Vision Res. Vol . 32, No. 12, pp. 2313-2329. t992Printed in Great Britain. All riehts reserved

0042-6989 92 $5.00 + 0.00Copyright O 1992 Pergamon Press Lrd

Visual Processing of Optic AccelerationPETER WERKHOVEN,*I HERMAN P. SNIPPE,* ALEXANDER TOET*Receiued 10 February 1992; in reuised form 15 May 1992

We present data on the human sensitivity to optic acceleration, i.e. temporal modulations of the speedand direction of moving objects. Modulation thresholds are measured as a function of modulationfrequency and speed for different periodical aelocity uector modulation functions using a localizedtarget. Evidence is presented that human detection of velocity vector modulations is zot directty basedon the acceleration signal (the temparal deriuatiue of the velocity yector modulation). Instead,modulation detection is accurately described by a two-stage model: a low-pass temporal filtertransformation of the true velocity vector modulation followed by a aariancJ detection stage. Afunctional description of the first stage is a second order low-pass temporal filter having acharacteristic time constant of 40 msec. In effect, the temporal low-pass filter ii an integration oflhevelocity vector modulation within a temporal window of 100-l40msec. A non-triviat link of thislow-pass filter stage to the temporal characteristics of standard motion detection mechanisms will bediscussed. Velocity vector modulations are detected in the second-stage, whenever the variance of thefiltered velocity vector exceeds a certain threshold variance in either t-he speed or direction dimension.The threshold standard deviations for this variance detection stage are estimated to be l7o/o for speedmodulations and 9oh for motion direction modulations.

Motion Acceleration Speedmodulation

INTRODUCTION

Man is capable of interacting successfully with complexdynamic environments. This abil ity is due primarily topowerful neural mechanisms that have evolved to pro-cess optical motion information (see Nakayama, l9g5bfor a survey). Therefore motion perception has beenstudied extensively. Psychophysical research has shownthat the human visual system contains highly sensitivemotion extraction mechanisms (DeBruyn & Orban,1988; McKee, l98l ; Werkhoven & Koender ink, l99l)that map spatiotemporal image structure into explicitmotion information (e.g. velocity and direction).

Motion perception has traditionally been studiedusing spatiotemporal invariant (uniform) motion stim-uli. Relatively few studies have aimed at the humansensitivity to the spatiotemporal structure of motionfields or uelocity uector modulations. Although previousstudies helped to define methods and stimuli, none ofthem allowed defrnitive statements concerning humansensitivity to acceleration or mechanisms for detectinghigher derivatives of motion (Regan, Kaufman & Lin-coln, 1986). This scarcity of studies is surprising, since innatural vision, optical motion on the retina is generailyvarying in both space and time even if environmentalobjects move at a constant speed and direction. Struc-tured motion fields are not just an inevitable burden for

tlnstitute of Perception TNo, Kampweg 5,3769 DE Soesterberg, TheNetherlands.

tPresent address: Utrecht Biophysics Research Institute (uBI), BuysBallot Laboratory, Utrecht University, princetonplein 5, 35g4 CCUtrecht, The Netherlands.

our visual system. In fact, it has been shown that thespatial structure (Koenderink, 1986) and temporal struc-ture (Arnspang, 1988) of optical motion fields are ofmajor importance to the visual agent and are closelyrelated to egomotion and 3D shape extraction.

The study presented here focuses on the humansensitivity to temporal velocity vector modulations, thatis, the abil ity to detect temporal variations in speed ordirection (called optic acceleration).

The paradigm

A fundamental and intriguing question to be answeredis: does the human visual system contain specific acceler-ation detectors? In other words, do human observersdirectly assess the optic acceleration of a moving object(the temporal deriuatiue of the velocity vector function)or do they indirectly infer optic acceleration from vari-ations in the perceived velocity along its trajectory (bysampling velocities at different times)?

This question strongly resembles a classic debate in thestudy of uniform motion perception: are human observ-ers able to directly sense optical motion, or do they infermotion indirectly lrom the variance in object positionover time? Nakayama and Tyler (1981) have answeredthe latter question using a target with a periodically(sinusoidally) modulated position in time. Theymeasured modulation threshold amplitudes as a func-tion of the frequency (inverse period) of the positionmodulation function. They argued that modulationthreshold amplitudes would be independent of themodulation frequency when motion was inferred fromthe variance in position. However, when motion was

23t3

2314 PETER WERKHOVEN er a/.

assessed directly (e.g. the temporal derivative of theposition modulation function) threshold modulationamplitudes were expected to decrease with increasingmodulation frequency.

For low modulation frequencies (<2Hz), Nakayamaand Tyler found strong experimental support for a directassessment of motion. Modulation thresholds did notshow an invariance when expressed in terms of dis-placement. For higher frequencies (>2Hz), Nakayamaand Tyler found deviations from the expected depen-dence of modulation thresholds on frequency, pre-sumably as a consequence of some finite temporalintegration of the motion signal in the human motionsystem.

To examine optic acceleration, we adopt this elegantparadigm used by Nakayama and Tyler substitutingvelocity modulations for position modulations. That is,we use a target with a velocity vector modulated in timearound a certain mean velocity vector and measurethreshold amplitudes for the detection of velocity vectormodulations as a function of the modulation frequency.We study velocity vector modulations both in the direc-tion of the velocitl, vector (speed modulation) andorthogonal to the velocity vector (direction modulation).Invariant modulation thresholds as a function of modu-lation frequency would indicate an indirecr detection ofmotion modulation or optic acceleration.

General stimulus considerations

The choice of an adequate stimulus to be used in astudy on motion modulation detection is not trivial. Itis important to design the modulation detection exper-iment such that detection cannot take place outside themotion system in other dimensions than speed or direc-tion. In the following we list a few considerations regard-ing some widely used stimuli in motion experiments.

Sine-v'at,e gratings. Sine-wave luminance gratings area powerful tool for studying l inear systems and also forstudying motion perception. However. the use of movingsine-wave gratings leads to several problems. First, localspeed and local temporal frequency are inherently con-founded. As a result, a speed modulation of a movingsine-wave grating might be detected outside the motionsystem as a local modulation of stimulus temporalfrequency. For example, a detector with a spatiotem-poral separable response function, thus not tuned tospeed at all, would be sufficient. Second, with a one-dimensional spatial pattern. such as a sine-wave grating,it is not at all obvious how one could study motiondirection modulations. Third. moving sine-wave gratingsallow extensive spatial integration by the motion detec-tion system. This property makes all spatially extendedmoving patterns especially unattractive to study spatiallylocal modulations in speed or direction. Fourth, spatiallyextended moving patterns inherently stimulate motiondetectors at a range of eccentricities. Thus, a study ofmotion sensitivity as a function of eccentricity cannot bespecific.

Random pixel arrays. Another visual stimulus oftenused in studies in motion perception is a random pixel

array or "Julesz pattern" (Julesz, l97l). An importantproperty of a Julesz pattern is that its power spectrumis flat. Therefore, a moving Julesz pattern with a modu-lated speed function would not yield the temporal fre-quency cue discussed above. However, human sensitivityto temporal modulations is l imited by the fl icker fusionfrequency. As a result of this cut-off lrequency fortemporal modulations. the sensed energy of a movingJulesz pattern decreases when speed increases. That is,when speed increases. an increasing proportion of thespectral components of the moving pattern wil l yieldtemporal frequencies beyond the fusion frequency, thusconceivably reducing the apparent contrast of the stimu-lus. Thus. speed modulation for Julesz patterns mayprovide the observer with an apparent contrast modu-lat ion as a cue.

Furthermore, Julesz patterns are spatially extended.Hence, they yield similar problems for the study ofmotion modulations as discussed above for sine-wavegrat ings.

Localized torgets. We have discussed a few extraneouscues associated with spatially extended stimuli. Many ofthese problems are circumvented when using stronglylocalized targets, such as dots. A moving dot allowsfor the study of local motion perception (restrictedspatial integration) and for the control of eccentricity ofpresentation. Furthermore. local temporal frequencymodulat ion is not a cue for mot ion modulat ion detec-t ion.

However. an increase in dot speed can yield anapparent spatial stimulus extent (if the visual systemintegrates the stimulus over a fixed window in time). andalso a decrease in apparent contrast (if the visual systemintegrates the stimulus over a fixed window in space). Toget some grip on the possible contributions of theseextraneous cues, we studied motion modulation sensi-tivity using moving dot targets and blob targets(spatially blurred dots).

METHODS

Method speed modulations

This section describes the method for our study on thehuman sensitivity to temporal modulations of motionspeed.

Stimulus specifcations. The stimulus consisted of amoving luminous dot (well above detection threshold) ofI mm dia. The dot projected on the screen of a CRT wasblurred by a sheet of diffusing material which was placeddirectly in front of the CRT screen. We estimate thestandard deviation of the resulting isotropic luminance"blob" at l.5cm, thus its full width at half maximum(FWHM) at about 3-4 cm. The dot moved horizontallyacross the screen at a variable (modulated) speed fromthe leftmost point to the rightmost point of a horizontaltrajectory across a distance do.This single left-to-rightmotion is called a sv;eep (the distance do is the sweep-length). When it reaches the right end on its trajectory,the dot returned to the far left position on the trajectoryand continued its motion (the next sweep). The time to

OPTIC ACCELERATION 2 3 1 5

f inish one sweep is called the sweep-time (lo). At aviewing distance d,., the average dot speed u0 was:

uo: t i t arctan(dof d,) . (1)

One motion stimulus presentation consisted of foursweeps. Thus, the presentation time was 4ro. The dotspeed was modulated in time yielding a non-uniformperiodic function r'. '(r) with modulation frequency ra.Speed modulation functions u,(r) were either periodic(symmetric) triangular functions A(t) or periodic blockfunctions l l(r) (see Fig. l). The amplitude du, of themodulation functions was varied but was always smallerthan the average speed uo, such that the dot speed wasalways positive. The phase of the periodic modulationfunction at the start of the stimulus presentation wasrandomized.

