Research Note 86-30

___ SENSITIVITY TO VISUAL MOTION IN STATISTICALLY DEFINED DISPLAYS

I R. W. Sekuler, D. W. Williams, and S. A. Harp

Cresap Vision Laboratory

Northwestern University

Contracting Officerts Representative

Robert M. Sasmor

OTIC

ELECTE

Basic Research MA K5U

Milton S. Katz, Director MY08U

U. S. Army

Research Institute for the Behavioral and Social Sciences

March 1986

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ARI RESEARCH NOTE 86-30

4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

Sensitivity to Visual Motion in Final

Statistically Defined Displays 1./

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R. W. Sekuler, D. W. Williams, and MDA 903-80-C-0154

S. A. Harp

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Northwestern University

Cresap Vision Laboratory 2QI61102B74F

2021 Sheridan Road, Evanston, IL 60201

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Contracting Officer's Representative, Robert M. Sasmor

119. KEY WORDS (Continue on reverse side if necessry end identify by block number)

'otion perception, Ternus illusion perception

visual sensitivity stroboscopic motion /psychophysics. -,

spatial frequency cinemtograms

spatial vision; vision

apparent motion visual perception

20. ABSTRACr rcombou a reverse afer if necessary strd Idenltf by block number)

.This project explored several related aspects of how uman observers perceive

moving targets, particularly targets whose motion is apparent'! rather than ."

continuous. . •

*ERIGINS OF GLOBAL MOTION PMRCEPTS. When several different, local motion vectors

are intermixed the rc,ult may be a percept of global, coherent motion. We

exploited this discovery in order to develop a better understanding of the

mechanisms that support the perception of motion. Our experiments used

specially developed s;timuli presented by computer control. (continued)

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The stimuli were random-dot cinematograms made up of 512 elements (bright dots

on a cathode ray tube). From one frame of the display to the next, each

element took an independent, random walk. All steps in the random walk were

of constant size and the directions of these steps were drawn from a uniform

distribution.

When shown stimuli in which different, local motion vectors were mixed,

observers tended to see a global, coherent flow along the mean of the uniform

distribution of directions. This perceptual tendency varied inversely with the

range of the distribution. Standard psychophysical techniques were used to

obtain psychometric functions for cinematograms having various step sizes and

spatial densities of their elements. A wide range of conditions produced

results that were consistent with a modified version of S. Ullman's "minimal

map theory" of motion correspondence.

SIZE FACTORS IN APPARENT MOTION. We tested the idea that the system creating

the percept of motion makes use, at an early stage, of information about size.Pi

The size information was presumed to arise from size-tuned mechanisms with

fairly broad bandwidths in the domain of spatial frequency. To test this

hypothesis, we used stimuli whose luminance profile was a difference of

Gaussians (DOGs). Such stimuli are spectrally band-limited and therefore

should differentially stimulate size-tuned mechanisms.

In one study, adjacent DOGs of varying size were alternated in a simple apparent

motion display. When DOGs were of the same size, they were more likely to

elicit reports of motion. This preference for size similarity was found under a

range of display conditions. The generality of this finding was tested under

other conditions. The central DOG stimulus alternated with two flanking DOGs.

The central DOG could participate in motion with one, both, or neither of the

flanking DOGs. The size relations among the three DOGs were varied and

observers' reports of apparent motion recorded. In general, the relations

discovered with the two DOG case held for this more complex display. A

mathematical model was developed to account for the results. The model

conceptualizes apparent motion as resulting from a competition between

simultaneous tendencies to see motion in several different directions. The

model supports the idea that an early stage in the processing of apparent motion

is the extraction of information about the visual size of the stimuli.

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COHERENT GLOBAL MOTION PERCEPTS FROM STOCHASTIC LOCAL MOTIONS

The com bination of several different motion vectors carn Produce a

Pe rcept of coherent .iotion in a single direction. For e:amle, if two

sin.cSoidzI Arstirs of similar spatial freauencies move in different

d:'--c:on-: they pay appear to cohere into a single moving

:,eker-bo'arolike Pattern (Adelson and Movshon, 1980). Also, if

rtr.t :s near threshold, two spatially interspersed random dot

s.tterns moving in orthogonal directions can generate a PercePt of

Lction along the mearn of the two directions (Levinson, Coyne and Gross,

1.o9 .

Ullman (197?) has demonstrated that many motion Percepts, including

t,.e result of c:bining several different motion vectors, can be

exslained ir termts of urely local interactions. The spatial freauericv .-

..tlectivitV of coherent unidirectional motion for moving 3insusoidal

sratirs Persuaded Adelsorn and Movshon (1980) that mechanisms which

5erer~te the Percept of coherent motion operate on the responses of

:7 ati 2 freouencv channels. Models of spatial vision that are

;,' ,i teo ir, terMs of spatial localized, spatial freoJencd channel

Ses-.. in visu l space have met with considerable success (e.-.

scr sr,. Berser,, 179). We were therefore interested in how a

:;.-er-r,t 5:tbal Percept could result from the combinatiorn of locslized

o e:Alore the role of s ptialiv localiced Processra ir, the

of lzba, coheret moIor*on? we used i ov na s an. a o- zt

,c:,ato r:.. keaatosrats r, art te serer3tec crdL, ts....

'- -. .*...*.'-J"- "-. -. ?.'-.'-..-...... "-."... .'-. -v..... ,-.-..-.........."."'....."'..""."".-......"."....--.-.....--- . .- . -. "- ,"'.-. "- ,--.'- -, 
PAGE2

Williams and Sekuler

.liverse r'ulest resulitin~g ira as man-s different twpes, of stimuli. In one

comon tupe? large subsets of the dots move in one direction. But such

stiauli would not be appropriate for our Purposes: the contribution of

thie local motion of individual dots to the global Percept is obscured

bv the reduridancv of multiple motion vectors in the same direction. 5.

* ~ :7.stead,.we developed a kinebatosraa in which the direction of motion of

Each dot is independentl' defined. The stimuli were constructed ia the

*follcwir,., cianne r. Initiallwt dots were distributed randoalj over our

cathode rav disp'law. Each dot then took an independent 2-dimensional

ran~dom walk. Though all dots travelled the same distacce from frame to

f'amet the d-4ectioa in which anv dot moved was independent of the

lirections in which the other dots moved. Furthery the direction a

gIven dot moved from one frame to the next was independent of the

directicn of its Previous displacements; the Possible directions in

which all dots moved were chosen from the same uniform r-robabilitv

* distrib~ution.

Not suprisirngN, if the range of the distribution of directions

*e::tenhJed over all 360 degrees, localy random movement of individual

*oswas evident. Put if the range of the distribution was less than

360 degrees- the Patterna could appear to flow eos-aasse in the

di.rection of the mean of the distribution, even though the individual

Perturbations of the dots were still evident.

* We Faraietrized the rrobabilitv of seeing a gl36al, coheren-t

-ercept of unidirectional flow fromt local motion vectors. To do thisq,

.e varied th e r ange of the distr:bution of vectors and mea-z:jred~ the

* robabiljti of ieeing unidirectional flow inr a direction a Iorns the
PAGE 3

.... m and Sekijler

distribution's mean. We then investisated the properties of local

*echanisms of motion bs e':amining how Perceived coherence of motion

changed with various local Parameters. These Parameters included

spatial factors, the step size in the random walk and the densitV of

dots scross the displav, as well as a temporal factor, the duration of PC

the movement.

detbods

The Patterns were generated bv a PDP 11/34 computer that passed

valzJas through a digital to analog converter for displaw on a Hewlett

P ckard 1321A X-Y displaw with a P31 Phosphor. The displaved stimulus

Wss confined to a sauare region with sides measuring '.8.5 degrees. A

'wrap around' scheme caused dots to 'disappear* when displaced beyond

the boundarv of the snuare and then 'reappear' at the opposite side of

the soijare. The Pattern was viewed through a cardboard mask with a

circular opening subtending IS degrees of visual angle. Subjects

fi::ated the center cf the screen; viewing was monocular with the other

ae-e occluded Lb translucent eve Patch.

Each dot n,easured 0." degree in diameter. Though frame .Iurat.on

Zs a la's S msec, the irterframe interval renuired to generate

3-1srent contiruous motion varied with step size; Table I lists those

,.erva.s for each of the step sizes used, Perception of coherent

,:nidirecticnal flow varied with the stimulus duration (i.e. the number

of frames Presented). Therefore, etcept when we measured Perceptiorn as

s f:jracticn of the number of frames Presented, the st;imulus duration was

74a*rta.ree' at ore second. The reason for choosing ths a'.'je wilil be

. . . . . . . . . . * * . . * * . . ... 

. . . . . . . . . . . . . . . . . .oo
to, PAGE 4

Williams ard Sekuler

made clear in a later sect:on, dealing with temporal Properties of the

st imulus,.

The X-Y display provided the orly luminance ir the room and

subjects adapted to these luminance conditions for five minutes before

startir g an e: perimental session. Contrast of the Patterns was

Ss'ntained at twice threshold contrast. At the beginning of each

sessi n the threshold contrast was reestablished using a von ?ekesv

track'.ni Procedure (Tynan and Sekuler, 1977). Preliminary e: periments

.ndic-te that a coherent motion Percept could be generated over a wide

rare of contrasts. However since the temporal conditions for

Producing ccherent motion varied with contrast, we decided to confine

tne formal study to a single contrast.

In Preliminary e: periments, a 2-alternative forced choice

Procedure determined the Probability of seeing unidirectional flow along

the Mear, of the uniform distribution of directions. These

- oabilities were measured as a function of the range of the

*striution. Steps covered 0.1 degree and the dot density was 1.6 dots

zer desree. The results were the same for different directions of the

mean (e.g. left, right, obligue, etc.). Therefore, with no sacrifice of

aereral::abilitv we subseauentlv concentrated only on the case in which

the iean direction was upward with respect to the subject. Data were ..-

3thered usir, a sirple Yes-no paradigm, it, which the observer

Indrc3ted whether or not a coherent unidirectional flow was evident.

Four subjects were tested, three of whom were naive as to the

z:ur-ose of the stuy. The fourth subject was one of the authors.

• .' Z. " . '. ° " ". " " " . ,.. . .. . .".". '. . ..- ..... "' . .' ... .' .' . -".' . , -. ,. - , .'- , , .- - -. . - .' ,
i 'li3ms and Sekuler

Sesul..s~aodDiscussoO .

EX-EEIENII :SteeSize

For various step sizes, we first measured the Probability of seeins

coherent flow in the mean direction (upward) as a function of the ranse --

of a uniform distribution of directions. Four subjects Participated in

the studdy, under conditions already described.

Figure 1 shows the data from subject SDT for 4 different step

si:es: 0.1, 0.9Y 1.1 and 1.4 degrees. The Percentage of trials on which

the subject reported coherent unidirectional flow 'upward' is Plotted as

a function of the rarge of the distribution. Note that 'upward' is the

mear, direction of the distribution of directions. Results fall into

tuc categories, depending on whether the step size is larger or smaller

t;*.-r, 1.0 degree. If the step size was treater than 1.0 degree,

unidirectional flew was reported only when the range of directions was

Leit :icse to the mean; directions of motion had to be within ,- .

sppro:;imately 4 degrees of the mean for these step sizes. For step

-:-,,ez s-,sller than 1.0 degree, a considerably larger range of

di.tribution of directions could generate a Percept of coherent flow.

r. ,!,rti:uarp when the total range of 180 degrees was used with small

ste2ps coherent motion was reported almost 100% of the time. Similar

r-?suits 4ere obtained for all four subjects Participating in the study.

7hose far subject AHA are shown in Figure 2. A striking feature of

'.ioth fi-ures is that a ssall, two tenths of a degree change in step

si:e. from 0.9 to 1.1p Produces a large lateral shift :n the

=s'chcmetric fu nctior, while other changes by as much as eight tenths

-. . . . . . .~ *..~*."* *"--
PAGE 6

Wi'llaLs snd Se~uler

cf 3 .eiree. 'rom 0.1 to 0.9, result in little or no shift. I.

* There is a conceptual impediment to a straight forward .

irterpretatior: of these results. One can riot assume that the Perceived

Fath a dot travels is the one which was determined bv the random walk

" Prescribed for that dot. It maw be that for a givers dot, its Perceived

path is a cozbination of its own random walk with those for intruding

:e:shbors. This Perceptual ambiguitv is common2v referred to as the

:orresponderce Problem' (Braddick, 1982; Marr, 1982). If such

confusions did occur, spurious directions of movement could be

Perceived that were inconsistent with the Predefined distribution of

Possible directions. The probability of confusion will depend of such

factors as the step siZe, spacing or densitv of dots, and the

interstirulus interval (Ullman, 1979). If the spacirng among dots is

increased while other factors remain constant, it seems reasonable to

e;..ect that the Probability of confusion amon Paths should be reduced.

ExEE:~EL2:Deosiy...f..fo±.s

:ri the Previous e-,periment the densitv of dots was constants 1.6

,.ots Per souare degree for all step sizes. We repeated the e..Perimert

at three additional densities 0.8, 0.4 and 0.2 dot Per sauare desree,

and several step sizes. Four subjects Participated in this experiment.

