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.

*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.

*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.

*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”).

^{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/m

^{2}). 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).

*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.

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 = P − D = 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) |

*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.

- 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. - 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. - 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. - 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. - 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.

*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.

*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.

*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.

*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*) to represent the probability of a certain categorization given the parameter value

_{x}, σ_{x}*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,

*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*). 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.

_{x}, σ_{x}*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.

Subject | μ _{ s} (ms) | σ _{ s} (ms) |
---|---|---|

VE | 83 | 11 |

EG | 86 | 8 |

JH | 122 | 45 |

WM | 77 | 15 |

Pooled | 88 | 19 |

*f*

^{c}(and its complementary

*f*).

*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.

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 |

*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).

*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.

*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).

*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.

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 |

*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.

*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.

*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).

*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.

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 |

*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

*u*

^{c}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*×

*h*

^{c}×

*v*

^{c}already fitted to the stimulus motion data. Thus, the only free parameters in the fit of the product function

*f*×

*h*

^{c}×

*u*

^{c}×

*v*

^{c}to the data were those of

*u*

^{c}. 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).

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 |

*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.

*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).

*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.

*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.

*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.

*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.

*μ*estimated based on our data are given in Tables 2 3 4 5– 6. 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.

*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.