Com

puta

tiona

l Vis

ion

U. M

inn.

Psy

503

6D

anie

l Ker

sten

Lect

ure

19: M

otio

nII

http

://vi

sion

.psy

ch.u

mn.

edu/

ww

w/k

erst

en-la

b/co

urse

s/P

sy50

36/S

ylla

busF

2000

.htm

l)

Init

ializ

e

Off@General::spell1D;

<<Graphics`

Ou

tlin

e

Las

t ti

me

• E

arly

mot

ion

mea

sure

men

t--t

ypes

of

mod

els

corr

elat

ion

grad

ient

feat

ure

trac

king

•Fun

ctio

nal

goal

s of

mot

ion

mea

sure

men

ts

• O

ptic

flo

w

Cos

t fun

ctio

n (o

r en

ergy

) de

scen

t mod

el

A p

oste

rior

i and

a p

rior

i con

stra

ints

Gra

dien

t des

cent

alg

orith

ms

Com

pute

r vs

. hum

an v

isio

n an

d op

tic f

low

-- a

rea

vs. c

onto

ur

To

day

‡M

otio

n p

heno

men

a

Nei

ther

the

area

-bas

ed n

or th

e co

ntou

r-ba

sed

algo

rith

ms

we'

ve s

een

can

acco

unt f

or th

e ra

nge

of h

uman

mot

ion

phen

omen

a or

psy

chop

hysi

cal d

ata

that

we

now

hav

e.

Loo

k at

hum

an m

otio

n pe

rcep

tion

‡L

ocal

mea

sure

men

ts

Rep

rese

ntin

g m

otio

n, O

rien

tatio

n in

spa

ce-t

ime

Four

ier

repr

esen

tatio

n an

d sa

mpl

ing

Opt

ic f

low

, the

gra

dien

t con

stra

int,

aper

ture

pro

blem

Neu

ral s

yste

ms

solu

tions

to th

e pr

oble

m o

f m

otio

n m

easu

rem

ent.

Spac

e-tim

e or

ient

ed r

ecep

tive

fiel

ds

‡G

loba

l int

egra

tion

Sket

ch a

Bay

esia

n fo

rmul

atio

n--t

he in

tegr

atin

g un

cert

ain

loca

l mea

sure

men

ts w

ith th

e ri

ght p

rior

s ca

n be

use

d to

mod

el a

va

riet

y of

hum

an m

otio

n re

sults

.

Hu

man

mo

tio

n p

erce

pti

on

Dem

o:

are

a-b

ased

vs.

co

nto

ur-

bas

ed m

od

els

Las

t tim

e w

e as

ked:

Are

the

repr

esen

tatio

n, c

onst

rain

ts, a

nd a

lgor

ithm

a g

ood

mod

el o

f hu

man

mot

ion

perc

eptio

n?

The

ans

wer

see

ms

to b

e "n

o". T

he r

epre

sent

atio

n of

the

inpu

t is

prob

ably

wro

ng. H

uman

obs

erve

rs s

eem

to g

ive

mor

e w

eigh

t to

cont

our

mov

emen

t tha

n to

inte

nsity

flo

w. H

uman

per

cept

ion

of th

e se

quen

ce il

lust

rate

d be

low

dif

fers

fro

m

"are

a-ba

sed"

mod

els

of o

ptic

flo

w s

uch

as th

e ab

ove

Hor

n an

d Sc

hunc

k al

gori

thm

. The

two

curv

es b

elow

wou

ld g

ive

a m

axim

um c

orre

latio

n at

zer

o--h

ence

zer

o pr

edic

ted

velo

city

. Hum

an o

bser

vers

see

the

cont

our

mov

e fr

om le

ft to

rig

ht--

beca

use

the

cont

ours

are

str

onge

r fe

atur

es th

an th

e gr

ay-l

evel

s. H

owev

er w

e w

ill s

ee in

Ade

lson

's m

issi

ng f

unda

men

tal

illus

ion

that

the

stor

y is

not

as

sim

ple

as a

mer

e "t

rack

ing

of e

dges

" --

and

we

will

ret

urn

to s

patia

l fre

quen

cy c

hann

els

to

acco

unt f

or th

e hu

man

vis

ual s

yste

m's

mot

ion

mea

sure

men

ts

219.MotionII.nb

size = 120;