In addition to the moving dot, we also provided theobserver with a stationary fixation dot (a green LED),placed at a distance equal to the sweep-length d0 abovethe center of the horizontal trajectory, thus makingeccentricity of presentation (e ) about equal to the lengthof the trajectory of the moving dot. The sweep-time fora particular experiment was taken to be such that onesweep contained a few cycles of the speed modulation.Hence. for low temporal modulation lrequencies

2 n

t . l I pu tse6 ( t ) | |

I T0

t l - - Trianqular^(t) r v_-Z,I \Z

t i m e

t l , -

a ( t ) ( \ s i ne

. l \ , ,t i m e 2;r

1

r ( t )

0

4

r ( r )

00 t ime Z i r

FIGURE l. A sketch of some modulation functions: puise shaped

d(r) , t r iangular A(r) , s inusoidal O(t) , b lock shaped t I ( r ) and sample

function )11), as a function of time l. For this i l lustration. all functions

are normalized such that their mean value over time is 0.5. their

temporal wavelength is 2n, and the modulation amplitude is 100%.

except for function (l). The function (l) is the velocity modulation

function as used in the "sample and hold" paradigm of Burr et al.(1986) (for this i l lustration also with mean 0.5, and temporal wave-

examined, a longer-sweep time was required. The par-

ameters as set in the different modulation experiments

are specified in separate parameter tables in the result

section.In the main experiments, speed covaried with the

eccentricity of the moving dot. To study the effects of

eccentricity and speed independently, we ran two control

experiments. In one, we varied eccentricity but kept the

viewing distance constant. In the second. we varied speed

but kept eccentricity constant.Apparatus. The speed modulation functions were gen-

erated by manipulating the position of the beam of a HP

1321A high speed graphic display (P31 phosphor).

The beam produced a I mm dia luminous dot on the

screen (well above detection threshold). A Wavetek 185,

5 MHz, function generator produced a saw-tooth

horizontal position signal n(/), as a function of t ime /,

which was fed to the X-channel of the HP 1321A.

The horizontal position of the dot was l inear with this

signal. Hence, the dot moved from left to right across

the screen unti l the saw-tooth reached its maximum(finishing one sweep) at which point it returned(invisibly) to the far left and started to traverse the

screen again (the next sweep). The amplitude of x(r)(and thus the sweep-length) across the screen was con-

stant. For a constant period l" of this saw-tooth signal

in time. the dot crossed the screen at a constant speed.

determined by the temporal derivative of ,r(r), and thusproportional with the reciprocal period j of .r(l). The

reciprocal period I of .r(r) (and thus the dot speed) was

modulated in time by a periodic modulation function

u,(t) with (modulation) frequency ar (using a HP 33254

synthesizer function generator). The modulation func-

t ion u.( t ) was ei ther a t r iangular funct ion A(t) or a block

function l l(r) (see Fig. l). Speed modulation L'y(r) varied

around an average speed t'o with an amplitude du- (see

Fig. 1). This set-up allowed easy adjustment of the

average speed. amplitude and frequency of the speed

modulat ion funct ion.The importance of t'isual -fixation. Pilot experiments

showed that visual f ixation during modulation detection

experiments is crit ical. Observers reported to have no

dfficulties in detecting modulations when tracking the

moving dot for conditions where detection failed under

visual f ixation. Obviously, pursuit eye movements facil i-

tate modulation detection. It is well-known that the

pursuit system is quite slow (cut-off frequency at about

I Hz). For speed modulation frequencies higher than

I Ha observers could not follow the exact speed modu-

lation, but might track the dot at its average speed. The

actual speed modulation would then become apparent as

a displacement in the retinal coordinate frame. Thus,

allowing the observers to track the dot would provide

them with a displacement cue. resulting in modulation

detection outside the motion system. In order to elimin-

ate this cue, visual fixation is crucial.

Note that in much of the older l i terature (e'g. Hick,

1950), but also in more recent l i terature (Burr, Ross &

Morrone, 1986) no mention of the observer's f ixation

condition is made.vR 32 i12-F

length 2z).

2316 PETER WERKHOVEN er a/.

Procedural information. Speed modulation thresholdswere measured in a modulation detection experiment. Inone session, observers viewed 18 stimulus presentationsof a modulated speed function u(r) (with an averagespeed u0 and a modulation amplitude dr,,) and 18presentations of an unmodulated (uniform) speed func-tion (having a constant speed uo). The order of presen-tation for these 36 trials in a session was randomized, aswas the phase of the modulation function for the trialsthat contained the speed modulation. The tasks of theobservers was to indicate for each stimulus presentationwhether they perceived a modulated or an unmodulatedmotion in t ime.

Usually 4-5 sessions with different adequately chosenmodulation amplitudes were sufficient to determine thespeed modulation detection threshold by data interp-olation. We defined the speed modulation threshold W,as the relative modulation amplitude dt.f uo at thresholdperformance (yielding 80% correct answers). Measure-ments were performed binocularly with natural pupils ina darkened room. No feedback was provided in eitherexperiment.

In one of the control experiments we studied speeddiscriminarion using the present experimental set-up. Ina session for speed discrimination, observers vieweduniform speed functions with a constant speed that wasei ther higher (us* dt , , ) or lower (uo- dt , , ) than theaverage speed uo of the ensemble of presentations. Ob-servers indicated whether the perceived speed was highor low. Before a session started, the motion stimuli wereshown on request to build an internal representation ofthe high and low speeds. The procedure for determiningspeed discrimination thresholds was otherwise similar tothe procedure for the modulation detection experiment.

Subjects. Five subjects with normal or corrected-to-normal vision participated in the experiments. Threesubjects. HS, PW, and AT are authors of this paper andhad foreknowledge of the design, and are experiencedobservers in psychophysical experiments involving opticmotion. The results of these main subjects are presented.The general findings were confirmed by two naive sub-jects, working on an hourly fee. There was no obviouscorrelation between subject experience and thresholdvalues.

M e thod direction modulations

This section describes the method for experiments ondirection modulation detection.

Stimulus. Similar to speed modulation functions, thedirection modulation functions were generated by ma-nipulating the position of the beam of the HP 13ZlAhigh speed graphic display. However, for directionmodulation functions, both the horizontal and verticalposition of the dot were manipulated.

The time dependent horizontal position x(r) of the dot(the X-channel of the HP l32lA) was driven by a Hp3325A synthesizer function generator. This generatorproduced a saw-rooth signal x(r) with a period A thatdetermined the sweeptime r0 and an amplitude thatdetermined the sweep-length d0 For this direction

modulation experiment, the period and amplitude ofx(r) were constant during a stimulus presentation, re-sul t ing in a constant hor izontal speed r ' . ( t ) - u0: dol to.

The time dependent vertical position 1,(r) of the dot(the )'-channel of the HP l32lA) was driven by aWavetek 185 function generator. The y(l) signal deter-mined the direction modulation. The position functions1'(r) were periodic with frequency cr.r. The vertical speedfunction r', (r) was simply the temporal derivative of vert-ical position t '(t). Thus, the modulation frequency wasro. The average vertical speed was zero. The amplitude ofthe vertical speed modulation function r,,(r) is written asdr',. As a result. the speed of motion 1'(r) of, the dot was:

[tmr ' ( l ) - r ' n / l + - .

V l ' o(2)

For a small vertical speed r', (r) relative to the horizontalspeed uo [ r ' , ( l ) ( r 'o ] . the average speed was approx i -mately constant [ r ' ( t ) I r ,o] . The direct ion 0(r) of morionas a function of t ime r is approximately l inear in r ',.(r)when r ' , ( r ) ( r 'n:

o( t ) :u r . ,un [ !1 ' ) l -u . (3)L l ' ' I l 'o

The average motion direction 0o in all experiments washorizontal: 0o : 0. The amplitude of the direction modu-lation function 0(r) is de - arctan(dt:,, lu).

Tr iangular posi t ion funct ions . ) ' ( / ) : ,1( l ) resul ted inblock shaped direct ion modulat ion funct ions 0(t) :fI(r). Sinusoidal functions .r '( l) : O(r) resulted in sinu-soidal d i rect ion modulat ions 0(r) : Q(r) [ integrated y( l )functions], but shifted a phase i backwards in time.Final ly, b lock wave posi t ion funct ions y( t ) : l l ( r ) re-sulted in pulse shaped direction modulation functions0(t) : d (r) . An i l lustrat ion of these posi t ion modulat ionfunctions and resulting direction modulation functions isshown in Fig. l .

Procedure. The procedure was identical to the pro-cedure for speed modulation detection experiments.Observers indicated for each motion stimulus whetherthe moton was modulated (non-uniform) or not. Twomain observers participated in the direction modulationexperiments (HS and AT). Two naive subjects confirmedthe findings for the main subjects.

Modulation direction thresholds are defined as thedirection modulation amplitude d0 yielding thresholdperformance (80% correct answers).

