the lawful perception of apparent motionmk9y/documents/papers/gepshteinkubov… · korte’s law...

1 The lawful perception of apparent motion2

3

4 Perceptual Dynamics Laboratory, Brain Science Institute,5 RIKEN, Wako, Saitama, Japan6 Sergei Gepshtein AQ17

8 Department of Psychology, University of Virginia,9 Charlottesville, VA, USA10 Michael Kubovy AQ1

11 Visual apparent motion is the experience of motion from the successive stimulation of separate spatial locations. How12 spatial and temporal distances interact to determine the strength of apparent motion has been controversial. Some studies13 report space–time coupling: If we increase spatial or temporal distance between successive stimuli, we must also increase14 the other distance between them to maintain a constant strength of apparent motion (Korte’s third law of motion). Other15 studies report space–time tradeoff: If we increase one of these distances, we must decrease the other to maintain a16 constant strength of apparent motion. In this article, we resolve the controversy. Starting from a normative theory of motion17 measurement and data on human spatiotemporal sensitivity, we conjecture that both coupling and tradeoff should occur, but18 at different speeds. We confirm the prediction in two experiments, using suprathreshold multistable apparent-motion19 displays called motion lattices. Our results show a smooth transition between the tradeoff and coupling as a function of20 speed: Tradeoff occurs at low speeds and coupling occurs at high speeds. From our data, we reconstruct the21 suprathreshold equivalence contours that are analogous to isosensitivity contours obtained at the threshold of visibility.

22 Keywords: apparent motion, perceptual equilibrium, Korte’s law, visual sensitivity, normative theory

23 Citation: Gepshtein, S., & Kubovy, M. (2007). The lawful perception of apparent motion. Journal of Vision 0(0):1, 1–15,24 http://journalofvision.org/0/0/1/, doi:10.1167/0.0.1.

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2627 Introduction

28 In this article, we propose a solution to a long-standing29 controversy about the perception of apparent motion.30 Apparent motion is produced by a sequence of frames31 portraying a stimulus at different locations (Nakayama,32 1985; Ullman, 1979; Wertheimer, 1912). To describe the33 problem, consider a display in which short-lived dots34 appear sequentially at three loci: o, a, and b (Figure 1A).35 Suppose the spatial distance between a and b is very long,36 and motion from a to b is unlikely. Then, dot at o has two37 potential matches: at a and b. Because dot at o has two38 potential matches, the display is ambiguous; one can39 perceive motion either from o to a or from o to b.40 We can find combinations of spatial and temporal41 components of the competing motion paths such that they42 are seen equally often. Under these conditions, we say that43 the two motion paths are in “perceptual equilibrium”.44 According to some measurements (Koffka, 1935/1963;45 Korte, 1915), equilibrium can be observed only when the46 spatial and temporal components of one motion path are47 longer than the spatial and temporal components of the48 other motion path. This has been called Korte’s third law49 of apparent motion. We call this result “space–time50 coupling”. According to other measurements (Burt &51 Sperling, 1981), equilibrium can be observed only when52 the spatial component of one path is longer than the53 spatial component of the other, whereas the temporal54 component of the first is shorter than the other. We call

55this result “space–time tradeoff”. In this article, we show56that by changing the conditions of stimulation, we can57cause the pattern of results to change smoothly from58space–time coupling to space–time tradeoff.59We develop our argument in three steps. (a) We60examine how the inconsistent results in the motion61literature are related. (b) We show that both coupling62and tradeoff are consistent with a normative theory of63motion measurement and with human sensitivity to64continuous motion, which suggests that both tradeoff65and coupling may occur, but under different conditions.66(c) We confirm this in two experiments and show that67coupling and tradeoff are special cases of a simple law.

68

6970Regimes of apparent motion

71In the motion display illustrated in Figure 1A, let us72denote the motion path from o to a by ma and from o to b73by mb. Let us denote the (potential) percepts of motion74along these paths by 2a and 2b and the strength of the75apparent motion experienced when 2a or 2b is seen in76isolation by Aa and Ab, respectively. Each path has a77temporal (Ta and Tb) and a spatial (Sa and Sb) component78(Figure 1A). The two motion paths are in perceptual79equilibrium for such combination of conditions (Sa, Ta)80and (Sb, Tb) that the two motions are seen equally often81and their strengths are equal: Aa = Ab. The controversy

Journal of Vision (2007) 0(0):1, 1–15 http://journalofvision.org/0/0/1/ 1

doi: 10 .1167 /0 .0 .1 Received November 28, 2006; published 0 0, 2007 ISSN 1534-7362 * ARVO

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82 concerns the relationship between the pair (Sa, Ta) and the83 pair (Sb, Tb) when the two motions are in equilibrium:

84 Coupling: Equilibrium is obtained when Sa and Ta are both85 longer or both shorter than Sb and Tb. This is Korte’s third86 law of motion (Koffka, 1935/1963; Korte, 1915).

87 Tradeoff: Equilibrium is obtained when Sb 9 Sa and Tb G88 Ta or Sb G Sa and Tb 9 Ta. This result was obtained by89 Burt and Sperling (1981).

90 Let us represent each motion path by a point in a plot of91 spatial and temporal distances: a distance plot (Figure 1B).92 Suppose we hold the spatial and temporal components of93 ma constant, so it is represented in the distance plot by a94 fixed point. Suppose also that we hold the temporal95 component of mb constant, twice as long as the temporal96 component of ma: Tb = 2Ta. Then, we can vary the spatial97 component Sb and find the value of rba = Sb/Sa for which 2a98 and 2b are in perceptual equilibrium: p(2a) = p(2b) = 0.5.99 This manipulation is represented in the distance plot by100 moving the point for mb along the line Tb = 2Ta (Figure 1B).