Fisure 3 shows the results for step si:es of 0.1 and 0.? desree

obtained frca subject SDT. For clarity of Presentation the data for

each step si:e have been PIotted asairst a separste abscissa.

I. .

.* . .* . . . .. .. . .- . . ..- **.-.-. .. * -~*~ **~ : ~ -.
- - - - - *%. . , -- -

PAGE 7

Williams and Sekuler

Nio significant change in Perceptibilitv occurs when dot densit,

changes b' a factor of eilht, from 1.6 to 0.2 dots Per sauare degree, .

for either step size. This constancv is not evident for step sizes

lzrger than 1.0 degree. As shown for subject SDT in Figures 4 and 5, 

for step sizes of either 1.1 and 1,4 degrees, decreasing the densitv of

dots increases the tendencv to Perceive unidirectional flow, permitting

* nidirectional flow to be seer, over a wider range of directions. The

.sshed line in each figure represents the psychometric function for step

size 0.1 degree and density 1.6 dots Per souare degree taken from

Figure 1. For a densitv of 0.2 dot Per souare degree the data for both

step sizes, I.' and 1.4 degrees, are almost congruent with this dashed

Slire. Thus for sufficiently small densitv of dots, Perceptibilitv for

step sizes greater than 1.0 degree is nearlv eouivalent to that for

*. step sizes less than 1.0 degree.

Two important Points follow from the results. First, the fact that

spa:ing of dots car alter Perception has important implications for the

spatial Properties of an hvpothesized local mechanisms of motion"

detection and the 'correspornderce Problem'. These implications are

decribed nelow, in the General Discussion. A second implication is

iore germane to the formulation of the remaining experiments and will be

*icscus.ed here. For ster sizes less thar 1.0 degree, the constancy of

results over a large range of dot densities suggests that spurious

r rectiors of dis lacemert do not significantly contribute to the

Percept. Thus for small step sizes, the Perceived random walks more

"a:thfull' reflect the Prescribed distribution of directions. Because

we wish to draw conclusions based on the ass: med Perceived distributior,

*f tirections, the remairin e:erimerts were conducted under conditions
PAGE 8

W.:liiams and Sekuler

N'~ which the Perceived distribution of directions would be most

consistent with the distribution of directions which define the random .

'aks.

ESER 17N-Z Stimulus-.Duratioo

Detectabilitv' of unidirectional flow was measured as a function of

4ti~liulus duration (i.e. the number of frames Presented). For two

subJectsi, the effect of stimulus duration was determined for a step siz-e

o 0.? degree with a dot densitv of 1.6 dots Per saouare degree.

Stimulus durations (number of frames Presented) used were 2 framesy arid

all odd numibers of frames ranging from, 3 to 13. For a third subject

%easurements were made for a step size of 0.1 degree at a dot densit-: of

.6dots Per souare degree. The stimulus durations considered in, this

case were 6y 12 arid 25 frames. The relationship Proved to be nonliniear:

lip to eleven framtes the Probabilitv of seeing unidirectional flow

increased with the number of frames Presented; Presentation of

ad-ioral frames bevorid eleven did riot further augmen~t perceptibilitv.

Fi--ulre 6 shows the data for two durations, two frames and eleven framest

with step size of 0.9 degree arid dernsitv 1.6 -dots Per sauare degree. It

shiid bie noted that the Previous ie,.eriaents discussed and those in the

%R . ealnde!n of the Paper were conducted using a stimulus duration for

which -encept.ibilitv is in the asvmptotic region.

aria lv i ig temiporal summuation for our displa, it is iatportant to

nczted that its 'ocali otion vectors are distributed in the visual field

a rd t-.~s distribultion, varies with time. We therefore wonidered whether

tai ;erception of coherent motion depended onilv or, the set of
PAGE 9

Williams and Sek.uler

dire:tions Present from frame to frame or if it also depended on the

Particular Path each dot took over time. For example, would consecutive

steps by the same dot in the same direction over a number of frames be

more significant to Perception than if these individual steps were

spatially separated over successive frames?

To ex:amine if spatial factors contribute to temporal summation, we

compared Perceptibilitw of coherent motion under two conditions. The

first corndition used stimulus Patterns consisting of two sets of

spatiallv interspersed random dots. For one set of dots (denoted as

'noisel) the distribution of directions was uniform over all Possible

360 degrees of directions; for the other set (denoted as 'signal'),

dots moved only in a single direction; upward on the display. The set

assignments of the dots remained the same over all frames Presented, so

that sc .e dots moved upward frame after frame while other dots ioved

7anac2i!v frame after frame. We'll call this condition the 'Separate'

The second condition was idertical to the first e::cept in one

aspect: in each frame, the Particular dots constituting the sisnal set

and those constituting the noise set were chosen independerntly of the

d3t assignerti to the two sets in Previous frames. Thoush the

ProDrtior, of dots constituting the signal remained constant over all

frames in this condition? there were rot two disjoint sets of dots, one

sisnal and one noise, as in the first condition. We'll refer to this

condition ai the 'Combined' case.

~ c--; ~-'&Q'.:-c~ :*~ w ~ KKK&'.7- v .:-~. .
' PAGE 10

W:lliam3 and Sekuler

'-s 7cth conditions? for a given Proportion of dots which made up

the si.Inal, the distribution of Possible directions from one frame to

th2 ne::t is the same. For the Separate case, the Probabilitv that ari

dot made N consecJte steps in the 'upward' direction is eoual to the

Proportion of dots which are signal; for the Combined case the

Pobabilitv that anj dot makes N consecutive steps in the upward

-,rectior, is the Proportion of dots in the signal raised to the Power N.

The Probabilitv of seeing unidirectional coherent flow upward for

loth conditions was measured as a function of the Proportion of the

total number of dots which were in the signal. The step si-e was 0.9

degree and densitw was 1.6 dots Per souare degree. Two subjects

Participated in the studw.

As shown in Figure 7, there is no significant difference in the

Perception of coherent unidirection flow upward between the separate

3nd combined cases. The results indicate that temporal summation over

frames is criticallv dependent only on the distribution of directions

of motion Present from frame to frame. We can conclude that temporal

u,;mmat or, does not depend on the spatial relationship between local

- :tior, vectors over time.

Geieraliscussioo

As we noted beforet it is not Possible to know aztiori whether the

Perceived path a dot travels is identical to the random walk Prescribed

f*r that dot. Consider two successive frames, N and N + 1. For a given

dot, At on frame N we can define its 'corresPondent dot', B, on frame N

................. ".. -..... ... "L. .' '' "q ' '%" .'"""' . . .'"- -,'""' '% "'' .- "'-.-.'".-'' .' ' .-. '..,- - -II
mii. i. !up

PAGE 11e

Will:ams and Sekuler AE

+ 1. B is the dot on the frame b + I to which a is rerceived to move

between frame N and 0 + 1. If a's correspondent dot is the one that was pR

determined by the random walk Prescribed for a, the correspondent dot is r.

said to constitute a 'match'; if A's correspondent dot is not the one

defined by A's random walk, the correspondent dot is said to constitute

a 'mismatch'. Such mismatches Produce spurious directions of motion

that could be inconsistent with the Predefined distribution of Possible

directions.

Mismatches are the result of the Perceptual confusion of random

walks Perscribed for different dots. Decreasing the spatial density of

dots should reduce the Possibility of such confusions. If spurious

direct:ans of motion due to mismatches do contribute to a Percept of

coherent, unidirectional motion then a change in the density of dots

aone should alter the Psychometric function. Our experiments showed

su.ch changes. For step sizes greater than 1.0 degrees, decreases in dot

density, increases the Probability of Perceiving coherent flow (see

Figures 4 and 5). In these conditions unidirectional flow was Perceived

over a wider range of distribution of directions at the lower dot

densities. This suggests that spurious directions of motion due to

Sismatches may contribute to the Percept of coherent motion.

Hcwever our e;;Perimerts also contained conditions in which the ..

%s'chc'etric function was not affected by changes in dot density. For

* step si:es less than 1.0 degree, a change in dot density by a factor of

eirht, from 0.2 to 1.6 dots Per souare degree, did not alter the

dete:tab~litv zf coherence (see Figure 3). This suggests that for the

smaller step si:as, onlj the directions of local motion determined b".

°°

............... .
PAGE 12

Williams arid Sekuler

the Predefined distribwtioi of directions significantlv contribute to

the perce.-tio. of the 'nidirectional, coherent mo*.ion. Mismatches

spcear to be minimized or nonie.-isterat for these small step sizes. it

should be rioted that at a density of 0.2 dots Per sauare degreet

Perception of coherert motion for steps greater than 1.0 degree is

eauivalent to that for the steps less than 1.0 degree(see Figures 4 and

J..

Since mi:smatches are minrimized for the smaller step sizes at all

* dot densities and at the lowest dot derisitv for the la-ger step sizeso

it seems reasonable to speculate that the correspondence between dots on

successive frames is based on a nearest neighbor relationship. In this

view? the correspondent dot will be the dot on the next frame that is

closest. If the correspondent dot constitutes a match then bw

definition the Perceived distance moved is the step size. Table 2

listst the -robabilitw that the distance from a given dot on a frame to

the nearest dot on the next frame is less than the step size. If a dot

is alwsvs Perceived to move to the nearest dot on the next framet the

Table lives the Probabilities that the correspondent dot will oot be

the ore Prescribed bv the random walk. This is the Probabilitv of a

iismatch occurinZ. For each of the step sizes 0.1, 0.9t 1.1 arid 1.4

Oeereest this Probabilitv Is shown for two different dot densities: 1.6

=snd 0.2 dots Per. snuare degree.

Je tried to determine whether the Probabilite of mismatch could

lathe /Briatior, in the Probabilitv with which c:)herent flow was

seen. For 3 step of 0.1 degree the Probathilitv of a mismatch is

e; tremel'v smallp less than 0.05 at dot densities of both 1..6 and 0.2
PAGE 13

W iIA.i ms nrd Sak Ae l r

,tots .er sal.;sre degree (see Table 2). This is consistent with the

;esults shown in Figure 3 for this step size, a decrease in dot density

from, 1.6 to 0.2 dots Per souare degree did mot alter the Psychometric

function. With step sizes of 1.1 and 1.4 desrees, the same decrease in

dot density reduces the Probability of mismatch from 0.99 to 0.53 and

0.7:, respectively (see Table 2). With such a large change in the

Probabilitj of mismatch one would articipate a significant alteration in

the Pschometric function with the same change in dot density. As shown

f FiJures 4 and 5, for both these step sizes this decrease in dot

dersitv rrod'ices a large increase in the tendency to see unidirectional

,. 2herent aotion (Figure 4 and 5). For a step size of 0.9 degree,

decreasing the dot density from 1.6 to 0.2 dots Per suare delree,

reduces the Probability of mismatch by an evern larger amount, from 0.98

tZ 0.40 (see Table 2), As for step sizes 1.1 and 1.4, with such a

substantial change ir the Probability of mismatch, one would e::pect to

find an alteration in the Psychometric function with the same change

ir dot density. However as shown in Figure 3s with the 0.9 degree stepsize, the Probability of seeing unidirectional coherent motion was

unaffected by this charge in dot density. We su~gested above that a

variatir ir, the confJsabilitv of various random walks could e.plain

'3 c har, ge ir, dot density affected the Probability of seeing coherent

flow. :f this e:.Planation is correctour results for step sizes of 0.9 .

s.'ee cr larger contradicts the hypothesis that confusabilitv - or its

.erse, Correspondence - is determined strictly by nearest neighbor

relatio-.shirs. ?rsddick(1973) and Ullman(1979) arrived at similar A.

crclus:ors regarding the utility of a nearest neighbor basis for the

.crress r1renrce Process.

I-L*
-T - , - - -1 -' - , ' - - - i- - V.7-.. ,

To deal with this inadeauac,, Ullman (1979) has Proposed a 'minimal

,ap theorv of iotion correspondence' to account for the Perceived PAGE 1

direction of motion of each element in multi-element motion stimuli.

Acc~rding to the theory, each element (in our case, each dot) is

assi ned a 'cost function' that determines the Probabilitv that a dot

will appear to move at a particular velocity. Since the temporal

char3cteristics of all of the elements in our stimuli are the same, we

can replace velocity with distance travelled by a dot to simplif' the

dicussior. Acccrding to Ullman, the cost function is identical for

each elesent. The distance each element or dot will be Perceived to

aove from one frame to the next will be the distance that minimizes the

'total cost' over all elements in the stimuli. Preliminary results

sus.est that the functional form of the cost function will be sigmoid

(Ullman, 1979. pzge 113).

on sider a silmoid cost function that increases with distance

travelled and has the sharplw rising Portion of the sigmoid between 0.9

and 1.1 degrees. For step sizes of 0.1 and 0.9 degree, 'he 'over all

cost' will be minimized by having the dots move from ?rame to frame the

.4istance Perscribed by the Predefined random walk. The Path each dot is

Perceived to travel would then be the one defined bv the Prescribed . -.