SPEED MODULATION DETECTION

Speed modulation detection dependence on modulationfrequency

Results. Speed modulation detection thresholds I,Z,. asa function of modulation frequency a-t for two differentspeed modulation functions and different speeds u0 arepresented in Fig. 2. The parameter settings for differentaverage dot speeds u0 are listed in Table l. Since the datawere very similar for the three main observers (PW, HSand AT). we averaged modulation detection thresholds

.........ts---t-

-+-#

...,....1.r

+

1.7 deg/s5 deg/s1 5 deg/s1 deg/s

2.5 deg/s5 deg/s

OPTIC .\CCELERATION

, l 1 1 0

Modulat ion Frequency, . , ; (Hz)

FIGURE 2. Threshold speed modulation amplitudes Wu as a functionof speed modulation frequency ro. Thresholds W, are the relative speedmodulation amplitudes (dt;,,,t,n) that yield 80%o correct answers. The(very similar) data of three observers (HS. PW and AT) have beenaveraged. Triangular symbols indicate (symmetric) triangular speedmodulation functions A(t1.The different sizes of the symbols indicatedifferent average speeds ro as given in the figure. Note, that the closedtriangles indicate a special condition in which the diffusing screen wasremoved such that the target was a luminous dot. Square symbolsindicate thresholds for (symmetric) block speed modulation functionstl(t). The * symbois indicate results obtained in a separate speeddiscrimination experiment. Observers indicated whether a uniformmotion stimuius moved at a high veiocity (r:o * dr',) or at a low velocity( t 'n - r / t ' . ) . As in the modulat ion exper iments. Wu: dt , . l t ,o. For the lef tspeed discrimination threshold in the figure the presentation time ofeach speed interval was I sec. for the right threshold it was l25msec.To facil itate a comparison with the speed modulation thresholds. thetwo speed dist'rininatiorr thresholds are plotted at a horizontal positionthat equals hal f their inverse presentat ion t ime. Hal f the length of theshown error bar for each data point corresponds to the square-root

var iance of the binominal orobabi l i tv d ist r ibutron for that ooint .

W, for this presentation. The modulation thresholds ofFig. 2 are presented as (relative) speed modulationthresholds (speed modulation Weber fractionsW,.: du,f uo). Triangular symbols indicate the triangularspeed modulation function A(t) and square symbolsindicate the block modulation function II(r). Opensymbols indicate that the moving target was blob l ike.Solid symbols indicate dot targets.

Consider triangular modulation functions. For lowmodulation frequencies (ro ( 2Hz) speed modulationthresholds for very different conditions ( I deglsec dottargets at 0.25, 0.5 and lHz: 1.7 deg,,sec blob targets atI and 2Hz: and l5deglsec blob targers at 2Hz) areidentical within measurement error (approx. 32%). Thissuggests that speed modulation detection thresholds atlow frequencies are constant and independent of fre-quency, speed and target shape. However. speed modu-lation detection threshold values do depend on the shapeof the modulation function used. Thresholds for thetriangular speed modulation functions ,4 (r) are approxi-

mately a lactor 1.8 higher (averaged over speed, modu-lation frequencies and subjects) than the thresholds(17%) for the block modulation function l l(r).

At high modulation frequencies (tt > 2Hz) the speedmodulation thresholds in Fig. 2 rise strongly with in-creasing modulation frequency for both triangular andblock-shaped modulation functions.

Discussion: threshold inuariance at low modulationfre-quencies. The frequency independence of modulationthresholds for low modulation frequency strongly sup-ports the hypothesis that modulation detection is basedon the magnitude of the speed modulation signal. Themodulation magnitude is independent of modulationflrequency. A detection mechanism based on the differ-ence in maximum speeds of the speed modulation func-tion is indeed expected to yield constant thresholds,independent of modulation frequency.

The invariance of thresholds for low frequencies rulesout the hypothesis that speed modulation detection isbased on the magnitude of the optic acceleratio,? signal.The optic acceleration signal is the temporal deriuatiu^e ofthe speed modulation signal. Hence. its magnitude islinear with the modulation frequency. Therefore. a de-tection based on the acceleration magnitude is expectedto improve with increasing modulation frequency. Ahypothetical acceleration detector (requiring a constantacceleration threshold for detection) would yield a hy-perbolic (inverse l inear) decrease of speed modulationthreshold in Fig. 2, which is not supported by the data.

The low lrequency plateau in Fig. 2 rules out anotherhypothesis saying that observers base detection on thespatial excursions of the moving dot with respect to itsaverage path (i.e. the path of constant speed ro). Accord-ing to this hypothesis, the speed modulations are de-tected whenever the excursions exceed a certainexcursion threshold. The magnitude of the spatial excur-sion is the temporal integral of the speed signal and islinear with speed and with the period of temporalmodulation. Thus, the "excursion" hypothesis predictsthat speed modulation thresholds decrease with decreos -

ing modulation frequency. This prediction is inconsistentwith the finding that thresholds are constant for lowmodulation frequencies (see Fig. 2).

In conclusion, the threshold invariance at low modu-lation frequencies strongly support the view that humanspeed modulation detection is based on the speedsignal itself [the relative magnitude du,, 'L'o of the speedmodulation function u(r)], and not on the temporalintegral of r '(r) (position), or the temporal derivative ofu(r) (acceleration).

I 1oo

B B osto8 6 0o-c

l 4 0Eo=E 2 0q)q)

A 0

TABLE L Parameter settings for speed modulation thresholds of Fig. 2

SPeedWave form uo (deglsec)

Sweep-length

4 (cm)Sweep-time

ro (sec)Distance Eccentricity

4 (m) e (deg)

A ( t )A ( t )A ( t )A ( t )I1(r )

1 . 01 . 75.0

15 .02 .5

+ J

30303030

4 . 1 01 .685.0415 .05.04

6.00r0.203.40t . t 23.40

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'/'2 /, NE^OH)UEA\ dEIAd8ttz

OPTIC ACCELERATION

TABLE 2. Parameter settings for speed modulation thresholds of Fig. 3

2 3 1 9

Wave form

Speeduo (deg/sec)

Distance

4 . ( m )Eccentricity

e (deg)Sweep-length SweeP-time

do 1cm; to (sec)

A ( t )A ( t )A ( t )A ( t )A ( t )

0.260.632.55 .07 .5

25.010.42.602.00r .35

0.521 .265.00

10 .015 .0

Z J

233535

22222

Results (see " * " symbols in Fig. 2) show that speed

-liscrimination thresholds (6%) are indeed much lower

rhan speed modulation detection thresholds (l7oh fot

block wave modulation functions). The 60/o speed dis-

crimination thresholds were independent of presentation

time.Discussion. At the longest presentation time each

uniform speed was shown 1000 msec in the speed dis-

crimination experiment, yielding a 60lo threshold. It is

interesting to compare this 6% speed discriminationthreshold with the 17o/o speed modulation threshold for

block modulation functions at 0.5 Hz modulation fre-quency. For a block shaped modulation function at thisfrequency, the presentation time of each speed intervalof the block function was also 1000 msec. Thus,

although the different speeds in both experiments werepresented at equal (long) time intervals. the thresholdsare markedly different.

The high thresholds for modulation detection may bea consequence of a fundamental problem observers havern segmenting the modulated motion stimulus into highand low speed intervals when speed itself is the onlysegmentation cue, as originally proposed by Snowdenand Braddick (1991). However, we propose an al terna-tive explanation (as discussed in detail in the Model andGeneral Discussion sections): high thresholds for modu-lation detection may be caused by the uncertainty (of

observers) about the phase of the speed modulationfunction.

The cut-off frequency dependence on speed

Motiuation. The cut-offfrequency @.(t'o) is defined asthe modulation frequenc,v yielding threshold detectionperformance (80% correct answers) for a given averagespeed ro and modulation amplitude dr'-. The data inFig. 2 suggest that this cut-off frequency is a functionof speed. For example. for the lowest speed tested(1.7 deg/sec), modulat ion detect ion thresholds increasesomewhat faster for increasing modulation frequency(e.g. at 4 Hz) than the threshold for an average speed ofI 5 degTsec.

To study this issue further, we measured cut-offfrequencies for a wide range of average speeds uo. Tofacil i tate a comparison of our data with the cut-offfrequencies for random-dot patterns used in the exper-iment of Snowden and Braddick ( 199 I ), we removed thediffusing screen yielding a luminous dot as a target. Thespatial power spectra of dot targets and random-dotpatterns are comparable.

Results. We measured cut-off frequencies @,(uo) for awide range of average speeds us by measuring the

percentage of correct responses as a function of modu-

lation frequency at modulation depth dt),:100% for

the triangular speed modulation function A(t). The

speeds and corresponding parameter settings are listed in

Table 2. It should be noted that target speed was varied

by varying the viewing distance d, (although for two

conditions the sweep-length d0 was slightly adjusted).

The solid symbols in Fig. 3 are cut-off lrequencies for

dot targets and show a clear increase of cut-off frequency

with stimulus speed. Open symbols are cut-off frequen-

cies for blob targets and are extrapolated from the

thresholds for triangular modulation functions in Fig. 2

using a temporal low-pass filter that is justified and

specified in the Model section. Because these extrapo-

lated data for observer AT and HS were very similar' we

averaged them for this presentation.

Discussion. We fitted the dependence of the cut-off

frequency o.(uo) for dot targets (solid symbols) on speed

uo to a power function:

@. (uo ) oc u6 (4)

and est imated the power exponent a :0.3 - 0.35. In the

General Discussion section, we discuss this power law in

terms of well known properties of elementary motion

detectors.A comparison of the cut-off frequencies for blob and

dot targets shows that only for the highest speeds used

(uo > 7.5 degisec) the cut-off frequency becomes pattern-

--.-..----l---*-

HS (do t )

AT (dol)

HS.AT {blob)

. 1 1 1 0 1 0 0

Speed, r'6 (degls)

FIGURE 3. Upper cut-off modulation frequency or0 as a function of

speed ro. Cut-off frequency r-r.t. is defined as the modulation frequency

yielding threshold performance (80% correct answers) for a modu-

lation amplitude of 100%. Cut-off frequencies are measured for

triangular speed modulation function A(t) and two individual observ-

ers (HS and AT). Parameter settings are listed in Table 2' Solid

symbols indicate cut-off frequencies for a iuminous target dot. Open

symbols are cut-off frequencies for a blurred (blob shaped) target and

are extrapolated from the data of Fig. 2 using the temporal low-pass

filter described in the Model section. Because these extrapolated data

for observers AT and HS were very similar. we averaged them for this

presentatlon.