Clear[y];

low = 0.2; hi = .75;

y[x_] := hi /; x<1

y[x_] := .5 Exp[-(x-1)^2]+.1 /; x >= 1

ylist = Table[y[i],{i,0,3,3/255.}];

width = Dimensions[ylist][[1]];

Let

's m

ake

a 2D

gra

y-le

vel p

ictu

re d

ispl

ayed

with

Lis

tDen

sity

Plo

t to

exp

erie

nce

the

Mac

h ba

nds

for

ours

elve

s. P

lotR

ange

al

low

s us

to s

cale

the

brig

htne

ss.

picture1 = Table[ylist,{i,1,width/2}];

picture2 = .9 - Transpose[Reverse[Transpose[picture1]]];

g1 = ListPlot[picture1[[size/2]],DisplayFunction->Identity,PlotStyle-

>{Hue[.3]}];

g2 = ListPlot[picture2[[size/2]],DisplayFunction->Identity,PlotStyle-

>{Hue[.6]}];

Show[g1,g2,DisplayFunction->$DisplayFunction];

50

100

150

200

250

0.2

0.4

0.6

0.8

ListDensityPlot[picture1,Frame->False,Mesh->False,

PlotRange->{0,1}, AspectRatio->Automatic];

ListDensityPlot[picture2,Frame->False,Mesh->False,

PlotRange->{0,1}, AspectRatio->Automatic];

19.MotionII.nb

3

Ap

ertu

re e

ffec

ts

niter = 8; width = 64;

theta1 = Pi/4.; contrast1 = 0.5;

freq1 = 4.; period1 = 1/freq1;

stepx1 = Cos[theta1]*(period1/niter); stepy1 = Sin[theta1]*(period1/niter);

grating[x_,y_,freq_,theta_] := Cos[(2. Pi freq)*(Cos[theta]*x +

Sin[theta]*y)];

‡C

ircu

lar

aper

ture

For[i=1,i<niter + 1,i++,

DensityPlot[If[(x-0.5)^2+(y-0.5)^2<0.3^2,grating[x+i*stepx1,y+i*stepy1,freq

1,theta1],0],{x,0,1},{y,0,1},

Mesh->False,Frame->None,PlotRange->{-2,2},PlotPoints->width];

];

‡Sq

uare

ape

rtur

e

For[i=1,i<niter + 1,i++,

DensityPlot[grating[x+i*stepx1,y+i*stepy1,freq1,theta1],{x,0,1},{y,0,1},

Mesh->False,Frame->None,PlotRange->{-2,2},PlotPoints->width];

];

Wha

t do

you

see

at th

e ve

rtic

al b

ound

arie

s? T

he h

oriz

onta

l bou

ndar

ies?

419.MotionII.nb

‡R

ecta

ngul

ar h

oriz

onta

l ap

ertu

re

For[i=1,i<niter + 1,i++,

DensityPlot[grating[x+i*stepx1,y+i*stepy1,freq1,theta1],{x,0,1},{y,0,.25},

Mesh->False,Frame->None,PlotRange->{-2,2},PlotPoints->width,

AspectRatio->Automatic];

];

‡R

ecta

ngul

ar v

erti

cal a

pert

ure

For[i=1,i<niter + 1,i++,

DensityPlot[grating[x+i*stepx1,y+i*stepy1,freq1,theta1],{x,0,.25},{y,0,1},

Mesh->False,Frame->None,PlotRange->{-2,2},PlotPoints->width,

AspectRatio->Automatic];

];

Ad

elso

n's

mis

sin

g f

un

dam

enta

l m

oti

on

illu

sio

n

We

firs

t m

ake

a sq

uare

-wav

e gr

atin

g.

real

squa

re[x

_,y_

,pha

se_]

:=

Sig

n[Si

n[x

+ p

hase

]];

And

mak

e a

four

-fra

me

mov

ie in

whi

ch t

he g

rati

ng g

ets

prog

ress

ivel

y sh

ifte

d L

EF

T in

ste

ps o

f P

i/2. T

hat

is w

e sh

ift

the

grat

ing

left

in 9

0 de

gree

ste

ps.

