0042-6989:91 53.00+ 0.00

Vivon Rcr. Vol. 31. No. 5. pp. 877-893. 1991 Printed ia Great Britain. All tights rc~r~cd

PERCEIVED

Copyright 0 1991 PergamoaPresspk

SPEED OF MOVING PATTERNS VINCENT P.

TWO-DIMENSIONAL

and HUGHR. WI~N FERRERA+

Department of Ophthalmology and Visual Science and Committee on Neurobiology, The University of Chicago, 939 E. 57th St, Chicago, IL 60637, U.S.A. (Receiued I5 June 1989; in reuisedform

16 April 1990)

Abstract--When two cosine gratings drifting in different directions are superimposed they can form a coherently moving two-dimensional pattern (plaid) whose resultant speed is related to the component vclocitcs by a geometric construction known as the intersection-of-constraints (IOC). When measured against a standard which has the same spatial frquency as its components, a plaid always appears to move slower than the IOC prediction. However, the perceived speed is generally faster than would be predicted if speed were judged based on the temporal frequency of either the components or the nodes of the plaid. On the other hand, when the standard has the same spatial period as the nodes, the plaid appears to move at the same rate as the predicted IOC resultant. Furthermore, a grating with the same spatial period as the nodes appears to move slower than a grating at the component spatial frequency. just the plaid does. It is therefore likely that speed is encoded similarly for both gratings and plaids, and that the perceived speed of both is determined by the spatial pcriodicity of the pattern. We have previously classified 2D moving patterns as either type I (resultant lies between component directions) or type II (resultant outside of components). We find that the perceived speed of both types can be accounted for on the basis of the nodal spatial period. Finally we present a model for velocity coding which is based on the responses of spatio-temporal mechanisms.

Motion

Velocity coding

Spatio-temporal

INTRODUCTION

One of the problems in early visual processing is to determine the velocity of moving surfaces in the two dimensions of the image plane. In order to completely specify velocity, both the speed (V) and dirction (4) of the surface must be determined. However, given a one-dimensional stimulus pattern (e.g. a grating), one cannot unambiguously determine V and 4, but only the product V COS(C$ - 4’). where 4’ is the direction orthogonal to the orientation of the grating. This is known as the problem of motion ambiguity, or the aperture problem (Wallach, 1935, 1976). Thus, in order for the visual system to recover both components of velocity, the stimulus must contain energy at two or more distinct orientations. One of the simplest stimuli which satisfies this condition can be produced by physically superimposing two drifting cosine gratings with different orientations (the speeds of the two gratings may or may not be different). If the two gratings are

Two-dimensional

Discrimination

similar enough in spatial frequency, contrast and drift rate, they will lock together to form a coherently moving two-dimensional pattern (Fennema & Thompson, 1979; Adelson 8c Movshon, 1982). It has been suggested that the motion of such a pattern can be accounted for by a geometric construction known as the infersection-ofconstraints (Adelson & Movshon, 1982). This construction is illustrated in Fig. 1. The thin vectors represent the velocities of the two gratings, sampled in the directions orthogonal to their orientations. The constraint lines, which are tangent to the component motion vectors, represent the range of surface motions which could produce each estimate. The point where the two constraint lines intersect (thick vector) gives the resultant pattern motion which is consistent with both components. This construction predicts that the speed of the pattern, VP, will be related to the speed of its components, V,, and the angle between the direction of the resultant and the components, 8, as follows:

*Present address: University of Rochester, Department of Physiology, Box 642. Rochester, NY 14642, U.S.A.

VP= vJcos(e). 877

(1)

878

Vwz~.s;r P. FERBERA

We tested this prediction by measuring perceived speed discrimination thresholds for two-dimensional moving patterns. We found that when a two~imensiona1 plaid was compared to a grating of the same spatial frequency as the components, the apparent speed of the plaid was underestimated relative to the intersection-of-constraints prediction. Furthermore, the degree of underestimation increased as a function of the angle between the components. However, a plaid has conspicuous nodes where the light and dark extrema of the individual gratings coincide. If the speed of a plaid is compared to that of a grating which has the same spatial period as the nodes of the plaid, then a match is made when the gratings is moving at the speed predicted by the intersection-of-constraint. In a previous study (Ferrera & Wilson, 1987), we made a distinction between type I and type

component i

and HUGH R. WIWN

II patterns (Fig. 2). For type I patterns, the res&nt ties between the two components, whereas for type If patterns, the resultant lies outside of the two components. We found that type I patterns (previously referred to as “plaids”) had a much greater ability to mask the detection of a localized test pattern moving in the direction of the resultant than type II (previously referred to as “blobs”). In a second study (Ferrera & Wilson, 1990). we found that the perceived direction of type I patterns agreed with the intersection-of-constraints prediction and that direction discrimination thresholds for type I patterns were as good as thresholds for one-dimensional patterns (about I .Odeg). However, direction discrimination thresholds for type II patterns were roughly 5 times greater than thresholds for type I patterns, and the perceived direction of type II patterns was significantiy biased toward the mean direction of

component

2

plald

Fig. 1. (A) The supposition of two gratings moving in different directions gives rise to a coherenfly moving plaids. (E) The intersection-of-constraints in velocity space. ?he axes labeled V, and V, represent the horizontal and vertical components of velocity, respectively. The thin arrows (CI and C2) represent the motion of the individual gratings which comprise the stimulus. Each grating gives rise to a constraint line which represents the family of physical motions which is consistent with the motion of that grating. The point where the constraint lines intersect yields the resultant motion (R) which is consistent with both gratings. Note that the constraint lines are at right angles to the grating motion vectors so that the magnitude (speed) of the resultant is equal to the speed of the grating divided by co@?).

