Visual apparent motion is the experience of motion from the successive stimulation of separate spatial locations. How spatial and temporal distances interact to determine the strength of apparent motion has been controversial. Some studies report space–time coupling: If we increase spatial or temporal distance between successive stimuli, we must also increase the other distance between them to maintain a constant strength of apparent motion (Korte's third law of motion). Other studies report space–time tradeoff: If we increase one of these distances, we must decrease the other to maintain a constant strength of apparent motion. In this article, we resolve the controversy. Starting from a normative theory of motion measurement and data on human spatiotemporal sensitivity, we conjecture that both coupling and tradeoff should occur, but at different speeds. We confirm the prediction in two experiments, using suprathreshold multistable apparent-motion displays called motion lattices. Our results show a smooth transition between the tradeoff and coupling as a function of speed: Tradeoff occurs at low speeds and coupling occurs at high speeds. From our data, we reconstruct the suprathreshold equivalence contours that are analogous to isosensitivity contours obtained at the threshold of visibility.

*o*,

*a*, and

*b*(Figure 1A). Suppose the spatial distance between

*a*and

*b*is very long, and motion from

*a*to

*b*is unlikely. Then, the dot at

*o*has two potential matches:

*a*and

*b*. Because the dot at

*o*has two potential matches, the display is ambiguous; one can perceive motion either from

*o*to

*a*or from

*o*to

*b*.

*shorter*than the other. We call this result “space–time tradeoff”. In this article, we show that by changing the conditions of stimulation, we can cause the pattern of results to change smoothly from space–time coupling to space–time tradeoff.

**and from o to b by**

*m*_{a}**. Let us denote the (potential) percepts of motion along these paths by**

*m*_{b}*μ*

_{a}and

*μ*

_{b}and the strength of the apparent motion experienced when

*μ*

_{a}or

*μ*

_{b}is seen in isolation by

*T*

_{a}and

*T*

_{b}) and a spatial (

*S*

_{a}and

*S*

_{b}) component ( Figure 1A). The two motion paths are in perceptual equilibrium for such combination of conditions (

*S*

_{a},

*T*

_{a}) and (

*S*

_{b},

*T*

_{b}) that the two motions are seen equally often and their strengths are equal:

*S*

_{a},

*T*

_{a}) and the pair (

*S*

_{b},

*T*

_{b}) when the two motions are in equilibrium:

- Tradeoff: Equilibrium is obtained when
*S*_{b}>*S*_{a}and*T*_{b}<*T*_{a}or*S*_{b}<*S*_{a}and*T*_{b}>*T*_{a}. This result was obtained by Burt and Sperling (1981).

*distance plot*( Figure 1B). Suppose we hold the spatial and temporal components of

**constant, so it is represented in the distance plot by a fixed point. Suppose also that we hold the temporal component of**

*m*_{a}**constant, twice as long as the temporal component of**

*m*_{b}**:**

*m*_{a}*T*

_{b}= 2

*T*

_{a}. Then, we can vary the spatial component

*S*

_{b}and find the value of

*r*

_{ba}=

*S*

_{b}/

*S*

_{a}for which

*μ*

_{a}and

*μ*

_{b}are in perceptual equilibrium:

*p*(

*μ*

_{a}) =

*p*(

*μ*

_{b}) = 0.5. This manipulation is represented in the distance plot by moving the point for

**along the line**

*m*_{b}*T*

_{b}= 2

*T*

_{a}( Figure 1B).

*r*

_{ba}> 1), tradeoff (

*r*

_{ba}< 1), and an intermediate condition that we call “time independence” (

*r*

_{ba}= 1). Each of these outcomes corresponds to a different slope of the line connecting the representation of the competing motion paths in the distance plot: positive slope for coupling, negative slope for tradeoff, and zero slope for time independence. We will use the notion of slope between equivalent conditions in the distance plot to relate results on apparent motion to predictions of a normative theory of motion measurement and to data on human spatiotemporal sensitivity. Each of the three regimes has played a role in the literature on motion perception, as we show next.

**and**

*m*_{a}**occurs when**

*m*_{b}**is longer than**

*m*_{b}**both in time (**

*m*_{a}*T*

_{b}>

*T*

_{a}) and in space (

*S*

_{b}>

*S*

_{a}). It is illustrated in the distance plot ( Figure 2) by the positive-slope line between 1 and 4.

*speed invariance*(e.g., Koffka, 1935/1963; Kolers, 1972). This would be the case only if the space–time pairs that correspond to the same speed of motion had also corresponded to the same strength of apparent motion. In our terms, spatial and temporal distances in the conditions of equilibrium would then be related directly:

*S*=

*vT,*where speed

*v*is a positive constant.