For

[i=0

,i<4,

i++,

Den

sity

Plo

t[re

alsq

uare

[x,y

,i P

i/2],

{x,0

,14}

,{y,

0,1}

, Fra

me-

>Fal

se,

Mes

h->F

alse

,Plo

tPoi

nts

-> 6

0, A

xes-

>Non

e, P

lotR

ange

->{-

4,4}

]];

Now

a s

quar

e w

ave

can

be d

ecom

pose

d in

to it

s Fo

urie

r co

mpo

nent

s as

:

rea

lsqu

are(

x) =

(4/p)

*{si

n(x)

+ 1

/3 s

in(3

x) +

1/5

sin

(5x)

+ 1

/7 s

in(7

x) +

...}

19.MotionII.nb

5

Now

sub

trac

t ou

t th

e fu

ndam

enta

l fr

eque

ncy

from

the

squ

are

wav

e

...le

avin

g (4

/p)*

{1/3

sin

(3x)

+ 1

/5 s

in(5

x) +

1/7

sin

(7x)

+ ..

.}

real

mis

sing

fund

amen

tal[

x_,y

_,ph

ase_

] :=

rea

lsqu

are[

x,y,

phas

e] -

(4.

0 / P

i) S

in[x

+ p

hase

];

And

mak

e an

othe

r fo

ur-f

ram

e m

ovie

in w

hich

the

mis

sing

fun

dam

enta

l gr

atin

g ge

ts

prog

ress

ivel

y sh

ifte

d L

EF

T in

ste

ps o

f P

i/2. T

hat

is w

e sh

ift

the

grat

ing

left

in 9

0 de

gree

ste

ps.

It is

wel

l-kn

own

that

a lo

w c

ontr

ast s

quar

e w

ave

with

a m

issi

ng f

unda

men

tal s

imila

r to

sam

e sq

uare

wav

e (t

here

is a

pitc

h an

alog

y in

aud

ition

). O

ne r

easo

n is

that

we

are

mor

e se

nsiti

ve to

sha

rp th

an g

radu

al c

hang

es in

inte

nsity

. If

you

look

at t

he

lum

inan

ce p

rofi

le w

ith th

e m

issi

ng f

unda

men

tal,

you

wou

ld p

roba

bly

gues

s th

at th

e pe

rcei

ved

mot

ion

for

this

seq

uenc

e w

ould

app

ear

to m

ove

to th

e le

ft, a

s be

fore

. But

it d

oesn

't. S

urpr

isin

gly,

the

mis

sing

fun

dam

enta

l wav

e ap

pear

s to

mov

e to

th

e ri

ght!

For

[i=0

,i<4,

i++,

Den

sity

Plo

t[re

alm

issi

ngfu

ndam

enta

l[x,

y,i

Pi/2

],{x

,0,1

4},{

y,0,

1}, F

ram

e->F

alse

,M

esh-

>Fal

se,P

lotP

oint

s ->

60,

Axe

s->N

one,

Plo

tRan

ge->

{-4,

4}]

]

Play

the

abov

e m

ovie

. It t

ypic

ally

app

ears

to b

e m

ovin

g to

the

righ

t. Y

ou c

an g

ener

ate

mov

ies

with

dif

fere

nt c

ontr

asts

by

adju

stin

g th

e Pl

otR

ange

par

amet

ers.

In f

act

the

mis

sing

fun

dam

enta

l fr

eque

ncy

mov

es t

owar

ds t

he le

ft a

s yo

u ca

n s

ee b

y pl

ayin

g th

e m

ovie

bel

ow.

For

[i=0

,i<4,

i++,

Plo

t[Si

n[x

+ i

Pi/2

],{x

,0,1

4},

Plo

tPoi

nts

-> 6

0, A

xes-

>Non

e]];

619.MotionII.nb

Wha

t in

the

sti

mul

us d

oes

mov

e to

the

rig

ht?

Why

mig

ht th

is b

e? P

roba

bly

the

best

exp

lana

tion

com

es f

rom

look

ing

at th

e do

min

ant f

requ

ency

com

pone

nt in

the

patte

rn, w

hich

is th

e 3r

d ha

rmon

ic. I

t tur

ns o

ut th

at th

e th

ird

harm

onic

is ju

mpi

ng in

1/4

cyc

le s

teps

to th

e ri

ght,

even

th

ough

the

patte

rn a

s a

who

le is

jum

ping

in 1

/4 c

ycle

ste

ps (

rela

tive

to th

e m

issi

ng f

unda

men

tal)

to th

e le

ft, a

s sh

own

in th

e fi

gure

bel

ow:

Mak

e a

mo

vie

wit

h P

lot[

] th

at s

ho

ws

the

thir

d h

arm

on

ic.