Perceived speed of moving plaids

TYPE I

879

TYPE II

Fig. 2. St:ltic versions of type I and type II patterns. The diagrams to the upper right of cxh pattern are v&city space rcprcscntations showing the rclativc directions and speeds of the components (thin arrows) as well as the spczd and direction of the resultant (thick arrow). The double hcadcd arrows adjacent to cxh pattern indicate the cllixxivc pattern sprrtiA period mcasurcd in the dircxtion pnrallcl to the resultant.

the two component gratings by an average of 7.5 deg. With respect to perceived speed, however. we have been unable to find any systematic differences between type I and type II patterns that cannot be accounted for on the basis of the internode spatial period. In addition, speed discrimination thresholds are similar for both pattern types. Finally, we tried lo model our perceived speed and speed discrimination data using two theoretical approaches. To model the discrimination thresholds, we considered the effect of noise in the component motion signals on the distribution of resultant speeds predicted by the intersection-of-constraints. This was accomplished by measuring the speed and direction discrimination thresholds for individual components (i.e. obliquely moving gratings). and then using these estimates of component uncertainty in Monte Carlo simulations of the intersection-of-constraints transformation. In the case of type I plaids. we verified the simulations by deriving an analytic approximation for the distribution of resultant speeds. To make predictions for the dependence of perceived speed

on pattern spatial period, we constructed a model based on an array of spatio-temporally separable mechanisms (Thompson & Movshon, 1975; Wilson, 1980a). The point of this model was to test two different schemes for using the output of such mechanisms to encode speed. Specifically, one approach was to pool the outputs of mechanisms which have the same ratio of preferred temporal to preferred spatial frequency, as these units would lie along a line of constant speed in frequency space (McKee, Silverman & Nakayama, 1986). The other approach was simply to pool over units which have the same preferred temporal frequency, without compensating for spatial frequency. Surprisingly, the latter approach makes better predictions when some realistic assumptions about the relative sensitivities and contrast response of the mechanisms are incorporated into the model (Wilson, 1980b; Lehky, 1985). METHODS

Patterns were generated on a Macintosh II microcomputer and displayed on an Apple

880

VINCENT P. FERRERI and HUGH R. WILSDN

high-resolution monochrome monitor with a P4 experiments. Subjects sat facing the display with (white) phosphor and a 66.7 Hz vertical scan- their heads comfortably positioned in a chin ning frequency. The spatial resolution of the rest. Viewing was monocular, and the unused display was 76 pixels,‘in., and the luminance of eye was covered with a translucent occluder. each pixel was resolved with &bit accuracy. The Subjects were instructed to fixate the center of display had a mean luminance of 17.7 cd/m2, the circular field, which was 8.0 deg in diameter and was viewed through a circular aperture in a for most experiments, but 4.0 deg for spatial cardboard surround that was illuminated at the frequencies greater than 2.0c/deg. A central same mean level and approximate hue as the fixation spot, approx. 5 min in diameter, was monitor. The nonlinear intensity response of the used to minimize voluntary eye movements. phosphor was compensated for in software by We used a two-interval forced-choice method using a specially calibrated color look-up table. of constant stimuli paradigm in which one The largest possible distortion product from temporal interval contained a two-dimensional residual nonlinearities was calculated to be less test pattern and the other contained a onethan 0.4% contrast. dimensional standard moving in the direction of All patterns were composed of two one- the intersection-of-constraints resultant for the dimensional cosine gratings, both of which had test. The test and standard were presented in the same spatial frequency and contrast but random order, and the two intervals were separdifferent orientations and drift rates. Except ated by 200 msec during which the screen was where otherwise noted the contrast of each blank. For each trial, a small speed increment component was 40%. and thus the plaid con- (or decrement) was added to each component of trast was 80%. Pattern motion was achieved the test. However, the ratio of the components using the technique of color table animation speeds, and hence the resultant direction (Shoup. 1979: Sheets, 1988; Apple, 1988; remained vertical. The direction of the test and Knastcr. 1988). A pixel map consisting of a standard (up vs down) varied randomly from lincnr ramp of gray Icvcls was first loaded into trial to trial, The proportion of trials on which scmxn memory, and then a cosine of the approthe test was seen to be moving faster than the priate frequency was scrolled through the color one-dimensional standard was plotted as a funclook-up table, thereby transforming the ramp tion of test speed and fitted with a Quick (1974) into a drifting cosine grating. The look-up table function using a maximum likelihood cstiwas changed once per frame (66.7 times/set) and mation procedure. The speed of the test (I’,. the phase shift of the cosine between successive expessed in terms of the inters~tion-of-conframes, and hence the spatial step size, AX, straints resultant) which gave a perceptual could be made in sub-pixel increments. The two match to the standard (V,) was calculated by stimulus components were displayed on alter- determining the speed which corresponded to natc pixels, with the alternations forming a the 50% level of the best-fitting Quick function. checkerboard pattern. This method does not The ratio k’,/V, therefore, expresses how fast the avoid distortions which can be introduced by test appeared to move relative to the intersecthe inability of the electron guns to follow large tion-of-constraints prediction. Speed discrimisteps in intensity between horizontally adjacent nation thresholds, AL’, were determined by pixels. However, we found the same results in taking half the difference of the 25% and 75% controls experiments where the two components levels. These thresholds were converted to were displayed on alternate raster lines. The two Weber fractions by dividing AY by the mean of components also used different sets of color the set of test speeds (Y,,,). table entries, and were therefore temporally independent. Both stimulus components were RESULTS present simultaneously. The one-dimensional We first consider the case of type I symmetric gratings used as standards were displayed in the same manner, except that both components had plaids. We measured perceived speed for plaids the same orientation. The contrast of the stan- whose components ranged in spatial frequency dard was 80%. Each pattern was presented in a from 0.5 to 4.0 c/deg and in temporal frequency from 2.0 to 16.0 Hz, although we did not try all 450 msec interval and the contrast was constant possible combinations. Relative perceived speed throughout the presentation. A total of six subjects participated, four of (V,/ Y,) is plotted as a function of 6 in Fig. 3. whom were unaware of the purpose of the When the spatial frequency of the one-dimen-

Pertxivcd speed of moving plaids

sional standard was qua1 to that of the components of the plaid (solid symbols), the plaid always appeared to move slower than the IOC prediction (dashed line). As the angle between the component directions increased, so did the deviation of the perceived speed from the predicted speed. When 8 was qua1 to 75 deg, the perceived speed was only about 50% as fast as predicted. However, the perceived speed was always faster than would be predicted if subjects were simply matching the temporal frequency of either the components or the nodes (where the light and dark extrema of the components intersect) of the plaid to that of the standard. One can easily verify that the temporal frequency of the nodes is the same as that of the components,

