is f × h c × v c with parameters estimated directly from the data. Each data point corresponds to 12 “virtual” measurements (4 subjects × 3 repetitions), whereby each virtual measurement is the mean of two measurements (optimal motion rating and part motion rating). Error bars represent one SD in each direction.
Figure 10
 
Mean pooled stimulus motion ratings. The gray line is a “prediction” based on the flicker data and the shadow motion data (see text). The red line is f × h c × v c with parameters estimated directly from the data. Each data point corresponds to 12 “virtual” measurements (4 subjects × 3 repetitions), whereby each virtual measurement is the mean of two measurements (optimal motion rating and part motion rating). Error bars represent one SD in each direction.
While this prediction describes the data fairly well at the lower SOA levels, a systematic deviation becomes apparent at the higher SOA levels: The prediction increases monotonically with ISI, but the data curves descend again after the initial increase in the ratings. Referring back to Figure 2, it can be seen that this fall-off occurs in regions of the plots (highest SOA levels, positive ISIs) where basically only optimal motion is reported, but not with the maximal rating. A possible reason for this sub-additivity may be that some other percept not recorded in our study competes with optimal motion in these cases. Informal observations do however not suggest that this is the case. Instead, the impression of optimal motion remains throughout, but the motion is very slow and jerky at the longest ISI levels. Considering that the pause between the two stimulus elements (ISI) is almost one second at the greatest ISI level (the greatest ISI level at each SOA level increases with SOA), it is clear that the motion percept should be “slow” in this parameter range. Thus, this “slowness” of the motion impression may explain the low ratings. In the fit of the data, we described this fall-off as the complementary v c of a cumulative Gaussian v in terms of ISI. Accordingly, the product function fitted to the data was f × h c × v c. The best fit is shown as a red line in Figure 10. The parameters yielding the best fit are given in Table 5. The interobserver variability of the parameters μ i and σ i for v c reflects the fact that the size of the fall-off varied very much across observers. It was virtually absent in the data of subject VE and much steeper in the data of WM. 
Table 5
 
Parameters of the product function f × h c × v c yielding the best fit to the stimulus motion data (see Figure 10).
Table 5
 
Parameters of the product function f × h c × v c yielding the best fit to the stimulus motion data (see Figure 10).
Subject μ s (ms) μ s (ms) μ δ σ δ μ i (ms) σ i (ms)
VE 71 0 0.58 0.04 1559 1073
EG 9 60 0.68 0.20 381 427
JH 94 0 0.53 0.02 796 272
WM 91 27 0.60 0.05 202 85
Pooled 78 11 0.57 0.05 533 333
Part motion
Figure 11 shows the individual mean ratings in the part motion task. As is readily apparent, there is a substantial interobserver variability. A general feature of the data is that part motion is much more frequent at the lower SOA levels (except at the two lowest SOA levels, where we have already found flicker to dominate almost exclusively). At the higher SOA levels, part motion is generally restricted to a narrow range of ISI values around or somewhat below an ISI of zero, but at the lower ones it extends further into the domain of positive ISIs. The latter effect is much more prominent in the data of observers VE and JH than in the data of the other two observers, though. 
Figure 11
 
Individual mean ratings from the part motion task.
Figure 11
 
Individual mean ratings from the part motion task.
Figure 12 shows the mean data pooled across observers. The modeling of the data was guided by the idea that visual persistence may be expected to influence the categorization of stimulus motion percepts as either part motion or optimal motion. Visual persistence is the phenomenon that visual stimuli are perceived as lasting much longer than their objective duration (Coltheart, 1980). In the present motion stimuli, visual persistence could lead to the experience that both stimulus elements are simultaneously present, even when this is not objectively the case. This is illustrated in Figure 13. All three space–time diagrams show the same apparent motion sequence with a positive interstimulus interval. The gray regions in the space–time diagrams are schematic representations of the subjective temporal elongation of the stimuli. From left to right, the duration of the “visual persistence trace” increases, being exactly equal to the interstimulus interval in the middle space–time diagram. In the right diagram, where the visual persistence trace is longer than the interstimulus interval, the visual persistence trace of the first stimulus is still visible as the second stimulus appears. Thus, for a brief fraction of the stimulus cycle, the two stimuli should be perceived as simultaneously present. This simultaneous presence would presumably justify a classification of the motion percept as part motion instead of optimal motion since the critical difference between part motion and optimal motion is that two stimuli are perceived to move instead of just one. Accordingly, we assume that stimulus motion is categorized as part motion if the visual persistence trace is longer than the interstimulus interval and as optimal motion if not. The duration of the visual persistence trace is known to be inversely related to the duration of the stimulus in a roughly linear fashion for stimuli of comparatively brief durations, and constant, in some studies even absent at longer stimulus durations (Coltheart, 1980). Based on this reasoning, one would expect the critical ISI at which part motion turns into optimal motion to be a function of the stimulus duration. We modelled the transition from part motion to optimal motion by searching in a family of cumulative Gaussians u(ISI). All members of this family were identical except that their mean μ was taken to be a function of the stimulus duration D. Specifically, we assumed that μ = a × D + b. Accordingly, the function uc used to describe the transition from part motion to optimal motion has the three free parameters a, b, and σ. Since part motion is a subset of stimulus motion, we reused the parameters of the function f × hc × vc already fitted to the stimulus motion data. Thus, the only free parameters in the fit of the product function f × hc × uc × vc to the data were those of uc. The best fit is shown as a solid red line in Figure 12. For comparison, the thick gray line shows the fit to the stimulus motion data from Figure 10. The parameters of the fit for the individual and the pooled data are given in Table 6. There are considerable differences between observers, but all estimates of the slope a are negative, in accordance with the inverse-duration law of visual persistence (Coltheart, 1980). 
Table 6
 
