Viri~ Res. Vol. 30, No. 2, pp. 273-287, 1990 Printed in Great Britain. All rights reserved
0042-6989/90 SUM + 0.00 Copy6gbI Q 1990 Pergamon Prest plc
PERCEIVED DIRECTION OF MOVING TWO-DIMENSIONAL PATTERNS VINCENT
P.
FERRERA+and HUGH R. WILSON
Department of Ophthalmology and Visual Science, and Committcz on Neurobiology, The University of Chicago, 939 E. 57th St. Chicago, IL 60637, U.S.A. (Received 27 December 1988; in revised fotm 24 May 1989) Am-When two drifting cosine gratings arc superimposed, they will, under appropriate conditions, form a coherently moving two-dimensional pattern whose resultant direction of motion may either be between (type I), or outside (type II) the directions of the two components. We have previously shown that type I patterns produce much stronger masking than either of their components, while type II patterna do not. In this study, we measured perceived direction of motion and thresholda for d&rMnation of motion direction. We found that type II patterns had a peraivcd bias of about 7.5 dcg toward the direction of their components, and bad discrimination thresholds around 6.5 dcg, whereas type I pattuns had discrimination thresholds around 1.Odcg and no significant bii. We conclude that the neural me&a&ms which compute two-dimensional image motion do not strictly implement the inters&ion-of-consMnts construction proposed by Adelson and Movshon (1982). Aperture problem
Direction selective
Intersection-of-constraints
Motion
Plaid
a way that the individual gratings form a twodimensional pattern which may be perceived to The direction of motion of a one-dimensional move as a single coherent entity. Adelson and pattern such as a drifting cosine grating is Movshon (1982) noted that the perceived direcambiguous, and the direction which is actually tion of such a two-dimensional pattern might be perceived may vary depending on the shape accounted for by the ambiguity inherent in the and orientation of the aperture through which motion of its one-dimensional components, the grating is viewed. This phenomenon was and they captured this idea in a geometric observed by Wallach (1935,1976) and has come construction known as the intersection-ofto be known as the aperture prohlem or the constraints or velocity-space construction. This problem of motion ambiguity. Another way of construction assumes that the two onestating the problem is that the perceived motion dimensional components are attached to a rigid of a single drifting grating is consistent with a surface undergoing uniform translation in family of physical motions which are related in the frontoparallel plane, and then derives the that all members have the same magnitude or motion of that surface. speed in the direction orthogonal to the orienThe intersection-of-constraints is illustrated tation of the grating. Thus, if each member of in Fig. 1. The stimulw is composed of two this family were represented by a vector in cosine gratings with difTerent orientations and velocity? space, then the heads of all the vectors drift rates which are physically superimposed. would fall along a straight line, known as a The thin vectors represent the velocity comconstraint line (Fennema & Thompson, 1979; ponents of the two gratings, in the direction Adelson & Movshon, 1982). orthogonal to their orientations. The constraint On the other hand, if two one-dimensional lines, which are perpendicular to the component gratings are physically superimposed, the visual motion vectors, represent the range of physical system can resolve the motion ambiguity in such motions which could produce each ‘component. The point where the two constraint lines intersect (thick vector) gives the resultant pattern lFVcscxlt addras: University of Rochester, Dcpnrtment of velocity which is consistent with the velocities of phyriology Box 642, Rochester, NY 14642, U.S.A. tin keeping with the d&&ion in physics. we shall use both component. This construction has two important features velocity to denote a vector quantity comprised of a magnitude or apeed and a direction of motion. which are of interest in the current study. First, INTRODUCTION
273
VINCENT P.
214
FERREFU and HUGHR. WILSON
Fig. 1. The intersection-of-constraints in velocity space. Vx and Vy rqmsent the vertkal pod horizontal component of motion, respckly. Thin aruxvs repmmat the motion of the two ooaywwsnt ~tiq6, whii #ve rise to two constraint l&as. Where the constraint lb intamect dateaaiwr the motion of the msultant two-dimensional pattern. 8, and t$ give the directions of the components rsbtive to the rasultont.
for any pair of components there is a unique resultant. This means that subjects should be able to uniquely specify the direction of motion of these hinds of two-dimensional pat@rns rclative to a single drifting grating. Howcwcr, this discrimination depends on the subject’s ability to pcrccive the pattern as moving coherently. Addson and Movshon (1982) found that twodimensional patterns do not appear coherent when the two components difkr in spatial frcqucncy by more than 1.5-2 octaves or in contrast by a factor greater than 3. In tlmse cases, the gratings appear as two transparent su&ccs sliding past one another. In other words, the two gratings are not treated by the visual system as belonging to the same surface, and hence the intersection-of-constraints does not apply. In this study, WCavoided the probkm of transparency by always using components which had the same spatial frequency and contrast. The second important feature of the inters&on of constraints is that all pairs of components which lie along the ci& indkated in Fig. 1 have the same resultant. This means that we can construct different two-dimensionaI patterns, using components of various oriantations and drift rates, which will all have the same predicted diraxion of motion as lore as the dircctionandsp&ofcachcomponentobeythc following relation: V ----c-; cw ) V,, is the pattern speed, V, is the VP=
where
component speed, and 0 is the di@&cnce bewccn the pattern’s direction and the component’s direction. WCtested this predktion by measuring perceived dire&on and direction discrimination thresholds for the three types of patterns d@mmmcd in Fig. 2. For type I patterns, the resultant lies between the directions of the two components, whemas for typcIIpattenls,themsultantlksoutsidethc two components. We have constructed type I asymmetric and type II patterns as cumpkmentary pairs, such that they have a sin& component in common (component 8 in Fii. 2) and their other components arc mirror images of each other, refketcd about the axis of the resultant (components b and h’ in Fig. 2). All three patterns types have the same intersection-of-constraints resultant. In a previous study (Ferrera t Wilson, 198?), we found that type I patterns (previously rcferred to as “plaidC) produce much strong&r maskillg &ects than type II (previously ret&Ted to as “blobs”). Spcdficouy, we mcasumd the ability of a two-dimensional mask to in&Mere with the detection of a onc-dimcnsional test pattern moving in the same direction as the ma#s resultant. We then compared the threshold akvation (ratio of masked to unmasked contrast threshold) obtained with the two-dimensional mask to that produced by a single component of the mask. We found that type II patterns produce about the same amount of threshold elevation as the singk component which is closest in direction to the resultant.
Moving two-dimensional pattcms
TYPO 1 Symmctrfe
Tm
275
I
Asymmetric
Fig. 2. Velocity-space diagrams for the types of stimuli used in these experiments. Thin arrows represent the component motions, while the thick arrow represents the resultant motion. The exact component directions and relative speeds arc given in Table 1.