(A) l.7,

provided that both components have the same speed. If component speed is held constant, the predicted resultant speed increases in proportion to l/cos(f?). Therefore, temporal frequency matching predicts that the ratio VJV, should fall off in proportion to cos(0). The predictions for temporal frquency matching are shown as the dotted lines in Fig. 3, and the data clearly lie above the predictions. Each set of solid symbols in Fig. 3 represents a set of patterns all of which have the same cu~p~~e~r speed. Thus, as the angle 0 increases, the predicted pattern speed gets faster. It is therefore possible that the faster moving patterns appear relatively slower simply because the response of velocity coding mechanisms

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Fig. 3. Perceived speed as a function of 0 for three sujccts. WC solid symbols represent the case where standard was equal to that of the components of the two~imensiona~ test pattern. Arrows indicated pairs of patterns which have nearly identical predicted resuitant velocities (for solid arrows, resultant = 8 degi’sec;for open arrows, resultant = 16 deg/sec). The !he spatial frequency of the one-dimensional

open symbols represent the case where the spatial frequency of the standard was equal to that of the nodes of the plaid. The dashed line indicates agreement with the inter~tion of constraints modti. Xx dotted lint indicates the speed at which the temporal frquency of the standard matches that of the components or nodes of the plaid. The component speeds are indicated in the figure legends.

882

VISCEIVT P. Fw

saturates at the higher velocities. However. this hypothesis is untenable first because the fastest resultant speed used was only 16.0 deg sec. whereas good speed disc~mination for gratings has been obtained at speeds as high as 15.0.24.0 and even 64.0deg/sec (McKee et al.. 1986; Smith, 1987; DeBruyn & Orban, 1988). A saturating response nonlinearity would predict increasing disc~mination thresholds. Second. one would expect the relative perceived speed to fall off much faster for higher component speeds. In fact, two subjects (TF and KMB) show small moderate effects of component speed, while a third (VPF) shows no effect. Finally, one would expect patterns with the same predicted resultant speed to have the same perceived speed. For example, a pattern with 0 = 60deg and component speed = 8 deg/sec should have the same perceived speed as a pattern with 0 = 75 deg and component speed = 4 deg:‘sec (see open arrows in Fig. 3B and C). When all possible pairs were compared (i.e. 0 = 60, v, = x vs 0 = IS, I-, = 2.r; 0 = 75, v, = I vs 0 = 60. V,. = 2x; and 0 = 75. V, = .r vs 0 = 15, Vc = 4.~) there wcrc signilicant differences in perceived speed (P c 0.05 for all three LXX%).

Apparent contrast has frequently been cited as a factor which can confound relative speed judgments (McKee et al., 1986; Smith, 1987). Apparent contrast decreases as temporal frequcncy is increased above about 8 Hz (Robson, 1966). Since the task we gave our subjects was to choose the interval containing the faster moving pattern. it is possible that they could instead have chosen the pattern of lower apparent contrast. This would have resulted in responses that were biased in favor of the standard (grating), as its temporal frequency was generally higher than that of the plaid. This efTcct would result in an artifactual reduction of the ratio V,/V, because it would require a higher plaid speed to match any given grating speed. We therefore performed a control experiment in which the plaid components were reduced to 70% contrast, while the grating was kept at 80%. Under these conditions, the apparent contrast of the plaid was at all velocitcs decidedly lower than that of the grating. Thus any bias toward choosing the pattern of lower contrast would artificially increase our measurement of the perceived speed of the plaid. In fact. we found no significant difference (P > 0.40) in perceived speed between plaids at the two different contrast levels.

and HCGH R. WI-N

It has been reported that spatial frequency affects perceived speed for drifting and rotating gratings such that higher spatial frequency gratings (up to about 4.0 c/deg) appear to move faster (Diener, Wist, Dichgans & Brandt, 1976; Campbell & Maffei. 1979). For a plaid, the spatial period of the nodes increases as the angle between the component directions broadens, We therefore remeasured the relative perceived speed of the plaid as described above except that we replaced the standard with a grating whose spatial period was the same as that of the nodes (and therefore lower than the spatial frequency of the components). We found that the ratio V,/ V, clustered around a value of 1.0 for four subjects (open symbols in Fig. 3). In other words, the perceived speeds matched when the grating was moving at the speed predicted by the intersection-of-constraints. We also measured perceived speeds for the case where both standard and test were one-dimensional gratings and found the same effect, i.e. that gratings of lower spatial frequency match gratings of higher spatial frequency which are moving significantly slower (open symbols in Fig. 4). In Fig 4, the relative perceived speed for both gratings and plaids is plotted as a function of relative spatial period, i.e. the spatial period of the standard divided by that of the test. These plots can be fit with a high degree of correlation (r > 0.98) by a straight line in log-log coordinates, indicating a power law relationship, The exponents for four subjects clustered around a value of 0.5 (0.45 + 0.03 I SEM), which suggests a square root law for perceived speed, as follows:

SP,,undr*d O5. $=k ___ sp,cs, I

( >

(2)

The highest spatial frequency we have tested is 4.0c/deg. There is evidence that the effect reverses, i.e. higher spatial frequencies appear slower, above 4.0c/deg (Campbell & Mafiei, 1979). Since the speed of a grating can be derived from its temporal (0,) and spatial (0,) frequency by the relation V = w,,h,, it is possible that the shift in perceived speed might be caused either by an increase in perceived spatial frequency or a decrease in perceived temporal frequency. There is some evidence that perceived spatial frequency increases at higher rates of temporal modulation (Gelb & Wilson, 1983). However, under the present conditions, we

Peraivcd

speed of moving plaids

found that when subjects were asked to match the spatial frequency of a drifting grating to that of a stationary grating flashed for the same duration, they always made a veridical match. Likewise, subjects could always make a veridicial temporal frequency match when presented with two counterphase flickering gratings of different spatial frequencies. These observations are consistent with the notion that speed is extracted by combining the output of mechanisms whose individual responses are separable in

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of relative

standard

1 spatial

was a ?.Oc/deg

grating and the tests were either gratings of various spatial frcqucncics components.

or

plaids

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with

different

the plaids,

angles

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between

the

spatial

frc-

qucncy was 2.0 c/deg. The drift rate for each test pattern was either 4.0 or 8.0 Hz. as indicated of the best-fitting around

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resent the expected temporal

in the legend. The slopes

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results for true velocity

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of spatio-temporally

anisms.

dashed

obtained

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line in (A)

when a constant-ratio mechanism

separable

mcch-

is the prediction

rule is used for pooling the responses.