Parameters of the cumulative Gaussian u estimated based on the part motion data (left) and the optimal motion data (right) (see Figures 12 and 14). The mean of this Gaussian is a × D + b, where D is the duration of the stimulus elements.
Table 6
 
Parameters of the cumulative Gaussian u estimated based on the part motion data (left) and the optimal motion data (right) (see Figures 12 and 14). The mean of this Gaussian is a × D + b, where D is the duration of the stimulus elements.
Subject a b (ms) σ i (ms) a b (ms) σ i (ms)
VE −0.55 228 43 −0.63 272 72
EG −0.40 71 1 −0.06 −4 9
JH −1.90 381 134 −0.75 91 41
WM −0.12 39 255 −0.10 28 30
Pooled −0.53 146 104 −0.47 56 298
Figure 12
 
Mean pooled ratings from the part motion task. Error bars represent one SD in each direction.
Figure 12
 
Mean pooled ratings from the part motion task. Error bars represent one SD in each direction.
Figure 13
 
Illustration of how visual persistence may influence the distinction between part motion and optimal motion. The three space–time diagrams represent the same two-element apparent motion sequence with a fixed positive interstimulus interval. Only the duration of the visual persistence trace (gray) varies. When the duration of visual persistence exceeds the interstimulus interval, the visual persistence trace of the first stimulus element is still visible when the second appears, so that the two stimulus elements should be perceived as simultaneously present for some fraction of the animation cycle.
Figure 13
 
Illustration of how visual persistence may influence the distinction between part motion and optimal motion. The three space–time diagrams represent the same two-element apparent motion sequence with a fixed positive interstimulus interval. Only the duration of the visual persistence trace (gray) varies. When the duration of visual persistence exceeds the interstimulus interval, the visual persistence trace of the first stimulus element is still visible when the second appears, so that the two stimulus elements should be perceived as simultaneously present for some fraction of the animation cycle.
Optimal motion
The pooled mean data on optimal motion are shown in Figure 14. The optimal motion data were fitted in the same way as the part motion data, except that the function u c was replaced by its complementary, the cumulative Gaussian u. The estimated parameters of u are also shown in Table 6. As with the part motion data, the individual estimates are rather variable, but the slope a is generally negative. Ideally, the parameter estimates from the part motion data and the optimal data should be equal. The deviations from this ideal expectation may be related to super-additivities, which were particularly large for observer JH. 
Figure 14
 
Mean pooled ratings from the optimal motion task. The error bars represent one SD in each direction.
Figure 14
 
Mean pooled ratings from the optimal motion task. The error bars represent one SD in each direction.
Complementary relationships between the different percepts
The data on the different percepts can be fairly well summarized in terms of the hierarchical classification scheme illustrated in Figure 4. The first classification into flicker vs. non-flicker depends on SOA, and the transition from flicker to non-flicker is modelled by the cumulative Gaussian f (and its complementary with the same parameters). Estimates of the corresponding transition point μ s for SOA were obtained from three different data sets (flicker, shadow motion, and stimulus motion). Panel a in Figure 15 compares the estimates of the corresponding transition point μ s for SOA for three different data sets (flicker, shadow motion, and stimulus motion). Each bar shows the mean of the four individual estimates, the open symbols represent the median and the error bars are one SEM in each direction. The p-value refers to a one-way ANOVA. The second classification into “stimulus motion” and “no stimulus motion” depends on the local temporal duty cycle δ and is modelled by the cumulative Gaussian h. Panel b in Figure 15 shows an analogous comparison of the different estimates of the transition point μ δ. The sub-classification of “no stimulus motion” into shadow motion and disappearance–appearance depends on the local interstimulus interval LII and is modelled by the cumulative Gaussian g. Panel c shows a comparison of the estimates of the transition point μ l. In this case, the p-value refers to a two-samples t-test (two-tailed). The sub-classification of stimulus motion into part motion and optimal motion depends on ISI and the duration D of the stimulus elements. We assumed that the transition point in terms of ISI depends on the stimulus duration according to the equation μ = a × D + b. Panel d compares the different estimates of the slope a, while panel e compares those of the intercept b. Again, the p-values refer to a two-sample t-test (two-tailed). The final classification in our model ( Figure 4) is the distinction between “convincing” and “unconvincing” impressions of optimal motion, which is modelled by the complementary cumulative Gaussian v c, and was introduced to describe the sub-additivities at the larger ISI levels of the largest SOA levels. The parameters were only estimated based on the stimulus motion data and vary strongly across observers ( Table 5). 
Figure 15
 
Comparison of the parameter estimates based on data from different tasks. Each bar represents the mean of the four individual estimates, the open symbol the median, and the error bars show one SEM in each direction.
Figure 15
 