1 0 0N

3jo

3 1 0ctotJ-

o

o1

2320 PETER WERKHOVEN et a/.

dependent (see also Watson, Ahumada & Farrell 1986)'

This is consistent with introspective reports saying that,

for sti l l higher average dot speeds of the modulation, the

percept was a "string of beads". The "beads" pre-

sumably correspond to the places where the stimulus

comes to an instantaneous standsti l l , thus allowing a

significant luminance build-up over time in a small

spatial region.

Disentangling t'iev'ing distance and eccentricitt'

Moti,^ation. In this section we report on a control

experiment to test our claim that the high frequency

cut-off we find for our speed modulation thresholds

(see Fig. 3) is caused by low-pass temporal f i l tering. [n

Fig. 3 we showed that the temporal cut-off frequency

depends only weakly on stimulus speed (4-8 Hz for

speeds < 7 deglsec). Therefore, it is tempting to assume

a temporal frequency l imit for the speed modulation

detection system. However, to support this conclusion

we have to tackle the following problem.

In the above experiment, speed was varied by varying

the viewing distance (see Table 2), thus covarying the

eccentricity of presentation with stimulus speed. The

spatial grain size of the visual system increases approxi-

mately l inear with increasing eccentricity (Watson,

1987), such that the spatial resolution for the spatial

speed variations of our stimuli decreases as a function of

eccentricity. Therefore. one could claim that the (near)

invariance of the temporal frequency cut-off can also be

explained by a constant spatial frequency l imit with

respect to the grain size of the visual system at the

eccentricity of presentation of the motion stimulus.

To disentangle the effects of viewing distance and

eccentricity we measured the cut-off frequency at a fixed

viewing distance and stimulus speed, but at different

eccentricit ies.Results. We measured the cut-off frequency at a fixed

viewing distance (d, :2.40 m) and stimulus speed

(4 deg/sec) for different eccentricit ies of presentation

(e :0 .5 , 5 , l0 and l5 deg) . The sweep- length do is 42cm

and the sweep-time /o is 2.5 sec. Cut-off frequencies for

a triangular speed modulation function with 80% modu-

lation depth are presented in Table 3' Table 3 shows that

the cut-off frequencies are virtually identical at all eccen-

tricit ies.Discussion. A correct explanation for the approxi-

mately invariant cut-off frequencies is indeed in terms of

TABLE 3. Cut-off frequency a.r. obtained for the triangular

speed modulation function ,,1(l ), with relative amplitude

80% and mean velocitY uo : 4 deg/sec, for three observers at

four eccentricities t

Frcentricity Cut-off HS Cut-off AT Cut-off PW

e (deg) a" (Hz) a, (Hz) a, (Hz)

Speed, r '6 (deg/s)

FIGURE 4. The dependence of upper cut-off frequency (rr, on speed

r, , for a t r iangular speed modulat ion funct ion,4( l ) wi th { ixed 80%

speed moduiat ion ampl i tude. Eccentr ic i ty was. f i red at l0 deg' Indiv id-

ual data for two observers (AT and HS are plot ted).

a temporal high-frequency cut-off. and not in terms of

an eccentricity-scaled spatial resolution l imit. Of course

this temporal frequency l imit cln be described as a

spatial l imit in units that scale with stimulus speed.

However, because of the scal ing in human mot ion v is ion

of the spatial grain size with stimulus speed. we believe

that such a descr ipt ion is equivalent to our explanat ion

in terms of a temporal resolution l imit.

Disentangling riev'ing distanc'e and speed

Motiration. The cut-off frequencies in Table 3 at

constant speed but varying eccentricity are invariant.,

whereas the cut-off frequencies in Fig. 3 at covarying

stimulus speed and eccentricity do show a slight (though

systematic) variation. We hypothesized that this small

variation in cut-off frequencies depends on the stimulus

speed. We tested this hypothesis explicit ly by measuring

cut-off frequencies at fixed viewing distance and eccen-

tricity but different speeds.Results. We measured cut-off frequencies for a tri-

angular speed modulation function with a modulation

depth of 80% at a .f ixed l0 deg eccentricity and flxed

viewing distance (240 cm). but at different speeds

uo. Because sweep-t ime /0 was constant (1.25 sec). the

sweep-length 4 was directly proportional to the dot

speed.Cut-off modulation frequencies are shown in Fig. 4 as

a function of the average speed uo (at f ixed eccentricity!).

As expected, we find a dependence of cut-off modulation

frequency on speed.Discussion. We fitted the dependence of the cut-off

frequency on speed to a power function [see equation(a)1. The exponent e that f its the data of Fig. 3 best is

est imated to be a :0.25 for HS and u : 0.30 for AT.

The absolute values of the cut-off frequencies at a fixed

speed in Fig. 4 can be compared with Fig. 3. The cut-off

frequencies of this experiment (measured with 80%

modulation amplitudes) are roughly 0.8 times the cut-off

frequencies in Fig. 3 (measured for 100% modulation

amplitudes). This can be explained by the fact that the

filtered modulation signal is proportional to the modu-

lation amplitude dt, times an attenuation function (see

Model section). This filtered signal has to exceed a

certain internal threshold for detection to take place.

1 0N

3

>o

o

ll-

o

o

1

0 .55

l 0l 5

n.a.5 .35.2n.a.

6 .05 .55 .35. ' l

6 .85.44.95 .8

The data for observer HS at 0.5 and l5 deg eccentricity were

not avai iable.

+-{---1-t-

1 deg/s2d.glg1.7 deg/s

OPTIC ACCELERATION 2321

Modulation Fr"f,r"n.y, ,u (Hz)

FIGURE 5. Threshold direction modulation detection amplitudes d0

as a llnction of modulation frequency cu. Weber fractions Wo are

simply related to d0 by the expression: Wo--tand?. Data of two

observers (AT and HS) have been averaged. Solid circles are data

obtained with a sinusoidal motion direction modulation function Q(l);

Open squares indicate a block shaped motion direction modulation

function ll(l) [corresponding to a vertical position modulation ,4(r)] '

Thus a higher modulation amplitude yields higher

modulation frequencies.Note, that the cut-off frequencies at the lowest average

speeds in Fig. 3 were measured at much smaller eccen-

tricit ies than the l0 deg eccentricity for this experiment.Thus. it seems that eccentricity of presentation is of small

relevance to speed modulation detection, even for speeds

that barely exceed the motion detectio,n threshold at the

eccentr ic i ty of presentat ion.Finally, it is of interest to note that human modulation

detection sensitivity at the lowest speed tested(0.5 deglsec at l0 deg eccentricity) is excellent when

expressed in terms of the spatial excursion of the modu-

lated motion path from the average motion path at

average dot speed 1'0. These spatial excursions did not

exceed 0.9 arcmin. which is approximately the hyperacu-

ity threshold that was found at this eccentricity for static

stimuli with an explicit nearby spatial reference available(Westheimer. 1982)!

DIRECTION MODULATION DETECTION

In this section. we study velocity vector modulations

orthogonal to the average velocity vector (direction

modulations) resulting in curved trajectories. Instead of

measuring speed modulation thresholds. we measure

direction modulation thresholds. The precise generation

of the direction modulation functions is specified in the

Method section. as is the definit ion of direction modu-

lation thresholds. Otherwise. the procedure and organiz-

ation of these experiments are quite similar to those of

the speed modulation detection experiment described

above.

Direction modulation detection dependence on modulation

frequencyResults. In Fig. 5, we present direction modulation

thresholds as a function of modulation lrequency for

two different direction modulation functions' Circles

represent sinusoidal direction modulations O(l), squares

block shaped direction modulations l l(r). Since direc-

tion modulation thresholds were very similar across

the main observers, we presented the averaged

thresholds for observers AT and HS. The parameter

settings for different velocities and functions are listed in

Table 4.Direction modulation detection thesholds for an aver-

age speed L)o : I deg/sec and sinusoidal modulation func-

tions (small solid circles in Fig. 5) are approximately

invar iant (d0:10.2- l2.3deg) for the range of f re-

quencies tested (0.25-l Hz).At high flrequencies, thresholds rise strongly as a

function of frequency rr.r for both sinusoidal and block

shaped direction modulation functions. The average

ratio of threshold amplitudes for sinusoidal and block

shaped modulation functions at high frequencies

a 2 2 H z i s 1 . 2 6 .Discussion: threshold int'ariance at low' modulation fre -

quencies. The threshold invariance at low frequencies

indicates that human direction modulation detection is

based on the amplitude of the direction modulation

function and not on its temporal deriuatiue (directional

acceleration); direction modulation amplitude is inde-

pendent of modulation frequency, whereas the derivative

is l inear with modulation frequency. Therefore, a detec-

tion based on directional acceleration would yield in-

creasing thresholds for decreasing low modulation

frequencies. which rs not observed.

On the other hand, direction modulation detection

might be based on the magnitude of vertical dot position

(the temp oral integral of the direction modulation func-

tion). However, a constant vertical position threshold

(relative to the mean position) would yield decreasing

threshold direction modulation amplitudes for decreas-

ing modulation lrequencies in Fig. 5, which is also not

observed.At I Hz modulation frequency, the 6.6 deg threshold

direction amplitude at t 'o : 2 deg/sec is smaller than the

l0'2 deg threshold found at t 'o : I deglsec' Although

here we compare only two data points we speculate that

the asymptotic level is speed dependent. Increasing

direction modulation thresholds at low speeds are not

surprising given the relatively low sensitivity of the

human motion system at low velocities (DeBruyn &

Orban. 1988).

1 5

1 0

ltooc)

F o ). 0 )

E ! to v

E q

.9oo.=

TABLE 4. Parameter seltings for direction modulation thresholds in Figs 5 and 6

Wave form

Speedr,o (degisec)

Sweep-length Sweep-time

do (cm) to (sec)Distance EccentricitY

4 (cm) e (deg)

O( / )o(r )i l ( , )d ( r )

1 . 02.0t . 7t . 7

6.006.006.006.00

4 . 1 04 . 1 03.433.43

43433636

4222

2322

In conclusion, the observed asymptotic behavior ofdirection modulation thresholds as a function of modu-lation frequency supports the hypothesis that detectionis based on the amplitude of the direction modulationfunction. The observed absence of mechanisms tuned tovisual acceleration seems consistent with a study onmotion after effects (MAE) by Schwartz and Kaufman(1987), who reported that "there is no MAE specific toadaptation for changing directions as distinct fromsimple mot ion".