101This manipulation can give rise to the three outcomes102illustrated in Figure 2: coupling (rba 9 1), tradeoff (rba G 1),103and an intermediate condition that we call “time indepen-104dence” (rba = 1). Each of these outcomes corresponds to a105different slope of the line connecting the representation of the106competing motion paths in the distance plot: positive slope107for coupling, negative slope for tradeoff, and zero slope for108time independence. We will use the notion of slope between109equivalent conditions in the distance plot to relate results on110apparent motion to predictions of a normative theory of111motion measurement and to data on human spatiotemporal112sensitivity. Each of the three regimes has played a role in the113literature on motion perception, as we show next.114

115116Space–time coupling

117Space–time coupling means that the equilibrium between118ma and mb occurs when mb is longer than ma both in time119(Tb 9 Ta) and in space (Sb 9 Sa). It is illustrated in the distance120plot (Figure 2) by the positive-slope line between 1 and 4.

Figure 1. Perceptual equilibrium in apparent motion. (A) A stimulus for ambiguous apparent motion is depicted in two spatial coordinates(x, y) and one temporal coordinate. The spatial projection of the stimulus is shown in the inset. Element o has two potential matches: a(giving rise to motion path ma) and b (giving rise to motion path mb). Each motion path has two parameters: temporal distance (Ta or Tb)and spatial distance (Sa or Sb). (B) In the distance plot, each motion path is represented by a point {Ti, Si}. We constrain the parameterssuch that Ta and Sa are fixed and Tb = 2Ta. The spatial distance Sb is allowed to vary to determine the point of perceptual equilibriumwhere the competing motions are equally likely to be seen (Figure 2). Note that the distance plot does not represent information about thedirection (and, therefore, about velocity) of motion.

Journal of Vision (2007) 0(0):1, 1–15 Gepshtein & Kubovy 2

121 An example of coupling is Korte’s third law of motion122 (henceforth “Korte’s law”; Koffka, 1935/1963; Korte,123 1915; Lakatos & Shepard, 1997; Neuhaus, 1930). Using124 a tachistoscope, Korte presented his observers with two125 brief visual stimuli separated by variable spatial and126 temporal distances. First, he found a space–time pair that127 gave rise to a compelling experience of motion (“good128 motion”). Then, he varied the spatial and temporal129 distances between the two stimuli and recorded the130 observers’ descriptions of the motion. From these descrip-131 tions, he derived a rating of motion strength (Figure 3). He132 found that when conditions for good motion held, he133 could not change just the spatial distance or just the134 temporal distance without reducing the strength of motion.135 To restore the experience of good motion, he had to136 increase or decrease both.137 Sometimes, Korte’s law is portrayed as evidence of138 speed invariance (e.g., Koffka, 1935/1963; Kolers, 1972).139 This would be the case only if the space–time pairs that140 correspond to the same speed of motion had also141 corresponded to the same strength of apparent motion. In142 our terms, spatial and temporal distances in the conditions143 of equilibrium would then be related directly: S = vT,144 where speed v is a positive constant.145 Note that if v denotes physical speed, Korte’s data do146 not provide support for speed invariance. However, if147 perceived speed is meant, and if perceived speed is related

148to physical speed nonlinearly, then Korte’s data could be149consistent with perceived speed invariance. In that case,150the physical spatial and temporal distances in the151conditions of equilibrium would be related by S = )(v)T,152where v is physical speed and )(v) is a nonlinear function.153Korte’s law has been surprising researchers ever since it154was proposed. Koffka (1935/1963, p. 293)VKorte’s155dissertation supervisorVwrote many years later:156

157This law has puzzled psychologists more than the158othersIat the time of Korte’s work one was still159inclined to think as follows: if one separates the two160successively exposed objects more and more, either161spatially or temporally, one makes their unification162more and more difficult. Therefore increase of163distance should be compensated by decrease of time164interval and vice versa.165

166They expected tradeoff but found coupling.

167The low-speed assumption

168A special case of coupling is the “low-speed assump-169tion”, motivated by a phenomenon first reported by170Wallach (1935), known today as the “aperture problem”171(Adelson & Movshon, 1982; Ullman, 1979). When you172look at a moving line through an aperture that occludes its173terminators, you are likely to see it moving orthogonally174to its orientation, even when it is moving in other175directions, and because a line moving along the orthog-176onal path is moving at the lowest speed consistent with its177displacement, Wallach (1976) attributed the phenomenon178to the visual system’s preference for slower motion. This179assumption has been used to specify the prior distribution180of motion estimates in a Bayesian model of motion per-181ception (Hurlimann, Kiper, & Carandini, 2002; Stocker &182Simoncelli, 2006; Weiss & Adelson, 1998; Weiss,183Simoncelli, &Adelson, 2002; see also Heeger& Simoncelli,1841991, and van Hateren, 1993).185A preference for slow physical speeds implies that186perceptually equivalent space–time pairs should fall on187the line of constant speed (1 and 5 in Figure 2): If mb had188corresponded to locations above 5 on the 2Ta line, the189speed of ma would be slower than the speed of mb, and mb

190would lose the competition with ma. But if mb had191corresponded to locations below 5 on the 2Ta line, then192the speed of mb would be slower than that of ma, and mb

193would win the competition. Thus, a preference for slow194physical speeds implies coupling. 195

196

197198Space–time tradeoff

199Space–time tradeoff means that the equilibrium200between ma and mb occurs when mb is longer than ma in

Figure 2. Hypothetical conditions of perceptual equilibriumbetween two paths of apparent motion, ma and mb, in Figure 1.The temporal and spatial distances {Ta, Sa } between successiveelements in ma are represented by Point 1, which is fixed. Thetemporal distance of mb is also held constant at Tb = 2Ta. We varythe spatial distance Sb to determine its value when mb is inequilibrium with ma, that is, p(2a) = p(2b) = 0.5. Space–timetradeoff occurs whenever Sb G Sa, in the dark gray region (e.g., atPoint 2). Space–time coupling occurs whenever Sb 9 Sa, in thelight gray region (e.g., at Point 4). Time independenceVor“shortest spatial path”Voccurs at Sb = Sa (Point 3).