Comparison of the parameter estimates based on data from different tasks. Each bar represents the mean of the four individual estimates, the open symbol the median, and the error bars show one SEM in each direction.
Figure 3 shows a reconstruction of the pooled data in Figure 2 based on this hierarchical model. The model parameters used in the construction of Figure 3 are those estimated based on the pooled data. Whenever parameter estimates were available based on more than one task, the average parameter value was used. There are two sub-additivities in the reconstruction. The fall-off of the top of the stacked ratings at the larger ISI levels of the largest SOA levels is due to the complementary cumulative Gaussian v c. The other sub-additivity at temporal duty cycles about 0.75 at the SOA levels of 235 ms or less is not a property of the model. Since data on the disappearance–appearance percept were not collected at these SOA levels and are thus absent in the data plot ( Figure 2), we also left the corresponding estimated tendencies out in the reconstruction ( Figure 3). This sub-additivity should therefore be regarded as a mere consequence of insufficient data. The proposed model is reasonably simple and provides a rather good summary of the data even though they span a considerable parameter range (SOAs from 47 to 941 ms). One systematic failure to describe the data at the highest SOA levels should be noted, however. As can be gleaned from Figures 7 and 10, the transition related to the cumulative Gaussian h (or h c) in terms of the temporal duty cycle seems to occur at slightly lower duty cycles than predicted by the model fit. Also, the transition in the data seems to be steeper than the model fit. 
The special case of ISI = 0
The subset of our data for which the interstimulus interval is zero is amenable to comparison with the findings of Tyler (1973). Figure 16 shows both data sets plotted against SOA. Tyler's data were read by eye from his Fig. 1, and the frequency values at the horizontal axis of his plots were converted to SOA according to the relation F = 1/P, where F is the frequency, P is the period, and P = 2 × SOA. Tyler viewed the stimulus for a longer period of time, pressing a button whenever he experienced the target percept, and used the fraction of the total presentation time during which the target percept was experienced as a measure of the tendency to perceive it. Our ratings on a scale from 0 to 5 were multiplied by a factor of 20 in order to make them comparable to his percentage vales. 
Figure 16
 
Comparison of the present data with those of Tyler (1973). Our rating data in the range from 0 to 5 have been multiplied by 20 for comparison with Tyler's percentage values.
Figure 16
 
Comparison of the present data with those of Tyler (1973). Our rating data in the range from 0 to 5 have been multiplied by 20 for comparison with Tyler's percentage values.
The solid lines represent the same model fit as the one used in the construction of Figure 3, with the same parameter estimates. Accordingly, the parameter estimates are consistent across all percepts and based on our entire data set, not just the subset of our data shown in Figure 16
As is evident in the two left panels, there is a good quantitative agreement between our and Tyler's data sets for the optimal motion task and the non-motion task. A difference between this study and that of Tyler is that he used a single category for non-motion, while we made a distinction between flicker and a percept of appearance and disappearance. At ISI = 0, the latter did however not occur, so it is likely that Tyler's non-motion percepts correspond directly to the flicker percepts in the present investigation. 
In the upper right panel, it can be seen that our data on shadow motion clearly differ from those of Tyler in an absolute sense. The general shapes of the data curves seem to be rather similar, though. This similarity can be even more clearly seen by regarding our model fit (solid curve). Scaling this curve with a factor of 10 along the vertical axis yields the dashed line, which describes Tyler's shadow motion data rather well. Approximately the converse can be said for the data on part motion (lower right panel). Again, the general shapes of the data curves are similar, but the absolute tendency to perceive part motion is much higher in the present data. Possible reasons for these deviations will be discussed below. 
Discussion
Although the total data set shown in Figure 2 may appear rather complex at first glance, it turns out that it can be well described by the fairly simple model visualized in Figure 4. Apart from the flicker percept, which is only subject to the constraint that the SOA must be less than a critical value μ s, the occurrence of the other percepts requires that several temporal constraints are simultaneously met. For shadow motion to occur, for instance, SOA must exceed a critical value μ s, the temporal duty cycle must exceed a critical value μ δ, and the LII must be less than a critical value μ l. This set of constraints corresponds to the lower branch of the “decision tree” in Figure 4. In an analogous way, the constraints for the other percepts can be read from the decision tree by following the path from “start” to the corresponding label. The decision tree visualizes the extent to which different percepts share common constraints. “Shadow motion” and “disappearance–apperance,” for instance, are subject to the same constraints in terms of SOA and temporal duty cycle. 
Each decision in the decision tree is assumed to be made probabilistically (and independently). Accordingly, the critical values μ mentioned in the diagram are to be understood as the means of cumulative Gaussians assumed to describe the probabilistic transition between different categories. The additional parameter σ required to describe the smoothness of the transition is not shown in the diagram. 
The critical values μ estimated based on our data are given in Tables 2 3 4 56. Since critical temporal values in apparent motion are known to depend on other variables such as spatial separation (Neuhaus, 1930; Zeeman & Roelofs, 1953), they are presumably of limited general value. Of greater theoretical interest are the suggested links between perceptual categories and specific temporal variables as well as the relations between different percepts. 
Theoretical significance of the different motion percepts
Treatment of apparent motion in two-element displays is often explicitly avoided in models of elementary motion detectors of the Reichardt type because it is not obvious how they can be applied to this situation (Adelson & Bergen, 1985; Sperling, van Santen, & Burt, 1985). We shall not deal with this difficult issue here because we are not primarily interested in the mechanism underlying the motion signal but rather in the question how temporal relations in the input signal determine to which objects the motion signals—whatever their origin—are attached: It is this assignment of motion signals to different perceptual objects that underlies the categorization into “shadow,” “part,” and “optimal” motion. 
The phenomenal characteristics of the three motion categories are as follows: In “optimal motion,” the motion signal is assigned to a single moving object corresponding to the two stimulus elements. In “shadow motion,” the motion signal is not assigned to the stimulus elements—which are perceived as stationary—but to a different perceptual object of unclear status moving in front of them. In “part motion,” a motion signal is assigned to both stimulus elements that are perceived as distinct entities. 
Figure 17 illustrates that the three motion categories observed in discrete apparent motion stimuli can be related to corresponding situations involving objects in real continuous motion. The analogy is particularly evident by considering multiple-element displays, but the general ideas are easily translated back to the case of two-element displays: Space–time diagram A1 represents a real object moving with constant speed from left to right, while diagram B1 represents a corresponding discrete motion stimulus, which typically evokes the impression of optimal motion. A space–time diagram as in A2 would arise if an object of the same color as the background moves exactly as in A1 but in front of four stationary objects. Space–time diagram B2 shows the corresponding discrete case, which typically evokes the impression of “shadow motion” (Steinman et al., 2000). The comparison of A2 with B2 suggests that shadow motion is related to the motion of an occluder in front of stationary objects, and the comparison of B1 and B2 suggests that the temporal duty cycle, which is small (1/4) in B1 and large (3/4) in B2, may serve to distinguish between optimal motion and shadow motion. A possible interpretation of “part motion” is shown in panels A3 and B3. As in panel A1, a single object moves with constant speed from left to right, but in this case, it is assumed that the object moves fast enough to produce a significant amount of “motion smear” due to visual persistence (Burr, 1980; Coltheart, 1980). In panel A3, the “motion smear” is schematically represented by the dark gray area. Assuming the same amount of visual persistence in a corresponding discrete stimulus leads to the situation illustrated in panel B3. Note that the phenomenal consequences of visual persistence are quite different in the continuous and the discrete case (consider for instance the moment in time indicated by the red lines). In the continuous case, the visual persistence trace is spatially contiguous with the moving object and thus makes it appear elongated in space. In the discrete case, on the other hand, the visual persistence trace is spatially separate from the currently presented stimulus element and leads to the perception of multiple objects being simultaneously present. 
Figure 17
 