883

spatial and temporal frequency (Wilson, 1980a, b; Adelson & Bergen, 1985). Speed discrimination thresholds (expressed as Weber fractions) are plotted as a function of 8 in Fig. 5. For each set of data points, component speed was held constant, but pattern speed increased with 0. In general, speed discrimination thresholds are lowest for component drift rates between 4.0 and 8.0 Hz, which is consistent with the temporal frequency dependence found by other measurements of speed discrimination using single gratings (McKee et al., 1986; DeBruyn & Orban, 1988), and plaids (Welch, 1989). Welch (1989) has found that speed discrimination thresholds for moving plaids depend on component speed. but not on pattern speed. If this were the case, then one would expect the curves in Fig. 5 to be perfectly flat, as each data set consists of patterns which had the same component speed. However, most of the curves show significant deviations from flatness, especially those measured with the lower component speeds. Both component speed and pattern speed were found to bc significant factors (P < 0.05) in determining discrimination thresholds when a two-way analysis of variance was performed on the data for the three subjects in Fig. 5. There is, however, a significant problem in the interpretation of these discrimination thresholds as on each trial the subject is comparing two different patterns, i.e. one interval contains a grating and the other contains a plaid. In addition, as 0 increased, the speed of the grating was also increased in order to keep pace with the speed of the plaids. It is therefore possible that an improvement in discrimination thresholds could be due to improved discrimination of the grating, rather than the plaid. To control for this effect, we performed an additional experiment in which we measured discrimination thresholds using a plaid in both intervals. Furthermore, we intentionally chose a component drift rate (I .O Hz) for which single grating speed discrimination thresholds were relatively poor, on the assumption that if there is a difference between component and pattern discrimination thresholds then the measured threshold should correspond to the lower of the two. The results for two subjects are plotted as a function of pattern speed (determined by the intersectionof-constraints) in Fig. 6, and clearly deviate from a flat line even though component speed is constant across all angles. This result can be explained by considering the uncertainty in the

884

VINCENT P. FERMICA and HUGH R. WIWN

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I VERSUS

TYPE

speeds, there is a clear tendency for

thresholds IO dccrcase as resultant

component motion signals and its effect on the ability to estimate the speed of the resultant (see Monte Carlo Simulations, below). TYPE

speed (see legend). Thus, the resultant speed within

of 0. At the lowest component

II

We have previously reported differences between type I (resultant direction lies between components) and type II (resultant direction falls outside components) patterns with regard to masking, perceived direction and direction discrimination thresholds (Ferrera & Wilson, 1987, 1990). Specifically, type II patterns generate weaker masking effects, have higher direction discrimination thresholds, and, unlike type I patterns, their perceived direction does not agree with the intersection-of-constraints, but is biased toward the direction of the components. We were therefore interested in determining

speed increases.

whether differences existed between type I and type II patterns with regard to perceived speed and speed discrimination. Stationary versions of type I and type II patterns are shown in Fig. 2 along with their respective velocity space representations, and the parameters are listed in Table I. Type I and type II patterns were divided into three groups (A, B and C), as indicated by the letters in parentheses. We constructed the patterns in groups A and B as complementary pairs, such that they have a single component in common (component a in Fig. 2) and their other components are mirror images of each other, reflected about the axis of the resultant (components b and 6’ in Fig. 2). The intersection-of-constraints resultant for all pattern types was upward (0 deg), and the component directions are given relative to the resultant with the convention that rightward is

Porccived sped of moving plaids

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Fig. 6. Plaid speed Weber fractions as a function of pattern speed for two subjects, measured under conditions where the test and standard were identical patterns. Component spatial frequency is indicated in the legend. Component drift rate was I.0 Hz for all data points, while pattern speed increases with 6. The range of plaid angles went from 0 to 80.5 deg.

+90 deg and leftward is -90 deg. Note that component speed is given as a fraction of resultant speed. not as the actual speed used in the experiments. Perceived speed and speed discrimination thresholds were measured as a function of pattern type, component spatial frequency and resultant speed for three subjects. In all cases, the spatial frequency of the one-dimensional standard was the same as the spatial frequency of the test components. Perceived speed for type I and type II patterns is plotted as a function of component spatial frequency in Fig. 7. The perceived speed for all patterns types was significantly different from the intersection-ofconstraints prediction (P < 0.05 for all subjects; note: the IOC predicts YJV, = I .O, as in Fig. 3). There was no consistent dependence of perceived speed on component spatial frequency over the range tested. Only one subject (KMB) showed an effect of component speed, i.e. V,/ V, was generally lower for faster component speeds. This subject showed the same effect for

both type I and type II patterns, and also for symmetric plaids, as discussed in the last section (see Fig. 3C). Pattern spatial period was a highly si~i~cant factor in dete~ining perceived speed. For groups A and B, the pattern which had the smaller spatial period (i.e. type I) also appeared significantly faster (P c 0.05 for all subjects). Patterns I(C) and II(C) had the same spatial period and also had similar perceived speeds (P >0.05). We tried to predict the perceived speed of type I and type II patterns based on the pattern spatial period by using the same power function relating spatial period to perceived velocity for one-dimensional gratings discussed in the last section. The estimates we obtained are shown as the straight lines running across the graphs in Fig. 7 (the figure legend indicates which prediction goes with each set of data). There are two cases where the predictions deviate signifiantly from the data, namely pattern II(C) for VPF and pattern I(A) for TF. However. the overall agreement between the actual and predicted values is reasonable (r = 0.82) considering that there were several factors which complicated the predictions. The most significant of these is the fact that the spatial period of asymmetric patterns can be measured along two orthogonal axes, neither of which is aligned with the direction of the resultant. We found that, empirically, the distance between the nodes measured in the direction most parallel to the perceived resultant (Ferrera & Wilson, 1990) gave the best predictions (this distance is indicated by the double-headed arrows in Fig. 2). We also measured speed discrimination thresholds for the patterns in Table I. When we compared disc~mination thresholds within each of the three groups, we found no significant differences between type I and type II patterns (P > 0.2 for all subjects). We therefore concluded that there are no fundamental differences in the way that speed is encoded for type I and type II patterns, and that differences in