I .

rsrdom walk, as such the number of mismatches would be minimized for

311 the dot densities considered. For step sizes of 1.1 and 1.4

.etrees, it will be more castr efficient to have the dots move distances

:ess than 0.9 degree from frame to frame where ever Possible. At the '."

iisher -ot .Jerisities this would result in a significant num'er of

ai5..,.h.s. As dot density is decreased, the Possi3i:itv of having a

................ - .".............,..__.......................
Ui:11afs and Sekller Pk

dot closer that 0.9 degree as the correspondent dot would be reduced,

therbw reducing the number of mismatches. At the lowest dot densitv,

each dot would be Perceived to travel according to its Predefined random

walk. Lt can be seen that bw the appropriate choice of cost function,

the results of the first two experiments would be consistent with the .F

.inimal map theorv of motion correspondence Proposed bw Ullman (1979).

T'e Parameters of the cost function will Provide constraints for

s-a' ia:1 loc3lized mechanisms of motion Perception.

Irrespective of the mechanism of correspondence between the dots on

sucessive frames, the correspondence Process alone is not sufficient to

e;::lair, the generation of a unidirectional coherent Percept of motion

fro. local motion vectors. Our data suggest that for step sizes less

than 1.0 degree and dot densities of 1.a dots Per souare degree or less,

orlv the directions of local motion determined bv the Predefined

distribution of directions significantlw contribute to the Perception of

coherent flow. We also found that although temporal summation occured

in a nonlinear manner over framesp it depended onl' on the set of

directions of motion Present from frame to frame. Taken together,

these two results are consisternt with the idea that directions of the

ridividual steos are independentl' detected and that these responses are

ther, Pcoled over time and space to generate the Perception of coherent

Moti on.

*rom the results of Experiment 1, we know that for a step size less

thin 1.0 degree and dot densitv 1.6 dots Per sauare desree, a uniform

-:stribution of directions with range 130 desrees generates a Percept

o unidirectional, coherent motion along the mean for nearl'I 100% of

--' :--: : , ,; :; ;;i "" " .". -" .. -_:2 . .. . . .". . . " " .-*..:. . . .*. ". -" -.-. .. .).::' :-. )": '";" .. ;'. .: : ""J:. . " ".": X 2 : . : ..
PAGE 16

Wi'l1Isms and Sekuler

thie trials (see Figures 1 and 2). Consider, as usual, the mean of the

distribution to be upward with respect to the subject. For this

stiauus, on eech sucessive framer each dot will be above or at least

level to its position on the Previous frame. (The majority of the dots

will of course be translated horizontally on sucessive frames as well.)

if the direction of the individual steps are independently detectedand

then the res.-onses Pooled, then the simple failute to Perceive a dot

below its previous Positiorn maw be sufficient to generate the Percept of

coherent, unidirectional flow in the upward direction. We tested the

idea. For the distribution of directions with a range of 180 degrees,

the Probability of seeing unidirectional flow along its mean was

measured as a function of the range of a uniform distribution of

directions that was deleted from the center of the original

distribution. For each of the distributions constructed in this manner,

every dot will be above or at least level with its Position on the

previous frame. The step size used was 0.9 degree and the dot density

was 1.6 dots Per snuare degree. Dhata were obtained for two subjects and

the results are shown in, Figure 8.

The Percentage of trials on which the subJat sees coherent,

,unidirectional upward flow is Plotted as a function of the range of the

distribution of directions deleted. As shown in Figure Sp if the

directions of motion within 20 degrees of the mean were removed from the

initial distribution, the freauencw of seeing coherert flow along the

,.esn~ is reduced to 50%. It should be roted that for this Part-cular

distribution, iore thar, ?9% of the dots will be above their Position or

the Previous framer wh:le less than 2% will be level with its Previous

Position. It is :'lear th-t the Presence of local ,,orion vectors all of

%F

• '. -,--'.-?.1; -...'.", -, "-",'.2.L- 4...- ."-. ..- "-, .'- .- .''"% "'' '." .-'. -. "•." '..---,-.• " --. 1- , -
PAGE 17

ias m an d -e u Je r

6hich hIsve a componert in the direction mean is not sufficient to ensure

a Percept of coherent unidirectional flow. To generate the Percept,

directions of local motion vectors in the neighborhood of the mean must

also the Present. This suggests that the Percept results from the

nonlinear spatial 4oolins for responses of direction selective

mechanisms that are tuned to the mean direction of the distribution.

M ar v eople have used cross correlation as a framework within

which to model direction selectivity; in this scheme spatial samples

- displaced by distance d- with a time lag dt are cross-correlated

(Reichardt and Varju, 1959; Pogio and Reichardt, 1973). For the

sti lus Patterns we have considered, with directions of motion chosen

from a uniform distribution, the cross correlation between two

successive frames is the same in all directions Present in the

distribution. Thus, anv correlation mechanism that could account for

the e",perirmental results must be selectively sensitive to a raose of

directions. In the visual spatial domain this implies that the cross

correlation is applied to the output of spatially localized mechanisms

each of which has a receptive field that is orientation selective.

°.

.....- "'........".....-... -"....-... ,m..l"' " ... . -...... ... .. .,. ,
PAGE 13

I aman sekuler

AcLoowledgement

1 .hus resEarch was supported bv grant number MDA9O3-80-C-0154 f'rom the

United States Arims Research Institute.

7,7
1 11 a aT a n d S e'uer PAGE 19 A

.4.

References

Acelson, E. H. and Movshong J. A. (1980) Phenomenal coherence of movins

IratirnSs. Journal cf t.be Oa±.ical, Societu of America.70? 1605.

P raddick, 0. (1973) A short-range Process in apparent miotion. Visi.on

£4raddickq 0. (1992) Correspondence Problemas in motion and stereor-sis

comp~ared. EerceEtioo.ii, A6.

Levinsont E.9 Cowner A. and Gross, J. (1980) Synthesis of visually

Perceived mnovem~ent. Suaslemeo±. to Investigative 02btbalmoo1v and

harro, D. (198) Vion. W. H. Freeman and Co.r Sari Francisco.

Posgia, T. and Reichardtr W. (1973) Considerations on models of m~ovemenit

Kdetection. 4vberroe"ik,13Y 223-2217,

FReichardt, W. and Varjup D. (1959) Uberbragungsei~erischaften in K Auswerteswstent fur das Bewegungssehei. Z.. daturf.14by 674-689.

Tvnranr P. and Sek'4lery R. W. (1977) Rapid measurement of contrast

sensitivity fu~nctions. kiet,. J4 Gatogaet.rw ard Ebvsiolosical 0&tics.SA,
r~ r U! r r Z-:_1 r r -. --.U . . w .- -;.- -NI. I . - -.- -

PAGE 20

W:iliamas ard Sekuler

-U"

Ullar,, S. (1979) The ThtI eraretatioo of Visual do±ior, Caabridge, Mass.

MIT Press.

Wilsor,, H. R. and Bergen, J. R. (1979) A four mechanism model for

spatial vision. isiaon Res,12, 18-23. I

U...

.4--

I.

,. ..

-p °
PAGE 21

:J:11 ams and Se ,ler - * .'" 0 .

IABLE_1i.

DURATION OF INTERFRAME INTERVAL REQUIRED TO GENERATE

APPARENT CONTINUOUS MOTION FOR A GIVEN STEP SIZE

'-

STEP SIZE INTERFRAME INTERVAL

(degrees) (msec.)

0.1 35.

0.3 50.

0.5 70.

0.9 or greater 90, e"

. ',

...-.
* --. - -.-. -- - , , ,- r "r -- , -. - . --- . , . .- -5 ,

PAGE 22

'ia3is I and Sekuler PAE"

IABLE_2 '

p

THE PROBABILITY THAT THE DISTANCE FROM A GIVEN

DOT IN A FRAME TO THE NEAREST DOT IN THE

NEXT FRAME IS LESS THAN THE STEP SIZE

STEP SIZE DENSITY OF DOTS

(degrees) (dots Per sauare degree)

1.6 0.2

I

0.1 0.05 0.01

0.9 0.98 0.40

1.1 0.99 0.5 3

1.4 0.99 0.71

The distribution of dots on each frame is Foissor

with Parameter, do the densits of dots Per sauare

degree. The .robabilitv that the distance from a

siven dot or, a frame to the nearest neighbor on the

r, e;:t fraa, e is less thar the step size, s, is given

1- EXP ( - i d s s)
PAGE 23

Wil'Aiams and Sekuler PE

EIGUEE-CAMIONS

EIGUEEI: The Percentage reports of unidirectional, coherent flow in

the upward direction as a function of the range of a uniform

distribution of directions. The mean of the distribution was in the ft...

upward direction; the range is given in degrees. Data were obtained for

4 different step sizes 0.1, 0.9, 1.1 and 1.4 degrees.The dot densitv in

each case was 1.6 dots Per souare degree. The results fall into 2

categories, depending on whether or not the step size is larger or '

su.aller than 1.0 degree. (Data for subject SDT). aft.°

EIURE_2' Same as Figure 1, except data for subject AHA.

E:GUE_-3: The Percentage reports of unidirectional, coherent flow in

the upward direction as a function of the range of a uniform

distribution of directions. The mean of the distribution was in the

upward direction; the range is given in degrees. Data were obtained for

2 different step sizes 0.1 and 0.9 degree. For both step sizes the

r easure~ents were obtained at two dot densities 0.2 and 1.6 dots Per

sauare degree. For step size 0.9 degree, measurements were also

zL"tained Ft dot densit' 0, dots Per souare degree. The Psychometric

function for each step size remains essentiallw unchanged with a change

ir, dot ensity. (r[ata for subject SDT).

E:^UBEJ: The Percentage reports of uridirectional, coherent flow in

the upward direction as a function of the ranse of a uniform;

distribution of directions. The mean of the distribution was in the

- , .]

, % v ,..,' ,, . :-,' '.'- -,'- ",. . - - .- , .. . ' . ' -- . ' . ' "' .....•" ,' "-", ' . '.". -. . "ft ". f -t : ''
PAGE 24

Williams and Sekuler

upIard direction; the range is given in degrees. Data were obtained for

step size 1.1 degrees at three different dot densities 0.2, 0.8 and 1.6

dots Per sauare degree. For this step size, Perceptibility does change

with dot density. With a decrease in dot density, unidirectional

coherent flow was Perceived over a wider range of distribution of

directions. For a density of 0.2 dots Per degree the data for a step

size of 1.1 detrees is almost congruent with those for step size 0.1

deiree and density 1.6 dots Per sauare degree (taken from Figure 1)

represented by the dashed line in the figure. (Data for subject SDT)

EIGURE_5: The Percentage reports of unidirectional, coherent flow in

"the upward direction as a function of the range of a uniform

II-stribution of direction~s. The mean of the distribution was in the

upward direction; the range is given in degrees. Data were obtained for

step size 1.4 degrees at three different dot densities 0.2, 0.4 and 1.6

dots Per sciuare degree. For this step sizep Perceptibility does change

with dot density. With a decrease in dot density, unidirectional

:oherent flow was Perceived over a wider range of distribution of

* directions. For a density of 0.26 dots Per degree the data for a step

si:e of 2.4 degrees is almost congruent with those for step si:e 0.1

desree and density 1.6 dots Per sauare degree (taken from Figure 1)

rePresented by the dashed line in the figure. (Data for subject SDT).

* GSEi: The Percentage reports of unidirectional, coherent flow in

the uipwpi'd directioni as a function of the range of a uniform

dIstr.bution of ;irections. The mean of the distribution was in the

*iward direction; the range is given in, degrees. Data are shown for two

..
PAGE 25

Will:azs and Sekuler

i,.fferert stimulus durations; 2 frames and 11 frames. The step size

was 0.9 degree and the dot density was 1.6 dots Per souare degree.

.erceptibilitv increases with the number of frames Presented. (Data for

subject SDT),

E.7GUEE': The Percentage reports of unidirection coherent flow in the

upward direction as a function of the Percentage of dots in th'e 'signal

set'. [lots in the 's:gnal set' moved only in the upward direction;

dCts in the 'noise set' took their directions from a uniform

distribution covering 360 degrees. The curve labelled 'Separate

D:stributiorn' denotes a condition in which dot allocation to the

'siarial set' and 'noise set' did not change for all frames Presented.

The :urve labelled 'Combined Distribution' denotes a condition in which

:ots are allocated to each set on each frame independentlv of

a&:ocatiar, or, Previous frames. There is essentially no difference in

Percep'ti r between the two conditions. (Data for subject SDT),

E;-U-S: The rercentage of reports of unidirection coherent flow in

-e upward direction ss a function of the range in degrees of a uriform

* .stributior, of directions deleted from the center of a urf~rm

.- stributicn. The distribution, before deletion, covered 190 degrees,

(Data for s:ubject SDT) .

I.