Space–time diagrams illustrating how the three motion categories observed in discrete apparent motion stimuli (B1: “optimal motion”; B2: “shadow motion”; B3: “part motion”) can be related to corresponding situations involving objects in real continuous motion (A1–A3). Each space–time diagram shows two cycles of linear motion from left to right. In A1, the black region corresponds to the moving object. In A2, a similar object of the same color as the light gray surround moves in the exact same way in front of four stationary black objects. A3 depicts the same situation as A1, but the influence of visual persistence is taken into account. The height of the dark gray region depicts the duration of visual persistence (see text for further details).
Figure 17
 
Space–time diagrams illustrating how the three motion categories observed in discrete apparent motion stimuli (B1: “optimal motion”; B2: “shadow motion”; B3: “part motion”) can be related to corresponding situations involving objects in real continuous motion (A1–A3). Each space–time diagram shows two cycles of linear motion from left to right. In A1, the black region corresponds to the moving object. In A2, a similar object of the same color as the light gray surround moves in the exact same way in front of four stationary black objects. A3 depicts the same situation as A1, but the influence of visual persistence is taken into account. The height of the dark gray region depicts the duration of visual persistence (see text for further details).
With these analogies between the phenomena of continuous and discrete motion illustrated in Figure 17 in mind, we will now explore the possible theoretical meaning of the temporal parameters in more detail. 
Shadow motion
The obvious similarity between the discrete apparent motion sequence (panel B2) typically evoking the impression of “shadow motion” and the continuous movement of an occluder moving in front of stationary objects (panel A2) suggests that the enigmatic pure objectless motion is not really objectless but that the motion signal is assigned to a virtual occluding object constructed by the visual system. This idea is well captured by Petersik and McDill's (1981, p. 564) characterization of the phenomenon as a “sort of spatiotemporal Kanizsa figure.” 
The “shadow motion” percept has two important aspects that need to be accounted for: First, the movement of the “shadow,” and second, the stationarity of the stimulus elements. The comparison of panel B1 and B2 suggests that the motion of the “shadow” can be explained in much the same way as the motion of the stimulus elements in “optimal motion.” This is confirmed by our finding that the minimal necessary SOA is comparable for both kinds of motion. It remains to be considered which temporal cue indicates stationarity of the stimulus elements. Stationarity means that a stimulus element is perceptually identified only with the next ipsilateral stimulus presentation (and never with the contralateral stimulus element). An obvious candidate cue for this kind of identification is briefness of the temporal gap between successive ipsilateral stimulus presentations. There are basically two ways of specifying the duration of this time interval. It can be measured in absolute terms or relative to the duration of one stimulus cycle. The local interstimulus interval is an absolute measure, while (1 − δ), the complement of the temporal duty cycle δ, is a relative measure of the gap duration. Our results indicate that both of these variables are important. Basically our results suggest that the absolute gap duration (LII) must not exceed a value of about 150 ms and the relative gap duration (1 − δ) must not exceed 40% of the stimulus period for shadow motion to be perceived. Violations of these two constraints have different consequences: Increasing LII above the critical value leads to the appearance–disapperance percept, while increasing (1 − δ) above the critical ratio evokes one of the stimulus-motion percepts. 
As has been noted previously (Petersik & McDill, 1981; Steinman et al., 2000), the shadow motion percept is in many ways similar to the occlusion-related figure-ground reversal of apparent motion reported by Sigman and Rock (1974). Figure 18 shows simplified versions of their stimuli. Animation sequence a is just a classical apparent motion stimulus which evokes an impression of optimal motion of the black discs (given that the temporal parameters are appropriately chosen). Animation sequence b is identical to sequence a except that some 90° sectors have been cut out from the gray discs. This minor modification has dramatic perceptual consequences, though. Instead of a single moving black disc, one now perceives a white square moving in front of two stationary black discs. The occluder motion evoked by sequence b and shadow motion have many characteristic features in common, which also suggests that the two phenomena may be intimately related: The stimulus elements are perceived as stationary, while something else moves in front of them. Also, the shadow or the occluder, respectively, move in a direction opposite to that of optimal stimulus motion (in the displays considered here). The major difference between the phenomena is that there are no spatial cues to the existence of an occluder in the case of shadow motion. If the occlusion interpretation of shadow motion is actually correct, this would mean that the visual system also exploits purely temporal cues to occlusion. 
Figure 18
 