Table I. Parameters for type

Pat tern IfA) II(A) I(R) II(R) I(C) II(C)

Description Type I asymmetric Type Type Type Type

II I asymmetric II I symmetric

Type II

Component I Direction (deg) gpetd 0.67 0.67 0.83 0.83 0.92 0.92

-48.2 48.2 -33.6 33.6 -23.5 23.5

885

I and type 11 patterns Component 2 Direction gpced fdeg) 0.33 0.33 0.33 0.33 0.92 0.33

10.5 70.5 70.5 70.5 23.5 70.5

Resultant Direction (deg) %=d 1.0

0.0

t ::

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t :: I.0

0.0 0:o 0.0

VINCENT P. FEFZRERAand HUGH R. WIWH

886

(A)

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TF

I

Fig. 7. Rclativc

pcrccivcd

speed for type

I and type

10

I

Component rpaliol

II patterns

as a function

frequency tc/deg)

of component

spatial

frcqucncy for three subjects. The lines running across each graph are predictions based on the relationship between percieved speed and spatial period for one-dimensional prediction

for pattern I(B), the dotted line is for II(B).

line is for II(C).

For TF.

the solid line is the prediction

for I(A) and the dashed line is for II(A).

For KMB.

gratings.

For VPF,

the solid line is the

the dashed line is for l(C) and the dashed-dotted for pattterns

I(C) and II(C).

the solid line is the prediction

the dotted line is

for pattern I(A) and

the dashed line is for II(A).

perceived speed can be accounted for on the basis of the spatial period of the nodes of the pattern. MONTE

CARLO

SIWJLATIONS

We have considered the possibility that our results could be explained by uncertainty in the estimation of the speed and direction of the individual component gratings. This technique has been used by Nakayama and Silverman (1988) to predict thresholds for the perception of rigid motion in drifting sinusoidal lines. Figure 8 shows what happens to the intersection-of-constraints when small deviations in speed and direction are taken into account, as indicated by the gray circles around the heads of

the component velocity vectors. The shaded area surrounding the resultant vector shows the effect of a small amount of variance in the component velocity signal on the distribution of resultants. Thus, by the geometry of the intersection-of-constraints we can estimate the effects of a give level of uncertainty in the initial estimates of component velocity on the mean and variance of the distribution of resultants. To determine the relationship between component noise and the distribution of resultants, we ran Monte Carlo simulations in which, on each trial, random components were sampled from Gaussian distributions centered around the actual stimulus component. We used an elliptical error function in which the standard deviation of the Gaussian distribution was

Perceived speed of moving plaids

Fig. 8. E&t

of uncertainty

component resultants.

speed

on

of

the distribution

in the component

by the gray circles around

of

velocity signal is the heads of the

velocity vectors (thin arrows). The intersection-

of-constraints distribution

in the internal representation

direction

Uncertainty

represented component

and

translates

the component

uncertainty

into a

of resultant velocities (gray region surrounding thick arrow).

detcrmincd by measuring speed and direction discrimination thresholds for obliquely moving gratings (45”). We measured a direction discrimination threshold of I.0 deg and a speed discrimination Weber fraction of 10%. We also compared speed discrimination thresholds for upward drifting gratings vs oblique gratings. WC found that there was no oblique effect for speed discrimination as there is for direction discrimination (Ferrera & Wilson, 1990). In other words, there were no significant differences between speed discrimination thresholds for oblique and vertically moving gratings (paired I-test P > 0.15 for all subjects, estimates were paired by grating speed and spatial frequency). The probability of obtaining a component with a particular speed and direction is given by the following:

xexp

II

(0 -&)*+(u -uJ2

I[ -

2a:

20;

;

(3)

where 0, and L: are the direction and speed, respectively, of the stimulus components, and ug and Q, were derived from the direction and speed discrimination thresholds for an obliquely moving grating by the following formula: (4)

887

where AV is the discrimination threshold and z(c) is the z-score corresponding to a proportion, c, of the area under distribution between -cc and z. For 75% thresholds, c = 0.75. The relationship between two-interval forced-choice thresholds and the underlying signal and noise probability distributions can be derived from signal detection theory (McNicol, 1972). Once each random component was selected, a resultant velocity was computed using the intersection-ofconstraints. By running a large number of trials (typically 10,000) we obtained a distribution of resultants which could be used to predict both perceived direction and discrimination thresholds. The results of these simulations failed to provide quantitative agreement with the data for perceived speed and also predicted significantly higher speed discrimination thresholds for type II patterns relative to type I. First, the simulations failed to predict the dependence of perceived speed on 0 for plaids. In fact, the mean of the distribution of simulated resultants was never significantly different from the intersection-of-constraints resultant for either type I or type 11 patterns (i.e. the dilference was less than I% of the mean). Second, the Monte Carlo method predicts that speed discrimination thresholds for type II patterns should be 1.6-2.3 times greater than the complementary type I pattern (see Table I). The data of the previous section show that there is no significant difference in speed discrimination thresholds between these two pattern types. Finally, the predicted speed discrimination Weber fractions were always a constant fraction (approx. 0.71) of the component speed for Weber fractions, and did not depend on 0. This helps to explain how speed discrimination Weber fractions for plaids can be lower than for their components. The reason being that the variance in the resultant speed does not increase as fast as the mean-a result which can be derived analytically, as shown below. The prediction that speed discrimination thresholds are independent of the angle of the plaid also agrees with Welch’s (1989) finding that speed discrimination depends on component speed, rather than pattern speed (which covaries with 0). Unfortunately, the lack of dependence on 0 is essentially an artifact as the simulation assumes that the plaid components are always detected no matter how close their independently,