Discussion: waue forms, Because the fundamental fre-quency of sinusoidal functions has a smaller amplitude(a factor of f) than that of block functions (at a given

modulation amplitude). thresholds are expected to be afactor of | : 1.27 higher for sinusoidal than for blockfunctions at relatively high modulation frequencies. Theaverage rat io (1.26) of threshold ampl i tudes for s inu-soidal and block shaped modulation functions found forthe frequency range ).-2 Hz, is in good agreement withthe ratio, suggesting a relatively low cut-off frequency.The strong increase of direction modulation thresholdsfor both sinusoidal as for block functions, suggests atemporal frequency l imit for the direction detectionsystem of approx. 2Hz. We found a similar temporall imit for speed modulation detection.

Discussion : direction modulations us speed ntodulations.To compare the (absolute) direction modulation detec-tion thresholds d0 with the Weber fractions for speedmodulations. we use the following Weber fraction Wofordirection modulations Wo: d"^,,lt,o: tan d0. Now. bothWeber fractions Wa: du; f uo and W,: dr,f t^o are ele-gantly expressed in terms of velocity vector modulationsand can be compared.

For example, one can compute that the Weber frac-tion Wofor block shaped direction modulation detectionat I Hz and speed Do : 1.7 deg/sec is approx. 9%. Thisis about a factor 2 lo*-er than the Weber fractionW,: l7o/o for block shaped speed modulation detection.Furthermore, if one takes into account the effects ofprobabil ity summation (or variance detection of thevelocity vector modulation), very similar ratios (a factorof 2) for speed and direction modulation thresholds canbe shown to hold for other modulation functions used(see General Discussion section).

The fact that the human visual motion system is moresensitive to direction than to speed is a well-knownphenomenon in motion discrimination experiments,where Weber fractions for speed discrimination aretypically twice the Weber fractions for direction discrimi-nation (Nakayama, 1985b; DeBruyn & Orban, 1988).Thus,-although absolute Weber fractions for modu-lation detection are a factor of 3-4 higher than Weberfractions for motion discrimination experiments-theratio of Weber fractions for speed and direction modu-lations is very similar (about a factor of 2) in both typesof experiments.

The invariant ratio of sensitivity of speed and direc-tion reveals a fundamental characteristic of the humanvisual motion system. Let's consider a motion systemconsisting of an ensemble of Reichardt correlators that

have spatially rotational-invariant prefi lters (i.e. spatialinput filters that are not onentation selective) (see Gliin-der, 1990). For such a motion system, the higher sensi-tivity to direction than for speed is l ikely to be relatedto the fact that in the motion direction both spatial andtemporal prefi lters contribute to a broadening of theensemble response correlation peak, whereas in theorthogonal direction the width of the correlation peak isdetermined solely by the spatial prefi ltering.

Discussion: lov' -pass tentporal fltering at high.frequen-cies. We proposed a low-pass temporal f i l ter to explainthe cut-off frequency for speed modulation detection.The temporal l imit found for speed modulations (2 Hz)was similar to the cut-off frequency of approx. 2Hzobserved here for direction modulation detection. In theModel section, we model this temporal behavior ofdirection modulation detection analogous to the way wemodeled that of speed modulation detection: a temporalfi l tering of the direction modulation signal which corre-sponds to a temporal f i l tering of the speed signal r ', (r)or thogonal to the mean veloci ty vector [see equat ion (3)] .

Spatial cues at t,ery high modulation frequenciesMotication. For modulation frequencies that far ex-

ceed the temporal l imit (2Hz) for detection of modu-lation by the motion system, thresholds are determinedby other cues outside the motion system. such as spatialcues. Although these thresholds do not reveal thecharacteristics of the motion system, they are of interest.

Results. We measured direction modulationthresholds at 8 and 100 Hz modulation frequency for aL7 deg/sec speed frequency. Thresholds were foundto level off for this frequency range, suggesting thatobservers made use of weaker spatial cues to detectmodulations, which were independent of modulationfrequency.

The threshold for the remaining spatial excursion cueat high frequencies is a measure of the dynamic spatialacuity orthogonal to the motion direction of a movingdot. We found it to be about 8 arcmin for our observers,which is a few times the 2-3 arcmin acuity limit for staticstimuli (Wertheim, 1894) at the eccentricity used in thisexperiment (3-4 deg).

Pulse -shaped direction modulation functionsMotittation. Here we make a short study of pulse-

shaped direction modulation functions d(r), yieldingsquare wave vertical position modulation functions II(r)(see Fig. l). Detection of pulse-shaped direction modu-lation functions is interesting because the correspondingvertical velocities are too high to be sensed by the motionsystem. Hence, modulation detection must be based oncues outside the motion system. Therefore, the detectionof pulse-shaped modulation functions is l ikely to makeuse of a spatial cue: the spatial excursion from the meanvertical position.

Filtered pulse-shaped direction modulation functionshave constant amplitudes for low modulation frequen-cies (when the response functions in the time domain forconsecutive pulses are well-separated), yielding constant

PETER WERKHOVEN er a/.

OPTIC ACCELERATION : ) z )

spatial excursions from the mean vertical position.

Therefore, if the spatial excursion is the cue for modu-

lation detection, thresholds are expected to be indepen-

dent of modulation frequency for low frequencies. We

tested this prediction by measuring modulation detection

thresholds (expressed as spatial excursions) as a function

of modulation frequency for pulse-shaped modulation

funct ions d (r) .

Results. The parameter settings for this pulse detection

experiment are l isted in Table 4.

Figure 6 shows that the spatial excursions yielding

threshold performance are approximately invariant

(2.3-2.8arcmin) for the range of modulation fre-

quencies tested (l_4Hz). Similarity in performance

for subjects AT and HS allowed averaging over these

observers.Discttssion. The finding that the spatial thresholds at

the I Hz modulation frequency are slightly higher than

those for 2 Hz may be explained by probability sum-

mation or by the nature of variance detection. At the

fixed stimulus presentation time used there are twice as

many "events" (pulses) at a 2 Hz than at a I Hz modu-

lation frequency. Consequently, probabil ity summation

or variance computation can take place across more

events yielding lower thresholds. Obviously, there is a

limit to summation such that no improvement occurs at

even higher modulation frequencies.Note that the spatial modulation thresholds (spatial

excursions\ (2-2.5 arcmin) are considerably lower than

the dynamic acuitt ' (8 arcmin) determined earlier in

this paper at the same speed (1.7deg1sec). However,

they are sti l l higher than the hyperacuity thresholds

(0.2-{).5 arcmin) that have been measured for stationary

spatial configuration at similar eccentricit ies (West-

heimer. 1982). A more elaborate experiment that was

performed on pulse shaped speed modulation functions

will be presented elsewhere.The fundamental frequency components of block

shaped and triangular shaped position modulation func-

tions (at a given modulation amplitude) have relative

amplitudes i. With a low cut-off frequenc.v of I Hz. these

fundamental frequencies are expected to dominate the

detection thresholds at high modulation frequencies.

Therefore, we expect relative thresholds of I ry 0.64. This

can be verified by comparing thresholds for block

shaped position functions (see pulse shaped direction

modulations in Fig. 6) with the triangular shaped pos-

ition functions (see block shaped direction modulations

in Fig. 5) at, for example. 4 Hz modulation frequency'

This average ratio across observers is 0.67 supporting

our claim that we deal with positional cues and a

low cut-off frequency such that the fundamental fre-

quencies dominate detection of high lrequency modu-

lation functions.

MODEL

we present a model for the detection of velocity vector

modulations in the human visual system. The model

consists of two stages. The first stage is an nth order

low-pass temporal filter that operates on the velocity

vector modulation function. This fi l ter is characterized

by its order n and a characteristic t ime constant z. The

second stage is a decision stage based on the filtered

modulation function.In this section, we wil l show that our data provide

strong experimental evidence that the decision stage is a

uariance detection stage. The single parameter that

specifies the variance detection stage is a variance

threshold and can be estimated from the data. Further-

more, having knowledge about the decision stage, we

can estimate the parameters that characterize the first

(low-pass temporal f i l ter) stage.

The decision stage: cariance detection

The amplitude detection thresholds presented in this

paper have been interpolated based on detection proba-

bil it ies as measured using a method of constant stimuli.

Detection thresholds are the modulation amplitudes at

threshold performance (80% correct answers), and thus

form one parameter to characterize the full psychometric

functions available. However, the shape of the psycho-

metric functions reveal the parameter used in the

decision stage (e.g. velocity, squared velocity, etc').

Therefore. it is of interest to examine the shape of the

psychometric functions.Due to the noise, associated with the stochastic (bino-

mial) nature of the observer decision process. we need a

large number of elementary decisions for each modu-

lation amplitude in order to discriminate small shape

differences of different psychometric functions. There-

fore. we averaged psychometric functions, for all par-

ameter settings used in the modulation detection

experiments described above, as a function of normali:ed

velocity vector modulation amplitudes i. With a normal-

ized vector modulation amplitude (, we mean a modu-

lation amplitude for a given parameter setting, divided

by the modulation amplitude yielding threshold per-

formance for that part icular set t ing: C:@tlu1JlW' 'Assuming that velocity vector modulation detection

is ruled by a single detection process' psychometric

goooL J

E

F ^. Y Co ' = t= 6( ' ' ; ) L

x G l

( ! I'-

a

1 2 4

Modulat ion FrequencY, * (Hz)

FIGURE 6. Modulation thresholds for the detection of pulse shaped

direction modulation functions d (l ), expressed as vertical spatial

excursiotts of the block shaped vertical position modulation fl(l ).