201 time (Tb 9 Ta) but is shorter than ma in space (Sb G Sa). It is202 illustrated in the distance plot (Figure 2) by the negative-203 slope line between 1 and 2.204 As we mentioned earlier, Burt and Sperling (1981)205 obtained evidence for tradeoff. These authors used206 apparent-motion displays that were ambiguous: Observers207 could see motion along one of several directions. The208 display consisted of a succession of brief flashes of a209 horizontal row of evenly spaced dots. Between the flashes,210 the row was displaced horizontally and downward, so211 that observers could see the row move downward and212 to the right or downward and to the left, parallel to one213 of several orientations, three of which are shown in214 Figure 4A (paths p1, p2, and p3). The interstimulus215 interval, T, was constant within a display. To measure216 strength of motion, Burt and Sperling derived new stimuli217 from the one shown in Figure 4A by deleting subsets of218 dots. For example, when every other dot in the row was219 deleted, the spatial and temporal separations along path p2220 doubled without affecting path p1 (Figure 4B). Now, the221 dominant path becomes p1. Burt and Sperling used two

222methods to measure strength of apparent motion: rating223and forced choice:

224Rating: On each trial, observers saw two displays (such225as those shown in Figure 4) in alternation and used a226seven-point scale to rate the strength of the AQ2motion227along p1 in one display compared with the other, R(p1):2280 = I can’t see p1 in the first display, 5 = p1 is equally229strong in both displays, and 6 = p1 is stronger in the230second display. They also recorded ratings for the231strength of motion along p2, R(p2). They plotted R(p1)232and R(p2) as a function of T. As T increased, R(p1)233increased and R(p2) decreased. They called the value of234T at which these two functions crossed, TH1,2, the235transition time between the two paths.236

237Forced choice: They presented each stimulus at several238values of T. They designed the stimuli so that one239motion was to the left of vertical and the other to the240right. On each presentation, observers reported the241direction of motion. Using the results of the rating

Figure 3. Data reconstructed from Korte (1915). The rows contain data for different observers; and the columns, for different interstimulusintervals (ISI). The highest rating corresponds to good motion. To maintain the experience of good motion, the temporal distance betweensuccessive stimuli and their spatial distance should be both increased or both decreased.


242 experiment, they selected spatial parameters so that for243 each stimulus, the selected values of T were smaller and244 larger than a transition time, to find the transition time245 by interpolating the proportions of responses across the246 tested magnitudes of T.

247 They used the rating method to study the effect of T and248 the forced-choice method to study the interaction of T249 with spatial parameters. Across all conditions, they found250 tradeoff. The authors concluded that their data were251 incompatible with Korte’s law and conjectured that252 Korte’s methodology had been faulty.

253

254255 Time independence

256 Between coupling and tradeoff lies time independence,257 or the preference for the “shortest [spatial] path” (Wallach,258 1935; see also Wallach, 1976). Wallach (1935) proposed259 that the visual system prefers the shortest spatial path in260 perception of apparent motion before he offered the low-261 speed assumption. Wallach thought that orthogonal motion262 is preferred because it allows the line to travel the shortest263 spatial distance. Such a preference for the shortest spatial264 path has been assumed in many other studies that used265 ambiguous motion displays (Hock, Schoner, & Hochstein,266 1996; Kolers, 1972; Ramachandran & Anstis, 1983;

267Ullman, 1979; von Schiller, 1933). The shortest path268hypothesis is a straightforward generalization of the269spatial proximity principle to space–time: It simply270ignores temporal parameters.

271

272273Prospect of resolving the274controversy

275Data on human spatiotemporal sensitivity and a norma-276tive theory on motion measurement suggest that both277regimes of tradeoff and coupling might exist, but for278different parameters of stimulation.279Many studies of human and animal vision have explored280spatiotemporal contrast sensitivity (Burr & Ross, 1982;281Nakayama, 1985; Newsome, Mikami, & Wurtz, 1986; van282de Grind, Koenderink, & van Doorn, 1986; van Doorn &283Koenderink, 1982a, 1982b). A comprehensive summary of284human contrast sensitivity was obtained by Kelly (1979). In285Figure 5A, we plot Kelly’s estimates of human sensitivity286to drifting sinusoidal gratings converted into the format287of our distance plot. (See Gepshtein, Tyukin, & Kubovy,288in press, for the details of conversion.) The colored contours289are the isosensitivity contours: Each contour connects the290conditions for which human observers are equally sensitive291to drifting sinusoidal gratings. Kelly and others found that292for every speed, there exists a condition for which contrast293sensitivity is maximal. The data about maximal sensitivity294for motion from many studies are consistent with Kelly’s295data, as summarized by Nakayama (1985). Note that Kelly296(1979) obtained his estimates of spatiotemporal sensitivity297using an image stabilization technique that allowed him to298precisely control motion on the retina. As Kelly and299Burbeck (1984) observed, those data had the same form300as the data obtained with no image stabilization (Kelly,3011969, 1972; Kulikowski, 1971; Robson, 1966; van Nes,302Koenderink, Nas, & Bouman, 1967). Thus, the data303obtained by Kelly (1979) can be used to interpret results304from studies that did not use image stabilization.305Gepshtein et al. (in press) proposed an equilibrium306theory of motion perception. They investigated how307limited neural resources (a limited number of neurons308that can be tuned to speed) should be allocated to different309conditions of stimulation and found that human spatio-310temporal sensitivity approximately follows the optimal311prescription. In the theory of Gepshtein et al., resources312should be allocated according to the degree of balance313between measurement uncertainties and stimulus uncer-314tainties. The authors estimated measurement uncertainties315using the uncertainty principle of measurement; they316estimated stimulus uncertainty from measurements of317speed distribution in the natural ecology (Dong & Atick,3181995). Using these estimates, Gepshtein et al. derived319optimal conditions and uniformly suboptimal conditions320for speed measurement.