VISIT

888

P. FERRERAand HUGH R. H’IWN

directions are. If both components are within the direction bandwidth of a single motion detector, then the noise will be correlated and they will effectively be “seen” as a single grating. Stone (1989) has also used the correlation between detectors to explain direction discrimination of plaids. Thus, a more realistic model would predict a gradual improvement as plaid speed increases up to some optimum, in agreement with the data (Fig. 6). The reason that discrimination thresholds then get worse at higher pattern speeds could have to do with the selectivity of pattern direction selective mechanisms. These mechanisms may be insensitive to higher pattern speeds or incapable of combining components whose directions are outside a given range (this would also account for the more fluid appearance of plaids with the broadest angles). As a means of checking the results of the Monte Carlo simulations, one can also compute the standard deviation of the resultant speed directly using perturbation methods. This is done by first expressing the resultant speed as a function of the component speeds and directions and then determining the effects of small changes in component speed and direction by considering the linear term in the Taylor series expansion of that function. This analysis has been used to calculate the standard deviation of rcsultunt direction by Stone (1988. 1989). The expression for the standard deviation of the resultant speed (g,,) that one obtains is the following:

J2 cos 6’

where V, is the components speed, tI is the angle of the plaid (see Fig.lB), a,, is the standard deviation of the component speed. As the mean resultant speed increases in proportion to cos(O)-‘, this predicts that the Weber fraction for plaid speed discrimination should be l/J2 times the Weber fraction for the components, which agrees the factor of 0.71 obtained in the Monte Carlo simulations. IMODEL

We now present a model for speed coding which attempts to explain quantitatively the manner in which perceived speed depends on spatial frequency. This model assumes that perceived speed is encoded by the output of an array of spatio-temporally separable

mechanisms. The purpose of the model is to determine how the outputs of the mechanisms should be combined and also to investigate the effects of nonlinearities in the responses of the mechanisms and differences in their relative sensitivities. The main conclusion drawn from this model is that the visual system does not pool the responses of mechanisms which have the same ratio of preferred temporal to preferred spatial frequency to achieve constant speed tuning, as such pooling leads to predictions which are drastically incorrect. There is considerable evidence that both psychophysical mechanisms and single units in area 17 have responses which are separable in spatial and temporal frequency (Wilson, 1980a; Thompson & Movshon, 1975; Lehky. 1985). Furthermore, it has been estimated that there are about six such mechanisms spanning the range of visible spatial frequencies (Wilson, McFarlane & Phillips, 1983) and 3~4 spanning the visible temporal frequency spectrum (Mandler & Makous 1984; Lehky, 1985). The bandwidths of these mechanisms at half-height range from 1.25 to 2.5 octaves in spatial frequency (Wilson et al., 1983) and 2.0 to 2.5 octaves in temporal frequency. Thus. we start out with a simple model consisting of 12 mcchanisms tuned to four diffcrcnt preferred spatial frequencies and three preferred temporal frcquencies. The sensitivity of each mechanism to a stimulus of spatial frequency w, and temporal frequency w, is given by a two-dimensional Gaussian: 5, (a,, QJ,)= A, xexp

(o,-co;)yw,-oJ;): a,Z af

I[ -

1 1 xIn2.

(6)

The parameters A, determined the relative sensitivity of each mechanism and a, and a, were scaled appropriately so that the mechanisms had spatial bandwidths between I.5 and 2.5 octaves and temporal bandwidths of 2.5 octaves. The peak spatial and temporal frequencies of each mechanism were represented by w: and w] resepectively (see Table 2 for model parameters). The outputs of the filters are then sent through a contrast nonlinearity which is compressive at medium to high contrasts. The output of this stage is a function of filter sensitivity, S,, and stimulus contrast, C, which we modeled

Pmcived spdofmovingplaids

using the transducer function derived by Wilson (1980b): R,(S,C) =

A

([I + (S,C)Q]'l' - I) k[E + sic]” *

(7)

A, B, and E are given by: A

=H(l

k -r)Sf-’

E =3+c-1 Q B xA”E-

1

(10)

N and c are empirical constants which are derived from contrast increment threshold data (see Wilson, 1980). H is positive, 6 ranges between zero and unity. A value of 4.0 was used for Q, which is the exponent in Quisk’s (1974) formula for probability summation, and k = 0.2599, To summa~ze, we can express the response of each mechanism, R,, to a stimulus of any spatial and temporal frequency, as a function of filter sensitivity and stimulus contrast. We can now compute a speed index, VI, by taking a weighted average of the mechanism responses:

where R, is the response of the ith mechanism, and V, is the speed which is signaled by that

mechanism. However, we have yet to determine exactly what speed is to be signaled by each mechanism. One approach which seems plausible is to make V, proportional to o;/o;; in other words, to make the preferred speed for each mechanism proportional to the ratio of peak temporal frequency to peak spatial frequency. This leads to the prediction that perceived speed should decrease with increasing spatial frequency because a low spatial frequency grating will tend to stimulate mechanisms signahng high and moderate veIocities whereas a high spatial frequency grating will tend to stimulate mechanisms signaling moderate and low velocities. On the other hand we could simply make V, proportional to 0;. This leads to the prediction that perceived speed should increase in direct proportion to the spatial frequency of the stimulus, provided that the physical speed is held constant. In fact, we have shown that perceived speed for gratings which have the same physical speed is approximately proportional to the square root of the spatial frequency, at least

889

below 4.0c/deg. Second, two gratings with the same temporal frequency would be predicted to appear at the same speed, when in fact, the grating of lower spatial frequency appears somewhat faster. Thus, this scheme predicts shifts in perceived speed as a function of spatial frequency which are in the right direction, but of the wrong magnitude, i.e. the shifts are too large. However, we have yet to consider the effect of differences in the relative sensitivities or in the characteristics of the contrast nonlinearities of the different mechanisms. All available evidence suggests that at low spatial frquencies, higher temporal frequency m~hanisms are more sensitive, whereas the reverse is true at high spatial frequencies (see Lehky, 1985). When the relative sensitivities of the various mechanisms are adjusted accordingly, the result is that the perceived speed of lower spatial frequency patterns is increased somewhat, while that of higher spatial frequency patterns is reduced. Likewise, data on contrast increment thresholds (Wilson, 1980b; Swanson, Wilson & Giese, 1984) and masking as a function of contrast (Wilson et al., 1983; Lehky, 1985; Ferrera & Witson, 1985) suggest that the exponent t is greatest at low spatial and high temporal frequencies (e.g. 0.5 c/deg and 8.0 Hz), with values around 0.8-0.9, and decreases to a value between 0.25 and 0.4 at high spatial and low temporal frequencies (e.g. 4.0-8.0 c/deg and 1.0 Hz). The model parameters are Iisted in Table 2. This model was used to make speed matching predictions in the following manner. First, a perceived speed was calculated for a test pattern of a given spatial frequency and speed (V,). Then, for a standard of a different spatial frequency, the program iteratively modified the temporal frequency until the perceived speed calculated for the standard was equal to that of Table 2. Parameters for 12 s~tiotem~ral mechanisms Pzc