- .•
Page 26

Percent Seen Upwrd

0 00

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CCA

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yEIL~~!. ~ q-Ft tFU tq j .- .17 .----...-.-.-

Page 27

Percent Seen "Upward"

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L Page 28

Percent Seen "Upward"

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Percent Seen "Upward"' Page 29

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Page 30

Percent Seen "Upward"

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Page 31

MW

Percent Seen "Upwards"

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Page 32

Percent Seen "'Upward '_

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Page 33

Percent Seen "Upward"

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of
Page 34

Size Factors In Apparent Motion

General Introduction

The nature of the sensation of motion has been debated

since at least the time of Zeno whose mathematical paradoxes

have taken centuries to tame. Zeno's assertions on motion

perception are still interesting. Specifically, he suggested

that objects are detected in different places at different

times; memory, bridging the gap of time, connects the objects

of past and present by inferring motion to -.resolve their

spatial discrepancy. Some light was shed on the subject by

demonstrations in the late 1,9th century (Exner,1875);

adjacent electrical sparks could give rise to a sensation of

motion even though they occurred so close in time that their

order could not be reliably reported. Clearly, memory could

not have any role in the sensation of motion here, or could

it? The argument assumes a view of how the mind operates

that is different from that prevalent today. Now, it Is

certain that when past events influence present ones in an

orderly fashion, then something analogous to a memory is

operating, even If it is not available for introspective

interrogation.

The phenomenon described by Exner, in which two

* stimuli presented in succession give rise to a sensation of

motion from the first to the second, has become known as

* *.-.
Page 35 I

Size Factors in Apparent Motion

apparent motion (AM). This is to be distinguished from

real" motion in which the stimulus follows a continuous

path. Apparent motion is amenable to a remarkable variety of

experimental manipulations, many of which have been done by ,...

now. One of the largest and earliest contributions to this

literature is the work of Wertheimer. Considered a seminal

paper in Gestalt psychology, Wertheimer's (1912) article

opened up the questions of limits on spatial separation, I

timing, duration and intensity necessary for the production

of this illusory motion. Gestalt psychology realized the

value of such illusions--that they are not so much defects as

logical consequences of the underlying rules employed by the

perceptual systems, and as such offer the opportunity to

discover those rules.

In regard to apparent motion, one of the key issues of

interest to these psychologists was that of phenomenal

identity. Without the continuity provided by real motion,

the objects In the two frames of an apparent motion display

are somehow matched, or identified as a single object in

motion. This identification task has come to be known as the

"correspondence problem." In displays consisting of single

fla.shes of light, this would hardly seem to be a problem; but F

the Gestalt psychologists were adept at designing more 4.

ambiguous displays, which were nonetheless readily perceived

as motion.

..

. . . . .- * ' . 4'~ . . .
Page 36

Size Factors in Apparent Motion

An example where the correspondence problem is really

more of a problem is the display of Ternus (1926). Fig 1

shows the two frames of this display, the first on the top

and the second on the bottom. When presented in alternation

at the appropriate rate, the two dots appear to move as a

group, although the central dot actually never changes

position. According to the Gestalt thinkers, the fact that

this result would not have been predicted by observing either

pair of dots in isolation is Indicative of the "globality" of

the correspondence process. Observations such as this

evolved into the principles of Pragnanz (Koffka,1935),

stating essentially that, within the constraints of the

. information present, the percept formed will maximize such

properties as symmetry, simplicity, regularity and so forth.

Due partly to the advances in physiological psychology

and to the growing body of research on "real" motion,

speculation on apparent motion has grown increasingly

mechanistic. Divers researchers have realized the power of

relatively simple, physiologically realizable local processes

° to detect and analyse motion. The basic scheme is

exemplified by Barlow and Levick's (1965) model for the

directionally selective cells in the rabbit retina (Fig 2).

Two spatially separated detectors (Ri & R2) are logically

*- conjoined via an AND gate; one is connected directly (R1) and

-.

. . . . . . . * . . .* , . . . . .

.... ... *-.ailnm.... .
Size Factors in Apparent Motion Page 37

the other is connected via a time delay, with its sense

inverted (not R2). Objects passing from R2 to R1 will send,

simultaneously, signals "true" (from Ri) and "false" (the

delayed inverted R2) to the AND gate, which will not respond.

Stimuli moving the opposite direction do not have this

inhibition problem, so the output of the AND gate is ..

effectively directionally selective.

Such a scheme has the obvious advantage of explaining

the sufficiency of discontinuous stimuli to produce a

sensation of motion. It is also amenable to modifications

such as replacing the time delay by an element with low-pass

temporal properties, or adding characteristics to explain the

course of adaptation or aftereffects. Yet, this sort of

motion detector can only suffice at the most primitive levels

of any working motion system; without a rather sophisiticated

algorithm to interpret an ensemble of these simple units,

their outputs In response to complex moving scenes would be

rather ambiguous. The heart of the problem is that in order

to deduce motion from the response of such a detector, one is

forced to make the naive assumption that the two inputs were

stimulated by the very same physical object on a continuous

path; this is not always a reliable assumption. To make the

response of a simple motion detector a reliable indicator of

motion requires additional information from other motion

detectors, memory, etc. In other words, the simple units

I;
Page 38

Size Factors In Apparent Motion

must somehow be wired to solve the correspondence problem.

A number of explicit theories have been suggested for

accomplishing this, notably the vector model of Brown and

Voth (1937) and more recently an iterative model by Caelli

* (1980); all are distinguished by working well, but only for a

modest range of displays. Despite the failure to produce

quantitatively useful models, some progress has been made in

understanding the problem at a higher level of abstraction.

One line of progress has been a computational theory

outlining the goals and resources of the system and some of

the logic needed to connect them, but without reference to

particular algorithms. An example of such an approach in

motion perception is the work of Ullman (1979), and similar

work on other visual problems can be found in Marr (1982).

After deciding that motion perception can be conceived

as a correspondence problem, the preeminent remaining issue

is what is being put in correspondence. It is on this Issue

* that I will present some new empirical evidence. One

suggestion for the domain of the correspondence process is

raw, gray-level data from the images (Anstis,1970). Two

* frames of an apparent motion display could be compared point

*2 by point (or perhaps by small "windows" of points) using S

*: cross-correlation or differencing techniques, and the match

yielding the minimal error would be the one perceived. As

• -.

~ ' . ~ *; > . : ~.*. '*.-**....*.**.* a .. .- ~ **
.Page 39

Size Factors in Apparent Motion

Ullman points out, a simple gray-level match could not

account for what is perceived when figures such as those in

Fig 3. are shown. Fig. 3a shows the intensity profile for

the first and second frames (top and bottom); each contains a

smooth gradient and a fairly sharp edge (marked by a * here).

The smooth gradient in the first frame Is in registration

with the sharp edge on the second, and vice-versa. A

correspondence based on gray-levels alone will show a maximum

correlation centered on zero, i.e. no motion should occur

when these two frames are alternated. Human observers,

however, report motion between the two sharp edges.

The conclusion drawn from experiments such as the one

above is that some higher level organization of the raw

intensity data takes place prior to the correspondence

process. A good deal of attention has been paid to the

figural aspects of the stimuli in apparent motion, mostly

with null result. Orlansky (1940) made a more or less

systematic investigation of the correspondence of various

geometrical figures shown in alternation. Squares, circles,

triangles were matched with one another. Perhaps

surprisingly, the disparate pairings produced good apparent

motion; a modest amount of selectivity was demonstrated,

however, by measuring the range of interstimulus intervals

that would support apparent motion for each pair. Kolers and

* Pomerantz (1971) performed a somewhat cleaner version of

. * .. "* * *
rr r v..-.r r -=- -, r

k Page 40

Size Factors in Apparent Motion

essentially the same experiment, using the percent seen in

smooth motion as a dependent variable, and found again almost

no effect due to figural similarity.

An ingenious study by Navon (1972) presented a subject

with several possible paths on which to see the object

present in the first frame to move to in the second frame;

the objects were different Hebrew letters. Again, no

reliable differences were found to suggest that the motion

system preferred to match like letters. A similar result was

reported by Burt and Sperling (1981) who also used a

competing paths technique and simple geometric figures. On

the other hand, Frisby (1972) has shown that difference in

the orientation of line segments can affect the likelihood of

seeing apparent motion. Ullman (1980) provided corroborating

evidence along with data indicating that the length of

vertical line segments also Influenced their correspondence.

Thus it would seem that the analysis of motion must precede

the organization of the image into complex forms, but -.

probably follows certain more primitive levels of analysis.

Marr (1982) has divided visual organiation into

several hierarchical levels. This hierarchy starts with the

raw intensity values present in the image. Changes in the

sign of the slope of these intensity arrays are then noted as

zero crossings". Zero crossings refer to places where the

........... ..... ................................ ................................
Page 41

Size Factors in Apparent Motion

non-directional" second derivative, or Laplacian

a2 + 2

3X 2 ay

2

is zero. This may seem like an unlikely operation, but it

does have nice properties and can indeed be computed by

biologically realistic hardware. Consider that if a Gaussian

function were first convolved with the image before computing

the Laplacian, it would be mathematically equivalent to

taking the Laplacian of the Gaussian and then convolving this

with the image. The Laplacian of a Gaussian is a Mexican

hat-shaped affair, very similar to the difference of

Gausslans curve (DOG) shown in Fig. 4 a. It also looks

remarkably like the weighting functions of the receptive

fields of retinal ganglion cells as described by

Enroth-Cugell and Robson (1966), with an excitatory center

and an inhibitory surround.

By using different sizes of receptive field, zero

crossings can be obtained at varying scales or levels of

coarseness (equivalent to blurring the image by convolving

with different sized Gaussians). The reports of these

zero-crossings can then be combined to form , Marr suggests,

the signals of the next level of the hierarchy, the raw ,e7

primal sketch. Items such as bars, edges and blobs with

• attributes of position, length, width, orientation and

contrast are the primitives of the raw primal sketch. It is

this level that Marr explicitly suggests forms the domain of

. .

" 

.. .

-.-.-.. .. .. . . ................ . . . .".-..---...-"-"-.. . . . ..- . .. . ............-... . . . . . . . . . . .....-.- . . ,--.
Page 42

Size Factors in Apparent Motion ,1

*' the correspondence problem in motion.

Ullman, in his book (1979), has elaborated on Marr's

primal sketch suggestion, and has provided some evidence

supportive of it. One such piece of evidence is a

demonstration that suppposedly whole forms can be broken into

simpler constituents to satisfy the solution of the

correspondence problem. The "broken wheel" demonstration is

a variation on the well known wagon wheel effect seen in

motion pictures wherein upon reaching a certain angular

velocity a spoked wheel appears to change direction. Fig. 5 1.

shows such a wagon wheel in which every other spoke is broken

by having a piece removed from its center. By presenting

slightly rotated versions of the pattern on successive

frames, the wheel will appear to rotate. The solid and

dotted lines in Fig. 5 show the first and second frames of

such a sequence. At a certain rate of succession however,

what is oberved is that the wheel seems to split up into

three concentric wheels. As indicated by the arrows, the

outermost and inermost segments will appear to rotate

clockwise, while the central (broken) segment will appear to

rotate in the oppposite direction; this is consistent with

the "minimum distance" principle observed in many ambiguous

AM displays, but is surprising in that it suggests that, for

the purposes of motion analysis, a whole figure such as the

.. '.: ...- _., .. .. . . . . . ..... -... ...-... -.. . -... . . . . _ ..
Page 43

Size Factors In Apparent Motion 0,e

spoke really consists of a number of components.[lj

Experiment One: Introduction & Methods

Although It has been suggested that size is an

attribute available in the primal sketch and hence a

candidate for influencing the correspondence process in

motion, evidence for this is scant. As mentioned before,

Ullman did look at the effect of length on vertical line

segments in horizontal apparent motion, and noted that paths

pairing segments of equal length were preferred to paths

pairing segments of unequal length. Kolers (1972) mentioned

anecdotally that a square figure, measuring 0.3 deg of visual

angle/side was just as likely to appear to move to a

like-sized circle as to an identical square; but, when all

figures were scaled up to 1 degree, the similar figures

predominated In motion. Fernberger (1934) used "wide" and

narrow" bars (no dimensions given) In the Ternus

configuration as shown in Fig. 6. When the relative

positions of the two sized bars remained the same from first 

to second frame, the bars appeared to move as a group just as

Ternus had reported (6a). When, however, the bars reversed

1. The minimum distance principle dictates that the

mapping chosen minimizes the total distance moved, summing

over all the indidual paths.

6-.°"
Page 44 1"

I Z

Size Factors in Apparent Motion

Experiment One: Introduction and Methods

position so that bars of the same size occupied the central

position (Fig. 6b), the central bar remained stationary

while the flanking bars moved from side to side. This *

behaviour is consistent with a size selective correspondence

process.

If the size information available in the raw primal

sketch is based upon the responses of different sized

center-surround operators, then the simplest sort of size

difference would be represented by the differential

stimulation of the various classes of these size tuned

mechanisms. Wilson and Bergen (1979) have suggested that

only four classes of size tuned mechanisms are necessary to

account for human psychophysical performance for patterns

concentrated below 16 c/d. Their conclusions are based on

threshold data for various patterns that is consistent with a

* small number of broadly tuned (bandwidths between I and 2

'. octaves) detectors and certain assumptions about their joint

probability distribution. While the number of such classes,

and even whether there is a finite number, are still highly

"" debatable topics, there is a vast pool of evidence pointing

to size tuned mechanisms. Besides the detection data of N.

Wilson and others, studies of adaptation (Williams, Wilson &

Cowan,1982) and visual masking (Mostafavi and Sakrison,1976) 'S

draw similar conclusions.