*v*denotes physical speed, Korte's data do not provide support for speed invariance. However, if perceived speed is meant, and if perceived speed is related to physical speed nonlinearly, then Korte's data could be consistent with

*perceived speed invariance*. In that case, the physical spatial and temporal distances in the conditions of equilibrium would be related by

*S*=

*η*(

*v*)

*T,*where

*v*is physical speed and

*η*(

*v*) is a nonlinear function.

The law has puzzled psychologists … at the time of Korte's work one was still inclined to think as follows: if one separates the two successively exposed objects more and more, either spatially or temporally, one makes their unification more and more difficult. Therefore increase of distance should be compensated by decrease of time interval and

vice versa.

**had corresponded to locations above 5 on the 2**

*m*_{b}*T*

_{a}line, the speed of

**would be slower than the speed of**

*m*_{a}**, and**

*m*_{b}**would lose the competition with**

*m*_{b}**. But if**

*m*_{a}**had corresponded to locations below 5 on the 2**

*m*_{b}*T*

_{a}line, then the speed of

**would be slower than that of**

*m*_{b}**, and**

*m*_{a}**would win the competition. Thus, a preference for slow physical speeds implies coupling.**

*m*_{b}**and**

*m*_{a}**occurs when**

*m*_{b}**is longer than**

*m*_{b}**in time (**

*m*_{a}*T*

_{b}>

*T*

_{a}) but is shorter than

**in space (**

*m*_{a}*S*

_{b}<

*S*

_{a}). It is illustrated in the distance plot ( Figure 2) by the negative-slope line between 1 and 2.

**p**,

_{1}**p**, and

_{2}**p**). The interstimulus interval,

_{3}*T,*was constant within a display. To measure strength of motion, Burt and Sperling derived new stimuli from the one shown in Figure 4A by deleting subsets of dots. For example, when every other dot in the row was deleted, the spatial and temporal separations along path

**p**doubled without affecting path

_{2}**p**(Figure 4B). Now, the dominant path becomes

_{1}**p**. Burt and Sperling used two methods to measure strength of apparent motion: rating and forced choice:

_{1}- Rating: On each trial, observers saw two displays (such as those shown in Figure 4) in alternation and used a seven-point scale to rate the strength of the motion along
**p**in one display compared with the other,_{1}*R*(**p**): 0 =_{1}*I can't see**p*_{ 1}*in the first display,*5 =*p*_{ 1}*is equally strong in both displays,*and 6 =*p*_{ 1}*is stronger in the second display*. They also recorded ratings for the strength of motion along**p**,_{2}*R*(**p**). They plotted_{2}*R*(**p**) and_{1}*R*(**p**) as a function of_{2}*T*. As*T*increased,*R*(**p**) increased and_{1}*R*(**p**) decreased. They called the value of_{2}*T*at which these two functions crossed,*T*^{*}_{1,2}, the*transition time*between the two paths. - Forced choice: They presented each stimulus at several values of
*T*. They designed the stimuli so that one motion was to the left of vertical and the other to the right. On each presentation, observers reported the direction of motion. Using the results of the rating experiment, they selected spatial parameters so that for each stimulus, the selected values of*T*were smaller and larger than a transition time, to find the transition time by interpolating the proportions of responses across the tested magnitudes of*T*.

*T*and the forced-choice method to study the interaction of

*T*with spatial parameters. Across all conditions, they found tradeoff. The authors concluded that their data were incompatible with Korte's law and conjectured that Korte's methodology had been faulty.

*equilibrium theory*of motion perception. They investigated how limited neural resources (a limited number of neurons that can be tuned to speed) should be allocated to different conditions of stimulation and found that human spatiotemporal sensitivity approximately follows the optimal prescription. In this theory of Gepshtein et al., resources should be allocated according to the degree of balance between measurement uncertainties and stimulus uncertainties. The authors estimated measurement uncertainties using the uncertainty principle of measurement; they estimated stimulus uncertainty from measurements of speed distribution in the natural ecology (Dong & Atick, 1995). Using these estimates, Gepshtein et al. derived optimal conditions and to equally suboptimal conditions for speed measurement.

*increase*the spatial coordinate (arrowhead up), the expected system's sensitivity should

*decrease,*manifested by cooler colors of contours that indicate less favorable conditions for motion measurement. Thus, the stimulus should be perceived less often than the stimulus on the left side of the pair. But if we

*decrease*the spatial coordinate (arrowhead down), then the system's sensitivity should

*increase*(warmer colors of contours), and this stimulus should be perceived more often than the stimulus on the left side. Therefore, in this region of the distance plot, we expect to find space–time coupling.