Figure 4. Stimulus of Burt and Sperling (1981). (A) A horizontalrow of dots is flashed sequentially while it is displaced horizontallyand downward. The rows of unfilled dots are occupied by the rowof dots across time: Every row is occupied once, at time Ti,designated on the right. Under appropriate conditions, oneperceives a flow of motion along a path pi. The three pathsmarked by arrows are most likely. (B) Removing every second dotin the row affects the spatial and temporal parameters of paths p2and p3, but not of path p1.


321 By the equilibrium theory, more resources should be322 allocated to the optimal condition for every speed than to323 other conditions, yielding a normative prediction of the324 maximal sensitivity for that speed (the grey curve in325 Figure 5B). Similarly, equal amount of resources should326 be allocated to the uniformly suboptimal conditions,327 yielding normative predictions for the isosensitivity sets328 (the colored contours in Figure 5B). By the theory, the329 maximal sensitivity set invariantly has the roughly hyper-330 bolic shape, and the isosensitivity sets invariantly form331 closed contours, whose shapes follow the roughly hyper-332 bolic shape of the maximal sensitivity set. Along every333 isosensitivity contour, its slope changes smoothly from334 negative to positive, such that negative slopes dominate at335 low speeds (the right part of the distance plot) and positive336 slopes dominate at very higher speedsAQ4 (the left top part),337 similar to the shapes of the empirical isosensitivity sets in338 Figure 5A. The similarity in theoretical and empirical339 equivalence contours illustrated in Figure 5 suggests that340 human motion sensitivity has the shape it does because341 the visual system distributes its spatiotemporal sensitivity342 according to the balance of uncertainties.343 From the data on human spatiotemporal sensitivity and344 from the equilibrium theory ofAQ3 Gepshtein et al. (in press),

345it follows that some conditions of stimulation are more346favorable for motion measurement than others. Suppose347that equally favorable conditions for motion measurement348correspond to the conditions equally preferred by the349visual system in competition between alternative appa-350rent-motion paths, as in Figure 1. In other words, suppose351that the conditions of perceptual equilibrium in apparent352motion lie on the same equivalence contour: empirical353(Figure 5A) or theoretical (Figure 5B). Then, the shapes of354equivalence contours predict different regimes of motion355perception under different conditions of stimulation:356coupling when the slopes of the isosensitivity contours357are positive, and tradeoff when the slopes are negative, as358we illustrate in both panels of Figure 5.359Take the pair of points labeled “coupling” in Figure 5A360or 5B. The two points belong to the same equivalence361contour and therefore mark two conditions equally362sensitive to motion (in Panel A) or equally suitable363(equally suboptimal) for speed measurement (in Panel B).364Let us vary the spatial coordinate of the stimulus365represented by the right point in the pair, as we did in the366thought experiment in Figure 1B. If we increase the367spatial coordinate (arrowhead up), the expected system’s368sensitivity should decrease, manifested by cooler colors of

Figure 5. Empirical and theoretical equivalence classes. The format of both panels is similar to that of Figure 2, except that the two axesare logarithmic. (A) Human isosensitivity contours. Each contour is a spatiotemporal isosensitivity curve. The color of each curvecorresponds to the normalized magnitude of sensitivity, as explained in the color bar on the right. (B) Theoretical equivalence contourspredicted by the normative theory of Gepshtein et al. (in press). AQ3The roughly hyperbolic curve connects conditions expected to be optimalfor speed estimation across speeds. The colored curves connect conditions equally suboptimal for motion measurement and henceexpected to be equally preferable: The warmer the color, the more resources should be allocated to the corresponding contour and thehigher sensitivity is expected. The pairs of connected circles in Panels A and B demonstrate that the regimes of coupling (at a high speed)and tradeoff (at a lower speed) are expected from normative considerations (in Panel B) and are consistent with data on humanspatiotemporal sensitivity (in Panel A). The vertical arrows correspond to the arrow in Figure 1B; they illustrate the procedure we use toreveal the regimes of coupling and tradeoff.


369 contours that indicate less favorable conditions for motion370 measurement. Thus, the stimulus should be perceived less371 often than the stimulus on the left side of the pair. But if372 we decrease the spatial coordinate (arrowhead down),373 then the system’s sensitivity should increase (warmer374 colors of contours), and this stimulus should be perceived375 more often than the stimulus on the left side. Therefore, in376 this region of the distance plot, we expect to find space–377 time coupling.378 By applying this reasoning to other regions of the379 distance plot, we expect to find space–time tradeoff in380 some regions, where the slopes of equivalence contours381 are negative.382 By this argument, the regime of tradeoff should smoothly383 change to the regime of coupling as a function of stimulus384 parameters, for example, by increasing the spatial distance385 of stimuli from the bottom left region labeled “tradeoff” to386 the top left region labeled “coupling” in the distance plot in387 Figure 5. This manipulation also leads to change in speed388 of motion; thus, we expect the smooth transition between389 regimes also as a function of speed. Specifically, as speed390 grows from the bottom right corner of the distance plot to391 its top left corner, we expect tradeoff to be found at low392 speeds and coupling to be found at high speeds.393 Note that by this argument, the regime of time394 independence (not shown in Figure 5) should be found395 only accidentally: It is obtained whenever the line396 connecting equivalent conditions in the distance plot397 happens to be exactly parallel to the abscissa.