Wh

ich

way

do

es it

mo

ve?

For

[i=0

,i<4,

i++,

Plo

t[Si

n[3

(x +

i P

i/2)]

,{x,

0,14

},P

lotP

oint

s ->

60]

]

24

68

10

12

14

-1

-0.5

0.51

24

68

10

12

14

-1

-0.5

0.51

19.MotionII.nb

7

24

68

10

12

14

-1

-0.5

0.51

24

68

10

12

14

-1

-0.5

0.51

And

her

e is

the

mov

ie w

ith

jus

t th

e th

ird

har

mon

ic. W

hich

way

doe

s it

mov

e?

For

[i=0

,i<4,

i++,

Den

sity

Plo

t[Si

n[3

(x +

i P

i/2)]

,{x

,0,1

4},{

y,0,

1}, F

ram

e->F

alse

,M

esh-

>Fal

se,P

lotP

oint

s ->

60,

Axe

s->N

one,

Plo

tRan

ge->

{-4,

4}]

]

819.MotionII.nb

The

mai

n co

nclu

sion

dra

wn

from

this

dem

onst

ratio

n is

that

hum

an m

otio

n m

easu

rem

ent m

echa

nism

s ar

e tu

ned

to s

patia

l fr

eque

ncy.

How

can

the

infe

rred

bio

logi

cal m

echa

nism

s be

pie

ced

toge

ther

to c

ompu

te o

ptic

flo

w?

We

can

cons

truc

t th

e fo

llow

ing

roug

h ou

tline

. (Fo

r a

rece

nt a

lgor

ithm

for

opt

ic f

low

bas

ed o

n bi

olog

ical

ly p

laus

ible

spa

tiote

mpo

ral f

ilter

s se

e H

eege

r, 1

987)

. Ass

ume

we

have

, at e

ach

spat

ial l

ocat

ion,

a c

olle

ctio

n of

filt

ers

tune

d to

var

ious

ori

enta

tions

(q)

and

spe

eds

(s)

over

a lo

cal r

egio

n. (

Alr

eady

we

run

into

pro

blem

s w

ith th

is s

impl

e in

terp

reta

tion,

bec

ause

man

y V

1 ce

lls a

re k

now

n to

be

tune

d to

spa

tial a

nd te

mpo

ral

freq

uenc

y in

suc

h a

way

that

the

spat

io-t

empo

ral

filte

r is

the

prod

uct o

f th

e sp

ace

and

time

filte

rs. T

his

mea

ns th

at th

ere

is a

fav

ored

tem

pora

l fre

quen

cy th

at is

the

sam

e ac

ross

spa

tial f

requ

enci

es, s

o th

e fi

lter

will

be

tune

d to

dif

fere

nt s

peed

s de

pend

ing

on th

e sp

atia

l fre

quen

cy).

In th

is s

chem

e, th

e op

tic f

low

mea

sure

men

ts a

re d

istr

ibut

ed a

cros

s th

e un

its, s

o if

we

wan

ted

to r

ead

off

the

velo

city

fro

m

the

patte

rn o

f ac

tivity

, we

wou

ld n

eed

som

e ad

ditio

nal p

roce

ssin

g. F

or e

xam

ple,

the

optic

flo

w c

ompo

nent

s co

uld

be

repr

esen

ted

by th

e "c

ente

rs o

f m

ass"

acr

oss

the

dist

ribu

ted

activ

ity. B

ecau

se th

ese

mea

sure

men

ts a

re lo

cal,

we

still

hav

e th

e ap

ertu

re p

robl

em. W

e w

ill lo

ok a

t pos

sibl

e bi

olog

ical

sol

utio

ns to

this

pro

blem

in th

e ne

xt le

ctur

e.

Mo

tio

n p

laid

s

Tw

o ov

erla

ppin

g (a

dditi

ve tr

ansp

aren

t) s

inus

oids

at d

iffe

rent

ori

enta

tions

and

mov

ing

in d

iffe

rent

dir

ectio

ns a

re, u

nder

ce

rtai

n co

nditi

ons

seen

as

a si

ngle

pat

tern

mov

ing

with

a v

eloc

ity c

onsi

sten

t w

ith a

n in

ters

ectio

n of

con

stra

ints

. Und

er o

ther

co

nditi

ons,

the

two

indi

vidu

al c

ompo

nent

mot

ions

are

see

n.