However, type I patterns produce up to 4.0 times as much masking as either of their components, even when the contrast of the single component mask is doubled. At the time we noted that the motion of type II patterns appeared more fluid or “blob-like” as opposed to the very rigid appearance of type I patterns. The results of the current study support the distinction between type I and type II patterns found in the masking study and also provide a means of quantifying our intuition that the motion of type II patterns is less well-defined. In particular, we found that the perceived direction of type I patterns agrees with the intersection of constraints prediction and that direction thresholds for type I patterns are as good as thresholds for one-dimensional patterns (about 1.0 deg). However, discrimination thresholds for type II patterns are roughly 5.0 times greater than thresholds for type I patterns, and the perceived direction of type II patterns is significantly biased toward the mean direction of the two component gratings by an average of 7.5 deg. METHODS
Patterns were generated by a Macintosh II microcomputer, passed through a GW Instruments D/A converter, and displayed on a Tektronix 608 monitor with a P31 phosphor. Each frame contained a one-dimensional pattern defined by a list of 512 luminance values. Individual luminance values were resolved with ll-bit accuracy. The display had a mean luminance of 15.0 cd/m’, and was viewed through a circular aperture in a cardboard surround that was illuminated at the
same mean level and approximate hue as the monitor. All patterns used were composed of two one-dimensional cosine gratings, of the same spatial frequency and contrast but diErent orientations and drift rates. For all experiments the contrast of each component was 40%. Presentation of the components at various orientations was accomplised with an electronic raster rotator which was controlled by a 15-bit D/A converter, so that component orientation could be specified to within 0.1 deg. The computer presented the two differently oriented gratings interleaved in alternate frames. The overall frame rate was 200 frames/set, so that the two components appeared to be physically superimposed with no perceptible Ilicker. The one-dimensional patterns used as standards were presented in the same manner, except that both components had the same orientation. Each pattern was presented in a 1.0 set interval, during which the pattern contrast was temporally modulated using a trapoidal envelope with on- and off-ramps of 0.25 set and a 0.5 set plateau at full contrast. These ramps have been shown by Bergen (1981) to minimize the effects of transients at stimulus onset and offset. Some control experiments required the use of a 200 msec presentation, in which case the pattern contrast was constant throughout the presentation. Other controls required the use of stationary patterns which were either presented with the same temporal trapezoid as the moving patterns or were counter-phase Bickered using a 1.0, 5.0 or 8.0 Hz sinusoid. A total of five subjects participated, three of whom were unaware of the purpose of the experiments. Subjects sat facing the display with
VINCENTP. FERRERAand HUGH R. WILSON
276
Table 1. Parameters for two-dimensional Pattern I-S(A) I-A(A) II(A) I-S(B) I-A(B) II(B) I-S(C) II(C) ID-O lD-45
Description Type I symmetric Type I asymmetric Type II Type I symmetric Type I asymmetric Type II Type I symmetric Type II 1-D Grating I-D Grating
moving patterns
Component 1 Component 2 Resultant Speed Direction (deg) Speed Direction (deg) Speed Direction (deg) 0.67 0.67 0.67 0.83 0.83 0.83 0.92 0.92 I.0 1.0
-48.2 -48.2 48.2 -33.6 - 33.6 33.6 -23.5 23.5 0.0 45.0
their heads comfortably positioned in a chin rest. Viewing was monocular, and the unused eye was covered with a translucent occluder. Subjects were instructed to fixate the center of the circular field, which was 8.0 deg in diameter for most experiments, but 4.Odeg for spatial frequencies greater than 2.0 c/deg. A central fixation mark was used to minimize voluntary eye movement. We used a two-alternative forced-choice method of constant stimuli paradigm in which one temporal interval contained a onedimensional standard and the other contained a two-dimensional test pattern. The test and standard were presented in random order. For each presentation, a small angular offset was added to each component of the twodimensional test pattern and the subject indicated by moving a mouse which interval had a greater rightward component of motion. The motion of the standard was identical to the ablated intersection-of-constraints resultant of the test pattern with zero offset. Note that adding a given angle to both components of a two-dimensional pattern changes the diion of the resultant by exactly the same amount. The frequency with which the test was seen to be moving to the left of the one-dimensional standard was plotted as function of angular offrct and fitted with a Quick (1974) function using a maximum likelihood estimation procedure. The perceived direction of the test was calculated by determining the o&et which corresponded to the 50% level of the bestfitting Quick function, Direction d&rin&ation thresholds were determined by taking half the di&mnce between the o&ets corresponding to the 25% and 75% levels, respectively. We were thus able to determine both the perceived direction and discrimination threshold in a single experiment. Additional experiments were performed to
0.67 0.33 0.33 0.83 0.33 0.33 0.92 0.33 1.0 1.0
48.2 70.5 70.5 33.6 70.5 70.5 23.5 70.5 0.0 45.0
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 45.0
verify our measurements of perceived direction using a random double staircase procedure described by Cornsweet (1962). These experiments were also run using a two-interval forced-choice paradigm and the subject’s task was exactly the same as before. The only difference was that after each trial, the computer either increased or decreased the angular offset added to each of the test pattern components, depending on the subject’s response. This procedure was repeated until the perceived direction of the test approximately matched that of the standard. RPSULTS The main results were obtained using the patterns described in Table 1, which gives the component directions and relative speeds for each pattern, as well as for the one-dimensional gratings used for comparison. Two-dimemGona1 patterns were divided into three groups (A, I3 and C), as indicated by the letters in parentheses. Within each group, any two patterns had exactly one component in common. The intersection-of-constraints (ICE) resultant for all pattern types (except lD-45) was upward (Odeg), and all component directions are given relative to the resultant with the convention that rightward is + 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 direction and direction discrimination thresholds were measured as a function of pattern type, component spatial frequency and resultant velocity for three subjects. Perceived direction is given relative to the IOC p&&ion so that a positive bias means that the perceived direction had a greater rightward component than the prediction. Thus, the perceived bias was equal in magnitude, but opposite in sign to the angular offset which corresponded to the
Moving hvodimensional patterns
211
0.25 for all. subjects, see Figs 3 and 4). However, for type II patterns, the perceived direction was signitkantly biked (P < 0.0005) toward the mean direction of the two components. In general, the magnitude of the bias for type II patterns was between 5.0 and lO.Odeg, with an average of 7.5 deg. Results obtained using a staircase procedure to match the direction of a two-dimensional pattern to that of a one-dimensional grating showed excellent quantitative agreement with the method of constant stimuli data (see Table 3, in Discussion). Discrimination thresholds for type I patterns were generally between 0.5 and 1.5 deg (see Figs 5 and 6, and Table 2). Thresholds for type I asymmetric patterns were only slightly higher than symmetric patterns, but the difference was statistically significant (P < 0.0025). Thresholds for symmetric patterns were not signifkantly different (P > 0.85) from thresholds obtained with one-dimensional gratings moving in the same direction as the intersection-of-constraints resultant of the two-dimensional pattern (compare pattern TYPE I with lD-0 in Fig. 7). However, direction discrimination thresholds for one-dimensional gratings showed a significant oblique effect (P < 0.0005) in that thresholds for obliquely moving gratings were Q! >
m
TYPE I --+
SPATIAL FREQUENCY (CPD)
Fig. 3. perceived bias as a function of component spatial frequency for two subjects. For type I, data from pattern typca I-s(A) and l-A(A) (see Table 1) were averaged. Type II corresponds to pattern II(A). For VPF, two resultant ~docitks (2.7 and 5.4deg/sec) are shown. For TF, the velocity varied with component spatial frequency so as to produce a constant temporal frequency. (The numbers in the legend indicate the temporal frequency of the faster moving component.) The dashed line indicates the intersection-ofconstraints prediction. The solid line indicates the least mean squares regression for the type II data.
50% level of the psychometric function. We tested a range of component spatial frequencies from 0.25 to S.Ocpd, and resultant velocities ranging from 2.7 to 21.6 deg/sec. Perceived biases for two subjects are plotted as a function of component spatial frequency and resultant velocity in Figs 3 and 4, respectively. Direction discrimination thresholds are similarly plotted in Figs 5 and 6. One-way analyses of variance revealed that neither perceived direction or discrimination thresholds had a significant dependence on either spatial frequency or resultant velocity (P > 0.05 for all subjects). Therefore, we will focus on differences due to pattern type, which were highly significant (P < 0.005 for all subjects). Unless otherwise indicated, all statistical P-values indicate the results of paired ttests, where measurements were paired by spatial frequency and resultant velocity (or temporal frequency for stationary patterns). The perceived direction of type I patterns (either symmetric or asymmetric) was not significantly different from the IOC prediction
0
o-
B
‘TYPE
IId
“p” s
a \
4.
8 h ti’ E
TYPEIR 0 _.........” .. . .. .. . .. .. .. .. . . .. ..._.. . . ..Y. .. .. .. .. .-...-g .._ .. .._. .. . .
-2-l 0
.
, 2
.
.
4
.
1
2
1
2
.
RESULTANT v5LCCrlY
*
10
-
12
.
(Dps)
Fig. 4. Perceived bias as a function of resuhant velocity for two suw. For type I, data from pattern types I-S(A) and I-A(A) (see Table 1) were averaged. Type II cormsponds to pattern II(A). Compownt spatial f~wncy was I.Oqxl. The dashed line indiates the intwneuioll-of-coastrllfconrtrsiatr prediction. The solid line indicates the but mean squares regression for the type II data.
VINCENTP.
278
FERRERA and HUGHR. WKSON
Table 2. Pattern versus component direction discrimination thresholds Pattern
Description
VPF
Subject TF
KM3
ID-O ID-45 I-S I-A
Upward grating Oblique grating Trpe 1 symm. Type I assym.