Left: a two-frame animation sequence (a) leading to classical apparent motion. A single black disc is typically perceived to move. Right: a very similar animation sequence (b) producing a rather different percept. Two black discs appear to be stationary while a white (subjective) square appears to move in front of them. Adapted from Sigman and Rock (1974).
Figure 18
 
Left: a two-frame animation sequence (a) leading to classical apparent motion. A single black disc is typically perceived to move. Right: a very similar animation sequence (b) producing a rather different percept. Two black discs appear to be stationary while a white (subjective) square appears to move in front of them. Adapted from Sigman and Rock (1974).
Sigman and Rock's (1974) analysis highlights the inherent onset–offset ambiguity of classical motion stimuli and shows that motion interpretations often stand in conflict with alternative interpretations and that the visual system needs to evaluate appropriate cues in order to choose between them. This does not necessarily mean that general purpose high-level mechanisms akin to problem solving are involved. Rather, as the present findings suggest, the visual systems may rely on relatively simple temporal criteria. 
Part motion
In our data analysis, we initially conflated the optimal motion and the part motion percepts into the category “stimulus motion” because—as reflected by the large interobserver variability—the subjects found it difficult to clearly distinguish between them. 
The dependency of the critical interstimulus interval at which part motion turns into optimal motion on stimulus duration supports the link between part motion and visual persistence suggested in Figure 17: Just as visual persistence, it seems to obey an inverse-duration law (Coltheart, 1980). The large individual variability with respect to the distinction between part and optimal motion mentioned above may be related to the finding that some measures of visual persistence have been found to be highly variable across subjects (Georgeson & Georgeson, 1985). 
Optimal motion
Not surprisingly, our data show that optimal motion is predominantly perceived at positive interstimulus intervals. It is also not very surprising that the optimal motion ratings declined as the interstimulus interval becomes very large. It is interesting, though, that this decline in the optimal motion ratings is not accompanied by an increase in the disappearance–appearance ratings. Informal inspections of the stimuli suggest that this decline in the ratings is not related to a qualitative change in the percept. Rather, as ISI becomes very large the motion impression assumes a very “viscous” character, which presumably may instill a certain reluctance to give high ratings of optimal motion. The large interobserver variability with respect to this decline suggests that the subjects differed in their inclination to let this viscous quality of the motion influence their ratings of optimal motion. 
An upper limit on optimal motion in terms of SOA is sometimes suggested in the literature (Palmer, 1999), but such a limit is not discernable in our data. Actually, it is difficult to see why such an upper limit should exist. If one stimulus element is on for a long time after which the other is immediately turned on, one would surmise that the sudden change in position should evoke an impression of motion however long the stationary time intervals may be. What can be expected, though, is that periods of rest are interspersed between brief moments of movement. 
Comparison with previous findings for ISI = 0
The present data are in fair agreement with the findings of Tyler (1973), who also used two-element displays. While the data sets on optimal motion and non-motion are in good quantitative agreement across the data sets, the agreement is but qualitative in the case of shadow motion and part motion (see Figure 16). The differences between the two data sets are roughly complementary with respect to shadow motion and part motion, suggesting that the “balance point” between part motion and optimal motion differs in the two studies. Many factors, such as stimulus size, luminance, and distance (Korte, 1915; Zeeman & Roelofs, 1953), may be responsible for this difference. Two possible explanations that we find particularly plausible are as follows: First, as is evident in our data, the tendency to perceive shadow motion in stimuli with an ISI of zero is much lower as compared with negative interstimulus intervals. Thus, the observers in our experiment may have been particularly reluctant to assign very high ratings to the shadow motion percepts at zero interstimulus intervals because they were also shown the much more compelling stimuli with negative ISIs, while this comparison was not possible for Tyler, who viewed only stimuli with a zero ISI. Secondly, Tyler viewed the individual stimuli for a considerably longer time period than our subjects (60 vs. 5 s, respectively). In the course of our experiments, we noted that the shadow motion percept typically needs more time to develop than other alternative percepts. This was also noted by Zeeman and Roelofs (1953). Even in cases where very convincing and stable impressions of shadow motion were perceived, there was often a brief initial period during which part motion was perceived. Thus, the differences in viewing time may be responsible for the differences between the data. 
Caveats
In this study, we investigated a set of five different target percepts which were intended to be exhaustive. Nevertheless, the sums of the mean ratings were not always equal to the maximal possible rating. We have discussed possible reasons for some of these sub-/super-additivities, but the sharp “trough” in the stacked data of Figure 2, which is most prominent at an SOA level of 282 ms, remains to be commented on. The reason for this sub-additivity cannot be the lack of data on disappearance–appearance, as may be the case at the lower SOA levels, because disappearance–appearance data were collected at this SOA level. It is possible that this sub-additivity reflects a sixth percept for which we did not collect data, namely the percept of tunnel motion (see Zeeman & Roelofs, 1953). The reason why we did not collect data on this percept was that it occurred extremely seldom in pilot experiments. 
Conclusions
The present findings show that stable percepts of shadow motion cannot only be obtained in the multiple-element sequences of Steinman et al. (2000) but also in two-element apparent motion sequences, provided that the temporal duty cycle of the stimuli exceed a value of about 0.6. It is suggested that the temporal duty cycle is a temporal variable that the visual system relies on for assigning motion to different perceptual objects. 
Acknowledgments
This research was supported by a grant (EK 72/1-1) from the Deutsche Forschungsgemeinschaft to V. Ekroll. We are indebted to two anonymous reviewers for helpful suggestions. 
Commercial relationships: none. 
Corresponding author: V. Ekroll. 
Email: vekroll@psychologie.uni-kiel.de. 
Address: Olshausenstr. 62, D-24118 Kiel, Germany. 
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Figure 1
 