Spatial thresholds are shown as a function of modulation frequency cr.t.

The average speed t'0 was l.Tdeglsec and the eccentricity of presen-

tation was 3.4deg. Data for two observers (AT and HS) have been

averaged.

PETER WERKHOVEN er a/.2324

functions for all parameter settings will be identical ifplotted as a function of normalized amplitudes!

The resulting psychometric curve that describes all

experiments reported in this paper is shown in Fig. 7.

The horizontal axis in Fig. 7 represents normalized

modulation amplitudes (. The ordinate represents thepercentages correct for a small range of normalizedmodulation amplitudes clustered around a range ofplotted normalized amplitudes of the data points. Halfthe length of the shown error bar for a normalizedamplitude i corresponds to the sqare-root variance (or)

of the binomial probabil ity distribution for that point:

o, :1t .11 -p51., ,6, , wi th p, the ordinate for modu-lation i. and ni the number of elementary observerdecisions. Each individual data point is based on about700 elementary (yes/no) observer decisions (n,: 700).

We find an excellent fit of this psychometric functionusing a standard error function or (scaled) standardnormal distributi on Erf(z):

r f',':

Erf( : ) : + | , - t iz 'zt 4 i , (5). \ / L t L J

- q

that has the square normalized modulation amplitude(: : (2) as i ts argument (Xt :5.2 wi th 7 d. f . ) . The scalefactor ̂ i rs constant: ̂ l :0.84, and causes Erf(z) to be80% for z : | (a constraint, set by our threshold defi-nit ion). Note that the function Erf((2) used for the fit hasno free parameters! The evidence for this particularshape of psychometric function is very strong since a fitwith functions Erf(O or Erf((t) yields unacceptable

chi-square values of X' : 152 and 7?: 55 respectively.Thus, assuming that a standard error function is a validdescription of the psychometric function associated withthe final observer decision process (Green & Swets,1966), the empirical function shown in Fig. 7 stronglysuggests that modulation detection is based on thesquare modulation amplitude. i.e. the uariance of the(temporally f i l tered) velocity vector modulation.

A variance detection process is certainly not exclusivefor velocity vector modulation detection and has. forexample. also served to explain human sensitivity totemporal fluctuations in the luminance domain (Rash-bass. 1970: Koenderink & van Doorn. 1978). The choiceof variance detection for these visual tasks is not surpris-ing since variance detection has been shown to beoptimal for a statistical point of view in a number ofinstances in which the visual system is uncertain aboutsome aspects of the stimulus to be presented (Green &Swets. 1966), e.g. the phase of per iodical modulat ionfunctions. This uncertainty may force the observer to usethe autocorrelation (variance) of the velocity modulationsignal. instead of the more efficient cross correlation ofthe modulation signal received with the signal expected(Green & Swets. 1966; Burgess & Ghandeharian. 1984).

We argued that the uncertainty of the observer aboutthe phase of the modulation function accounts forvairance detection and thus for our finding that speedmodulation detection thresholds (l7o/o) are much higherthan speed discrimination thresholds (6%). However. analternative explanation for variance detection is in termsof the diff iculty of the visual system in segmentingmodulated motion paths in temporal segments of differ-ent velocities in the absence of cues other than in themot ion dimensions (Snowden & Braddick. l99l) . Todecide on this issue, experiments are useful in which theobserver is provided with explicit cues that define thephase of the motion modulation signal andlor allow fora segmentation of the stimulus.

Interestingly, such experiments have been performedin another domain. In stereo vision (disparity process-ing), for example, the absence of an explicit segmenta-tion cue (i.e. defined outside the stereo domain) can leadto dramatic increases in the disparity discriminarionthreshold (McKee, 1983; Fahle & Westheimer, 1988). Inthe domain of two-dimensional luminance pattern per-ception. on the other hand, an explicit segmentation cuehas been observed to force the visual system in aprocessing mode yielding higher discriminationthresholds than obtained without segmentation cues(Watt , 1985)!

Rashbass (1976) has shown that a uariance detectionprocess based on the square modulation amplitude yieldsidentical detection performance to an alternative detec-tion model consisting of peak detection based on thelinear modulation amplitude, with a detection prob-ability summation that is governed by a psychometricfunction similar in form to the empirical curve shown inFig. 7.

We have shown that modulation detection is accu-rately described by a variance detection. based on a

1 0 0

90

Ec 8 0oo

3 7 0o-

l n

0 . 6 0 . 8 1 . 0 1 . 2 ' 1 . 4

Norma l i zed Amp l i t ude , €

FIGURE 7. Probabilit ies P of correct answers as a function of thenormalized velocity vector modulation amplitude (. With a normalizedvector moduiation amplituded i, we mean a modulation amplitude fora glven parameter setting divided by the modulation amplitudeyielding threshold performance for that particular setting. The datapoints are collected using all speed/direction modulation detectionexperiments described in this paper. The horizontal axis representsnormalized modulation amplitudes (. The ordinate represents thepercentages correct P for a small range of normalized modulationamplitudes clustered around a range of plotted normalized amplitudes

{, of the data points (solid circles). Half the length of the plotted errorbar for a normalized amplitude (, corresponds to the square-rootvariance or standard deviation (o,) for that point (see text). Becausewe normalized the amplitudes to the threshold amplitude, the curve isexpected to reach threshold (807o correct answers) for i : I and reachchance ievel (50%) for { :0. The solid curve is a best fit of thepsychometric function Erf(z) to the data, taking the squared modu-lation amplitude z : 4, as its argument. The dotted and dashed curvesare the expected psychometric curves for arguments ; : ( and z : g I

respectivelv. and fit less well.

OPTIC ACCELERATION

low-pass transformation of the velocity vector function,

that is. velocity vector modulations are detected

whenever the variance (after filtering) exceeds a certain

internal threshold. The internal thresholds are most

easily derived from threshold modulation amplitudes for

block shaped modulation functions at a low frequency,

when the modulation function is nearly unaffected by the

low-pass fi l ter. This is because the variance of block

shaped functions is equal to the square modulation

amplitude. Therefore. the internal threshold for variance

detection is equal to the square of the threshold modu-

lation amplitude measured for these functions. For

example. the minimum standard deviation (square-root

variance) yielding modulation detection is W,: 17o/o for

speed modulations (see Fig. 2) and W :9o/o for direc-

tion modulations (see Fig. 5)'

The filter stage: a second-order lov'-pass .filter

The data presented in this paper suggested that the

detection of velocity vector modulat.ion functions is

based on a temporal low-pass fi l tered version of the true

(physical) stimulus velocity vector function. The effects

of such a low-pass temporal f i l ter on detection perform-

ance depends on the type of decision stage that follows

temporal f i l tering. Now we have specific knowledge

about this decision stage (see above), we can estimate the

temporal characteristics of the first (f i l ter) stage'

We model the temporal low-pass fi l ter as a standard

temporal n th order low-pass fi l ter and estimate its order

and its characteristic t ime constant from the dependen-

cies of modulation thresholds on modulation tiequency.

Methocl. A standard temporal nth order low-pass fi l ter

has a pulse-response function:

8 ( r ) : - f * ( l ) ' e - ' ' ( n > o ) . ( 6 )r l n - l ) ' \ r /

and a transfer function A(c,r ) of modulation frequency 1a:

S@t) : [ + (2non) : ]

This f i l ter reduces the ampl i tude of the modulat ion

functions. For example. a sinusoidal modulation func-

tion with frequency o and amplitude .4 passing the l i l ter

g(r) wi l l have a reduced ampl i tude A!(ot) . The detect ion

of such a modulation signal takes place in a variance

detection stage as described above. At threshold. the

variance of this fi l tered sinusoidal signal is equal to a

th resho ld var iance o i : A ' 'E t (o t )12 : oa ( reca l l tha t the

var iance of a s inusoidal s ignal equals hal f i ts square

ampl i tude).This threshold oi can be estimated from an empirical

modulation detection threshold w(an) at a low modu-

lation frequency @o ( l lQnt) for which g(r'to) = l ' For

example. for block shaped modulation functions the

variance of the function at threshold amplitude W(algo)

is exactly wt(') ' Thus' oi: w'(con) for block shaped

modulation functions.To estimate the time constant r, we consider the

threshold amplitudes at modulation frequencies r.r.r, and

co, for which the higher order spectral components of

the rno,Julation function can be ignored- In those cases,

2325

the modulation functions are approximated by their

fundamental (sinusoidal) components. For example, for

block shaped functions with (threshold) amplitude

Wko,), the amplitude of the |undamental sinusoidal

component is f W(a,). At threshold, the variance of the

filtered fundamental equals the detection threshold oi:

: o i . (8)

For frequencies |i. i) (2m )-

"

we could use the asymp-

totic behavior E@) = (Zna,t)- ' to estimate r and n

analytically from two data points at high frequencies.

However, we can not use this approximation a priori and

use a numerical approach. First, we rewrite equation (8)

to solve r? as a function of n and crr,:

I [ / . , ,Ew,1 . , ) \ t ' _ , - l:,^;iL(H) -rl (e)

The t ime constant t is constant. Thus. r(co, , n) is ex-

pected to be the same for any two different (sufficiently

irigft) modulation frequencies cDr and crr' ' Hence, the

order n of the fi l ter is the solution of the equation:

( l 0 )

Il| ""t', l] t | + (2nw'r')l -'

t ( ( t r . n ) : t ( o t r , n ) .

W e c h o s e t h e l o w e s t n f o r w h i c h l r ( o ' , n ) - t { a t , n ) l i

[ r ( ro , , n ) + r (@t ,n) ] i s <20o/o . The t ime cons tan t t i s

iaken to be the average of the two values r(@r,n) and

r ( r o 2 , n ) a t t h i s n .