0.79 f 0.07 1.41f 0.05 0.71 io.04 1.13 f 0.05
1.09f 0.21 2.10 * 0.22 1.20&0.11 1.51f 0.22
1.12* 0.28 2.98 & 0.80 1.33f0.15 2.08 f 0.20
consistently around 2-3 times higher than thresholds for vertically moving gratings (compare patterns lD-0 and lD-45 in Fig. 7 and Table 2). It is therefore interesting to note that thresholds for type I patterns (both symmetric and asymmetric) were significantly lower than thresholds for the obliquely moving gratings of which they were composed (P < 0.005, compare pattern lD-45 with TYPE I with in Fig. 7 and with I-S or I-A in Table 2). Conversely, when type I patterns were moving obliquely, discrimination thresholds were elevated to the same level as obliquely moving gratings, even though the components of the two-dimensional pattern were close to horizontal and vertical where
2-l - ..I
thresholds for moving gratings are lowest. For example, when pattern I-S(A) was rotated 45 deg to the right so that its components w&e at 3.2 and 93.2 deg, respectively, the direction disc~~nation threshold was 1.46 f 0.05 deg (VPF, l.Ocpd). These results suggest that the precision in the estimate of pattern direction cannot be predicted by considering the noise in the component motion signals propagated through the intersection-of-constraints construction, as will be demonstrated below (see Monte Carlo Sim~a~ons). Type II patterns had discrimination thresholds that were on average about 4.5 times higher than the complementary type I asymmetric patterns, and the difkence between the two thresholds was highly significant (P < 0.0005, see Figs 5 and 6). It is interesting to note that, with type I patterns, subjects
I .,
PRlt~lRtRGY (GPO)
r-0
0
SPATIAL
Fig. 5. Dimotion ~~~~~~O~Of . cmnponent q&si frcquenty for two s&jwts. For type I, duta from pottern type X-S(A)and I-A(A) (MC-IWe 1) were avera&. Type II cosespond8 to piwr41 U(A). For
0
*
.,.......z.c.
0
s
4
*
RCSULTmY
Fig. 6. Wmction discrimhtiou
8
10
12
VaLoGiTY (DPS)
tkasholds M a fun&o0 of
VPF, two W wlodda (2.7 and 5.4&&a@ UC rat&ant v&c&y for two subjwa. For type i, dasa from shown.For TF. the &ocity varisd w&h pwrn typr, I-S(A) pad I-A{A) (EMT&k i)wem ~wuqpdw fmqwcncysoutopreducuaconshat WfpIQpDDFy. (TftenumbcminthekloadhdkatccBeteqemlfbqtwncy of the fastermovlnp component.) The solid liawiedbtea the least mean squares rcgfhon for the type II data.
Type II oormqoud8 to pattern II(A). Component q&l frapmcy WM1.Ocpd. The dot&d be indiwtuu the intersection-of-censtraintsptedktioa. The solid lb indhea the
VINCENTP.
280
Type
FFRRERA and
1
HUGH
R. WILSON
Type
II
Fig. 8. Efkct of component uncertainty em thrsinMa-c~f++ts. Thin arrows represent stimulus component direction. The thick arrow wta exact m&ant motions. Shad& circles repmsent the uncertainty associated with each stimulus Mponcnt. Sh&d areas around resultant vectors represent the distribution of resultants ariaing from compontrlt unartajnty.
discrimination thresholds were much smaiiet. In difbena particukr there was no sign&ant betweenorientation disckrination thr&olds for type II patterns and obhquely oriented gratings (compare stationary pazterns ID-45 and type II in Fig. 7). Thus, the rehitkly poor ability to discriminate the direction of type II patterns cannot be based on any deficiency in discriminating the orientation of a stationary pattern with components at the same ~~~~ons as the moving pattern. It is somewhat surprising that observers apparently never learn to use orientation as a cue when performing the direction dia&mination task with type II patterns, since cornponent orientation and pattern direction are perfectly correlated.* This me&y ~~ the perceptual difkrences between direction of motion and orientation. It also agraeo with the observations of Wailach (1935) which irrdieate that perceived direction of motion is readily dissociated from orientation when moting.hnes are viewed through apertures of difkrent shapes.
We considered the possibility that our results could be explained by uncertainty in the estimatian of the direction of the individual component gratings. This approach has been usad by Nakayama and Silverman (“1.988)to predict thmsholds for the perception of rigid motion in drifting sinusoidal lines. Fiwre 8 shows what happens to the intcrsection-ofconstraints when small deviations are added to the component directions, as indicated by the two dikent constraint lines for each component. The shaded areas surroundi~ each resultant vector show the e&t of a small amount of Yariance in the component direction signal on the ~bu~~n of resultants, which is much broader for type II than for type f patterns. Thus, the geometry of the intersr&on of constraints sug&ests that a given. level of unfxrknty in the initial:estimates ofcomponent dkvction can theoretically have a more d&r&c a&et on ~~~buti~ of resuhants in the case of type II patterns than in the case of type I patterns* To determine whether this e&ct couid quantitatively aocount for the. perceived d&@on and axon thresholds of either type I or type II pattern we fkst had to determine the mppnitude of the uncertainty. We did this by measuring both direction and speed discrimi-
Moving two-dimensional patterns
nation thresholds for a one-dimensional cosine grating which was oriented at + 45 deg. We used +45 deg because it was close to the average of the actual component directions, and because direction discrimination thresholds are largest at this angle, thus biasing the results in favor of this conjectured explanation. The speed, spatial frequency and contrast of the test were the same as those of the standard. Direction discrimination thresholds for obliquely moving gratings are shown in Fig. 7 (see pattern lD-45), and were generally between 1.5 and 2.0 deg. Speed discrimination thresholds, given as Weber fractions (AV/V), were 7.5% and 10% for subjects VPF (l.Ocpd, 10.8dps) and TF (l.Ocpd, 5.4 dps), respectively. McKee, Silvermann and Nakayama (1986) reported slightly lower speed discrimination thresholds for vertically oriented gratings using as somewhat different method and a display with twice the mean luminance used in our study. We then ran Monte Carlo simulations in which, on each trial, random components were sampled from Gaussian distributions centered around the actual stimulus component (see the shaded areas around the heads of the component motion vectors in Fig. 8). We used an elliptical error function in which the standard deviation of the Gaussian distribution was determined by the results of the onedimensional direction discrimination experiments described above. Thus, the probability density as a function of speed and direction is given by the following relation: 1
281
a resultant direction was computed using the intersection-of-constraints. 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 either perceived direction or direction discrimination. First, the simulations failed to predict the rather large perceived bias for type II patterns. 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 II patterns. Second, simulated direction discrimination thresholds did not fit the pattern of results in the data. Simulated thresholds are compared to experimental data for two subjects in Fig. 9 (different component discrimination thresholds were used for the two subjects, as mentioned above). The simulations predict that thresholds for type I symmetric patterns should always be greater than component thresholds, even though the data show that for type I patterns moving vertically, pattern discrimination thresholds are actually lower than component thresholds. For type I asymmetric patterns, the simulated thresholds are significantly lower than
I}
(6 - Q2, (0 - rQ2 ; exp - 24; 26: 27wf4 I[ where 6, and v, are the direction and speed, respectively, of the stimulus components, and cre and a, were derived from the direction and speed discrimination thresholds for an obliquely moving grating by the following formula: P(t9, u) =-
where d’ is the discrimination threshold and z(c) is the z-score corresponding to a proportion, c, of the area under the normal distribution between - 00 and z. For 75% thresholds, c P 0.75. This 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,
1YCEMW PATTERN
Fig. 9. Monte Carlo siatulated discrimination thresholds cmnpmd to experinmttd data for two subjects. Real and simulated thmholds am paired by pattern type and subja% of pattcms). For VPF, the (see Tabk I for dachphn umponent spatial fmquency was l.Ocpd and the multant velocity for all patterns was 10.8dpr. For TF. the cmnponcnt spatial frapmcy was O.Scpd and the resultant velocity for all patterna was 5.4dps.
282
VINCENTP. FERRERAand HUGH R. WILSCIN
the simulated thresholds for type I symmetric patterns. However, the data indicate that thresholds for type I asymmetric patterns should be the same if not higher than for symmetric patterns. The simulated thresholds for type II patterns are in good agreement with the data. However the difference between the simulated thresholds for type I and type II patterns is only a factor of 2, whereas the difference between the measured thresholds is greater than a factor of 5. The overall correlation between the simulated thresholds and the data for the two subjects shown in Fig. 9 is rather poor (r = 0.55).