BanZkhh

0.5

2.5

1.0

2.0

2.0

1.5

4.0

I.5

r& 1.0 4.0 8.0 1.0 4.0 8.0 1.0 4.0 8.0 1.0 4.0 8.0

Bandwidth

A

c

2.5 2.5 2.5 2.5 2.5

35.0 59.0 96.0 60.0 75.0 85.0 75.0 75.0 65.0 80.0 51.0 30.0

0.31 0.78 0.95 0.37 0.66 0.95 0.38 0.67 0.89 0.30 0.39 0.40

;:: 2.5 2.5 2.5 2.5 2.5

VISCEXT P. FERRERAand HUGH R. WLSON

890

the test. The true speed of the standard (V,) was then calculated from its spatial and temporal frequency and the ratio VJV, was calculated and plotted vs the relative spatial period of the test with respect to the standard. The predictions of the model for two subjects are shown in Fig. 4 (heavy solid lines). The same set of parameters was used to make predictions for both subjects at two temporal frequencies (4.0 and 8.0 Hz) each and the overall correlation was good (r = 0.95). The model tends to predict lower relative perceived speeds for higher temporal frequency patterns, a tendency which can be seen in the data for two subjects (Fig. 3B and C). Thus, pooling responses over units which have the same preferred temporal frequency can explain the dependence of perceived speed on spatial period. However. when the model was modified so that responses were pooled over mechanisms which had the same ratio of preferred temporal frequency to preferred spatial frequency, the predicted trend was opposite to that found in the data, i.e. the model predicted that lower spatial frcqucncy patterns should appear faster (Fig. 4, heavy dashed line). The constant-ratio version of the model could only be made to fit the data if the sensitivities of the mechanisms were adjusted so that low spatial frequency mechanisms were most sensitive to low temporal frequencies and high spatial frequency mechanisms were most sensitive to high temporal frequencies, which is contrary to a large body of psychophysical and physiological evidence (see Lehky, 1985). We included the contrast transducer in the model in order to compare the predictions over different contrast levels. When we compared predictions for stimulus contrasts of 40% and 80% we found a good correlation (r = 0.93). These predictions are therefore consistent with data which indicate that velocity matching is independent of contrast for contrasts greater than 5-10% (Campbell & Maffei, 1981; Thompson, 1982). DISCUSSION

There is considerable physiological and psychophysical evidence that the processing of twodimensional moving patterns occurs in two stages (Adelson & Movshon, 1982; Movshon, Adelson. Giui & Newsome, 1986; Ferrera 8c Wilson, 1987; Welch, 1989). Units at the first stage are selective for component motion and respond in a linear manner to the superposition

of two moving gratings. Units at the second stage combine the estimates of component motion in a nonlinear manner so that their response varies as a function of pattern, rather than component, direction. One hypothesis as to how the component motion signals are combined is the intersection-of-constraints rule, which can be used to derive the true physical motion of a surface undergoing rigid translation in the fronto-parallel plane. We have found that the perceived speed of type I symmetric plaids can be up to 50% slower than the speed predicted by the intersection-ofconstraints when the plaid is compared to a grating of the same spatial period as the components of the plaid (Fig. 3, solid symbols). However, if the spatial period of the comparison grating is the same as the nodes of the plaid, then the perceived speed of the plaid agrees with the intersection-of-constraints (Fig. 3, open symbols). It has previously been reported that the perceived speed of a one-dimensional grating depends on spatial frequency such that higher spatial frequencies appear relatively faster (Diener et al., 1976; Campbell & MafTei, 1979). We have replicated this result, and we find that when relative perceived speed (perceived speed divided by actual speed) is plotted as a function of relative spatial period (spatial period of standard divided by spatial period of test) on log-log coordinates, the data fall on a straight line with a slope close to 0.5 (Fig. 4). Furthermore, the data for both one- and twodimensional patterns describe nearly identical functions. It thus appears that speed is encoded in a similar manner for gratings and plaids, and that in both cases perceived speed varies as a function of pattern spatial period. The intersection-of-constraints model may be extended by taking into account uncertainty in the component velocity signal. However, Monte Carlo simulations reveal that this causes less than a I% change in the predicted pattern speed, which is not enough to account for the rather large shifts in perceived speed. Furthermore, the simulations predict that discrimination thresholds for type I patterns should be about twice as good as for their type II counterparts, whereas the data show no significant difference between pattern types. However, the simulations correctly predict that speed Weber fractions for two dimensional patterns should improve by up to a factor of J2 over component speed Weber fractions (Figs 5 and 6).

Perceived speed of moving

In a previous study (Ferrera & Wilson, 1990) we looked at the perceived direcrion and direction discrimination thresholds for two-dimensional patterns. Comparing the results of that study with those of the current study, we find several notable differences. First, there is no evidence for an oblique effect for speed discrimination, although there is an oblique effect for direction discrimination. Thus, while there is no difference in speed discrimination thresholds for upward vs obliquely moving gratings, direction discrimination thresholds are approximately twice as high for the oblique pattern. Second, whereas we previously found significant differences between type I and type 11 patterns with respect to direction processing, we found no fundamental differences for speed processing. In particular. the perceived direction of type I patterns was found to agree with the intersection-of-constraints. while the perceived direction of type 11 patterns was biased by about 7.5 deg toward the direction of the components. The intersection-of-constraints could not account for the perceived speed of either type of pattern. however pattern spatial period was a fairly accurate predictor of perceived speed for both type I and type II patterns. Furthermore, while direction discrimination thresholds for type II patterns were generally five times greater than for type I patterns, there was no significant diflerencc between the two types with regard to speed discrimination thresholds. Other investigators have examined related aspects of plaid motion perception. Welch (1989) measured speed discrimination for moving plaids and came to the conclusion that performance is limited by component speed, but not pattern speed. One would thus predict that the data in Figs 5 and 6 should lie along flat lines. which they clearly do not. However, Welch used lower contrast patterns which we have found to produce weaker effects (i.e. shallower functions for speed discrimination vs 0). Furthermore, she only looked at two angles (0 = 0 deg and 0 = 78.5 deg) which are on either side of the minimum shown in Fig. 6. We feel that the method of maintaining a constant component speed while varying the angle is a more sensitive measure of the pattern speed effect, especially since the Monte Carlo Simulations suggest that plaid speed discrimination thresholds will be at most 30% lower than component thresholds. Furthermore, this difference between plaid and component thresholds is not inconsistent with Welch’s conclusion that