Estimation oJ'Jilter parameters for speecl modulation

thresholcls. Consider the speed modulation thresholds tor

block shaped functions in Fig. 2. We use the threshold

at low modulation frequency @o: I Hz for the esti-

mat ion o f oo "

o0 : W, ( l \ : 17o /o ' Fur thermore ' we use

the two thresholds at high modulation frequencies

()) t : 4Hzand cr.r ' : 8 Hz with thresholCs W,(a,) :30o/o

and W,(ro,) : 8l%'

Using the method described above, we find that a

value n :2 and a t ime constant r : 33 msec adequately

model the dependence of speed modulation detection

thresholds w,(a) as a funct ion of modulat ion f requency

@. In fact . we found r(cor.2) : 30'6 msec and

r ( @ : . , 2 ) : 3 6 . 1 m s e c .With r :33 msec. th is second order low-pass f i l ter

corresponds to a value of approx. 90 msec for the full

width at half maximum (FWHM) of the pulse-response

of the speed integration fi l ter yielding an integration

(smoothing) of the physical speed signal in the human

visual system within roughly a 100-140 msec temporal

window.Estintation of ftter parameters for direction modulation

thresholds. Similar to the previous section. we estimated

the order n and time constant r from the direction

modulation thresholds for block shaped functions as

presented in Fig. 5. We used the threshold^Wo(a) - -8 '7o /o

fo r (D0: lHz ' and Wakt t ' ) : l l ' 4o /o

(ar ' : 2Hz) and W6(ror) : 24'9% (ar:4Hz) '

Using the method described above' we find that a

value n :2 and a time constant r : 42 msec adequately

model the dependence of direction modulation detection

threshold s w6(a) as a function of modulation frequency

2326

()). In fact, wer( :@2,2) : 49 msec.

This time constant for the low-pass filtering of the

direction modulation signal is only slightly higher than

the time constant (r :33 msec) estimated from the

low-pass filtering of the speed modulation signal. How-

ever. as we showed in Fig. 3, this small discrepancy may

be a consequence of the different speed ranges used for

the determination of the temporal filter characteristicsfor direction and speed modulations.

The overall similarity of the characteristics of human

detection of speed and direction modulations and the(near) equality of the integration time-constants derived

strongly suggests a detection system that monitors the

full (temporally filtered) velocity uector.

GENERAL DISCUSSION

Euidence .for indirect optic acceleration detection

We presented a study of human sensitivity to optic

acceleration and have been unable to find any evidence

for a visual mechanism that directl l 'detects optic accel-

eration. i.e. the temporal derivative of the velocity vector

modulations. Instead we find strong evidence that modu-

lation detection is based on the amplitude or modulation

depth of a temporally filtered velocity vector modulation

signal. The temporal characteristics of the temporal f i l ter

are adequately described by a second-order low-pass

filter with a time constant r ry 40 msec. Effectively, this

fi l ter corresponds to a temporal integration of the vel-

ocity signal of at least 100 msec. This is consistent with

the upper temporal l imit of about 100 msec for the

integration of velocity information (improving signal-to-noise ratios) in motion discrimination experiments(DeBruyn & Orban, 1988; Snowden & Braddick, 1991).

Thus, the lower and upper l imits for temporal inte-gration in the human visual motion system are equal,

suggesting a single hard-wired temporal filter in the

motion processing system. This view is further supportedby the close quantitative correspondence between the

increase of cut-offfrequency with speed (as reported here

for motion modulation detection) and the decrease of

temporal integration time with speed found in motion

discrimination studies (van Doorn & Koenderink, 1982,

l e85).This leads to the intriguing question: which stage in

the stream of visual motion processing accounts for the

characteristic temporal filtering found in our exper-

iments?

Temporal filtering: mechanistic considerations

A functional description of the phenomenology of our

experiments consists of a temporal integration of anunsmoothed internal representation of the true velocity

signal [Fig. 8(a)]. At this point we will try to link thisfunctional description to an actual implementation in thevisual system in terms of well-known motion detectionmechanisms.

An abstract description in terms of a smoothedmotion signal does not necessarily mean that the visual

PETER WERKHOVEN

f o u n d r ( @ r , 2 ) : 3 4 m s e c a n d

et al.

FIGURE 8. Two similar (but fundamentally different) modulation

detection processing streams. The first stream (a) consists of the

extraction of a true uelocitt' signal. followed by iow-pass temporal

fi ltering and by a final detection stage (e.g. peak or variance detection).

The second stream (b) consists of a velocity extraction that does not

yield a true velocity signal but a lov'-pass transformation of the true

t'elocitl ' . followed by a final detection stage as in the first stream.

system actually extracts (unfi ltered) velocity signal to

subsequentially low-pass fi l ter it in t ime. In flact, thefollowing rhetorical questions make such an implemen-

tation unlikely: (l) How does the visual system arrive at

the representation of the true (unfi ltered) velocity in the

first place? (2) lf such a representation exists. should this

signal be low-pass fi l tered given the great advantages of

having access to a velocity signal with high temporalresolution (Arnspang, 1988; Gli inder. 1990)? Because of

the above puzzles. we believe that the temporal inte-gration is inherent to the mechanism that arrives at a

velocity representation and that it takes place effectivelybefore the final estimate of the velocity vector. akin tothe scheme of Fig. 8(b).

We il lustrate some of the possible stages of temporalfi l tering by adopting a specific but plausible basic motion

detector: the Reichardt-correlator (van Doorn & Koen-

der ink. 1985), see Fig. 9.A plausible implementation of such a correlator typi-

cally contains three temporal f i l tering stages:

l. A temporal prefi lter f(t) for each input l ine.2. A temporal delay fi l ter (with time constant t ) in

one of the input l ines.3. A temporal low-pass fi l ter J. of the correlatorinput.We will discuss each of these fi l ters as candidates to

account for the temporal low-pass fi l tering of the vel-

ocity vector modulation functions found in our exper-iments.

Temporal lsvt-pass filtering of correlator outpur. Intui-

t ively, it is tempting to associate the psychophysically

observed intergration of velocity with the temporalintggration J. of the correlator output. Perhaps suripris-ingly, however, temporal f i l ter [, is not equivalent with

FIGURE 9. A standard motion detector (Reichardt correlator). Stan-

dard motion analysis consists of two input l ines (receptive fields) with

temporal f i l ters / ( l ) , a delay f i l ter (wi th t ime constant r ) for one of the

input l ines. a correlat ion stage and a temporal integrat ion f i l ter Jr

atemporal in tegrat ionof themodulat ionofspeedorupperr ightcornerofF ig. l0 .Tomakeourargumentasdirection. To show this, we will consider an ensemble of strong ai possible, we ?:::t:d :T::T,

temporal inte-

motion detectors (Reichardt "orr"iuto..),

ideally tuned gration t;kes place within a temporal window that

to a continuum of velocities. r. ,"lr"r1ii"i" -i, poini J*""ra, the period of the modulation function a few

we will focus on speed modulati"; f*;ii;;t Assuming times such that it flattens the profile'

that detecrors tuned to identical'J""iii*" "."

p."r"i Figure t0 also shows the ensemble activation profiles

(Glilnder, 1990), this ensemble "un

i" pu'"-at'i"a- ty for a unknown (unmodulated) speed function be/ore

tuning velocity u, only. At t,." r,-ittt 'ioning target hai (bottom left) and after temporal integration (bottom

velocity u(r) and wilt thus *"""," ".rr

i"i"!,;";* a iigt t). ouuiourty a ionstant profile in time is invariant

it"rtg'""r".r,y ,, = u(r1' Therefore'-t# type of activated under temporal integration'

detectors (parameterized Uy ,,; witiin the ensemble will As a result of temloral integration Jr (blurring along

vary in time, yielding time depende'nt )nsemble artiuation the horizontal time axis), both the ensemble activation

profiles. Forexample' ensemble ;;;t;;tt*';;;;;; for a profiles for modulated and unmodulated velocitv func-

iriangular speed modulation iun.iion ur"'giu.n in the tions. are constant it time' However' the shape of the

left upper corner of Fig. 10. ptofilet io' the modulated speed function (upper right)

In th is f igurethetypeofdetector(parameter izedbyi tsandthat for theunmodulatedspeedfunct ion(bot tomtunins speed t',, along the *"# #;;;;;

"tirtit"J righQ differ stronglv even for infinite blurring'

by the moving dot is given as u iun"tion oiti-" r iutong A true integrat'ion of the speed modulation signal'

the horizontal axis). wten tne oeiecio" utt u"'y 'hutply howeuer' tnou-id blut the ensemble profile along the

tuned, only one type or a.t."tor'i, -u.tiu"

oi u ti-", vertical6Deed) axis yielding blurred profiles that become

dependent on the sPeed "r,n.