398To summarize, predictions of the equilibrium theory399and the data on human spatiotemporal AQ5sensitivity suggest400that both regimes could hold: coupling (Korte’s law) at401high speeds and tradeoff at low speeds, and the con-402troversy would be resolved. These considerations motivate403the following two experiments on apparent motion, in404which we ask whether tradeoff and coupling are obtained405under the different conditions of stimulation.

406

407408Methods

409Experiment 1 AQ6410Stimuli

411To create competing paths of motion with inde-412pendently manipulable spatial and temporal parameters,413we use alternating dot patterns called motion lattices414(Gepshtein & Kubovy, 2000; Kubovy & Gepshtein, 2003),415which are a generalization of the stimuli used by Burt and416Sperling (1981). To create a motion lattice (Figure 6), we417take a lattice of spatial locations, whose columns are called418baselines, and split it into six frames fi, i Z {1,I, 6},419so that each frame contains every sixth baseline. In420Figure 6A, dot locations in six successive frames are dis-421tinguished by six levels of gray, as explained in Figure 6B422(screen shots of the stimuli are shown in Figure 7 and the423animated demonstrations in the Appendix).

Figure 6. The design of a six-stroke motion lattice M6. (A) The six successive frames of M6 are shown superimposed in space. Graylevels indicate time. (B) The time course of M6. To differentiate dots from different frames, in this illustration (but not in the actualstimulus), they are shown in different shades of gray. The three most likely motionsValong m1, m2, and m3Vcan occur because dots inframe fi can match dots in either frame fi+1 or frame fi+2. (C–D) Conditions in which different motion paths dominate: m1 in Panel C and m3

in Panel D. (In all our experiments, we chose conditions in which m2 would never dominate.)


424 Observers viewed the frames in rapid succession425 (f1, f2,I), separated by time C, the temporal scale of each426 motion lattice. (All other temporal parameters of the427 stimulus are integer multiples of C.) When spatial and428 temporal distances between the frames are chosen appro-429 priately, the motion lattice is perceived as a continuous430 flow of motion. (Six-stroke lattices are notated M6, to431 distinguish them from two-stroke ones, M2, used by432 Gepshtein & Kubovy, 2000.)433 A dot that appears in frame fi can match a dot that434 appears in frame fi+1 (after a time interval C) or in frame435 fi+2 (after a time interval 2C). As a result, motions parallel436 to three paths can be perceived: m1, m2, and m3. We chose437 conditions such that m2 would never dominate, so that438 observers saw motion only along m1 or m3 (red arrows in439 Figure 6A).440 Other motions can be seen in motion lattices. For441 example, when temporal scales are short, motions along442 paths with t 9 2C become perceptible, in which case the443 matching process can skip more than one frame (Burt &444 Sperling, 1981). In this study, we used temporal scales for445 which neither motions along paths with t 9 2C nor zigzag446 motions were perceived. We imposed another constraint447 on the design of our stimuli: We chose spatial parameters448 to prevent observers from seeing the motion of spatial449 groupings of dots (Gepshtein & Kubovy, 2000). We450 did this by making the baseline distance b (Figure 6A)451 much longer than the spatial distances of m1, m2, and452 m3.453 We denote by Sk and Tk, respectively, the spatial and454 temporal distances of mk. The temporal components of m1

455 and m2 are equal (T1 = T2 = C); the temporal component of456 m3 is twice as long (T3 = 2T1 = 2C). Although the ratio of457 the temporal components of m1 and m3 is fixed at 2, we458 can vary the relative magnitudes of their spatial compo-459 nents, as we did in Figure 2B. When m3 is much longer460 than m1 both in space and in time (S3 d S1 and T3 = 2T1;461 Figure 6C), 21 is seen more often than 23. (Note that

462S2 d S1; thus, 21 is also more likely than 22.) But when463S3 ¡ S1 (Figure 6D), 23 is often seen, even though T3 =4642T1. (Here, S2 d S1 d S3; thus, 23 is also more likely465than 22.) For 23 to be seen, the visual system must have466matched elements separated by interval 2C even though467other elements appeared in the display during this interval468at time C.469We defined all spatial parameters of the motion lattice470(S2, S3, b) relative to S1 its spatial scale. The radii of the471dots were 0.3S1. To minimize edge effects, we modulated472the luminance of dots according to a Gaussian distribu-473tion, with the maximal luminance of 88 cd/m2. The spatial474constant of the Gaussian luminance envelope of the475lattices was A = 1.5S1.476In Figures 6 and 7, we show two extreme configurations477of the motion lattice: In Figure 6C, S3 d S1 and m1

478prevails (see also Figure 7A). In Figure 6D, S3 ¡ S1 and479m3 prevails (see also Figure 7B). As we vary the spatial480ratio r31 = S3/S1 between these extremes, we find a ratio481r31* = s3*/S1, the equilibrium point, at which 23 is as likely482as 21.483

484Procedure

485We presented the stimuli on a computer monitor486(1,280 � 1,024 pixels, refresh rate = 75 Hz) in a dark487room. All stimuli were viewed binocularly (dioptically).488The observer’s head was stabilized using a chin-and-head489rest.490Each trial began with a 498-ms fixation point, followed491by 12 frames of a motion lattice at a random orientation,492and ended with a response screen consisting of pairs of493circles connected by radial lines with orientations parallel494to m1, m2, and m3 (Gepshtein & Kubovy, 2000).495Observers clicked on one of the circles to indicate the496direction of motion they perceived. This triggered a mask497(an array of randomly moving dots) and initiated the next498trial.