(A)
RELATIVE ANGLE
It may be noted that in the case of type II patterns, as the relative angle between the components is small, interactions or “cross-talk” between units tuned to similar directions of motion might affect the perceived angle between the components and thus bias the perceived direction of the resultant. In this regard, there are two effects worth considering. First suppose that component direction is encoded as an average of the responses of an array of units tuned to a range of different directions, with the response of each unit weighted by its preferred direction. If there is inhibition between neighboring units in direction space, then the effect might be to increase the relative angle between the perceived component directions. This is a simple extension of the idea that inhibition between neighboring orientation columns in primary visual cortex might be responsible for the phenomenon of acute angle broadening which underlies many familiar perceptual illusions (see Coren & Girguql978). From the geometry of the intersection of constraints (Fig. lOA), it is obvious that a perceived broadening of the relative angle between the two component directions will lead to a bias which is in agreement with what we observe for type II patterns, i.e. toward the mean direction of the components. In order to produce the observed bias of about 7.5 deg the relative angle must be increased by about 7-10 deg (the actual amount depends on the difference between the stimulus components). It is interesting to note that as the angle between the stimulus component directions increases, the interaction between the components, and hence the amount of perceptual broadening, should decrease. In addition, any
Fig. 10. (A) Efkts of perceptual broadening (or, conv@rs@y, nan~wing) on perceived direction. The thin shaded vc&tors rqcsent the stimulus components and yield a resultant moving straight upward (tbiclc shaded vectors)_ %r&ved component directions are rcpnsantcd by thin dark vectors which prod= a xsultant (thick dark vector) biased in the dire&on of the components. (B) Effect of averaging both component direction and speed. Again, tbc thin shaded arrows represent the stimulus components and yield a resultant moving straight upward (thick sbadul arrow). Pcrcdwd component directions ara represented by thin dark vectors which produce a resultant (thick d&c -or) biased in the direction of the components.
given amount of perceptual broadening should be less e&ctive in producing a bias as the relative aqk between the stimuius coqonents increases, simply because the angle at which the cons-t lines inttrssct will be cloacr to perpendictkr. Hence, the perceived bias in direction should decrease as the angk between the compononta increases. We teated this pmdiction by comparing the bias for three &&r&t type II patterns for which the relative angles between the components were approximately
Moving two-dimensional patterns
283
to a reduction of the relative angle, there should also be a reduced difference 0 t#mllF66@ between the relative speeds of the two comI A lmmIIQlO.lq6 12 m yr6wfhIgr ponents. Together, these two effects could, in . @p66Wfla6~ a^ # 10 principle predict the perceived bias (Fig. 10B). 0 *plTFOSrpd We decided to test this idea quantitatively by convolving the stimulus with a set of directionally tuned filters before applying the intersection-of-constraints. The filters all had Gaussian sensitivity profiles of the same bandwidth, but varied with respect to preferred B ....._. o . . . . . . . ;..&...A_ ...._.... ._......_....._....._...... direction, covering the entire 360 deg range of t possible directions. This kind of filtering has , , -2 . , . , , 120 160 1.0 0 60 60 60 been used as a front end for a cooperative model RELATIVE ANGLE (DEO) of direction encoding (Phillips, Williams & Fig. 11. F%xccivcdbias as a function of relative angle Sekuler, 1986; Williams & Phillips, 1987). between stimulus components for three subjects. For VPF, This procedure yields perceived component the component spatial frequency for both patterns types was directions, 6; and &, which are related to the 1.Ocpd. For TF and KMB, the resultant velocity for both actual component directions, 8, and t&, in the pattern types was 10.8 dps. The dashed line india~tes agreefollowing manner: ment with the IOC prediction. The solid line lab&d rkory additia~
1I
I
I
is the prediction of a model which reduces the ditTcrcncein component direction and speed before applying the intcrsectionsf-ccmstraints.
22, 37 and 47 deg, respectively [see patterns II(A), II(B) and II(C) in Table 11. Even if the magnitude of the broadening effect were exactly the same in all three cases, the intersection-ofconstraints predicts that the bias for pattern II(C) should be over 3.Odeg less than the bias for pattern II(A). We found that even though the average bias decreased slightly with increasing angle, the differences between the three patterns were not sign&ant (P > 0.5, see Fig. 11). Thus, it is unlikely that the bias in the perceived direction of type II patterns can be explained by inhibitory interactions between mechanisms tuned to similar directions of motion. On the other hand, if there are units which are strongly stimulated by both components of the stimulus, as is likely to be the case when the relative angle is small, then the effect might be to reduce the perceived angle between the two components. Unfortunately, this effect cannot account for the perceived bias of type II patterns because if the angle between the components is reduced, then applying the intersection of constraints to the perceived components predicts that the resultant will be biased in the wrong direction, i.e. away from the mean direction of the components (Fig. 10A). However, if speed and direction are encoded within the same set of mechanisms, then in
e;= e,+we,-61.
1+k
’
~;=e*-k1e~-e21~ I+k
’
where: k = exp[ - In 2(0, - e&h2]; and where h is the direction half-amplitude, half-bandwidth of the mechanism which encodes velocity. The perceived component speeds, v; and vi, were then determined as follows: , _ WV: +
VI
-
w,+w,
w92. ’
where: W,
3:
exp[ -In 2(8, - e;)2/h2];
W,
3:
exp[ - In 2(e2- 8 ;)2/h2].
A similar set of expressions was used for vi, except that t9; was replaced by 6;. The predictions from this model with h = 25 deg are shown in Fig. 11 as the curve labeled theory. The value of h was derived from oblique masking data using stationary patterns which were counterphase flickered at 8 Hz (Phillips & Wilson, 1984).We repeated these measurements using the same technique but with moving patterns. These results agree almost perfectly with the stationary oblique masking data. The predicted bias is 4.7 deg for a relative angle of 47 deg, which agrees with the average bias for
VINCENTP. FERREWand HUGH R. WILSON
Fig. 12. ~~~~ti~n thresholds as a function of relative angle betwwn stimulus components for three subjscts. For VPF, the component spatial freqwtcy for both patterns types was 1.O cpd. For TF and KMB, the resultant velocity for both pattern types was 10.8dps. The solid line is the result of Monte Carlo simulations for type I symmetric patterns.
ft is unclear how component interactions, as discussed above, would affect direction discrimination thresholds. According to Monte Carlo simulations, broadening the angle between the stimulus components by a small amount should have a negligible effect on discrimination thresholds for type II patterns. Nevertheless, we ran additional experiments in which we measured discrimination thresholds for tyJJe I symmetric patterns which encompassed the range of relative angles seen in the type ?I patterns. As can be seen in Fig. 12, there is still a large discrepancy between type I and type II thresholds, even when the relative angles are similar. Furthermore, thresholds for both pattern types do not show any significant dependence on relative angle, Thus interactions between ~mponents, if they occur, do not degrade discrimination of typ I patterns even when the angle between the component directions is small, and therefore are not a plausible explanation for the higher thresholds seen with type II patterns. We also tried to predict dissemination thresholds as a function of relative angle for type I symmetric patterns using the Monte Carlo technique. This model predicts that discrimination thresholds should increase approximately IO-fold as the angle between the component directions decreases from 150 to 30 deg. These predictions are shown as the solid curve in Fig. 12. It is clear that the predictions are not borne out by the data (Fig, 12, open symbols). Stone (1989) has derived a formula for the dainty in the direction of a type I pattern using perturbation methods. The agreement between Stone’s formula and Monte Carlo simulations is within 5Oio.
three subjects of 5.4 deg f 0.39. However, the model predicts that the perceived bias should increase ~~ti~Jly as the relative angle dtmwscs. As mentioned above, this sensitivity to relative angle is not supported by the data. These calculations were done with a fixed direction half-bandwidth, h. However, it is possible to vary h in order to obtain a best fit of the model. It was found that the value of k needed to predict the perceived bias for any given pattern was always a constant fractian (0.58) of the angle between the component directions, i.e a different value of h was needed for each pattern. For the pattern with the DISCUS~ON smallest relative angle, the IlcccsIsBly halfIn order to emphasize the di%renees between bandwidth was 13 deg* which, as the masking data indicate, is too narrow for either stationary type I and type II patterns, perceived dktion or direction selective mechanisms at low to and discrimination thresholds averaged over all subjects and conditions are given in Table 3. moderate spatial frequencies. Table 3. Fwcbwl Pattern TypeIsymm&c Type I asymmetric Type if 1D Vertical grating ID Obliwe -- ttraliae_ *SignScantly diit
(matbd
dirsctioa and disuWnation
h(61)
ofctwtw
thresholds
~GW stimuli)
0.09 f 0.07 -0.06f0.11 7.34 f 0.53* O.Off 0.13 -0.69f0.31 from IOC pm&tkm (P
0042-6989/90 SUM + 0.00 Copy6gbI Q 1990 Pergamon Prest plc
PERCEIVED DIRECTION OF MOVING TWO-DIMENSIONAL PATTERNS VINCENT
P.