Space–time diagrams of 5 different apparent motion sequences with a fixed stimulus onset asynchrony (SOA) and interstimulus intervals (ISI) in the range from −SOA to SOA. The middle space–time diagram represents a stimulus with a zero ISI, i.e., stimulus A is turned off as stimulus B is turned on and vice versa. Consequently, the temporal duty cycle at either of the stimulus locations equals 0.5, i.e., the local stimulus is on half of the cycle time. Toward the right, the ISI increases. The rightmost space–time diagram shows the purely notional, limiting case ISI = SOA, where the stimuli would always be absent. Toward the left, the ISI decreases below zero. The leftmost space–time diagram shows the limiting case ISI = −SOA where the stimuli would never be turned off. As is evident by comparing the upper and the lower scale, varying the ISI from −SOA to SOA is equivalent to varying the local temporal duty cycle of the stimulus elements from 1 to 0. D is the duration of the stimulus, the local interstimulus interval (LII) is the time interval for which either stimulus element is blanked.
Figure 1
 
Space–time diagrams of 5 different apparent motion sequences with a fixed stimulus onset asynchrony (SOA) and interstimulus intervals (ISI) in the range from −SOA to SOA. The middle space–time diagram represents a stimulus with a zero ISI, i.e., stimulus A is turned off as stimulus B is turned on and vice versa. Consequently, the temporal duty cycle at either of the stimulus locations equals 0.5, i.e., the local stimulus is on half of the cycle time. Toward the right, the ISI increases. The rightmost space–time diagram shows the purely notional, limiting case ISI = SOA, where the stimuli would always be absent. Toward the left, the ISI decreases below zero. The leftmost space–time diagram shows the limiting case ISI = −SOA where the stimuli would never be turned off. As is evident by comparing the upper and the lower scale, varying the ISI from −SOA to SOA is equivalent to varying the local temporal duty cycle of the stimulus elements from 1 to 0. D is the duration of the stimulus, the local interstimulus interval (LII) is the time interval for which either stimulus element is blanked.
Figure 2
 
Stacked mean ratings pooled across all subjects. The stacking order corresponds to the vertical order of the legend. Each panel shows the mean ratings plotted against the interstimulus interval (ISI) for a fixed level of stimulus onset asynchrony (SOA). Each data point is the mean of 12 single observations (3 repetitions × 4 subjects). The lower horizontal axis ranges from ISI = −SOA to ISI = SOA in all plots. This range corresponds to local temporal duty cycles δ between 1 and 0 (upper horizontal scale, increasing from right to left). Note that data for the flicker task were only collected for SOAs less or equal to 235 ms, while data for the disappearance–appearance task were only collected for the longer SOA levels. The solid line is the model fit to the shadow motion data.
Figure 2
 
Stacked mean ratings pooled across all subjects. The stacking order corresponds to the vertical order of the legend. Each panel shows the mean ratings plotted against the interstimulus interval (ISI) for a fixed level of stimulus onset asynchrony (SOA). Each data point is the mean of 12 single observations (3 repetitions × 4 subjects). The lower horizontal axis ranges from ISI = −SOA to ISI = SOA in all plots. This range corresponds to local temporal duty cycles δ between 1 and 0 (upper horizontal scale, increasing from right to left). Note that data for the flicker task were only collected for SOAs less or equal to 235 ms, while data for the disappearance–appearance task were only collected for the longer SOA levels. The solid line is the model fit to the shadow motion data.
Figure 3
 
Model fit to the data in Figure 2.
Figure 3
 
Model fit to the data in Figure 2.
Figure 4
 
Overview of the hierarchical categorization of the input into different percepts as suggested by our data analysis. The colors correspond to those used in Figures 2 and 3. The letters in the brackets denote the cumulative Gaussians used in our model.
Figure 4
 
Overview of the hierarchical categorization of the input into different percepts as suggested by our data analysis. The colors correspond to those used in Figures 2 and 3. The letters in the brackets denote the cumulative Gaussians used in our model.
Figure 5
 
Individual mean ratings in the flicker task.
Figure 5
 
Individual mean ratings in the flicker task.
Figure 6
 
Individual mean ratings in the flicker task, pooled across all ISI levels and plotted against SOA. Best-fitting complementary cumulative Gaussians f c, scaled with a factor of 5 (maximal possible rating) are also shown.
Figure 6
 