*motion lattices*(Gepshtein & Kubovy, 2000; Kubovy & Gepshtein, 2003), which are a generalization of the stimuli used by Burt and Sperling (1981). To create a motion lattice (Figure 6), we take a lattice of spatial locations, whose columns are called

*baselines,*and split it into six frames

*f*

_{i},

*i*∈ {1,…, 6}, so that each frame contains every sixth baseline. In Figure 6A, dot locations in six successive frames are distinguished by six levels of gray, as explained in Figure 6B (screen shots of the stimuli are shown in Figure 7 and the animated demonstrations in the 1).

*f*

_{1},

*f*

_{2},…), separated by time

*τ,*the

*temporal scale*of each motion lattice. (All other temporal parameters of the stimulus are integer multiples of

*τ*.) When spatial and temporal distances between the frames are chosen appropriately, the motion lattice is perceived as a continuous flow of motion. (Six-stroke lattices are notated

*f*

_{ i}can match a dot that appears in frame

*f*

_{ i+1}(after a time interval

*τ*) or in frame

*f*

_{ i+2}(after a time interval 2

*τ*). As a result, motions parallel to three paths can be perceived: m

_{1}, m

_{2}, and m

_{3}. We chose conditions such that m

_{2}would never dominate, so that observers saw motion only along m

_{1}or m

_{3}(red arrows in Figure 6A).

*t*> 2

*τ*become perceptible, in which case the matching process can skip more than one frame (Burt & Sperling, 1981). In this study, we used temporal scales for which neither motions along paths with

*t*> 2

*τ*nor zigzag motions were perceived. We imposed another constraint on the design of our stimuli: We chose spatial parameters to prevent observers from seeing the motion of spatial groupings of dots (Gepshtein & Kubovy, 2000). We did this by making the baseline distance

*b*(Figure 6A) much longer than the spatial distances of m

_{1}, m

_{2}, and m

_{3}.

*S*

_{ k}and

*T*

_{ k}, respectively, the spatial and temporal distances of m

_{ k}. The temporal components of m

_{1}and m

_{2}are equal (

*T*

_{1}=

*T*

_{2}=

*τ*); the temporal component of m

_{3}is twice as long (

*T*

_{3}= 2

*T*

_{1}= 2

*τ*). Although the ratio of the temporal components of m

_{1}and m

_{3}is fixed at 2, we can vary the relative magnitudes of their spatial components, as we did in Figure 2B. When m

_{3}is much longer than m

_{1}both in space and in time (

*S*

_{3}≫

*S*

_{1}and

*T*

_{3}= 2

*T*

_{1}; Figure 6C),

*μ*

_{1}is seen more often than

*μ*

_{3}. (Note that

*S*

_{2}≫

*S*

_{1}; thus,

*μ*

_{1}is also more likely than

*μ*

_{2}.) But when

*S*

_{3}≪

*S*

_{1}( Figure 6D),

*μ*

_{3}is often seen, even though

*T*

_{3}= 2

*T*

_{1}. (Here,

*S*

_{2}≫

*S*

_{1}≫

*S*

_{3}; thus,

*μ*

_{3}is also more likely than

*μ*

_{2}.) For

*μ*

_{3}to be seen, the visual system must have matched elements separated by interval 2

*τ*even though other elements appeared in the display during this interval at time

*τ*.

*S*

_{2},

*S*

_{3},

*b*) relative to

*S*

_{1}its

*spatial scale*. The radii of the dots were 0.3

*S*

_{1}. To minimize edge effects, we modulated the luminance of dots according to a Gaussian distribution, with the maximal luminance of 88 cd/m

^{2}. The spatial constant of the Gaussian luminance envelope of the lattices was

*σ*= 1.5

*S*

_{1}.

*S*

_{3}≫

*S*

_{1}and m

_{1}prevails (see also Figure 7A). In Figure 6D,

*S*

_{3}≪

*S*

_{1}and m

_{3}prevails (see also Figure 7B). As we vary the spatial ratio

*r*

_{31}=

*S*

_{3}/

*S*

_{1}between these extremes, we find the equilibrium point: a ratio

*r*

_{31}* =

*S*

_{3}*/

*S*

_{1}, at which

*μ*

_{3}is as likely as

*μ*

_{1}.

_{1}, m

_{2}, and m

_{3}(Gepshtein & Kubovy, 2000). Observers clicked on one of the circles to indicate the direction of motion they perceived. This triggered a mask (an array of randomly moving dots) and initiated the next trial.

*S*

_{1}∈ {0.38°, 0.65°, 1.10°, 1.90°, 3.00°}, at a viewing distance of 0.39 m. The smallest spatial scale (

*S*

_{1}= 0.38°) was the smallest scale at which observers could reach perceptual equality between the competing motion paths. The temporal scale

*τ*was 40 ms.