FERRERA+and HUGH R. WILSON
Department of Ophthalmology and Visual Science, and Committcz on Neurobiology, The University of Chicago, 939 E. 57th St. Chicago, IL 60637, U.S.A. (Received 27 December 1988; in revised fotm 24 May 1989) Am-When two drifting cosine gratings arc superimposed, they will, under appropriate conditions, form a coherently moving two-dimensional pattern whose resultant direction of motion may either be between (type I), or outside (type II) the directions of the two components. We have previously shown that type I patterns produce much stronger masking than either of their components, while type II patterna do not. In this study, we measured perceived direction of motion and thresholda for d&rMnation of motion direction. We found that type II patterns had a peraivcd bias of about 7.5 dcg toward the direction of their components, and bad discrimination thresholds around 6.5 dcg, whereas type I pattuns had discrimination thresholds around 1.Odcg and no significant bii. We conclude that the neural me&a&ms which compute two-dimensional image motion do not strictly implement the inters&ion-of-consMnts construction proposed by Adelson and Movshon (1982). Aperture problem
Direction selective
Intersection-of-constraints
Motion
Plaid
a way that the individual gratings form a twodimensional pattern which may be perceived to The direction of motion of a one-dimensional move as a single coherent entity. Adelson and pattern such as a drifting cosine grating is Movshon (1982) noted that the perceived direcambiguous, and the direction which is actually tion of such a two-dimensional pattern might be perceived may vary depending on the shape accounted for by the ambiguity inherent in the and orientation of the aperture through which motion of its one-dimensional components, the grating is viewed. This phenomenon was and they captured this idea in a geometric observed by Wallach (1935,1976) and has come construction known as the intersection-ofto be known as the aperture prohlem or the constraints or velocity-space construction. This problem of motion ambiguity. Another way of construction assumes that the two onestating the problem is that the perceived motion dimensional components are attached to a rigid of a single drifting grating is consistent with a surface undergoing uniform translation in family of physical motions which are related in the frontoparallel plane, and then derives the that all members have the same magnitude or motion of that surface. speed in the direction orthogonal to the orienThe intersection-of-constraints is illustrated tation of the grating. Thus, if each member of in Fig. 1. The stimulw is composed of two this family were represented by a vector in cosine gratings with difTerent orientations and velocity? space, then the heads of all the vectors drift rates which are physically superimposed. would fall along a straight line, known as a The thin vectors represent the velocity comconstraint line (Fennema & Thompson, 1979; ponents of the two gratings, in the direction Adelson & Movshon, 1982). orthogonal to their orientations. The constraint On the other hand, if two one-dimensional lines, which are perpendicular to the component gratings are physically superimposed, the visual motion vectors, represent the range of physical system can resolve the motion ambiguity in such motions which could produce each ‘component. The point where the two constraint lines intersect (thick vector) gives the resultant pattern lFVcscxlt addras: University of Rochester, Dcpnrtment of velocity which is consistent with the velocities of phyriology Box 642, Rochester, NY 14642, U.S.A. tin keeping with the d&&ion in physics. we shall use both component. This construction has two important features velocity to denote a vector quantity comprised of a magnitude or apeed and a direction of motion. which are of interest in the current study. First, INTRODUCTION
273
VINCENT P.
214
FERREFU and HUGHR. WILSON
Fig. 1. The intersection-of-constraints in velocity space. Vx and Vy rqmsent the vertkal pod horizontal component of motion, respckly. Thin aruxvs repmmat the motion of the two ooaywwsnt ~tiq6, whii #ve rise to two constraint l&as. Where the constraint lb intamect dateaaiwr the motion of the msultant two-dimensional pattern. 8, and t$ give the directions of the components rsbtive to the rasultont.
for any pair of components there is a unique resultant. This means that subjects should be able to uniquely specify the direction of motion of these hinds of two-dimensional pat@rns rclative to a single drifting grating. Howcwcr, this discrimination depends on the subject’s ability to pcrccive the pattern as moving coherently. Addson and Movshon (1982) found that twodimensional patterns do not appear coherent when the two components difkr in spatial frcqucncy by more than 1.5-2 octaves or in contrast by a factor greater than 3. In tlmse cases, the gratings appear as two transparent su&ccs sliding past one another. In other words, the two gratings are not treated by the visual system as belonging to the same surface, and hence the intersection-of-constraints does not apply. In this study, WCavoided the probkm of transparency by always using components which had the same spatial frequency and contrast. The second important feature of the inters&on of constraints is that all pairs of components which lie along the ci& indkated in Fig. 1 have the same resultant. This means that we can construct different two-dimensionaI patterns, using components of various oriantations and drift rates, which will all have the same predicted diraxion of motion as lore as the dircctionandsp&ofcachcomponentobeythc following relation: V ----c-; cw ) V,, is the pattern speed, V, is the VP=
where
component speed, and 0 is the di@&cnce bewccn the pattern’s direction and the component’s direction. WCtested this predktion by measuring perceived dire&on and direction discrimination thresholds for the three types of patterns d@mmmcd in Fig. 2. For type I patterns, the resultant lies between the directions of the two components, whemas for typcIIpattenls,themsultantlksoutsidethc two components. We have constructed type I asymmetric and type II patterns as cumpkmentary pairs, such that they have a sin& component in common (component 8 in Fii. 2) and their other components arc mirror images of each other, refketcd about the axis of the resultant (components b and h’ in Fig. 2). All three patterns types have the same intersection-of-constraints resultant. In a previous study (Ferrera t Wilson, 198?), we found that type I patterns (previously rcferred to as “plaidC) produce much strong&r maskillg &ects than type II (previously ret&Ted to as “blobs”). Spcdficouy, we mcasumd the ability of a two-dimensional mask to in&Mere with the detection of a onc-dimcnsional test pattern moving in the same direction as the ma#s resultant. We then compared the threshold akvation (ratio of masked to unmasked contrast threshold) obtained with the two-dimensional mask to that produced by a single component of the mask. We found that type II patterns produce about the same amount of threshold elevation as the singk component which is closest in direction to the resultant.
Moving two-dimensional pattcms
TYPO 1 Symmctrfe
Tm
275
I
Asymmetric
Fig. 2. Velocity-space diagrams for the types of stimuli used in these experiments. Thin arrows represent the component motions, while the thick arrow represents the resultant motion. The exact component directions and relative speeds arc given in Table 1.