.J"itg a"i ii";;;t;. indiscriminable for infinite blurring'

l a r m o m e n t i n t i m e , w e c a n w a l k a l o n g t h e v e r t i c a l a x i s l h e a b o v e r e a s o n i n g s h o w s t h a t t e m p o r a l l y f i l t e r i n gand find which detectors are active as a-function of their the speed sign-al is not iquivalent to-temporally filtering

tuning speed (the ensemble *U"",i"" pr"nf.l. Because the oulput "of

motion. d;teclors (correlaton) That is'

the dot moves with a single ;il;; !;;ffi;l"niti tr"mng in the speed dimension is g€n€rallv not equival-

rime. the ensemble profile is a single pulse thar shifts ent to ;lurring in _the time diT:1sio1-.we suggest that

along the verrical axis in tine.'- fritri sharply tuned the psychophisically observed integration of the speed

detectors. the ensemble activation profile is a perfect signal musi be inherent to a processing stage which

copy of the physical speed .oaotution signal u(i)' and comes before the correlation stage'

is thus triangular in time t"t'tpp"t ftft "o'n"'

of ,Tempiral

pre-fher' The shown temporal low-pass

F i g . l 0 ) ' c h a r a c i e r i s t i c f o i m o d u l a t i o n d e t e c t i o n m i g h t b e i n -N o w l e t . s c o n s i d e r t h e t e m p o l a l f i l t e r J r t h a t i n t e g - n " . " n t t o t h e t e m p o r a l f i l t e r / ( r ) a l . t h e i n p u t o f a

trares rhe output of rhe standard motio; detectors in standard motion d;tector (see Fig g)' The reasoning

time yielding an integration or-,r,'" in-r..Lr. u"tirution would be in terms of "window of visibility" arguments

p r o f i l e a l o n g t h e h o r i z o n t a l t i m e a x i s ' T h e r e s u l t i n g a s u s e d t o e x p l a i n t h e p e r c e i v e d e q u a l i t y o f a p p a r e n thorizontally blurred acrivatron^;;;i.; i io*n in th! motion with rial motion at adequate sampling frequen-

Prwrx! 'J

"i"t (wat'o"

"r o/ ' 1986: Burr et al'' 1986)' However' in

Bero,e Frterino Aner Firrerino fS:illm*:nf,i".-r."i: flt""::1ffiffi:l':.1l:ooral cut-off frequencies of 4-8 Hz' At these unrealistic

:'STi y"Hl iemporal cut-off frequencies' the motion system has not

I c"",l ,"."tt"d ootimal sensitivity (Burr & Ross' 1982)!

OPTIC ACCELERATION232'l

t p e e o s p e e o t e l l l p u l a l L t r L - ( - , r r r r v 9 s v r r v r v

t + .u.n ,.u.hed optimal sensitivity (Burr & Ross. 1982)!

T , ^ . , ^ . _ d o , ' , . q u . n ' t y , t h e t e m p o r a l p r e - f i l t e r e x p l a n a t i o n i sa - \ - / - - \ z { > l I i m p l a u i i b l e ., - + ^"!i{l:;!,!J!"{,{";i'y;JH:il11il'if 'TilHlirunins tunins to account for low-pass transformations of modulat ion

speed speeJ functions. The standard motion detector as sketched in

{ + Fig. 9 is optimally activated if an object traverses the

r - l+ sPatial inteI t t . - . - ^ -

' r ' Lo . noa . l ^ f t hc

: . , , > | - i l ; ; ; i l s i n t h e f i n i t e d e l a y t i m e t . T h e s p e e d o f t h ei I - - - r - - -

- , ^ : ^ ^ + ^ - . , ^ . l ^ n - , r c f h e

;bj;;, ;;y vary during its trajectory' as long as the. ' ^ : - ^ r - , + L . i ^ - ^ - . , l t o i n qrime

_ above conit.aini is satisfied. Intuitively, this results in a

FIGURE 10. Ensemble activatlon profi les as a funclion ol t lme lhe

ensemble of motion detectors considered rn this figure is Paramet€riz" :",Tp,":11-^11::aging (or temporal integration) of the

;;;;';;; the tr'rning velocitv r', lvertical axisl Activation speed.. tunctron'

profiles are given as a fun"tion tim" r (t'otiz'onJ axis) fo;a ffiangular Gliinder (1989) has recently presented an interesting

vetocirv modula[ion functior (upper lef! corner) and a constant speed mathematical analysis on this issue. His study focused on

A"-",i.ri".**.i. e;-irt igitirta" "" rrt"* rhe resulting activation ina qu.*ion of

-how velocity estimates through an

prohles for both functlons after temporal inteeration (lr) ot eacn

motion detector output in the ensemble (ie iitegtatioi along dlg €nS€mole of standard motion detectots depend on the

horizontal axrs) spatrar object function and on the impulse-response

function of the delay filter of the detector. For anensemble of bilocal correlators tuned to a continuum ofvelocity vectors. he showed that the estimated velocityfunction is the result of the convolution of the true(physical) velocity vector function with a time-invariantkernel which only depends on the integral function of theimpulse-response function. Hence, the estimated vel-ocity vector function is independent of the spatial objectfunction.

Ghinder's proof strongly supports our view that thephenomenological description of our results. in terms oflow-pass temporal f i l tering of the velocity vector func-tion, corresponds with a plausible implementation interms inherently non-ideal (realizable) band-pass delayfi lters in correlator detectors. Following this hypothesis,the cut-off frequencies for modulation detection dependinversely on delay value r: for longer delay values, thewidth of convolution kernel increases and yields strongertemporal blurring. Our finding that cut-off frequencieso)c are s l ight ly dependent on speed r .o[ra.(r .o)x1, f l35] thuslead to the conclusion that delay values r are speeddependent. Cut-off frequencies coc are expected to beinversely dependent on correlator delay r. This con-clusion corresponds closely to the empirical power func-t ion reported by van Doorn and Koender ink (19g2):z ocr ,o oao.

Speed dependence of cut-off frequenciesThe dependence of cut-off frequency on speed we

observe in Fig. 3 is consistent with what has beenreported in both psychophysical and electrophysiologi-cal l i terature: higher velocities correspond to somewhatfaster detectors.

However, this finding is at odds with the speeddependence of temporal velocity resolution obtained bySnowden and Braddick (1991). They found that rhecut-off frequency for speed modulation detecti on de -creases with increasing velocity in their experimentalset-up. This issue remains to be resolved by furtherexperimentation. The most noticeable difference betweenour experiments and those of Snowden and Braddickconcerns the spatial nature of the stimuli used. Snowdenand Braddick used a spatially extended random-dotpattern centered at the fovea. whereas we used a local-ized target moving at a trajectory with an approximatelyconstant eccentricity in our case.

Relation to sampled (apparent) motion experimentsApparent motion differs from continuous (smooth)

motion because it is characterized by a speed functionthat is modulated in time. Apparent motion is thus aspecial case of the modulated functions examined in thispaper. We wi l l d iscuss two studies (Watson et a l . , l9g6;Burr et al., 1986) that reported on the minimum tem_poral sampling frequency yielding perceptual equival_ence of apparent and real motion and compare themwith our study on the upper cut-off frequencies forvelocity modulation detection.

In Watson et al.'s stroboscopic paradigm the time_dependent speed is an i lr-defined iignar, butleriodic with

PETER WERKHOVEN er a/.

a frequency equal to the strobe frequency. They findminimal sampling frequencies (yielding perceptualequivalence of stroboscopic and real motion) that aremuch higher ( > 30 Hz) than the cur-off frequencies forvelocity modulation detection obtained in this study.However, this is probably due to the strong luminancecue at low velocities.

In Burr er al. 's "sample and hold" paradigm themoving dot is visible all the time and displaced stepwisein time. The time-dependent speed function )1r) is nowwell-defined (see Fig. l) and can be compared with theblock modulation functions used in our experimentswith 100% modulat ion ampl i tude. Figure I shows that"sample and hold" speed modulation function f(l)differs only in duty cycle from the block modulationfunction fl(r) at 100% modulation amplitude as used inour experiments. Furthermore. functions )Jr ) differ onlyin peak width f rom the tr iangular funct ions A(t) atl00o/o amplitude. However. the cut-off frequencies foundin th is paper (approx. 4Hz for t r iangular modulat ionfunctions &t lo : 2 degrsec. see Fig. 3) differ by at leasta factor of 4 from minimum sampling frequencies( l5-40 Hz, dependent on the drift rare) found by Burr elal. for this particular speed. We offer a number ofexplanations for this apparent discrepancy:

l . For a given average speed 1:n. the ampl i tude ofthe fundamental frequency in the ..sample

andhold" morion modulat ion funct ion )11) is muchlarger than for the triangular speed modulationfunct ion A(t)u,e used to obtain Fig. 3 ( the rat ioequals n214 = 2.5). This al lows for higher cut_offfrequencies in Burr's paradigm.

2. Burr et a l . do not ment ion v isual f ixat ion. Thestrong dependence of detection performance onpursuit eye movements was discussed in theMethod section. Our observers reported to haveno dfficulties in detecting modulations whentracking the mo', ' ing dot for conditions wheredetection .failed under visual f ixation.

3. We mentioned before that for modulation fre_quencies that far exceed the temporal l imit(2 Hz) for the detection of modulations by themotion system. thresholds are determined bycues orl/side the motion system. such as spatialcues. Thus it may be that experiments on theequivalence of apparent and real motion do notexclusively reveal the structure of the visualmot ion system.

Relation to experiments with controllecl eye molementsResults have been reported on frequency limits for

velocity modulation detection when a mouing referenceis provided (Funakawa, 1989) (contrary to our station-ar.v fixation dot). Providing a moving reference leads tocut-off frequencies (x25Hz) that are considerablyhigher than those obtained in our study. Interesting asthese results are, we believe them to be indicative of thetemporal resolution of visual subsystems concerned withthe spatial analysis of moving patterns. and not with the

2328

OPTIC .\CCELERATION

determination of velocity as such. We believe both types

of experiments (and visual subsystems) should be clearly

distinguished (although, of course, they may be inti-

mately intertwined).

CONCLUSION

In conclusion. human detection of velocity vector

modulations is not based on optic acceleration [thetemporal derivative of the velocity modulation functionu(r)1. The data presented in this paper strongly supportthe view that modulation detection consists of a ttariancedetection process, based on the magnitude of a low-passfilter transformation of the true modulation functionu(r). Effectively, the motion system integrates the vel-

ocity vector modulation signal for about 100 msec overt ime.

These results put severe constraints on viable theoriesaiming to explain human capacities in the extraction ofthree-dimensional environmental information frommotion paral lax cues (Nakayama, 1985a).

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Acknov'ledgements-This research was performed at the Institute for

Percept ion TNO. Soesterberg. The Nether lands. The research of Peter

Werkhoven was supported by the USAF Life Science Directorate.

Visual Informat ion Processing Grant 88-0140. Herman Snippe was

supported by the InSight project of the ESPRIT Basic Research

Actions of the European Community'. Alex Toet was supported by

NATO erant CRG 890970.