Individual mean ratings in the flicker task, pooled across all ISI levels and plotted against SOA. Best-fitting complementary cumulative Gaussians f c, scaled with a factor of 5 (maximal possible rating) are also shown.
Figure 7
 
Pooled data for the disappearance–appearance task. Each panel shows the data for one SOA level plotted against the local interstimulus interval LII (lower horizontal axis). The local temporal duty cycle δ corresponding to the values of the local interstimulus interval LII is given on the upper horizontal axis. The thick gray line is the product g × h of two cumulative Gaussians g and h, the former in terms of LII (dashed blue lines) and the latter in terms of temporal duty cycle δ (solid red lines). Note that multiplication of the cumulative Gaussians is performed before scaling to the maximal possible rating. Each data point is based on 12 observations (4 subjects × 3 repetitions). Error bars represent one SEM in each direction.
Figure 7
 
Pooled data for the disappearance–appearance task. Each panel shows the data for one SOA level plotted against the local interstimulus interval LII (lower horizontal axis). The local temporal duty cycle δ corresponding to the values of the local interstimulus interval LII is given on the upper horizontal axis. The thick gray line is the product g × h of two cumulative Gaussians g and h, the former in terms of LII (dashed blue lines) and the latter in terms of temporal duty cycle δ (solid red lines). Note that multiplication of the cumulative Gaussians is performed before scaling to the maximal possible rating. Each data point is based on 12 observations (4 subjects × 3 repetitions). Error bars represent one SEM in each direction.
Figure 8
 
Mean ratings from the shadow motion task, pooled across subjects. Each data point corresponds to 12 measurements (4 subjects × 3 repetitions). Error bars represent one SEM in each direction.
Figure 8
 
Mean ratings from the shadow motion task, pooled across subjects. Each data point corresponds to 12 measurements (4 subjects × 3 repetitions). Error bars represent one SEM in each direction.
Figure 9
 
Left: Estimates of μ s based on the shadow motion task plotted against the corresponding ones based on the flicker task. Middle: Estimates of μ l based on the shadow motion task plotted against the corresponding ones based on the disappearance–appearance task. Right: Individual estimates of μ δ based on the shadow motion task.
Figure 9
 
Left: Estimates of μ s based on the shadow motion task plotted against the corresponding ones based on the flicker task. Middle: Estimates of μ l based on the shadow motion task plotted against the corresponding ones based on the disappearance–appearance task. Right: Individual estimates of μ δ based on the shadow motion task.
Figure 10
 
Mean pooled stimulus motion ratings. The gray line is a “prediction” based on the flicker data and the shadow motion data (see text). The red line is f × h c × v c with parameters estimated directly from the data. Each data point corresponds to 12 “virtual” measurements (4 subjects × 3 repetitions), whereby each virtual measurement is the mean of two measurements (optimal motion rating and part motion rating). Error bars represent one SD in each direction.
Figure 10
 
Mean pooled stimulus motion ratings. The gray line is a “prediction” based on the flicker data and the shadow motion data (see text). The red line is f × h c × v c with parameters estimated directly from the data. Each data point corresponds to 12 “virtual” measurements (4 subjects × 3 repetitions), whereby each virtual measurement is the mean of two measurements (optimal motion rating and part motion rating). Error bars represent one SD in each direction.
Figure 11
 
Individual mean ratings from the part motion task.
Figure 11
 
Individual mean ratings from the part motion task.
Figure 12
 
Mean pooled ratings from the part motion task. Error bars represent one SD in each direction.
Figure 12
 
Mean pooled ratings from the part motion task. Error bars represent one SD in each direction.
Figure 13
 
Illustration of how visual persistence may influence the distinction between part motion and optimal motion. The three space–time diagrams represent the same two-element apparent motion sequence with a fixed positive interstimulus interval. Only the duration of the visual persistence trace (gray) varies. When the duration of visual persistence exceeds the interstimulus interval, the visual persistence trace of the first stimulus element is still visible when the second appears, so that the two stimulus elements should be perceived as simultaneously present for some fraction of the animation cycle.
Figure 13
 
Illustration of how visual persistence may influence the distinction between part motion and optimal motion. The three space–time diagrams represent the same two-element apparent motion sequence with a fixed positive interstimulus interval. Only the duration of the visual persistence trace (gray) varies. When the duration of visual persistence exceeds the interstimulus interval, the visual persistence trace of the first stimulus element is still visible when the second appears, so that the two stimulus elements should be perceived as simultaneously present for some fraction of the animation cycle.
Figure 14
 
Mean pooled ratings from the optimal motion task. The error bars represent one SD in each direction.
Figure 14
 
Mean pooled ratings from the optimal motion task. The error bars represent one SD in each direction.
Figure 15
 
Comparison of the parameter estimates based on data from different tasks. Each bar represents the mean of the four individual estimates, the open symbol the median, and the error bars show one SEM in each direction.
Figure 15
 
Comparison of the parameter estimates based on data from different tasks. Each bar represents the mean of the four individual estimates, the open symbol the median, and the error bars show one SEM in each direction.
Figure 16
 
Comparison of the present data with those of Tyler (1973). Our rating data in the range from 0 to 5 have been multiplied by 20 for comparison with Tyler's percentage values.
Figure 16
 
Comparison of the present data with those of Tyler (1973). Our rating data in the range from 0 to 5 have been multiplied by 20 for comparison with Tyler's percentage values.
Figure 17
 