. .S. . . . . . .... . .. .. . .. . . . . . . . .''
Page 45

Size Factors in Apparent Motion .,

Experiment One: Introduction and Methods

The ideal stimulus, then, would be both localized and

band-limited in spatial frequency. In this context, to be

localized means that the stimulus will occupy a finite area

on our display screen--its place will be unambiguous. To be

band-limited in spatial frequency space means that if we were

to decompose the stimulus into a sum of sine and cosine waves

of different frequencies and amplitudes via Fourier analysis,

then above a certain frequency the amplitudes would decrease

monotonically and likewise, below a certain frequency the

amplitudes would also decrease monotonically.

Although many classes of functions could be devised to

fulfill these requirements, the obvious choice is a

difference of two Gaussians (DOG). This pattern was used by ,. -

Wilson and Bergen in the formulation of their model. With

the parameters as shown in Eqn. I the DOG has a 1.8 octave -

half-amplitude full bandwidth. A typical DOG and its Fourier

transform have been plotted in Figs. 4 a and 4b.

(1) DOG(x) = 3exp(-x2 /o2) - 2exp(-x 2/2.25o 2 )

The center frequency of the DOG depends on the dispersion

parameter, sigma. This frequency can be shown to be

inversely proportional to sigma. Casual reference following

to the "frequency" of a DOG will actually refer to DOGs of

the form in Eqn. I whose Fourier transforms are centered at

.. -..-..-. ,
Page 46

Size Factors In Apparent Motion

Experiment One: Introduction and Methods

'a-.

the specified frequency. DOGs centered at lower spatial

frequencies will he larger than those centered at higher

spatial frequencies.

The most straightforward way to test whether the size

information Is used in the correspondence process for motion

perception Is to present pairs of DOCs of different sizes in

a situation known to produce apparent motion between two

identical objects. If observers report motion more

frequently between similar DOGs than between dissimilar DOGs,

then we would have some evidence indicating that this sort of

size Information Is used in the analysis of motion. If, on

the contrary, observers show little distinction between

different sized pairs and same sized pairs, (as Kolers'

subjects failed to do for different geometrical figures),

then we might conclude that size as defined in this fashion

Is not of primary importance in developing a correspondence.

This is the essential rationale for the first of two

experiments to be presented here.

... '

The factor of interest in this and in the subsequent

experiment is the spatial dispersion, sigma, of the stimulus.

It Is important to rule out other factors that may also

covary with sigma. The total area above zero (where zero

represents the mean luminance) is equal to the total area ,.bd1

,;below zero for all flOGs defined as in Eqn• 1, so the mean

%*I
Page 47

Size Factors In Apparent ?!otion

Experiment One: Introduction and Methods

luminance will not change with sigma. The issues of contrast

and total energy are not as simple. The contrast or

modulation level of a sine wave is fairly straightforward to

find because its amplitude is symmetric about the mean. Its

contrast is commonly expressed as the difference between the

maximum and minimum luminance levels (Lmax & Lmin) divided by

their sum.

(2) Contrast - (Lmax - Lmin)/(T.max + Lmin)

As you can see, contrast ranges over the closed interval

fO,1. With functions that don't show the symmetry across

the x-axis that sine waves do, the definition of contrast

will inevitably be somewhat arbitrary. The definition which

* Wilson and Bergen used Is the maximum luminance minus the

mean luminance (Lmean) all divided by the mean luminance,

(3) Contrast * (Lmax - Lmean)/(Lmean).

This function also ranges over 10,11 and, you can see that it

Is equivalent to (2) when applied to sine waves.

The formulae above were derived from mathematical

considerations and not psychophysical ones. It is not a

priori certain that patterns with the same calculated

contrast will appear to have equal contrast to observers. No

published reports are available on the suprathreshold

....................................................... *. *... .. ...
Page 48

-" Size Factors in Apparent Motion

Experiment One: Introduction and Methods

apparent contrast of DOGs, and how this may vary, If at all,

with their center frequency. Georgeson and Sullivan (1975)

carried out extensive experiments on the apparent contrast of

sine waves of different frequency and bars of different

width. The overriding conclusion of their work is that the

visual system shows remarkable contrast constancy over the

different sizes; subjective contrast matches between sine

waves of different frequency or lines of different width were

very nearly veridical. This constancy proved to be largely

independent of luminance and position on the retina. The

robustness of contrast constancy invites the conclusion that

different sized DOGs of the same calculated contrast will

share the same apparent contrast. To be certain that

Ceorgeson and Sullivan's results would generalize to DOGs

under the conditions of brief presentation to he used in the

apparent motion display, some exploratory data was collected.

Observers viewed binocularly the one-dimensional

vertical DOG patterns on a Tektronix 606A cathode ray tube

from a distance of one meter. At this distance, the screen

* osubtended 6 degrees of visual angle. Patterns were generated

by streaming luminance data from a digital computer; the

visible part of each frame was composed of 1100 lines, each

assigned its own luminance value from the computer's memory.

, The frame rate was 60 per seLond. Contrast levels could be

varied uniformly over the entire screen by an analog

-. ..... .. . .
Page 49

*- .

Size Factors in Apparent Motion

Experiment One: Introduction and Methods

multiplier, also under control of the computer. A block

diagram of the system with all parameters Is shown in

Appendix 1. b I

The mean luminance of the screen was kept at 12.3

candelas throughout; contrast could be varied from 0.0 to

0.44 while remaining well within the linear range of the

system. Observers were presented with a series of trials,

each starting with a single frame (16.7 ms.) presentation of

a particular DOG, a I second period of a uniform screen at

mean luminance, and finally the single frame presentation of

another DOG of a different size. The one second delay

between presentations was employed to minimize masking

efrects. The subject was asked to indicate by pressing a

lever which DOG had the greater contrast. The first DOG

maintained a standard contrast throughout a session, while

the contrast of the second DOG varied from trial to trial

with subject responses according to a staircase procedure

(Wetherill & Levitt, 1965). The staircase used was designed

to track the point at which the subject was just as likely to

pick one DOG as the other (50%). Initial changes in the

contrast of the second DOG were made in 1.5 dB steps. After

a certain number of trials were accumulated, the step size

was halved, and later halved again. Each staircase proceded

until 16 reversals (changes in the rank order of the two

stimuli) were obtained at the smallest staircase stepsize

• . . . . . *|
Page 50

Size Factors in Apparent Motion

Experiment One: Introduction and Methods X-

(0.38 dB). In fact, two independent staircases were randomly

interwoven for each stimulus pair as a check on convergence.

The levels at which the last ten reversals occurred were then

averaged to estimate the point of subjective equality. The

record of a typical staircase is shown in Fig. 7. DOGs

centered at 1/2, 1, 2, and 4 cycles/degree were taken two at

I

a time to form all six unique pairs. Two observers,

including the author, were recruited for testing on each

condition.

Results for the contrast matching sessions are shown

in Figs. 8a-f In most cases, a single regression line, shown

dashed, was found to provide an excellent fit to the data.

Regression parameters are shown at the top of each figure.

'" Almost all of the best-fitting lines have slopes close to one

and intercepts close to zero. The sample standard deviation

associated with nearly every point In Fig. 8 is

approximately 0.11 log units, (about 1dB), putting the

majority of points within one S.D. of a verldical match. In

a number of cases, there apppear to he small, consistent

shifts, favouring one DOG or another. Subject D.W. in Fig.

8 a and subject S.1. in Fig. Rf are the most prominent

examples of this. However, neither effect Is borne out by

the other subject under the same conditions. It is not known

at this time whether these data represent real individual

differences or systematic measurement error. At thresholds

9;

" . . ---r* 

• . • -. .- . . .. - .. ' ." .... * ,' . -. .*r, '. . .. . . . -' . . ". ..-. ' ..-. ' ,-.' . 2 .-. -"4 ' ' -J " '
Page 51

Size Factors in Apparent Motion

Experiment One: Introduction and Methods

for detection, (which are just below the lowest contrasts

measured here), there is known to be a 10:1 difference in the

.. V

just detectable contrasts of the two extremes in frequency,

1/2 and 4 c/d (Wilson and Bergen,1979). In light of this, it

was decided to keep the contrast of all DOGs fixed at the

highest feasible contrast for the subsequent motion studies.

Contrast as defined in Eqn. (3) depends on the peak

and mean luminances; the DOGs of Eqn.(l) all have the same

mean (zero) and all peak at 1.0 (when x-O), so as defined,

they all have the same contrast. It has been suggested that

the total energy of a figure is a factor in the

correspondence process (Burt and Sperling,1981). The total

energy in a waveform is proportional to the Integral of the

*square of its amplitude, which for DOGs makes It proportional

to sigma. It would be possible to match the different sized

DOGs for total energy, but at the cost of mismatching their

contrasts. As a check on the possible effects of total

energy differences as opposed to spatial dispersion of that

energy (size) it was decided to run some "energy-matched"

probe trials. If the correspondence process discriminates on

the basis of size, then pairs of DOGs differing substantially

in size should show the most pronounced effect. Therefore

the pairs (1/2 - 2) and (I - 4) were altered so that the 1/2

c/d DOG had the same total energy as the 2 c/d nOG and the

4 Ic/d DOG had the same total energy as the 4 c/d DOG. Should

.. .
Page 52

Size Factors in Apparent Motion

Experiment One: Introduction and Methods

differences in the likelihood of seeing motion In the (2 - 2)

pair versus the (1/2 - 2) pair be attributable to only to the

total energy discrepancy, no difference should be observed in Co.

between (2-2) and the energy-matched (1/2-2) pair.

The percentage of trials on which motion between the

two figures Is seen is the only dependent variable in the

first study; although It is perhaps the most common dependent

variable in simple apparent motion studies of this sort, it

is also susceptible to factors that are not of special

interest, particularly shifts in subject criterion. Several

stratagems have been adopted to help stabilize the results.

Instead of asking the subject to decide after a single pair

of frames, the two DOCs were shown in alternation for 8

complete cycles; under optimal conditions, an object will

appear to oscillate to and fro when presented in this manner.

Deciding whether the figure moves consistently for the R

cycles seems to be subjectively easier for subjects. Other

authors have employed this cycling technique with success

(e. g. Pantle & Picciano, lq76).

A second regimen employed for the stabilization of

results was randomization of the stimulus pairs. The four

different sizes combined to make 16 different pairs.

Ideally, all 16 pairs would he randomly distributed over the

entire period of a subject's participation. Practical
Page 53

Size Factors in Apparent Motion

Experiment One: Introduction and Methods

limitations on the subject's time and on the ability to

retrieve waveforms rapidly from the conputer's storage

allowed only a modest approximation of complete W..,

randomization. Stimuli from the 16 pairs of DOCs mentioned

plis the two energy-matched probes were chosen 6 at a time

without replacement. Each group of 6 was randomly permuted,

presented to the subject, shuffled again, etc. to accumulate

10 trials for each of the six pairs. (The process continued

until all of the pairs were chosen.)

A final step to improve the quality of our estimates

is a most traditional one: run more trials. The

randomization procedures described above were repeated 4

times for each subject. Using a different random sequence

for each occasion assured that each pair of DOGs was seen In

four different contexts. If being tested In a particular

random group of 6 had an effect on reports, we would expect

it to be neutralized by averaging over other random contexts.

The total exposure for each pair thus comes to 40 separate

t' r"-'a I s

trials.

,.* J.

A further factor that is known to affect the.

likelihood of reporting apparent motion Is the interstimulus

Interval (ISI), the time between successive presentations.

In the cycled display used here, this is both the time

between the presentation of the first and second DOG and the

W,°'
Page 54

Size Factors In Apparent Motion

* Experiment One: Introduction and Methods

time between the end of one cycle and the beginning of the

* next. This time parameter is known to affect the perception

of apparent motion in the same fashion as spatial separation

(Burt and Sperling, 1991; Caelli, 1981; Morgan,1980). Rather

than try to exec,,te a very lengthy full factorial design, I

decided to postpone manipulation of the distance parameter

for a future study. Instead, three different ISIs were

selected on the basis of published reports (Kolers,1972) and

preliminary observations, 67,100 and 133 ma. These ISIs were

likely to span unknown optimal IS! with the distance traveled

fixed at I degree.

The apparatus used for the presentation of the

patterns Is the same as described for the contrast matching

study. One DOG in each pair was presented on the line of

fixation, while the other was centered one degree to the

right. Subjects were asked to fixate on the central marker

and press a button to start the display when ready. If the

display appeared "to oscillate from left to right and back

for most of the duration of the show", subjects were asked to

indicate so by pressing a button; another button was reserved

for all other ca:3es, such as no motion or sporadic motion.

-. . . . "" . "" .." " "" " " .' "". . - " " ' - " . .. . . . . . ... . " " " .
Page 55

Size Factors In Apparent Motion .

Experiment One: Introduction and Methods

4..

Four subjects, including the atthor, completed the

regimen described above at each of the three ISIs. Three of

the subjects were naive to the purposes of the study,

although all had participated in other studies of visual

perception. Sessions were limited to approximately an hour

to avoid enilue fatIIJe. Subjects finished the entire programIn about 2 weeks. The Importance of careful fixation during ,.

presentation was stressed to subjects; eye movements can have

effects on the appearance of apparent motion displays. Two

of the subjects were emmotropes; the other two were fitted

with corrections for myopia during all runs.