plaids

891

plaid speed discrimination thresholds are limited by noise introduced at the component processing stage. In fact, the difference is predicted by the Monte Carlo simulations which observed this constraint in that noise was added only to the components. Our results can also be used to predict the perceived direction of plaids which have components of different spatial frequencies. If the two components are moving at the same speed, then the higher spatial frequency component should appear faster. The intersection-ofconstraints would therefore predict that the perceived direction of the plaid should be biased toward the direction of the higher spatial frequency component. Kooi, Grosof, DeValois and DeValois (1988) have found that the perceived direction is actually biased toward the direction of the lower spatial frequency component. These results conflict with ours only if one takes the intersection-of-constraints, which requires precise estimates of the speed and direction of the components, as a literal model for the combination of component motion signals. It is more likely that pattern motion mechanisms take as input the responses of low-level motion detectors which confound speed with spatial frequency and contrast, as do the motion energy units proposed by Ad&on and Bergen (1985). Thus, changing the response of these low-level mechanisms by changing the spatial frequency or contrast of one of the components could have the same effect as changing the speed of that component. In fact, in the experiment mentioned above Kooi et al. (1988) noted that increasing the contrast of the higher frequency component could eliminate or reverse the perceived direction bias. Finally, we have proposed a model for velocity coding which can explain the dependence of perceived speed on spatial period. Such a model must take into account the interdependence of spatial and temporal frequency in determining velocity. McKee et al. (1986) have addressed the issue of temporal frequency coding vs true velocity coding for drifting gratings. They compared perceived speed vs spatial frequency for gratings having the same objective speed and found that a 0.5 c/deg grating appeared about 85% as fast as a 1.5 c/deg grating. Thus, the fact that the gratings had different spatial and temporal frequencies seemed to have little effect on perceived speed. They also found that speed discrimination thresholds were not affected by random

892

VlNCE??T

P.

FERRERA and HUGH R. WIWH

variations in temporal frequency. They therefore argued for the existence of c&city detectors which are spatio-temporally inseparable in that they give the same response to different spatiotemporal frequency combinations which yield the same speed. In our speed matching results, we found shifts in perceived speed in the same direction as McKee et al. (i.e. lower spatial frequencies appear slower) but of much greater magnitude. For a direct comparison, we found that a 0.5 c/deg grating matched a 1.5 c/deg grating whose true speed was 55% slower. The discrepancy between the two sets of results is probably due to methodological differences. In the current study, we used a two-interval forcedchoice procedure so that the subject made a direct comparison between two gratings of different spatial frequencies. McKee et al., however, used a procedure in which each stimulus was compared against the perceived average of a group of stimuli. The latter technique may be more susceptible to criterion shifts over the course of an experiment, as McKee et al. (1986) have acknowledged. Other investigators have reported large shifts in perceived speed as a function of spatial frequency for drifting and rotating gratings (Diener et al.. 1976; Campbell & Maffci. 1979). Smith (1987) has also looked at speed discrimination in the presence of random variations in temporal frequency and found that thresholds were significantly elevated. These results challenge the notion that a true velocity signal is encoded in the response of non-separable velocity detectors. We have therefore developed a model in which perceived speed is encoded by an array of spatio-temporally separable units. The response of each unit is weighted by a factor which is proportional to its perferred temporal frequency and the weighted responses are averaged to yield perceived speed. Without any further embellishments, such a model would predict that two gratings with the same temporal frequency should always appear to have the same perceived speed, when in fact, the grating of lower spatial frequency will appear somewhat faster. This can be corrected for by two further properties of human spatiotemporal vision (see Lehky, 1985) which were incorporated into the model. First, at low mechanisms tuned to spatial frequencies, higher temporal frequencies are most sensitive, whereas at high spatial frequencies, mechanisms tuned to lower temporal frequencies are most

sensitive. Second, the contrast nonlinearity is associated with each mechanism is steepest for mechanisms tuned to low spatial and high temporal frequencies and becomes progressively shallower as preferred spatial frequency increases and preferred temporal frequency decreases. This model explains the dependence of perceived velocity on spatial period, and predicts that the relationship between the two should fall about midway between pure velocity coding and pure tetIIpOKii frequency coding (Fig. 4, heavy solid line). If the responses of the mechanisms in the model are pooled to produce constant speed tuning, then the model predicts that perceived speed should decrease with increasing spatial frequency, which runs counter to the trend found in the data (Fig. 4, heavy dashed line). When we compared predictions of the model for stimulus contrasts of 40% and 80%, we found a good correlation (r = 0.93). This agrees with the other reports that perceived speed does not depend on contrast, as long as the contrast is greater than 5-10% (Campbell & Maffei. 1981: Thompson. 1982). In conclusion, we have found that perceived velocity depends on pattern spatial period for both gratings and two-dimensional patterns. It thus appears that the speed of both one- and two-dimensional patterns is calculated in a similar manner, provided that the underlying mechanisms are capable of responding to the spatial period of the nodes of a two-dimensional pattern rather than its components. This could be accomplished by a spatial nonlinearity which produces a distortion product at the same spatial period as the nodes. We have also constructed a model for velocity coding based on an array of spatio-temporally separable mechanism. This model demonstrates that it is possible to account for the dependence of perceived speed on spatial period by taking a weighted average of the responses of an array of spatiotemporally separable mechanisms. This model suggests that the outputs of these mechanisms are not combined to produce speed-tuned detectors. AcknowMgemenr-This NIH

research was supported

grant no. EY02158

in part by

to H.R.W.

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