However, type I patterns produce up to 4.0 times as much masking as either of their components, even when the contrast of the single component mask is doubled. At the time we noted that the motion of type II patterns appeared more fluid or “blob-like” as opposed to the very rigid appearance of type I patterns. The results of the current study support the distinction between type I and type II patterns found in the masking study and also provide a means of quantifying our intuition that the motion of type II patterns is less well-defined. In particular, we found that the perceived direction of type I patterns agrees with the intersection of constraints prediction and that direction thresholds for type I patterns are as good as thresholds for one-dimensional patterns (about 1.0 deg). However, discrimination thresholds for type II patterns are roughly 5.0 times greater than thresholds for type I patterns, and the perceived direction of type II patterns is significantly biased toward the mean direction of the two component gratings by an average of 7.5 deg. METHODS
Patterns were generated by a Macintosh II microcomputer, passed through a GW Instruments D/A converter, and displayed on a Tektronix 608 monitor with a P31 phosphor. Each frame contained a one-dimensional pattern defined by a list of 512 luminance values. Individual luminance values were resolved with ll-bit accuracy. The display had a mean luminance of 15.0 cd/m’, and was viewed through a circular aperture in a cardboard surround that was illuminated at the
same mean level and approximate hue as the monitor. All patterns used were composed of two one-dimensional cosine gratings, of the same spatial frequency and contrast but diErent orientations and drift rates. For all experiments the contrast of each component was 40%. Presentation of the components at various orientations was accomplised with an electronic raster rotator which was controlled by a 15-bit D/A converter, so that component orientation could be specified to within 0.1 deg. The computer presented the two differently oriented gratings interleaved in alternate frames. The overall frame rate was 200 frames/set, so that the two components appeared to be physically superimposed with no perceptible Ilicker. The one-dimensional patterns used as standards were presented in the same manner, except that both components had the same orientation. Each pattern was presented in a 1.0 set interval, during which the pattern contrast was temporally modulated using a trapoidal envelope with on- and off-ramps of 0.25 set and a 0.5 set plateau at full contrast. These ramps have been shown by Bergen (1981) to minimize the effects of transients at stimulus onset and offset. Some control experiments required the use of a 200 msec presentation, in which case the pattern contrast was constant throughout the presentation. Other controls required the use of stationary patterns which were either presented with the same temporal trapezoid as the moving patterns or were counter-phase Bickered using a 1.0, 5.0 or 8.0 Hz sinusoid. A total of five subjects participated, three of whom were unaware of the purpose of the experiments. Subjects sat facing the display with
VINCENTP. FERRERAand HUGH R. WILSON
276
Table 1. Parameters for two-dimensional Pattern I-S(A) I-A(A) II(A) I-S(B) I-A(B) II(B) I-S(C) II(C) ID-O lD-45
Description Type I symmetric Type I asymmetric Type II Type I symmetric Type I asymmetric Type II Type I symmetric Type II 1-D Grating I-D Grating
moving patterns
Component 1 Component 2 Resultant Speed Direction (deg) Speed Direction (deg) Speed Direction (deg) 0.67 0.67 0.67 0.83 0.83 0.83 0.92 0.92 I.0 1.0
-48.2 -48.2 48.2 -33.6 - 33.6 33.6 -23.5 23.5 0.0 45.0
their heads comfortably positioned in a chin rest. Viewing was monocular, and the unused eye was covered with a translucent occluder. Subjects were instructed to fixate the center of the circular field, which was 8.0 deg in diameter for most experiments, but 4.Odeg for spatial frequencies greater than 2.0 c/deg. A central fixation mark was used to minimize voluntary eye movement. We used a two-alternative forced-choice method of constant stimuli paradigm in which one temporal interval contained a onedimensional standard and the other contained a two-dimensional test pattern. The test and standard were presented in random order. For each presentation, a small angular offset was added to each component of the twodimensional test pattern and the subject indicated by moving a mouse which interval had a greater rightward component of motion. The motion of the standard was identical to the ablated intersection-of-constraints resultant of the test pattern with zero offset. Note that adding a given angle to both components of a two-dimensional pattern changes the diion of the resultant by exactly the same amount. The frequency with which the test was seen to be moving to the left of the one-dimensional standard was plotted as function of angular offrct and fitted with a Quick (1974) function using a maximum likelihood estimation procedure. The perceived direction of the test was calculated by determining the o&et which corresponded to the 50% level of the bestfitting Quick function, Direction d&rin&ation thresholds were determined by taking half the di&mnce between the o&ets corresponding to the 25% and 75% levels, respectively. We were thus able to determine both the perceived direction and discrimination threshold in a single experiment. Additional experiments were performed to
0.67 0.33 0.33 0.83 0.33 0.33 0.92 0.33 1.0 1.0
48.2 70.5 70.5 33.6 70.5 70.5 23.5 70.5 0.0 45.0
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 45.0
verify our measurements of perceived direction using a random double staircase procedure described by Cornsweet (1962). These experiments were also run using a two-interval forced-choice paradigm and the subject’s task was exactly the same as before. The only difference was that after each trial, the computer either increased or decreased the angular offset added to each of the test pattern components, depending on the subject’s response. This procedure was repeated until the perceived direction of the test approximately matched that of the standard. RPSULTS The main results were obtained using the patterns described in Table 1, which gives the component directions and relative speeds for each pattern, as well as for the one-dimensional gratings used for comparison. Two-dimemGona1 patterns were divided into three groups (A, I3 and C), as indicated by the letters in parentheses. Within each group, any two patterns had exactly one component in common. The intersection-of-constraints (ICE) resultant for all pattern types (except lD-45) was upward (Odeg), and all component directions are given relative to the resultant with the convention that rightward is + 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 direction and direction discrimination thresholds were measured as a function of pattern type, component spatial frequency and resultant velocity for three subjects. Perceived direction is given relative to the IOC p&&ion so that a positive bias means that the perceived direction had a greater rightward component than the prediction. Thus, the perceived bias was equal in magnitude, but opposite in sign to the angular offset which corresponded to the
Moving hvodimensional patterns
211
0.25 for all. subjects, see Figs 3 and 4). However, for type II patterns, the perceived direction was signitkantly biked (P < 0.0005) toward the mean direction of the two components. In general, the magnitude of the bias for type II patterns was between 5.0 and lO.Odeg, with an average of 7.5 deg. Results obtained using a staircase procedure to match the direction of a two-dimensional pattern to that of a one-dimensional grating showed excellent quantitative agreement with the method of constant stimuli data (see Table 3, in Discussion). Discrimination thresholds for type I patterns were generally between 0.5 and 1.5 deg (see Figs 5 and 6, and Table 2). Thresholds for type I asymmetric patterns were only slightly higher than symmetric patterns, but the difference was statistically significant (P < 0.0025). Thresholds for symmetric patterns were not signifkantly different (P > 0.85) from thresholds obtained with one-dimensional gratings moving in the same direction as the intersection-of-constraints resultant of the two-dimensional pattern (compare pattern TYPE I with lD-0 in Fig. 7). However, direction discrimination thresholds for one-dimensional gratings showed a significant oblique effect (P < 0.0005) in that thresholds for obliquely moving gratings were Q! >
m
TYPE I --+
SPATIAL FREQUENCY (CPD)
Fig. 3. perceived bias as a function of component spatial frequency for two subjects. For type I, data from pattern typca I-s(A) and l-A(A) (see Table 1) were averaged. Type II corresponds to pattern II(A). For VPF, two resultant ~docitks (2.7 and 5.4deg/sec) are shown. For TF, the velocity varied with component spatial frequency so as to produce a constant temporal frequency. (The numbers in the legend indicate the temporal frequency of the faster moving component.) The dashed line indicates the intersection-ofconstraints prediction. The solid line indicates the least mean squares regression for the type II data.
50% level of the psychometric function. We tested a range of component spatial frequencies from 0.25 to S.Ocpd, and resultant velocities ranging from 2.7 to 21.6 deg/sec. Perceived biases for two subjects are plotted as a function of component spatial frequency and resultant velocity in Figs 3 and 4, respectively. Direction discrimination thresholds are similarly plotted in Figs 5 and 6. One-way analyses of variance revealed that neither perceived direction or discrimination thresholds had a significant dependence on either spatial frequency or resultant velocity (P > 0.05 for all subjects). Therefore, we will focus on differences due to pattern type, which were highly significant (P < 0.005 for all subjects). Unless otherwise indicated, all statistical P-values indicate the results of paired ttests, where measurements were paired by spatial frequency and resultant velocity (or temporal frequency for stationary patterns). The perceived direction of type I patterns (either symmetric or asymmetric) was not significantly different from the IOC prediction
0
o-
B
‘TYPE
IId
“p” s
a \
4.
8 h ti’ E
TYPEIR 0 _.........” .. . .. .. . .. .. .. .. . . .. ..._.. . . ..Y. .. .. .. .. .-...-g .._ .. .._. .. . .
-2-l 0
.
, 2
.
.
4
.
1
2
1
2
.
RESULTANT v5LCCrlY
*
10
-
12
.
(Dps)
Fig. 4. Perceived bias as a function of resuhant velocity for two suw. For type I, data from pattern types I-S(A) and I-A(A) (see Table 1) were averaged. Type II cormsponds to pattern II(A). Compownt spatial f~wncy was I.Oqxl. The dashed line indiates the intwneuioll-of-coastrllfconrtrsiatr prediction. The solid line indicates the but mean squares regression for the type II data.
VINCENTP.
278
FERRERA and HUGHR. WKSON
Table 2. Pattern versus component direction discrimination thresholds Pattern
Description
VPF
Subject TF
KM3
ID-O ID-45 I-S I-A
Upward grating Oblique grating Trpe 1 symm. Type I assym.