. -.
Page 56

Size Factors In Apparent Motion

Experiment One: Introduction and Methods

Experiment One: Results and Analysis

The percentages of trials on which subjects reported

, motion are plotted in Pigs. 10-13 for the four subjects

respectively. Each graph within a given figure represents

the results for all pairs of DOGs with a particular size DOG

shown on fixation. So each panel of Fig. 10 shows data for

subject S.A.H. at a particular center DOG with the frequency

of the off-fixation DOG plotted along the abscissa. The

filled figures (circle,square and triangle), represent points

for the three different ISis: 67, 10 & 133ms. respectively.

Compiled data is shown in Appendix 2.

The reader may have noted that two percentages have

have been plotted for every size pairing of DOGs, (A,B): one

*. with A on fixation (A-B), and one with B on fixation (B-A).

.; It is also true that the data could be replotted with the

frequency of the fixated DOG along the abscissa instead of

the frequency of the off-fixation DOG, as it Is shown now.

There Is little evidence in this data that the fixation of

one or the other member of a pair makes a difference in the

likelihood of seeing motion between them. This Is consistent

with the results of Kolers (1977). The data for subject S.T.

in the lOOms I5I condition has been plotted both ways In Fig.

L ....-

.. -. .......... -.,..., ......-. , .4..*...-*....*.... ...-.. .." *-,-: -",------ ., . . .- , - -- - -.. - - - -- . . . . .. . .. . ..... . .".. ::-
Page 57

Size Factors In Apparent Motion

Experiment One: Results & Analysis

14 for comparison; dotted lines represent data with the

center frequency on the abscissa, while represent data with

the off-fixation frequency on the abscissa as in Fig. 12.

The two most notable exceptions are the (1/2--l) pairs of

subjects S.H. and A.S. The percentages for the 1/2

on-fixation pair and the complementary pair (1/2

off-fixation) percentages for these two subjects are plotted

In Fig. 15. The differences are unexpectedly large for both

subjects. It is also true that the discrepancies are in

opposite directions for the two subjects; S.11. is more

likely to see motion with the larger DOG on fixation, while

A.S. is more likely to see motion with it off fixation. The

fact that these discrepancies are in opposite directions and

that the other two subjects fall to show such differences,

relegates this effect to the status of a curiosity until a

much larger number of subjects have been run. This is not,

however, the most interesting feature of the data.

The Intriguing aspect of the data in Figs. 10-13 is

the preference shown for movement between like-sized DOGs.

This may be made more obvious by averaging the results over

the four subjects. Since the question of fixational symmetry

is, momentarily, not an issue, we may also average the

complementary pairs, I.e. pairs that differ only in which

DOG is on fixation. The results of these manipulations are

shown in Fig. 16. Each curve peaks where the size

7--

-..-..-.... . .. . ....... . . - . .
Page 58%d

Size Factors In Apparent Motion

Experiment One: Results & Analysis

difference Is zero, and falls off monotonically elsewhere.

The preferences for (1-1) over (1-2) and for (2-2) over (2-1)

are both modest, biut significant at tile .05 level using

Wilcoxon's test For matched samples (T=14,m=12). This

preerece or ik-sized DOCs is consistent with the nto

*advanced earlier that size information is pertinent to the

correspondence process.

The prohe trial s that were energ~y matched Instead o f

contrast matched are shown as open symbols on Figs. 10-13;

the same ISI legend as used on the filled s ymb1)olIs pertains.

* The energy-matched I)OC pairs are (1/2-2) and (1-4) as

mentioned earlier. The total energy hypothesis wotild have

t th at equtalizing the to talI energy woulId r en de r (1/2-2)

equtivalent to the (2-2) pair in sponsoring motion, and

similarly equate the (1-4) pair to the (4-4) pair. A quitck

* glance at the data will reveal tIhat this position is

*untenable. WhIle the (2-2) and (4-4) pairs generate mot Io n

reports nearly 100%' of the tine, the energy matchied pairs are

*as depressed as their contrast matched equivalents. The

*single exception is the (1-4) energy matched p a Ir a t ll3ms

ISI for subject T.R.; at 100%., It fulfills the predictions of

the total energy hypothesis; hut, as a single point it holds

little weipht. Although energy factors may play some role In

the correspondence Process, they are not sufficient to

explain the effects demonstrated here.

% %1

*" *_ 4 * ** P- ..-. * * .- ... . . . .
Page 59

Size Factors 

I%

In Apparent Motion

Experiment One: Results & Analysis ."- ;

The question of ISI effects in these data is somewhat

more difficult to answer. At first glance (Fig. 16) they X.

would appear to he negligible. Indeed they are modest, but

there is some orderliness to their variation. Consider the

longest ISI (133ms,triangle) and the shortest IS"

(67ms.,circle). It appears that where lower frequencies are

involved, both on and off fixation, the circles dominate the

triangles; and, conversely, where higher frequencies are

involved, the triangles dominate the circles. There are

*. exceptions, of course. This sort of categorical interaction

is difficult to quantify without embracing dubious

assumptions. I will present a scheme that will hopefully be

intuitively pleasing despite being somewhat arbitrary. The

key to the scheme is to code each pair of DOGs by the sum of

their frequencies; instead of using actual centre

frequencies, the four values have been integerized (by taking

I plus lop, base 2 of the frequency ) to the values 0 through

3 for 1/2 through 4 respectively. The difference between

percent seen at the shortest IS! and percent seen at the

*longest IS (circles-triangles) will then hopefully be

predicted by the sum of the coded frequencies. Fig. 17

shows a scattergram with these sixteen points from the

composite data of Fig. 16. On the vertical axis is the

r difference in percentage seen at the shortest ISI and the

* longest ISI; on the horizontal axis is sum of the coded

o . . . . * . •..

- . ..* . . . . . . . . . ... . .. . -*.. ,. . .. . *. . • . . . . . - • " " ". • .". . " . • • , . . . . " - " " " 1

- * , . - . . ° ° .• . . . . . . .
Page 60

Size Factors in Apparent Motion .J

Experiment One: Results & Analysis

frequencies for each DOG in the pair. The visible tendency

of the points to cluster around a line of negative slope

suggests that the frequency codes are not such bad

predictors. A rank order correlation coefficient, Kendall's

tau, for this set is -0.78, a significant value at the 0.01

level.

This can he seen as partial confirmation of the

observation that at the shortest 1SI the pairs with larger

DOCs were more likely to be seen in motion than pairs with

smaller nOGs, while at the longest ISI, the inverse relation

holds. It is possible that this reflects differing temporal

properties of simple motion detecting units tuned to

different sizes. The slope of Fig. 17 would, in this view,

indicate that mechanisms arranging the correspondence of

smaller DOCs had a longer time constant than those reponsible

for the correspondence of larger DOGs. Given the small

absolute magnitude of the observed effect, these speculations

must await a more thorough -inipulation of stimulus timing

before being seriously considered.

J ,,

* **. . . . .. . . . . . . .. *.*~'**%* . '* -::
Page 61

Size Factors in Apparent Motion

Fxperiment One: Results & Analysis

U

Experiment Two: Introduction & Methods

The evidence presented so far suggests that the closer '

DOGs are in size the more likely they are to be seen In

apparent motion. The tentative explanation Is that whatever

mechanism Is responsible for developing the correspondence ._

for motion perception uses size Information to help determine

whether objects should he connected in motion. Ullman (1979)

has suggested that the various attributes of object pairs

such as colors, orientation, size, spatial and temporal

displacement are somehow aggregated in a single metric,

dubbed "affinity". This is taken as an index of goodness of

fit between any pair of primitives of the correspondence

process. The affinity of a given pair does not take into

regard the other possible pairs that either of its members

may participate In. Rather, it reflects that pair's quality

in isolation. Affinity is an aggregate In the sense that

changes made in one attribute of the pair, say distance, can

be made up for by changes In another attribute, say ISI. The

experimental results obtained here would, in this view,

reflect changes in the affinity of pairs due to size

discrepancy.%

U...

*. . . . . . . . . . . . . . . . . -

o. -... , °..... . ,. . . . .. , . . ....- ..... . .U U U. ':. . . - .,. - .... , . . .. .. . , . ..- ..°.. . . . .. . . . . . . . .. .. -.. . -.
Page 62 ISize Factors In Apparent Motion

Experiment Two: Introduction and Methods

In complex moving scenes, the affinity associated with

a given pair is not the only thing that determines whether

that pair is ultimately seen In motion. In all but the

simplest cases, a single primitive element may participate In

many pairs; it is the correspondence process that must decide

which of the many possible mappings will he selected.

Ullman's thesis is basically that the mapping which globally

maximizes affinity is selected. Various constraints on the

process are built-in, according to Ullman's scheme, such as a

preference for mappings that are both one to one and onto as

opposed to those that are merely one to one. Nonetheless,

the raw materials of the correspondence process are the -

affinities of the pairs.

If the measurements with isolated pairs of DOGs, made

In Experiment One, do Indeed reflect the underlying

affinities of the pairs, then they may he useful in

predicting the results of more complex displays using the

. same elements. A strategy suggested by Ullman for examining

affinity differences is to pit two (or more) pairs directly

against each other. This is the "competing paths" scheme

mentioned before. As illustrated in Fig. 18, it involves

the presentation of a single object (A) in the first frame,

and after a preset time, the presentation of two flanking

objects (B and C). Such an arrangement will hereafter be

• ...,.......... .... .,. .....,............................................-..........,.",,..,.-."..-...-.........*,-.-.._.....\x' ..."- ..... . - . . . ... ... """"".... . . ..
Page 63

Size Factors in Apparent Motion -

Experiment Two: Introduction and Methods

referred to as the triple, (B-A-C). The two frames can, of

course, be alternated as was done in the previous experiment.

* Several different percepts might result from such a display. 4

(A) might "split" and mc,-,? to both (B) and (C) . (A) might P$."

move to (B) exclusively or (A) might move to (C) exclusively. :,"

Finally, there may be no motion perceived at all. When the

spacing and timing are well adjusted and all objects are

identical, the predominant percept is splitting

(Kolers,19Y2;t'llman,1979). In cases where the affinity of

one of the pairs, e.g. (A-C) exceeds that of the other

(A-B), it is expected to draw more reports of motion since

cases that would have been labeled "splitting" would then be

(A-C) alone.

The configuration described above is the essence of

the second experiment I have conducted. The objects (A,B and

C) referred to in Fig. 18 were drawn from the kennel of DOGs

used in the last experiment. All possible pairs had been

measured in isolation previously, so it was decided to follow

up by considering all possible triples for this "3-DOG"

study; this comprises 64 different triples. One ISI was used

throughout, lfOms, the average value of the three previously

employed. The 1S1 was not manipulated because of the large

• .number of conditions already defined, and because of its

general impotence over the range previously investigated.

The stimuli were arranged such that the central DOG peaked on

- ". . ... .. .. -. . ... . . . . .. .. 
Page 64

PSize Factors in Apparent Notion

Experiment Two: Introduction and Methods

fixation, as in the last experiment, and the two flanking

DOGs peaked 1 degree of visual angle on either side of

fixation. The two frames, central and flanking DOGs,

alternated for 8 cycles.

This display system is as described for the last

experiment, but two additional buttons were nade available

" for subject responses. Subjects were instructed to press a

"Split" button if both flanking objects appeared to oscillate

from center to side for most of the duration of the show. If

only one of the flanking DOGs satisfied this criterion, the

subject was to press either a "Left" or "Right" button

indicating which side offered the motion. Finally, if no

consistent motion was observed, a "None" button was to be

* pressed. Stimulus presentations were randomized as before so

each subject saw each triple 40 times.

.1

Three subjects from the previous experiment served

again in this one; subject T.B. was unable to continue, so

another subject (N.L..) was recruited and run on the Initial

experiment, but in the lOOms. ISI condition only. Results

from this catch-up run are shown in Appendix 2, and are in

* general agreement with the data from the other subjects.

• .. o% . ,' N

. . ... . . . . - . . . . .. ,-. . -.- ' a'
-- .*. - .- -..- -,-

Page 65

Size Factors in Apparent Motion

Experiment Two: Introduction and Methods

Experiment Two: Results and Analysis

The results from the 3-DOG experiment were tabulated

and are available in Appendix 3. These data are displayed

graphically in Figs. 19-22. Each figure shows the results

for all triples with a particular size DOG at their center

(corresponding to object (A) in Fig. 18). Fig. 19 is for

the 1/2 c/d centers, Fig. 20 for lc/d centers, Fig. 21 for

2c/d centers and Fig. 22 for 4c/d centers. Figure

subscripts a..d index subjects S.4., A.S., S.T. and N.L.

For a given subject and center frequency, there are

sixteen individual bar graphs, arranged on a square matrix.

The vertical axis of the matrix indicates the spatial

frequency of the left-hand DOG in the triple; the horizontal

-. axis of the matrix indicates the frequency of the right-hand

DOG in the triple. From left to right and top to bottom,

these axes range from 1/2 to 4. Each of the 16 separate bar

graphs shows three bars. The left-most bar Indicates the

relative frequency (in percent) of the reports of motion

exclusively In the left-hand pair. Similarly the right-most

bar Indicates the frequency of reports of motion exclusively

in the right-hand pair. The central bar indicates the

percentage of reports indicating motion in both pairs

U-' .