19.MotionII.nb

910

19.MotionII.nb

Ori

enta

tio

n in

sp

ace-

tim

e

Rep

rese

nta

tio

n o

f m

oti

on

‡Mathematica

dem

o

size

=32;x0

=4;y0=4;pw

=12;xoffset

=1;

A1

=Table@Random@D,8size<,8size<D;H*A2

=A1;*L

A2=Table@Random@D,8size<,8size<D;

A2@@Range@y0,y0+pwD,Range@x0,x0+pwDDD

=A1@@Range@y0,y0+pwD,Range@x0-xoffset,x0+pw-xoffsetDDD;

ListDensityPlot@A1,MeshØFalseD;

ListDensityPlot@A2,MeshØFalseD;

05

1015202530

05

10

15

20

25

30 0

51015202530

05

10

15

20

25

30

19.MotionII.nb

11

nframes = 8;

xt = {};

For[i=0,i<nframes,i++,

A2[[Range[y0,y0+pw],Range[x0,x0+pw]]] =

A1[[Range[y0,y0+pw],Range[x0+i,x0+pw+i]]];

xt = Join[xt,{A2[[8]]}]

];

ListDensityPlot[xt, Mesh->False, Frame->False];

‡x-

y-t

spac

e

1219.MotionII.nb

Neu

rop

hys

iolo

gic

al f

ilter

s

‡Sp

ace-

tim

e fi

lter

s fo

r de

tect

ing

orie

ntat

ion

in s

pace

-tim

e

From

Wan

dell,

"Fo

unda

tions

of

Vis

ion"

19.MotionII.nb

13

‡A

pos

sibl

e m

echa

nsim

for

bui

ldin

g sp

ace-

tim

e fi

lter

s fr

om t

wo

spat

ial f

ilter

s w

ith

a te

mpo

ral d

elay

‡R

elat

ions

hip

of

the

grad

ient

con

stra

int

to o

rien

ted

spac

e-ti

me

filt

ers

vL x

vL y

L tx

y

∂ ∂∂ ∂

∂ ∂+

+=

0

vL x

L tx

∂ ∂∂ ∂

+=

0

Imag

e L

(x,y

,t) -

> b

lurr

ed in

spa

ce a

nd s

mea

red

in ti

me,

g(x

,y,t)

vg x

g tx

∂ ∂∂ ∂

+=

0

1419.MotionII.nb

Bay

esia

n m

od

el f

or

inte

gra

tin

g lo

cal

mo

tio

n m

easu

rem

ents

Glo

bal i

nteg

ratio

n.

19.MotionII.nb

15

Rec

all L

ore

nce

au &

Sh

iffr

ar's

dem

o

Gen

eral

pro

ble

m

Inte

rsec

tio

n o

f co

nst

rain

ts r

evis

ited

Gra

ting

plai

ds s

omet

ime

seen

as

cohe

rent

, oth

er ti

mes

as

two

over

lapp

ing

tran

spar

ent g

ratin

gs m

ovin

g se

para

tely

.

1619.MotionII.nb

Wei

ss &

Ad

elso

n's

Bay

es m

od

el f

or

inte

gra

tio

n

‡P

roba

bilis

tic

inte

rpre

tati

on o

f in

ters

ecti

on o

f co

nstr

aint

s

19.MotionII.nb

17

‡P

roba

bilis

tic

inte

rpre

tati

on w

ith

nois

y m

easu

rem

ents

‡G

ener

aliz

e to

oth

er t

ypes

of

mot

ion

stim

uli

Req

uire

men

ts f

or g

ener

aliz

atoi

n:

Bas

e lik

elih

oods

on

actu

al im

age

data

spat

iote

mpo

ral

mea

sure

men

ts

Incl

ude

“2D

” fe

atur

es

E.g

. cor

ners

Rig

id r

otat

ions

, non

-rig

id d

efor

mat

ions

Stag

e 1:

loca

l lik

elih

oods

Stag

e 2:

Bay

esia

n co

mbi

natio

n

- Pr

ior

slow

ness

--

wag

on w

heel

exa

mpl

e, q

uart

et e

xam

ple

smoo

thne

ss -

e.g

. tra

nsla

ting

rigi

d ci

rcle

1819.MotionII.nb

‡O

verv

iew

of

Wei

ss &

Ade

lson

the

ory

19.MotionII.nb

19

Tes

ts o

f th

eory

‡R

hom

bus

expe

rim

ent

‡A

pert

ure

effe

cts

2019.MotionII.nb

‡P

laid

s

19.MotionII.nb

21

From

Wei

ss a

nd A

dels

on, 1

998.