Figure 7. Single frames of motion lattices (not to scale). Only the filled dots appear in the actual frames; the open dots are shown in the figureto indicate the locations of dots in other frames. The frames in Panels A and B correspond to conditions in Panels C and D in Figure 6,respectively. Four animated demonstrations of motion lattices are available in the Appendix.


499 Five naive observers and one of the authors each500 contributed 100 trials per condition. We obtained equili-501 brium points in 25 motion lattices at five spatial scales,502 S1 Z {0.38-, 0.65-, 1.10-, 1.90-, 3.00-}, at a viewing503 distance of 0.39 m. The smallest spatial scale (S1 = 0.38-)504 was the smallest scale at which observers could reach505 perceptual equality between the competing motion paths.506 The temporal scale C was 40 ms.507

508

509 Experiment 2

510 In this experiment, we measured equilibrium points at511 four temporal scales, C Z {27, 40, 53, 67} ms, using the512 same five spatial scales S1 as in Experiment 1. Our513 apparatus did not allow us to present motion lattices at a514 temporal scale smaller than 27 ms. The upper limit on the515 temporal scales was perceptual: At the temporal scale of516 above 67 ms, observers started to experience fluctuations517 between motion along m1 and other motion paths within518 trials (i.e., they saw a zigzag motion).519 For each of the 20 combinations of spatial and temporal520 scales, we tested five magnitudes of r31 (as in Experiment 1)521 to obtain 100 lattices. Nine naive observers and one of the522 authors each contributed 24 trials per condition. Otherwise,523 this experiment was identical to Experiment 1.

524

525

526527Results

528Experiment 1

529For each spatial scale S1, we tested five magnitudes of530r31. In Figure 8A, we plot the log-odds of the probabilities531of 23 and 21,

L ¼ logpð23Þpð21Þ

ð1Þ

532533at one spatial scale (3-) for observer C.C. We found the534equilibrium points r31* as explained in Figure 8A. Perceptual535equilibrium holds when the competing percepts 23 and 21536are equiprobable, that is, when the log-odds of their537probabilities is zero. We found r31* by a linear interpolation538(the oblique solid line in Figure 8A) between the data539points that straddle L = 0 (the filled circles). In Figure 8A,540the equilibrium point is indicated by the vertical red line.541Here, r31* 9 0, indicating the regime of tradeoff, in support542of Korte’s law.543The results for all spatial scales are shown in Figure 8B,544a plot of the equilibrium points as a function of spatial545scale in six observers. For each, we found equilibrium546points that imply both tradeoff (r31* G 1) and coupling547(r31* 9 1).

Figure 8. Results of Experiment 1. (A) Computation of a single equilibrium point r31* . Perceptual equilibrium holds when the competingpercepts 23 and 21 are equiprobable, that is, when the log-odds of corresponding probabilities (the ordinate L) is zero. To find r31* , weperformed a linear interpolation (oblique solid line) between the data points that straddle L = 0 (filled circles). The equilibrium point ismarked by the vertical red line. (B) Equilibrium points plotted as a function of spatial scale for six observers. We observe equilibrium pointsin the tradeoff region (r31* G 1) and the coupling region (r31* 9 1). The vertical bars (visible only where they are larger than the data symbols)correspond to T1 SE. The plots for observers S.G., T.K., and C.C. contain only four equilibrium points because at the smallest spatialscale, they always saw 21 more often than 23 (i.e., L G 0).


548 The right-hand y-axis in each panel of Figure 8B shows549 the speed ratios:

v3v1

¼ S32T1

T1S1

¼ r312: ð2Þ

550551 Because the ratios are always less than 1, the speed of552 motion in m1 is always greater than in m3 when m1 and553 m3 are in equilibrium. Despite this, sometimes m1 is seen554 and sometimes m3 is seen. This finding is inconsistent555 with the low-speed assumption.556 We noted above that both tradeoff and coupling regimes557 of motion perception should be obtained in perception of558 apparent motion if predictions of the equilibrium theory559 hold and if data on human spatiotemporal sensitivity can560 predict perception of suprathreshold apparent motion.561 Now, we found in each observer that, indeed, both562 tradeoff and coupling regimes hold. Tradeoff holds at563 small spatial scales and coupling holds at larger scales.564 We also noted that, from the predictions of the565 equilibrium theory and from data on human sensitivity,566 we expect tradeoff to hold at low speeds and coupling to567 hold at high speeds. In Experiment 2, we ask how the568 regime of motion perception depends on the speed of569 motion.570

571 Experiment 2

572 In this experiment, we vary the temporal scale of our573 displays (which we held constant in Experiment 1).574 Figure 9A is a plot of the 20 equilibrium points,575 averaged across observers, as a function of spatial scale,576 S1. The equilibrium points obtained under different577 temporal scales follow different functions. However, if578 we plot the equilibrium points as a function of speed,

579S1/T1 (Figure 9B), they fall on a single function. Its value580varies from tradeoff to coupling, passing through time581independence at about 12-/s. Thus, speed (rather spatial582scale) determines the regime of motion (tradeoff or583coupling).584The function relating equilibrium points to speed is585nonlinear. However, if we plot our data against slowness586(i.e., reciprocal speed, T1/S1; Johnston, McOwan, &587Benton, 1999), we obtain linear functions (Figures 10A588and 10b).589As in Experiment 1, the results are inconsistent with the590low-speed assumption. We found that sometimes m1 is591seen and sometimes m3 is seen at equilibrium, despite the592fact that the speed of motion in m1 is always greater than in593m3, as indicated on the right ordinates in Figures 9 and 10.594From the isosensitivity contours in Figure 5, we595expected that tradeoff is obtained at low speeds and596coupling is obtained at high speeds. The results of597Experiment 2 confirmed this prediction. We found a598gradual transition between the regimes of tradeoff and599coupling. The transition could follow a variety of600functions. The fact that the observed function is linear601on the slowness scale is remarkable. 602

603

604605Discussion

606Summary

607We have reconciled allegedly inconsistent data on608apparent motion: Korte’s law and later results. In agree-609ment with prediction of the equilibrium theory and data on610continuous motion, our results indicate that previous611findings on apparent motion were special cases. The612allegedly inconsistent results are embraced by a simple

Figure 9. Results of Experiment 2. (A) Equilibrium points r31* (averaged across observers) plotted against spatial scale. The four curveswere derived from the fitted linear models in Figure 10A. (B) When equilibrium points are plotted against speed, they follow one function.The solid curve is a fit by the statistical model in Figure 10B. The model accounts for 98% of the variability in the data. The dashed curvecorresponds to the dashed line in Figure 10B.