0.79 f 0.07 1.41f 0.05 0.71 io.04 1.13 f 0.05
1.09f 0.21 2.10 * 0.22 1.20&0.11 1.51f 0.22
1.12* 0.28 2.98 & 0.80 1.33f0.15 2.08 f 0.20
consistently around 2-3 times higher than thresholds for vertically moving gratings (compare patterns lD-0 and lD-45 in Fig. 7 and Table 2). It is therefore interesting to note that thresholds for type I patterns (both symmetric and asymmetric) were significantly lower than thresholds for the obliquely moving gratings of which they were composed (P < 0.005, compare pattern lD-45 with TYPE I with in Fig. 7 and with I-S or I-A in Table 2). Conversely, when type I patterns were moving obliquely, discrimination thresholds were elevated to the same level as obliquely moving gratings, even though the components of the two-dimensional pattern were close to horizontal and vertical where
2-l - ..I
thresholds for moving gratings are lowest. For example, when pattern I-S(A) was rotated 45 deg to the right so that its components w&e at 3.2 and 93.2 deg, respectively, the direction disc~~nation threshold was 1.46 f 0.05 deg (VPF, l.Ocpd). These results suggest that the precision in the estimate of pattern direction cannot be predicted by considering the noise in the component motion signals propagated through the intersection-of-constraints construction, as will be demonstrated below (see Monte Carlo Sim~a~ons). Type II patterns had discrimination thresholds that were on average about 4.5 times higher than the complementary type I asymmetric patterns, and the difkence between the two thresholds was highly significant (P < 0.0005, see Figs 5 and 6). It is interesting to note that, with type I patterns, subjects
I .,
PRlt~lRtRGY (GPO)
r-0
0
SPATIAL
Fig. 5. Dimotion ~~~~~~O~Of . cmnponent q&si frcquenty for two s&jwts. For type I, duta from pottern type X-S(A)and I-A(A) (MC-IWe 1) were avera&. Type II cosespond8 to piwr41 U(A). For
0
*
.,.......z.c.
0
s
4
*
RCSULTmY
Fig. 6. Wmction discrimhtiou
8
10
12
VaLoGiTY (DPS)
tkasholds M a fun&o0 of
VPF, two W wlodda (2.7 and 5.4&&a@ UC rat&ant v&c&y for two subjwa. For type i, dasa from shown.For TF. the &ocity varisd w&h pwrn typr, I-S(A) pad I-A{A) (EMT&k i)wem ~wuqpdw fmqwcncysoutopreducuaconshat WfpIQpDDFy. (TftenumbcminthekloadhdkatccBeteqemlfbqtwncy of the fastermovlnp component.) The solid liawiedbtea the least mean squares rcgfhon for the type II data.
Type II oormqoud8 to pattern II(A). Component q&l frapmcy WM1.Ocpd. The dot&d be indiwtuu the intersection-of-censtraintsptedktioa. The solid lb indhea the
VINCENTP.
280
Type
FFRRERA and
1
HUGH
R. WILSON
Type
II
Fig. 8. Efkct of component uncertainty em thrsinMa-c~f++ts. Thin arrows represent stimulus component direction. The thick arrow wta exact m&ant motions. Shad& circles repmsent the uncertainty associated with each stimulus Mponcnt. Sh&d areas around resultant vectors represent the distribution of resultants ariaing from compontrlt unartajnty.
discrimination thresholds were much smaiiet. In difbena particukr there was no sign&ant betweenorientation disckrination thr&olds for type II patterns and obhquely oriented gratings (compare stationary pazterns ID-45 and type II in Fig. 7). Thus, the rehitkly poor ability to discriminate the direction of type II patterns cannot be based on any deficiency in discriminating the orientation of a stationary pattern with components at the same ~~~~ons as the moving pattern. It is somewhat surprising that observers apparently never learn to use orientation as a cue when performing the direction dia&mination task with type II patterns, since cornponent orientation and pattern direction are perfectly correlated.* This me&y ~~ the perceptual difkrences between direction of motion and orientation. It also agraeo with the observations of Wailach (1935) which irrdieate that perceived direction of motion is readily dissociated from orientation when moting.hnes are viewed through apertures of difkrent shapes.
We considered the possibility that our results could be explained by uncertainty in the estimatian of the direction of the individual component gratings. This approach has been usad by Nakayama and Silverman (“1.988)to predict thmsholds for the perception of rigid motion in drifting sinusoidal lines. Fiwre 8 shows what happens to the intcrsection-ofconstraints when small deviations are added to the component directions, as indicated by the two dikent constraint lines for each component. The shaded areas surroundi~ each resultant vector show the e&t of a small amount of Yariance in the component direction signal on the ~bu~~n of resultants, which is much broader for type II than for type f patterns. Thus, the geometry of the intersr&on of constraints sug&ests that a given. level of unfxrknty in the initial:estimates ofcomponent dkvction can theoretically have a more d&r&c a&et on ~~~buti~ of resuhants in the case of type II patterns than in the case of type I patterns* To determine whether this e&ct couid quantitatively aocount for the. perceived d&@on and axon thresholds of either type I or type II pattern we fkst had to determine the mppnitude of the uncertainty. We did this by measuring both direction and speed discrimi-
Moving two-dimensional patterns
nation thresholds for a one-dimensional cosine grating which was oriented at + 45 deg. We used +45 deg because it was close to the average of the actual component directions, and because direction discrimination thresholds are largest at this angle, thus biasing the results in favor of this conjectured explanation. The speed, spatial frequency and contrast of the test were the same as those of the standard. Direction discrimination thresholds for obliquely moving gratings are shown in Fig. 7 (see pattern lD-45), and were generally between 1.5 and 2.0 deg. Speed discrimination thresholds, given as Weber fractions (AV/V), were 7.5% and 10% for subjects VPF (l.Ocpd, 10.8dps) and TF (l.Ocpd, 5.4 dps), respectively. McKee, Silvermann and Nakayama (1986) reported slightly lower speed discrimination thresholds for vertically oriented gratings using as somewhat different method and a display with twice the mean luminance used in our study. We then ran Monte Carlo simulations in which, on each trial, random components were sampled from Gaussian distributions centered around the actual stimulus component (see the shaded areas around the heads of the component motion vectors in Fig. 8). We used an elliptical error function in which the standard deviation of the Gaussian distribution was determined by the results of the onedimensional direction discrimination experiments described above. Thus, the probability density as a function of speed and direction is given by the following relation: 1
281
a resultant direction was computed using the intersection-of-constraints. 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 either perceived direction or direction discrimination. First, the simulations failed to predict the rather large perceived bias for type II patterns. 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 II patterns. Second, simulated direction discrimination thresholds did not fit the pattern of results in the data. Simulated thresholds are compared to experimental data for two subjects in Fig. 9 (different component discrimination thresholds were used for the two subjects, as mentioned above). The simulations predict that thresholds for type I symmetric patterns should always be greater than component thresholds, even though the data show that for type I patterns moving vertically, pattern discrimination thresholds are actually lower than component thresholds. For type I asymmetric patterns, the simulated thresholds are significantly lower than
I}
(6 - Q2, (0 - rQ2 ; exp - 24; 26: 27wf4 I[ where 6, and v, are the direction and speed, respectively, of the stimulus components, and cre and a, were derived from the direction and speed discrimination thresholds for an obliquely moving grating by the following formula: P(t9, u) =-
where d’ is the discrimination threshold and z(c) is the z-score corresponding to a proportion, c, of the area under the normal distribution between - 00 and z. For 75% thresholds, c P 0.75. This 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,
1YCEMW PATTERN
Fig. 9. Monte Carlo siatulated discrimination thresholds cmnpmd to experinmttd data for two subjects. Real and simulated thmholds am paired by pattern type and subja% of pattcms). For VPF, the (see Tabk I for dachphn umponent spatial fmquency was l.Ocpd and the multant velocity for all patterns was 10.8dpr. For TF. the cmnponcnt spatial frapmcy was O.Scpd and the resultant velocity for all patterna was 5.4dps.
282
VINCENTP. FERRERAand HUGH R. WILSCIN
the simulated thresholds for type I symmetric patterns. However, the data indicate that thresholds for type I asymmetric patterns should be the same if not higher than for symmetric patterns. The simulated thresholds for type II patterns are in good agreement with the data. However the difference between the simulated thresholds for type I and type II patterns is only a factor of 2, whereas the difference between the measured thresholds is greater than a factor of 5. The overall correlation between the simulated thresholds and the data for the two subjects shown in Fig. 9 is rather poor (r = 0.55).