*τ*∈ {27, 40, 53, 67} ms, using the same five spatial scales

*S*

_{1}as in Experiment 1. Our apparatus did not allow us to present motion lattices at a temporal scale smaller than 27 ms. The upper limit on the temporal scales was perceptual: At the temporal scale of above 67 ms, observers started to experience fluctuations between motion along m

_{1}and other motion paths within trials (i.e., they saw a zigzag motion).

*r*

_{31}(as in Experiment 1) to obtain 100 lattices. Nine naive observers and one of the authors each contributed 24 trials per condition. Otherwise, this experiment was identical to Experiment 1.

*S*

_{1}, we tested five magnitudes of

*r*

_{31}. In Figure 8A, we plot the log-odds of the probabilities of

*μ*

_{3}and

*μ*

_{1},

*r*

_{31}* as explained in Figure 8A. Perceptual equilibrium holds when the competing percepts

*μ*

_{3}and

*μ*

_{1}are equiprobable, that is, when the log-odds of their probabilities is zero. We found

*r*

_{31}* by a linear interpolation (the oblique solid line in Figure 8A) between the data points that straddle

*L*= 0 (the filled circles). In Figure 8A, the equilibrium point is indicated by the vertical red line. Here,

*r*

_{31}* > 0, indicating the regime of tradeoff, in support of Korte's law.

*r*

_{31}* < 1) and coupling (

*r*

_{31}* > 1).

*y*-axis in each panel of Figure 8B shows the speed ratios:

_{1}is always greater than in m

_{3}when m

_{1}and m

_{3}are in equilibrium. Despite this, sometimes m

_{1}is seen and sometimes m

_{3}is seen. This finding is inconsistent with the low-speed assumption (see Discussion).

*S*

_{1}. The equilibrium points obtained under different temporal scales follow different functions. However, if we plot the equilibrium points as a function of speed,

*S*

_{1}/

*T*

_{1}( Figure 9B), they fall on a single function. Its value varies from tradeoff to coupling, passing through time independence at about 12°/s. Thus, speed (rather spatial scale) determines the regime of motion (tradeoff or coupling).

*T*

_{1}/

*S*

_{1}; Johnston, McOwan, & Benton, 1999), we obtain linear functions (Figures 10A and 10B).

_{1}is seen and sometimes m

_{3}is seen at equilibrium, despite the fact that the speed of motion in m

_{1}is always greater than in m

_{3}, as indicated on the right ordinates in Figures 9 and 10.

*k*is the (negative) slope and

*l*is the intercept. By multiplying both sides by speed

*v*

_{1}(i.e., by

*S*

_{1}/

*T*

_{1}), and noting that

*v*

_{3}=

*S*

_{3}/2

*T*

_{1}, we have

*f*(

*T*) at a very small value of

*T*on the left edge of the figure. Using these coordinates—[

*T,*

*f*(

*T*)]—we obtained

*f*(2

*T*) from Equation 7. We used the coordinates [2

*T,*

*f*(2

*T*)] to obtain

*f*(4

*T*) and thus iteratively propagated each solution to the maximal temporal distance in the figure. We repeated this procedure several times, starting at the same value of

*T*but with a linearly incremented value of

*f*(

*T*), to obtain all the solutions plotted in the figure.

- Kelly (1979) obtained the estimates of spatiotemporal sensitivity (Figure 5A) with narrowband stimuli (drifting sinusoidal gratings), whereas our stimuli are spatially broadband. Also, he used image stabilization to limit motion on the retina, whereas our observers were free to move their eyes during stimulus presentation; as a result, our stimuli cover a broader temporal-frequency band. Such increases in the width of the spatial and temporal frequency bands flatten the equivalence contours. We illustrate this in Figure 12, which shows the results of simulating the effects of widening of the frequency bands. We obtained the three panels by averaging Kelly's estimates of sensitivity across an increasing range of spatial and temporal frequencies and plotted the resulting equivalence contours in the distance plot.
- As Gepshtein et al. (2007) showed, a normative theory predicts that estimates of the sensitivity of the visual system depend on the task and the stimuli used in obtaining the estimates. Consider a task that uses an ambiguous stimulus for which motion matching is difficult. Such a task depends more on estimating stimulus frequency content than stimulus location. (See also Banks, Gepshtein, & Landy, 2004, who emphasized the role of stimulus spatial-frequency content for solving the binocular matching problem.) According to Gepshtein et al., an optimal visual system should change the distribution of its sensitivity across the parameters of stimulation so that its estimate of stimulus frequency content becomes more reliable than its estimate of stimulus location. On this view, there would be little reason to expect a quantitative agreement between the equivalence contours obtained using different tasks and different stimuli.

*The lawful perception of apparent motion*.