613 law in which a smooth transition from tradeoff to coupling614 occurs as a function of speed: Tradeoff holds at low615 speeds of motion (below 12-/s on average), whereas616 coupling (Korte’s law) holds at high speeds.617

618 Equivalence contours of apparent motion

619 Because the speed (or slowness) of motion determines620 the regime of motion perception, and because our data are621 linear on the slowness scale, we can summarize our results622 in terms of the speeds of the competing motions:

S3S1

¼ kT1S1

þ l; ð3Þ

623624 where k is the (negative) slope and l is the intercept. By625 multiplying both sides by speed v1 (i.e., by S1/T1), and626 noting that v3 = S3/2T1, we have

v3 ¼ l

2v1 þ k

2: ð4Þ

627628 If we substitute the fitted slope and intercept, we obtain a629 very simple summary of our data:

v3 ¼ 0:58v1 j 1: ð5Þ

630631 To plot this result in the format of the distance plot632 (Figure 2), we rewrite Equation 5 as

S3 ¼ 1:16S1 j 2T1; ð6Þ

633634 which leads us to a functional equation:

f ð2TÞ ¼ 1:16 f ðTÞj 2T: ð7Þ

635636Figure 11 shows our data (in red) superimposed on a637family of numerical solutions of Equation 7 (thin black638lines on the background). To obtain each solution, we639chose an arbitrary value of f(T) at a very small value of T640on the left edge of the figure. Using these coordinatesV[T,

Figure 10. Results of Experiment 2. (A) When equilibrium points are plotted against reciprocal spatial scale, they follow linear functions.(B) When equilibrium points are plotted against reciprocal speed (slowness), the data fall on a single linear function. The dashed line is afit to all the data; the solid line excludes two outliers (the two leftmost black dots) at the two largest spatial scales for C = 27 ms.

Figure 11. Results of Experiment 2 and the empirical equivalencesets in the space–time plot. The thin lines on the background arethe empirical equivalence sets we reconstructed from the resultsof Experiment 2 using the linear model in Figure 10. The pairs ofred connected circles represent the measured equilibrium points;each pair corresponds to a data point in Figure 10. The slopes ofboth the empirical equivalence sets and the lines connecting thecircles gradually change across the plot, indicating a gradualchange from tradeoff to coupling, as do the slopes of isosensitivitycontours measured for continuous motion at the threshold ofvisibility (Figure 5A). The oblique thick line indicates the boundaryspeed of 12-/s.


641 f(T)]Vwe obtained f(2T) from Equation 7. We used the642 coordinates [2T, f(2T)] to obtain f(4T) and thus iteratively643 propagated each solution to the maximal temporal644 distance in the figure. We repeated this procedure several645 times, starting at the same value of T but with a linearly646 incremented value of f(T), to obtain all the solutions647 plotted in the figure.648 The equivalence contours of apparent motion (Figure 11)649 are similar to the isosensitivity contours derived from650 measurements at the threshold of visibility (Figure 5A): In651 both cases, contour slopes change gradually from tradeoff652 to coupling as a function of speed. This result implies that653 there exists a monotonic relationship between human654 isosensitivity contours (Figure 5) and the equivalence655 conditions for apparent motion.656 We do not wish to suggest that the similarity between657 the equivalence conditions for apparent motion and658 human isosensitivity contours constitutes a reduction of659 the mechanisms of apparent motion to the mechanisms660 responsible for the detection of motion at the threshold.661 Rather, the similarity implies that the perception of662 motion is governed by the same constraints at the663 threshold of visibility and above the threshold.664

665 Why different regimes?

666 As we mentioned above, the equilibrium theory of667 motion perception (Gepshtein et al., in press) predicts that668 space–time coupling and tradeoff should hold at different669 speeds. The theory prescribes how a visual system should670 distribute resources to optimize the estimation of motion671 speed. The theory predicts uniformly suboptimal sets for672 speed estimation, which are shown in Figure 5B as the673 colored contours. The theory prescribes that equal amount674 of resources should be allocated to the uniformly675 suboptimal conditions, yielding normative predictions for676 the equivalence sets. The slopes of theoretical equivalence677 sets (Figure 5B) change smoothly, taking negative and678 positive values, which correspond to the tradeoff and679 coupling regimes of apparent motion, as it is the case in680 the equivalence contours of apparent motion (Figure 11).681 This similarity in the shape of the equivalence contours of682 apparent motion and the shape of theoretical equivalence683 sets suggests that the different regimes of motion684 perception emerge as an outcome of an optimization685 process that balances the measurement and stimulus686 uncertainties associated with speed estimation.687 Note that one might also predict the different regimes of688 apparent-motion perception at different speeds from the689 empirical isosensitivity contours (Figure 5A). To do so,690 one had to assume that equal motion sensitivities691 correspond to equal motion strengths. The equilibrium692 theory obviates that assumption because the theory tells693 which conditions are equivalent for motion measurement,694 regardless of whether it involves stimuli at the threshold695 or above the threshold.