(A)
RELATIVE ANGLE
It may be noted that in the case of type II patterns, as the relative angle between the components is small, interactions or “cross-talk” between units tuned to similar directions of motion might affect the perceived angle between the components and thus bias the perceived direction of the resultant. In this regard, there are two effects worth considering. First suppose that component direction is encoded as an average of the responses of an array of units tuned to a range of different directions, with the response of each unit weighted by its preferred direction. If there is inhibition between neighboring units in direction space, then the effect might be to increase the relative angle between the perceived component directions. This is a simple extension of the idea that inhibition between neighboring orientation columns in primary visual cortex might be responsible for the phenomenon of acute angle broadening which underlies many familiar perceptual illusions (see Coren & Girguql978). From the geometry of the intersection of constraints (Fig. lOA), it is obvious that a perceived broadening of the relative angle between the two component directions will lead to a bias which is in agreement with what we observe for type II patterns, i.e. toward the mean direction of the components. In order to produce the observed bias of about 7.5 deg the relative angle must be increased by about 7-10 deg (the actual amount depends on the difference between the stimulus components). It is interesting to note that as the angle between the stimulus component directions increases, the interaction between the components, and hence the amount of perceptual broadening, should decrease. In addition, any
Fig. 10. (A) Efkts of perceptual broadening (or, conv@rs@y, nan~wing) on perceived direction. The thin shaded vc&tors rqcsent the stimulus components and yield a resultant moving straight upward (tbiclc shaded vectors)_ %r&ved component directions are rcpnsantcd by thin dark vectors which prod= a xsultant (thick dark vector) biased in the dire&on of the components. (B) Effect of averaging both component direction and speed. Again, tbc thin shaded arrows represent the stimulus components and yield a resultant moving straight upward (thick sbadul arrow). Pcrcdwd component directions ara represented by thin dark vectors which produce a resultant (thick d&c -or) biased in the direction of the components.
given amount of perceptual broadening should be less e&ctive in producing a bias as the relative aqk between the stimuius coqonents increases, simply because the angle at which the cons-t lines inttrssct will be cloacr to perpendictkr. Hence, the perceived bias in direction should decrease as the angk between the compononta increases. We teated this pmdiction by comparing the bias for three &&r&t type II patterns for which the relative angles between the components were approximately
Moving two-dimensional patterns
283
to a reduction of the relative angle, there should also be a reduced difference 0 t#mllF66@ between the relative speeds of the two comI A lmmIIQlO.lq6 12 m yr6wfhIgr ponents. Together, these two effects could, in . @p66Wfla6~ a^ # 10 principle predict the perceived bias (Fig. 10B). 0 *plTFOSrpd We decided to test this idea quantitatively by convolving the stimulus with a set of directionally tuned filters before applying the intersection-of-constraints. The filters all had Gaussian sensitivity profiles of the same bandwidth, but varied with respect to preferred B ....._. o . . . . . . . ;..&...A_ ...._.... ._......_....._....._...... direction, covering the entire 360 deg range of t possible directions. This kind of filtering has , , -2 . , . , , 120 160 1.0 0 60 60 60 been used as a front end for a cooperative model RELATIVE ANGLE (DEO) of direction encoding (Phillips, Williams & Fig. 11. F%xccivcdbias as a function of relative angle Sekuler, 1986; Williams & Phillips, 1987). between stimulus components for three subjects. For VPF, This procedure yields perceived component the component spatial frequency for both patterns types was directions, 6; and &, which are related to the 1.Ocpd. For TF and KMB, the resultant velocity for both actual component directions, 8, and t&, in the pattern types was 10.8 dps. The dashed line india~tes agreefollowing manner: ment with the IOC prediction. The solid line lab&d rkory additia~
1I
I
I
is the prediction of a model which reduces the ditTcrcncein component direction and speed before applying the intcrsectionsf-ccmstraints.
22, 37 and 47 deg, respectively [see patterns II(A), II(B) and II(C) in Table 11. Even if the magnitude of the broadening effect were exactly the same in all three cases, the intersection-ofconstraints predicts that the bias for pattern II(C) should be over 3.Odeg less than the bias for pattern II(A). We found that even though the average bias decreased slightly with increasing angle, the differences between the three patterns were not sign&ant (P > 0.5, see Fig. 11). Thus, it is unlikely that the bias in the perceived direction of type II patterns can be explained by inhibitory interactions between mechanisms tuned to similar directions of motion. On the other hand, if there are units which are strongly stimulated by both components of the stimulus, as is likely to be the case when the relative angle is small, then the effect might be to reduce the perceived angle between the two components. Unfortunately, this effect cannot account for the perceived bias of type II patterns because if the angle between the components is reduced, then applying the intersection of constraints to the perceived components predicts that the resultant will be biased in the wrong direction, i.e. away from the mean direction of the components (Fig. 10A). However, if speed and direction are encoded within the same set of mechanisms, then in
e;= e,+we,-61.
1+k
’
~;=e*-k1e~-e21~ I+k
’
where: k = exp[ - In 2(0, - e&h2]; and where h is the direction half-amplitude, half-bandwidth of the mechanism which encodes velocity. The perceived component speeds, v; and vi, were then determined as follows: , _ WV: +
VI
-
w,+w,
w92. ’
where: W,
3:
exp[ -In 2(8, - e;)2/h2];
W,
3:
exp[ - In 2(e2- 8 ;)2/h2].
A similar set of expressions was used for vi, except that t9; was replaced by 6;. The predictions from this model with h = 25 deg are shown in Fig. 11 as the curve labeled theory. The value of h was derived from oblique masking data using stationary patterns which were counterphase flickered at 8 Hz (Phillips & Wilson, 1984).We repeated these measurements using the same technique but with moving patterns. These results agree almost perfectly with the stationary oblique masking data. The predicted bias is 4.7 deg for a relative angle of 47 deg, which agrees with the average bias for
VINCENTP. FERREWand HUGH R. WILSON
Fig. 12. ~~~~ti~n thresholds as a function of relative angle betwwn stimulus components for three subjscts. For VPF, the component spatial freqwtcy for both patterns types was 1.O cpd. For TF and KMB, the resultant velocity for both pattern types was 10.8dps. The solid line is the result of Monte Carlo simulations for type I symmetric patterns.
ft is unclear how component interactions, as discussed above, would affect direction discrimination thresholds. According to Monte Carlo simulations, broadening the angle between the stimulus components by a small amount should have a negligible effect on discrimination thresholds for type II patterns. Nevertheless, we ran additional experiments in which we measured discrimination thresholds for tyJJe I symmetric patterns which encompassed the range of relative angles seen in the type ?I patterns. As can be seen in Fig. 12, there is still a large discrepancy between type I and type II thresholds, even when the relative angles are similar. Furthermore, thresholds for both pattern types do not show any significant dependence on relative angle, Thus interactions between ~mponents, if they occur, do not degrade discrimination of typ I patterns even when the angle between the component directions is small, and therefore are not a plausible explanation for the higher thresholds seen with type II patterns. We also tried to predict dissemination thresholds as a function of relative angle for type I symmetric patterns using the Monte Carlo technique. This model predicts that discrimination thresholds should increase approximately IO-fold as the angle between the component directions decreases from 150 to 30 deg. These predictions are shown as the solid curve in Fig. 12. It is clear that the predictions are not borne out by the data (Fig, 12, open symbols). Stone (1989) has derived a formula for the dainty in the direction of a type I pattern using perturbation methods. The agreement between Stone’s formula and Monte Carlo simulations is within 5Oio.
three subjects of 5.4 deg f 0.39. However, the model predicts that the perceived bias should increase ~~ti~Jly as the relative angle dtmwscs. As mentioned above, this sensitivity to relative angle is not supported by the data. These calculations were done with a fixed direction half-bandwidth, h. However, it is possible to vary h in order to obtain a best fit of the model. It was found that the value of k needed to predict the perceived bias for any given pattern was always a constant fractian (0.58) of the angle between the component directions, i.e a different value of h was needed for each pattern. For the pattern with the DISCUS~ON smallest relative angle, the IlcccsIsBly halfIn order to emphasize the di%renees between bandwidth was 13 deg* which, as the masking data indicate, is too narrow for either stationary type I and type II patterns, perceived dktion or direction selective mechanisms at low to and discrimination thresholds averaged over all subjects and conditions are given in Table 3. moderate spatial frequencies. Table 3. Fwcbwl Pattern TypeIsymm&c Type I asymmetric Type if 1D Vertical grating ID Obliwe -- ttraliae_ *SignScantly diit
(matbd
dirsctioa and disuWnation
h(61)
ofctwtw
thresholds
~GW stimuli)
0.09 f 0.07 -0.06f0.11 7.34 f 0.53* O.Off 0.13 -0.69f0.31 from IOC pm&tkm (P