.****' ........ ... .... ........ ... .... 
Page 66

Size Factors in Apparent Motion

Experiment Two: Results and Analysis

simultaneously, i.e. splitting. Tic marks on the vertical

axes place 50% and 100%. Since subjects are forced to make a

response on each trial, percentages not accounted for by the

three bars are taken up by the "no motion" response category.

To help in Identifying the triple associated with a

particular bar graph, the three DOC center frequencies in the

triple, (left, center and right), are printed over the tops

of the bars from left to right.

This plotting scheme may seem confusing at first, but

it does offer certain advantages over most alternatives. By

means of an example, suppose we wished to know for subject

S.11. how well DOG pair (I-I) did when pair (1-2) was offered

as an alternative. Since both pairs have a 1c/d DOG on

fixation, the information required will be in the second

figure, Fig.20. Subject S.11. is represented in Ftg.20a

"" The particular match of pairs specified actually appears

twice: (1-1-2) and (2-1-1). The indices will help you find

, °them at the second row from the top in the third column and

in the third row from the top in the second column

respectively. Mirror image triples like these always lie

across the negative diagonal from each other. The first bar

graph mentioned, (1-1-2), shows that about 60% of the time,

the subject reported seeing motion only between 1 and I;

about 30% of the time, he reported seeing splitting-motion In

. both pairs (1-1) and (1-2). Motion exclusively between 1 and

7.--

t*....... a *......, .......... ,i. a 2L .2.... -.. ,. : .......... a
Page 67

Size Factors in Apparent Motion

Experiment Two: Results and Analysis

2 was reported less than 10% of the time. If S.!!. were an

ideal observer, the other bar graph would he a mirror Image

of the one already described. Actually, motion between 1 and

2 was reported less often in the (2-F-) cases. Motion

between 1 and 1 was seen just as frequently as in the (1-1-2)

case, as though (1-1) motion reports which were included in

the splitting category migrated to the (1-1) exclusive camp. .

A casual examination of the ."distribution of mass" on

Figs. 19-22 will reveal some of the grosser agreements of

* the data with the predictions advanced. Fig. 19 shows

results for triples with 1/2 c/d centers. As expected, most

of the motion reports were issued along the top and the left

flank, where other large DOGS were present in the triples.

Curiously, all subjects seemed to see less motion when all "Jthree members of a triple were 1/2 c/d. This holds true

vacuously for subject N.L. who reported almost no motion at

all for triples with 1/2 c/d centers. (Fig. 19d). At the

other end of the spectrum, triples with 4c/d DOG centers

(Fig. 22) show the converse effect. Most of the mass hugs

*i the bottom and right flank where the smaller DOGs are assured

to be in the triples. Figs. 20 and 21 representing the Ic/d

and 2c/d center triples respectively, show mass distributed

closer to the center of the matrix, as befits their

intermediate size status. If you blur your eyes, you might

even see the lc/d centered matrix curving slightly to the

o. * . . . . .*i* .
4 - . . .. -. WV -7 : - OY -- - - - *

Page 68

S Size Factors in Apparent Motion

Experiment Two: Results and Analysis

upper left corner and the 2c/d centered matrix curving to the

lower right corner.

In order to check the predictions involved in the

affinity hypothesis more rigorously, an index of preference

needs to be chosen that Is applicable to data from both

experiments. There are many ways to define such a measure,

and it is not clear a priori which should be chosen.

However, certain properties are desirable: it should be

continuous and well defined at the boundaries (where one pair

is preferred completely to the other); it should take into

account the magnitude of the indifference shown towards one

pair versus another, as expressed by splitting or null

reports; finally, hypothetical data from both experiments

ought to cover the same range. Eqn. (4) Is a simple metric

that meets these requirements. L and R are defined for the

3-DOC study as the percentages of reports of exclusive motion

In the left and In the right pairs respectively. In the data

of the original experiment, L and R are defined as the

*. percentages of trials on which motion was reported for each

*. of the two pairs that were joined in the latter experiment as

a triple.

(4) p 5*( 1 + (L-R)/100 ..

-* Harkening back to the example in Fig. 18, preference in the

...................................

S*****- . . . . ..-. . . . . '*' .. .
1,t . V r , ,L .* . "%-./ ' .*" * .- -- ' - -. - 4- - - - . - - -. - .

page 69

Size Factors In Apparent Motion

Experiment Two: Results and Analysis

triple (B-A-C) would he computed from the 3-DOG data by using

the percentages of left-only and right-only reports as L and

R in Eqn. (4). That preference would be predicted,

hopefully, by the percent seen in motion data from the first

experiment for pair (A-) as L and pair (A-C) as R • Both L

and R are restricted to the range [0,100]. If all reports of

motion on a given triple, (B-A-C) are left-exclusive, L will W.

be 100, R will be zero and so preference p-1. If, on the

other hand, all reports are exclusively to the right, L will

be zero, R will be 100 and so p-O. If L is equal to R , then

p-0. 5 . Although splitting and no-motion responses are not

explicitly referenced in Eqn. (4), they are not without

effect on p . If cnnslderable percentages of reports are

* either of splitting or no-motion or both, the maximum

absolute difference between L and R' is likewise restricted;

smaller differences will produce less deviation in p from

0.5.

The calculated preferences from both experiments are

* tabulated in Appendix 4. A scatterpgram (Fig. 23) shows the

predicted against the observed preferences for the entire

collection of data (256 pts *4subjects *64 triples).

Linear regression might be foolhardy; however, order

statistics are applicable. Kendall's tau is 0.35 for this

data set. With 256 points, this Is a highly significant

figure (p<0.01). The hypothesis that there is only chance
Page 70

i

Size Factors In Apparent Motion

Experiment Two: Results and Analysis

connection between the two variables must be rejected. As

with other correlation coefficients, however, a tau of 0.35

Is not exactly a howling success, so the question "Why wasn't

it better?" is well worth asking.

First, there Is some variation from subject to subject

In terms of the value of our predictor. Table 1 shows tau

for each of the four subjects, computed separately.

TABLE

Rank Order Correlation of Preference by Subject

Subject Tau

S.H. 0.37

A.S. 0.58

S.T. 0.31

N. L. 0.18

The relatively low tau associated with subject N.L. can he

attributed in large part to the extremely depressed responses

for triples with 1/2 and 4 c/d centers. Table 2 shows that

overall, these two extremes of center frequency showed the

highest correlation, while 2 c/d centers showed the lowest.

|4

4,,

: : d ' ' ": "." .' S.'
Page 71

Size Factors In Apparent Motion

Experiment Two: Results and Analysis

TABLE 2

Rank Order Correlation of Preference by Center Frequency

Center Freq. Ta i

1/2 0.60

1 0.34

2 0.12

4 0.46

Where else do the predictions break down? Fig. 20

(lc/d centers) shows some of the story. For all subjects but

A.S., the triples (1/2-1-1) (1/2-1-2) and their inverses

generated reports inconsistent with what was expected from A'

the earlier behaviour of the pairs (1-1/2), (1-1) and (1-2).

?. In particular, it was expected that the (1-1) pair in the " 2"

"- (1/2-1-1) triple would be seen in motion more often than the

* (1-1/2) pair; quite the contrary seems to be true for these

subjects. Similarly the (1-2) pair was previously judged to

be more consistently seen In motion than the (1-1/2) pair;

again the (1-1/2) pair dominates. Subject A.S. also showed

* this order in the 3-DOG data; her distinction is that theme

" results are more consistent with her performance in the first -,W

experiment. A few internal irregularities can also be

spotted in these figures. Most notably, subject S.H. seemed

• to prefer (1-1/2) motion in triple (1/2-1-1) but (1-1) motion

in triple (1-1-1/2); if he simply preferred seeing things

. . * * .

I * * .- I
- -.-

Page 72

Size Factors In Apparent Motion

Experiment Two: Results and Analysis

moving to the left, this asymmetry should have asserted

itself elsewhere, but this is not evident.

When one considers triples -with 2c/d centers (Fig.

. 21), the picture becomes even more confusing. With all

subjects, a major discrepancy was the general lack of motion

reports between (2-4) pairs in six of the seven triples

containing them (all but 4-2-4). But the (2-4) pair

engendered motion reports quite reliably when shown In

isolation. With subjects S.T. and A.S. (Figs. 21b and c),

* the (2-1/2) pair showed a curious strength, Inconsistent with

that demonstrated in the previous experiment. (This is

evident in the left-most columns of the two figures.) The

" least explicable aspect of the data is the asymmetry shown in

Fig 21b (subject A.S.). The pair (2-1) dominates In the

triple (1/2-2-1) while the pair (2-1/2) has the edge in

' symmetrical triple (1-2-1/2). The case is similar for the

* "triple (1/2-2-2) and its mirror image (2-2-1/2): (2-2) is

preferred in the former and (2-1/2) in the latter. Triples

. (1/2-2-4) and (4-2-1/2) offer another example of this

". asymmetry. It is difficult to explain. A fixational error

might alter the relative orders in this manner, but any such

eccentricity would manifest itself on other pairs in a like

manner since the triples were randomly interwoven. For

example, in the same figure, the triples (1-2-4) and (4-2-1)

ought to consist largely of (2-4) reports for the former and
Page 73

Size Factors in Apparent Motion

Experiment Two: Results and Analysis ~l$

(2-1) reports for the latter; on the contrary, the (1-2) pair

is more frequently seen in motion in both triples.
47-- -- . 77 -7 . -. . . . . . . . . . . . 7

Page 74

Size Factors In Apparent Motion

A Model Relating the Two Experiments

A Model Relating the Two Experiments

Ignoring for a moment the handful of internal

Inconsistencies with the 3-DOG data, a conclusion that can be

drawn from reported motion in triples is that the perceived

motion of one pair is not in general independent of the

perceived motion of the other. It is possible to look down

almost any row or column of the Figures 19-23 and observe

variation in the reports pertaining to the same pair. In

making the predictions from the pairs in isolation, however,

Independence was Implicitly assumed. Motion in each pair of

the triple was assumed to he an independent random process

akin to the toss of a biased coin. Each triple would have

two such coins. To stretch the analogy, if one coin turns up

heads, and the other tails, then motion would be seen on one

. side only. If both coins turn up heads then splitting is

• .reported; if both tosses produce tails, then no motion is

reported. Knowing how one coin turns up on a given pair of

tosses tells you nothing about how the other will turn up.

.. Since, under our assumptions, the tosses are independent, the

expected preference depends only on the biases of the two

* coins; supposedly we could measure those biases just as well

by tossing the two coins separately a Is the first

experiment. Clearly this is not the case; the nature of the

" interaction between the two motion paths will have to be

- ~..'.

... ' .. '.. ...........................
Page 75

b.

Size Factors In Apparent Motion

A Model Relating the Two Experiments

taken into account in order to improve on the predictions of

the first experiment. 4.

One hypothesis that has been advanced (Ullman,1979) to

explain the preference shown to one pair or another in

situations such as the second experiment is that of

competition. Put simply, if a given object could follow

various different paths in an ambiguous situation, these

alternatives will "compete"; the path offering the best , ,.,

overall features (e.g. maximal affinity, symmetry) will be

seen most frequently and at the expense of the other paths.

This postulates that paths interact, In a system that on any

given presentation must pick one path and exclude all the

rest. Many perceptual phenomena are like this; dramatic

examples are the histable illusions such as the Necker cube

and the "My Wife and My Mother-in-Law" cartoon (Boring,1930;

Attneave,1971).

There are certain drawbacks to the application of this

idea to the second experiment. The most obvious of these is

that the perception of motion in the left and right pairs is

not strictly exclusive. In some cases both are perceived at

once, and in some cases neither is perceived. If one path

were chosen to the strict exclusion of the other, triples

like (2-2-2) ought to draw 50% left and 50% right reports

instead of 100. splitting. tlowever, it may he possible to

* .

" - .. . . .. .. . ....
Page 76

Size Factors In Apparent Motion

LA Model Relating the Two Experiments

sidestep this problem while preserving the notion of

competition. I will advance a modest attempt to reconcile the

*pairwise percent-seen data of the first experiment with that

of the second by means of a modified competition scheme.

The hypothetical quantity which will be the currency

of competition shall be called "strength" , S Every pair

*5 (a-h) of objects that could potentially be seen in motion has

associated with it a strength S(a,b). Key to the derivation

of any predictions about observers' reports is the admission

that S is a random variable. This premise distinguishes

strength from affinity as previously discussed; the latter

quantity we may wish to associate with some parameter of the

distribution of strength. The triples used in the second

experiment can be considered as hearing two such variables:

S(a,b) and S(a,c). Suppose the difference between the two

variables were the arbiter of observers' reports. If the . '.

absolute difference I S(a,b)- S(a,c) I is less than some

constant, c, then one of two things will happen. If both

strengths are small, no motion will be reported. If both

"- strengths are large, then splitting will he reported. If the

difference between the two strengths is greater than c, we

will suppose that the pairs compete and the pair with the

larger strength will alone be reported in motion. This is

the essence of the interpretation of competition that will be

- considered. The shortcomings already apparent in the

.....................................................
Page 77

Size Factors in Apparent Motion

A Model Relating the Two Experiments

choosing between splitting and null responses I ask the 6

reader to excuse in order that the question of relating the

results of the two experiments in terms of preference alone

might be addressed.