696

697

698Shapes of equivalence contours

699In their quantitative details, the shapes of the empirical700equivalence contours of apparent motion are different701from the shapes of the empirical contours we derived from702the isosensitivity data (Kelly, 1979), as one can see by703comparing Figures 5 and 11. The empirical contours of704apparent motion are shallower in the region where we705could measure the equilibria of apparent motion. There706are two reasons we should expect such a discrepancy:

7071. Kelly (1979) obtained the estimates of spatiotemporal708sensitivity (Figure 5A) with narrowband stimuli709(drifting sinusoidal gratings), whereas our stimuli are710spatially broadband. Also, he used image stabilization711to limit motion on the retina, whereas our observers712were free to move their eyes during stimulus713presentation; as a result, our stimuli cover a broader714temporal-frequency band. Such increases in the width715of the spatial and temporal frequency bands flatten the716equivalence contours. We illustrate this in Figure 12,717which shows the results of simulating the effects of718widening of the frequency bands. We obtained the719three panels by averaging Kelly’s estimates of720sensitivity across an increasing range of spatial and721temporal frequencies and plotted the resulting equiv-722alence contours in the distance plot.7232. As Gepshtein et al. (in press) showed, a normative724theory predicts that estimates of the sensitivity of the725visual system depend on the task and the stimuli used726in obtaining the estimates. Consider a task that uses an727ambiguous stimulus for which motion matching is728difficult. Such a task depends more on estimating729stimulus frequency content than stimulus location. (See730also Banks, Gepshtein, & Landy, 2004, who empha-731sized the role of stimulus spatial-frequency content732for solving the binocular matching problem.) Accord-733ing to Gepshtein et al., an optimal visual system734should change the distribution of its sensitivity across735the parameters of stimulation so that its estimate of736stimulus frequency content becomes more reliable737than its estimate of stimulus location. On this view,738there would be little reason to expect a quantitative739agreement between the equivalence contours obtained740using different tasks and different stimuli.741

742Future research should address the question of how the743shape of equivalence contours depends on measurement744conditions by estimating contour shapes within observers745who perform a variety of tasks (e.g., stimulus detection746and motion-direction discrimination using unambiguous747stimuli, such as drifting gratings, and ambiguous stimuli,748such as random-dot kinematograms and motion lattices). 749

750The low-speed assumption

751Our results contradict the low-speed assumption as it was752formulated by Wallach (1935, 1976). The assumption was


753 motivated by observations that shorter motion paths are754 preferred to longer ones in apparent motion and by the755 aperture problem, as we mentioned the Regimes of756 apparent motion section. On this view, a low speed should757 always prevail in competition with a faster speed. This is758 not the case in our results. Under perceptual equilibrium,759 the competing motion paths were seen equally often, but760 the ratio of speeds in the two paths was always less than761 unity (Figures 8B, 9, and 10). That is, motion was not762 always seen along the slower path.763 Although our results are inconsistent with Wallach’s764 formulation of low-speed assumption, they are consistent765 with the evidence that low speeds prevail in the perceptual766 ecology. The equilibrium theory of motion perception767 (Gepshtein et al., in press) takes into account the768 distribution of speeds in the perceptual ecology (Dong &769 Atick, 1995), which implies that low speeds prevail in the770 natural stimulation. However, the theory does not predict771 that competition between alternative apparent-motion772 paths must be invariably resolved in favor of slower773 motion. It predicts that the outcome of competition774 depends on the degree of balance between measurement775 uncertainties and stimulus uncertainties. The degree of776 balance is reflected by the slope of the equivalence777 contours in the distance plot (Figure 5B). It is the sign778 of the slope that determines the outcome of competition.779 Thus, the equilibrium theory is consistent both with the780 fact that low speeds prevail in perceptual ecology and781 with our data showing that one motion path can dominate782 another, independent of which motion is slower.783

784 Conclusions

785 Korte’s counterintuitive law does hold under some786 conditions, but its claim to being a general law of motion

787perception is incorrect. The apparently inconsistent results788on apparent motion using suprathreshold stimuli are789special cases of a lawful pattern consistent with predic-790tions of a normative theory of motion perception and data791on continuous motion at the threshold of visibility. 792

793794Appendix A

795Demonstrations of motion lattices

796Four demonstrations of motion lattices M6 are avail-797able online at: http://pdl.brain.riken.jp/staffpages/sergei/798DEMOS/GepshteinKubovyM6.htm.799In Animation Movies 1 and 3, all lattice locations are800made visible for illustration, as in Figure 7. Only the filled801dots appeared in the actual stimuli, as in Animation802Movies 2 and 4.803

804805Acknowledgments

806We thank A. van Doorn, M. Landy, and C. van807Leeuwen for their comments on an earlier version of this808article; M. Rudd and I. Tyukin for stimulating discussions;809H. Hecht for help in translation from the German; and810E. Doherty for assistance in running the experiments. This811work was supported by NEI Grant 9 R01 EY12926.

812Commercial relationships: none.813Corresponding author: Sergei Gepshtein. AQ7814Email: [email protected]: Perceptual Dynamics Laboratory, Brain Science816Institute, RIKEN, Wako, Saitama, Japan.

Figure 12. An illustration of how the size of spatiotemporal frequency band of visual stimulus affects the shapes of equivalence contours. Wesimulated the effect of frequency band by averaging empirical estimates of spatiotemporal sensitivity (Figure 5A; Kelly, 1979) across a range ofspatial and temporal frequencies; this range grows across the panels, from left to right. The equivalence contours grow shallower as theaveraging range increases.


http://pdl.brain.riken.jp/staffpages/sergei/DEMOS/GepshteinKubovyM6.htm






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954


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AQ3: The phrase Bin press[ was added after Gepshtein et al. Please check if this is the correct reference.

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