Typ

e I

and

I pl

aids

. (Y

o an

d W

ilson

, 199

2)

Ap

pen

dic

es

Ref

eren

ces

Bar

low

, H. B

., &

Lev

ick,

R. W

. (19

65).

The

Mec

hani

sm o

f D

irec

tiona

l Sel

ectiv

ity

in th

e R

abbi

t's R

etin

a. J

ourn

al

of P

hysi

olog

y, 1

73, 4

77-5

04.

Bar

low

, H. (

1996

). I

ntra

neur

onal

info

rmat

ion

proc

essi

ng, d

irec

tiona

l se

lect

ivity

and

mem

ory

for

spat

io-t

empo

ral

sequ

ence

s.

Net

wor

k: C

ompu

tatio

n in

Neu

ralS

yste

ms,

7, 2

51-2

59.

Bra

ddic

k, O

. J. (

1974

). A

sho

rt-r

ange

pro

cess

in a

ppar

ent m

otio

n. V

isio

n R

esea

rch,

14,

519

-527

.

Has

sens

tein

, B.,

& R

eich

ardt

, W. (

1956

). F

unct

iona

l St

ruct

ure

of a

Mec

hani

sm o

f Pe

rcep

tion

of O

ptic

al M

ove-

men

t. Pr

ocee

ding

s Fi

rst I

nter

natio

nal

Con

gres

s on

Cyb

erne

tics,

797

-801

.

He,

S.,

& M

asla

nd, R

. H. (

1997

). R

etin

al d

irec

tion

sele

ctiv

ity a

fter

targ

eted

lase

r ab

latio

n of

sta

rbur

st a

mac

rine

cel

ls.

Nat

ure,

389

(664

9), 3

78-8

2.

Hild

reth

, E. C

. (19

84).

The

Mea

sure

men

t of

Vis

ual M

otio

n. C

ambr

idge

, MA

: MIT

Pre

ss.

Van

San

ten,

J. P

. H.,

& S

perl

ing,

G. (

1985

). E

labo

rate

d R

eich

ardt

Det

ecto

rs. J

ourn

al o

f th

e O

ptic

al S

ocie

ty o

f A

mer

ica,

2(

7), 3

00-3

21.

Ade

lson

, E. H

., &

Ber

gen,

J. R

. (19

85).

Spa

tiote

mpo

ral E

nerg

y M

odel

s fo

r th

e Pe

rcep

tion

of M

otio

n. J

ourn

al o

f th

e O

ptic

al

Soci

ety

of A

mer

ica,

2(2

), 2

84-2

99.

Koc

h, C

., T

orre

, V.,

& P

oggi

o, T

. (19

86).

Com

puta

tions

in th

e V

erte

brat

e R

etin

a: M

otio

n D

iscr

imin

atio

n G

ain

Enh

ance

-m

ent a

nd D

iffe

rent

iatio

n. T

rend

s in

Neu

rosc

ienc

e, 9

(5),

204

-211

.

Yui

lle, A

., &

Grz

ywac

z, N

. (19

88).

A c

ompu

tatio

nal t

heor

y fo

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rcep

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of c

oher

ent v

isua

l mot

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Nat

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, 71-

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J.A

. Mov

shon

, E.H

. Ade

lson

, M.S

. Giz

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.T. N

ewso

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he a

naly

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ovin

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sual

pat

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Wei

ss, Y

., &

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, E. H

. (19

98).

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w a

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moo

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otio

n si

gnal

s in

hu

man

vis

ion

(A.I

. Mem

o N

o. 1

624)

. M.I

.T.

© 2

000

Dan

iel

Ker

sten

, C

ompu

tatio

nal

Vis

ion

Lab

, D

epar

tmen

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Psy

chol

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Uni

vers

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f M

inne

sota

. (

http

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.psy

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mn.

edu/

ww

w/k

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en-la

b/ke

rste

n-la

b.ht

ml)

2219.MotionII.nb