Ignoring for the moment the constant c , which we will IL:

assume is the same for all triples, the competition scheme

described above bears considerable resemblance to Thurstone's

law of comparative Judgement (1927). The law was originally

constructed to help explain choice probabilities in social

psychology, but was defined in such generality that it has

since been used for problems as diverse as perception and

economics. The psychological variables that give rise to

decisions between choices are dubbed "discriminal processes"

in Thurstone's work. The law presumes the existence of means

" U(i), standard deviations s(i) and a correlation coefficient

'- r(i,j) the distributions of discriminal processes i and J

," In all practical applications, the distributions are presumed

to he normal, either by nature or through transformation done

by the experimenter. The law can then be stated as

*(5) UJ~i)-U(j -z(i.j) xI s2 (i) + s 2 (j) -2r(i,j)s(i)s(j)

where z(i,J) is the standardized score corresponding to the

proportion of Judgements observed where choice i is preferred

• . to choice J

I.

. . ................ * ** .. ,,...-.....................
L Page 78

Size Factors in Apparent Motion

A Model Relating the Two Experiments

.-

The statement of the law in (5) Is the most powerful

form that Thurstone considered; it is also the least

tractable in terms of analysis of experimental data.

Realizing this, Thurstone specified four additional forms of

the law (known as cases II through V) each making stronger

assumptions than the last. The simplest and most popular

case is V, which assumes that the correlation coefficients

are zero and that all standard deviations are equal. Case V

assumptions and analyses are very similar to those commonly

used in signal detection theory (Green & Swets, 1966).

In the current application, the discriminal processes

are the strengths associated with each pair of stimuli.

While it may be safe to assume, at least as a first

approximation, that the strengths themselves are

uncorrelated, the data suggests that the equal variance

* assumption would he foolhardy. This position is the one

outlined for Case III of the law of comparative judgement:

(6) U(1)-U(J) - z(ij) J s2 (i) + sZ(J)

" With minor modifications, this form of the law is readily

amenable to application to real data. In order to estimate

the parameters for each distribution, the data must be in the

form of probabilities of choosing one pair over another. The

. . . . . . ., .. 
Page 79

Size Factors in Apparent Motion

A Model Relating the Two Experiments

best guess we have of such a probability Is the preference,

p, as previously defined. If our measurements of preference

contained no error, the preference for an arbitrary triple

(A-B-C) would always be one minus the the preference measured

for triple (C-B-A). The fact that preferences for some

triples are not exactly the complements of their mirror

Images" preferences was reconciled by averaging. So if p-0.9

for triple (2-1-4) and p- 0 .3 for triple (4-1-2), the former

was adjusted to 0.8 and the latter to 0.2. The actual

estimation processes are fairly tedious and were carried out

with the aid of a small computer. A derivation of the

estimators is presented in Thurstone (1934) and in condensed

format in Appendix 5. A noteworthy feature of the process Is

the use it makes of approximation; the approximations may be

poor in data sets representing small numbers of stimuli or in

which ceiling or floor effects are apparent.

The results of applying a Case III analysis to the

data from the second experiment are shown In Table 3. The

estimates of means and standard deviations are tabulated for

the four subjects organized by center frequency. The 1/2 and

4c/d center triples for subject N.L. could not be modelled

successfuly in this manner, and so are omitted. This had

been expected in light of the extreme suppression of reponses

she exhibited at these two extremes.

• •o '. .°o . € •.. -.. °° . - .* -. *- . . . . .. .. . . . . .
.........-

Page 80

Size Factors In Apparent Motion

A Model Relating the Two Experiments

Table 3

Estimated Stimulus Distribution Parameters

On-Fixation DOG Frequency

Freq. lHalf One Two Four

Observer S.1.

mean S.D. mean S.D. mean S.D. mean S.D. .5 0.94 0.69 1.0 1.85 -. 56 2.92 -. 24 0.16 1 0.07 1.43 1.09 0.96 0.63 0.54 -. 30 2.23 2 -. 46 1.37 -. 23 0.28 0.57 0.30 0.42 1.60 4 -. 55 0.52 -1.83 0.91 -. 64 0.24 0.12 0.01

Observer S.T. mean S.D. mean S.D. mean S.D. mean S.D. .5 2.47 1.31 1.94 1.54 0.51 1.2 -. 25 .35 1 0.42 0.54 0.66 0.70 0.74 0.84 -. 21 1.06 " 2 -1.25 1.19 -. 28 .57 0.07 0.97 0.43 2.25

4 -1.63 0.96 -2.32 1.2 -1.31 0.99 0.03 0.03

mean Observer A.S. F S.D. mean S.D. mean S.D. mean S.D. .5 1.57 1.16 1.41 1.02 0.18 0.99 -. 50 0.48 I O.O 0.67 0.56 1.08 0.13 0.07 -. 31 1.35 2 -. 75 1.11 -. 26 0.81 0.05 2.77 0.38 1.76

4 -. 91 1.05 -1.71 1.09 -. 35 0.18 0.43 0.41

Observer N.L. mean S.D. mean S.D. mean S.D. mean S.D. 

.5 0.22 2.11 -. 58 2.11

1 0.57 0.30 0.60 0.4/4

2 -. 18 1.27 0.40 1.15 4 -.61 0.32 -. 41 0.31

• b;
Page 81 P'

Size Factors In Apparent Motion

A Model Relating the Two Experiments

Estimates of both parameters have a considerable range. .o

Consulting Table 3, means on the psychological scale vary

from -2.32 to +2.47; standard deviations vary from 0.01 to

2.92. The parameters for S.11. for the pair; making up

triples with Ic/d centers have been used to plot the strength

density functions (Fig. 24). Some expected relationships

between pairs are evident, such as the fact that the mean

strength of pair (1-1) is greater than that of (1-2), which

is in turn greater than that of (1-4). A less expected

relationship is that between pairs (1-1) and (1-1/2): their

means are very nearly equal. The strength of (1-1) should

exceed that of (1-1/2) only about half of the time and vice

versa. Furthermore, the mean strength of (1-1/2) exceeds

that of (1-2) despite the fact that S.H. in Experiment I

(see lob) reported motion more frequently for (1-2) than

*(1-1/2).'.7

One remaining task is to relate the estimates of

distribution parameters to the results of the first

experiment. flow didJ observers In the first experiment decide

whether or not to report motion when presented with a given

pair of DOGs? One proposal, in keepinr with the model just

* developed, is that they made their reports on the basis of.

the strength variable associated with the pair. The most

mundane example of such a decision rule is to set a fixed

threshold strength for reporting motion; observed strengths
W" .

-

.j Irv

Page 82

*- Size Factors in Apparent Motion

A Model Relating the Two Experiments

greater than or equal to the threshold will elicit a motion

rpport, while strengths less than threshold will not. Such a

scheme is Illustrated in Fig. 25. According to the model,

strengths are distributed normally for all pairs, but with

differing means and standard deviations. For each pair,

estimates of the mean and standard deviation are available in

Table 3. In order to calculate the probability that the

strength of a given pair will exceed a fixed threshold, one

simply has to convert the threshold to a standard normal

deviation and consult any table of the normal distribution

(flays & Winkler,1974) for the corresponding probability.

Thus, if the threshold level used by the subjects in the

first experiment were known, the probability of seeing motion

for any given pair of DOGs could be predicted from the -

" estimates of strength distribution obtained In the second

experiment.

The selection of a threshold value is a problem in

this line of reasoning. Without making further assumptions

. it cannot be extracted from the data in the second

experiment. In particular, the modelling of the "splitting" i.i

and "null responses would need to be elaborated. Since this

". is not my interest in this paper, a simpler assumption about

*the threshold strength will be adopted. The means of the

strength distributions, as a consequence of assumptions in

the estimation procedure vary more or less symmetrically
Page 83

Size Factors In Apparent ?Motion

A Model Relating the Two Experiments

around zero, (the origin of the psychological scale was

arbitrarily set there), so for the purpose of generating some

typical predictions, the threshold strength will also be

assumed as zero.

The predicted and observed percent-seen curves for all

16 pairs are shown in Figs. 26-29 for the four subjects

Individually. A set of curves showing the average of the

four subjects is shown in Fig. 30. The peaked curves

emerged from many of the predictions In more or less the

right places. Another look at the rank order statistics for

the 56 pairs of predicted and observed percentage-seen

reveals a Kendall's tau of 0.44 (p.01), Indicating that the

predicted levels are not in such poor agreement. It is

likely that thIs figure could be somewhat improved by

adjusting the threshold levels with a best fit procedure, but

this has not been attempted.

The simple model presented here should by no means be

considered a complete explanation of the data collected in

the second study. Rather, it was proposed with the intent of

supporting the size-specificity findings of the first study.

As such, it is notably incomplete and possibly inadequate In

explaining the splitting and no-motion cases arising with %.

triples. This aspect of the data was indeed more complicated

than expected. Acknowledging these shortcomings however, the

* -. . V
Page 84

Size Factors In Apparent Motion

A Model Relating the Two Experiments

notion that the perecent seen curves of the first experiment

Index some quantity relevant to perceived motion is generally

bolstered by the second experiment in conjunction with a

competition model.
Page 85

Size Factors In Apparent Motion 0

RReferences

ANSTIS, S. Phi movement as a subtraction process.

Vision Research, 1970, 10, 1411-1430.

ATTNEAVE, F. HultistahIlity In Perception. Scientific

American, 1971, 225, 62-71.

BARLOW, H.B. &LEVICK, W.R. The mechanisms of

directional selective units In rabbits' retina. 3

P hys iol1 (Lond.), 1965, 178 ,477-504.

BORING, E.G. Apparatus notes: a new amblg'ious figure.

Am. J . Psych., 1930, 42, 444.

BROWN, J.F. A VOTH, A.C. The path of seen movement as

a function of the vector field. Am. 'I. Psych.

1q37, 49, 543-563.

BURT, P. &SPERLING, G. Time, distance ind featue

trade-offs in vistial apparent motion. Psych.

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CAELLI, T. Visuial Perception: Theory and Practice.

Oxford: Perg~amon Press, 1981.

ENROTH-CUGELL, C. & ROBSON, P. The contrast

sensitivity of retinal gang~lion cells of the cat.

3. Pysil.,1966, 187, 517-552.

FERNBERGER, S.W. New p I Ie n om e it of ;~I pa rent v IituoI

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FRISBY, 3.?. The effect of stimuilus orient.,tion on the

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1145-11 66.

GEORGESON, M.A. &SULLIVAN, G.D. Contrast constancy:

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GREEN, D.H. &SWETS, 3. Strpnal detection theory and

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Page 86

Size Factors In Apparent Motion

References

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KOFKA, K. Principles of Gestalt Psycojjlogy New York:

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KOLERS, P. Fixation and attention in apparent motion.

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KOLERS, P. & POMERANTZ, J.R. Figural change in

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MARR, D. Vision San Fransisco: 1.11. Freeman, 1982.

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MOSTAFAVI, H. & SAKRISON, D. Structure and properties

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THURSTONE, L.L. A law of comparative judgement. Psych.

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%;
Page 87 I-o

Size Factors In Apparent Motion

References

THURSTONE, L.L. Stimulus dispersions in the method of

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WETHERILL, G. B. & Levitt, H. Sequential estimation

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WILLIAMS, D.W., WILSON, H.R., & COWAN, J.D. Localized

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19, 19-32.

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Page 89

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FIGURE 2 A SIMPLE MOTION DETECTION DEVICE AS PROPOSED BY BARL~w

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(TOP AND BOTTOM ROWS). PAIRS ( -i), LEFT COLUMNJ. &(-)

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FIGURE 25 FixED THRESHOLD DECISION RULE AS APPLIED TO PAIR

STRENGTH AND MOTION REPORTS. SHADED AREA SHOWS

PROBABILITY OF REPORTING MOTION.

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APPENDIX I

DESCRIPTION OF APPARATUS

The sustem used to disrlaw all Patterns described in the

Precedina experiments is illustrated in the diagram on the following

Page. The various difference of Gaussians (DOG) waveforms needed

are generated bv and stored in the memorw of a digi+al computer.

Sixtv times each second, a crwstal controlled clock (also Part of

the computer) initiates a new frame of the displav. The 'new-frame'

pulse has three main effects: the sweep generator is triggered to

start moving the scope's electron beam horizontallu across the

screen; the displaw contrast is selected from a temporal waveform

list in computer memorw and converted to a voltage level bw a 12-bit

digital to analog converter; the Points of the spatial waveform

are converted to voltages bv another 12 bit digital to analog

converter. All three Processes are initiated virtually simultaneouslw;

the selection of displaw contrast occurs onlu at the beginning of

each frame, and remains constant for the duration of the frame.

The conversion of the spatial waveform takes most of the duration of

the frame to complete. During this conversion, the sweep generator

moves the electron beam across the entire face of the scope, evenlw

distributing the Points of the spatial waveform. The vertical axis

of the displav score is modulated by a high-freouencw, free running

triangle wave generator; the line traced across the screen by the

sweep generator is therebu extended to a bar filling the entire

screen .

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