Space–time diagrams illustrating how the three motion categories observed in discrete apparent motion stimuli (B1: “optimal motion”; B2: “shadow motion”; B3: “part motion”) can be related to corresponding situations involving objects in real continuous motion (A1–A3). Each space–time diagram shows two cycles of linear motion from left to right. In A1, the black region corresponds to the moving object. In A2, a similar object of the same color as the light gray surround moves in the exact same way in front of four stationary black objects. A3 depicts the same situation as A1, but the influence of visual persistence is taken into account. The height of the dark gray region depicts the duration of visual persistence (see text for further details).
Figure 17
 
Space–time diagrams illustrating how the three motion categories observed in discrete apparent motion stimuli (B1: “optimal motion”; B2: “shadow motion”; B3: “part motion”) can be related to corresponding situations involving objects in real continuous motion (A1–A3). Each space–time diagram shows two cycles of linear motion from left to right. In A1, the black region corresponds to the moving object. In A2, a similar object of the same color as the light gray surround moves in the exact same way in front of four stationary black objects. A3 depicts the same situation as A1, but the influence of visual persistence is taken into account. The height of the dark gray region depicts the duration of visual persistence (see text for further details).
Figure 18
 
Left: a two-frame animation sequence (a) leading to classical apparent motion. A single black disc is typically perceived to move. Right: a very similar animation sequence (b) producing a rather different percept. Two black discs appear to be stationary while a white (subjective) square appears to move in front of them. Adapted from Sigman and Rock (1974).
Figure 18
 
Left: a two-frame animation sequence (a) leading to classical apparent motion. A single black disc is typically perceived to move. Right: a very similar animation sequence (b) producing a rather different percept. Two black discs appear to be stationary while a white (subjective) square appears to move in front of them. Adapted from Sigman and Rock (1974).
Table 1
 
Definition of symbols used to describe the temporal parameters of our stimuli (see Figure 1).
Table 1
 
Definition of symbols used to describe the temporal parameters of our stimuli (see Figure 1).
Symbol Meaning Relations
D The identical duration of the two stimulus elements A and B.
SOA Stimulus onset asynchrony. In all of our stimuli the SOA was symmetric (i.e., SOA AB = SOA BA).
P Duration of a single animation cycle. P = 2 SOA
ISI Interstimulus interval. ISI = SOA − D
LII Local interstimulus interval, i.e., the time interval between offset and the next onset of a given stimulus element (A or B). In all of our stimuli the LII was symmetric (i.e., LII A = LII B). LII = PD = SOA + ISI
δ Temporal duty cycle of either stimulus element (i.e., the fraction of the animation cycle during which the stimulus element is visible). δ = D / P = D / ( D + LII)
Table 2
 
Means μ s and standard deviations σ s of the complementary cumulative Gaussian f c fitted to the flicker data (see Figure 6) for each subject individually and for the pooled data.
Table 2
 
Means μ s and standard deviations σ s of the complementary cumulative Gaussian f c fitted to the flicker data (see Figure 6) for each subject individually and for the pooled data.
Subject μ s (ms) σ s (ms)
VE 83 11
EG 86 8
JH 122 45
WM 77 15
Pooled 88 19
Table 3
 
Parameters of the product function g × h yielding the best fit to the data from the disappearance–appearance task. Parameters are given for the individual data as well as the pooled data (see Figure 7).
Table 3
 
Parameters of the product function g × h yielding the best fit to the data from the disappearance–appearance task. Parameters are given for the individual data as well as the pooled data (see Figure 7).
Subject μ l (ms) σ l (ms) μ δ σ δ
VE 301 157 0.54 0.004
EG −7 284 0.59 0.044
JH 201 74 0.52 0.008
WM 182 81 0.62 0.058
Pooled 195 146 0.57 0.050
Table 4
 
Parameters of the product function f × g c × h yielding the best fit to the shadow motion data.
Table 4
 
Parameters of the product function f × g c × h yielding the best fit to the shadow motion data.
Subject μ s (ms) σ s (ms) μ l (ms) σ l (ms) μ δ σ δ
VE 85 6 165 61 0.63 0.00
EG 78 19 22 80 0.63 0.05
JH 94 0 347 159 0.58 0.08
WM 71 0 122 27 0.47 0.21
Pooled 81 11 154 142 0.59 0.07
Table 5
 
Parameters of the product function f × h c × v c yielding the best fit to the stimulus motion data (see Figure 10).
Table 5
 
Parameters of the product function f × h c × v c yielding the best fit to the stimulus motion data (see Figure 10).
Subject μ s (ms) μ s (ms) μ δ σ δ μ i (ms) σ i (ms)
VE 71 0 0.58 0.04 1559 1073
EG 9 60 0.68 0.20 381 427
JH 94 0 0.53 0.02 796 272
WM 91 27 0.60 0.05 202 85
Pooled 78 11 0.57 0.05 533 333
Table 6
 
Parameters of the cumulative Gaussian u estimated based on the part motion data (left) and the optimal motion data (right) (see Figures 12 and 14). The mean of this Gaussian is a × D + b, where D is the duration of the stimulus elements.
Table 6
 
Parameters of the cumulative Gaussian u estimated based on the part motion data (left) and the optimal motion data (right) (see Figures 12 and 14). The mean of this Gaussian is a × D + b, where D is the duration of the stimulus elements.
Subject a b (ms) σ i (ms) a b (ms) σ i (ms)
VE −0.55 228 43 −0.63 272 72
EG −0.40 71 1 −0.06 −4 9
JH −1.90 381 134 −0.75 91 41
WM −0.12 39 255 −0.10 28 30
Pooled −0.53 146 104 −0.47 56 298
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