The Gaussian Derivative Theory of Spatial Vision

Young, R. A. (1982a) The Gaussian Derivative Theory of Spatial Vision. Unpublished manuscript. 161 pages.

theory

.....,is i o n

Department of

and Neuroscience Institute

Psycholo~y

University of Oregon, Eugene, Oregon 97403 ABSTRACT

This stAgas

paper

proposes a new theory of the

spatial

of'

visual system. in

fields

through

the

processing

receptive

primary visual system

least

at

cortex can be characterized as

of a. Caussian function. and

the

in

The theory maintains that

striate

derivatives

information

early

confirmed for

the

The theory

various

up

is

tested

on

cat

and

monkey

receptive fields found in the literature

and

a sample of monkey cortical cells.

found

I t is

that the shapes of receptiv.e fields along one spatial dimension can be completely characterized by just one number

-- a

positive

integer

Gaussian derivative term. derivatives previous

shown

a

a

significant

Fourier-like

just as Fourier to

ba

improvement

analysis of

~nalysis

had been

closer to the visual l

the

I t is shown that Caussian

characterizations of the visual

performing scene,

offer

representing

over

system the

as

visual

previously

mechanisms

for

spa.t ia.l

vision

derivatives

than

Gaussian

are a.lso shown to be superior to several

other functions tested, which

detectors.

bar

are

a

except for Gabor

close

approximation

functions,

to

Ca.ussian

derivatives. Extensions

of Gaussian derivative theory led to

a number of predictions, all of which were confirmed, about the associations between a cell's peak frequency, width,

bandwidth,

and

providing

underlying

Gaussian

such

a

derivative

close fit

to

spatial

receptive number.

field Besides

neurophysiological

data, Gaussian derivatives were also shown to predict human

psychophysical

results

better

than

other

proposed functions. In

short,

the

derivative

provides

a

receptive

fields and human psychophysical.

He

close

Gaussian

approximation

to

theory

single

~nit

channels.

believe the Gaussian derivative theory represents

another

small

step

in

the

mathematical

characterization of the visual system. Key

words:

Gaussian

derivative

theory,

spatial

vision, receptive fields, Fourier, bandwidth, cortex, psychophysics, spatial frequency.

2

~--·····-··--···~----------------------

• Gaussian Derivative Theory

Introduction

J:NTRODUCTJ:ON

.L.. .I!Ul fou.rj1r Tb•orx .2f. Yl1t9n Fouri•r an

area

analysis of the visual system has of

scientists,

owing

application visual

chtafly

to

to

the

visual

successful

fi•lds

and

human

of

psychophysical

Indeed, one fundamental sign of our period

b•an

the

oscillating Ke

interest

of this technique to the description

rec•ptiv•

channels. has

considerable

been

high

amplitude

wave

of

around the meaning of such applications.

see the phasing in of terms such as

"spatial

research

frequency

sensitivity"

tuning",

"ba.ndwtd.th",

and

to describe the spatial

"contrast

properties

of.

the visual system, as had. been done with the temporal properties

since Iv1s C1922>.

An impr•sstve body of

literature

jn support of the utility and

predictive

power of the Fourier analysis of tha spatial of vision has arisen .

The system Many

question of Fourier analysis

ll.x the

has bean a co-sign of this wave of visual scientists,

Jfestheimer

,

hav• 3

visual

research.

despite aarly_cautioning by adopted what we

call

th•

-

·----------

~-

--~--

I ntr od.l.lc ti on

Gaussian Derivative Theory

"strong form" of the Fourier theory of vision,

which

maintains that the visual system itself carries out a Fourier transform of visual space .

Campbell stated. that the

neurophysiological findings in cat and monkey suggest that

the

"spatial frequency content

is ..• extracted.

stated:

masking visual the is

11

11

of

the

image

In the psychophysical a.rena, .. These

summation,

experiments .•. provide

Gr•ham

adaptation,

evidence

that

system does do a crude Fourier analysis." weak form" of the Fourier hypothesis,

made that the visual system itself

frequencies,

but

only

uses

and the In

no claim. spatial

that the visual system to

a

first approximation can be treated. as a linear system and Fourier analysis provides a valuable tool for its study. The hypothesis that there are "multiple frequency and

Robson,

form 11 been

channels" 1969>

in tha visual system is associated with

of Fourier theory. precisely

spatial

specified

frequency

Although it has what the

channels" 4

the

means,

term as

spatial

otherwise

Fou.rier spectra of Y,,ussi&n deriy&\iv11 What Fourier the

most

do

Gaussian

space?

derivatives

look

like

This question is important

reliable data so far collected

on

in

because visual

cells has been in the Fourier domain, due to the ease of

producing

multiple

24

repetitions

with

drifting

Introdu.c ti on

Gaussian Derivative Theory

gratings.

If visual eel l receptive fields really a.re

Gaussian

derivatives,

then

we

shou.ld be

able

to

recognize their shapes in the cell data collected

so

far. For Fou.r i er

mathematical convenience in solving for the transform,

definition

of

G

it

is

useful

to

·slightly by adding a

the

alter in

tr

the

0

denominator, as in Eq. . G (x)

o

The

= -1 - e-x

/2cr

2

(13)

er l2iT

derivatives 0¥ Eq.

are given by Eq.

2

C13> with respect to

( 1'4 >.

'') - - = - - 1 H {x) G \.)(, n n cr o • n cr WC

The Eq.

Fourier· transform of Eq.

( 15). \

is given

by

Details are given in Campbell and Foster 2

p. 84, pairs 707.0 to 708.4 with

imaginary value "i" in Eq.

8=2•~

>. The

is dropped to obtain

the absolute magnitude of the Fourier spectrum, which is what we are most concerned with here.

25

Introduction

Gaussian Derivative Theory

(15)

A six

plot

of the Fourier transforms of the

Gaussian

derivatives as given by

Eq.

first



is

presented in Fig. 2. insert Fig. 2 about hare These height

transforms have all baen sat to the

and

centered

at

f

which

is

the

same peak

0

frequency.

It

derivatives

can

be seen

that

the

get progressively narrower

higher-order

and

in

the

bandwidths

Gaussian

derivative domain are also predicted to

completely

independent of the size of the th the n Cau.ssian derivative has

field

bandwidth regardless of'

fl".

be

receptive the

same

This independence from absolute size might

be

u.sefu.l for a higher level 33

analysis

which

Introduction

Gaussian Derivative Thaory

must somehow deal with changing receptive field sizes as a function of retinal eccentricity. level

logarithmic

conjunction

mechanism

with

a

Such a micro-

might

operate

macro-level

logarithmic

retinotopic mapping scheme as postulated by

in

to explain overall object

Schwartz constancy

with changing image sizes. In sum, wa have given a theoretical development of Gaussian derivatives which suggests that they have a

number of interesting properties which would

them

useful

machanisms

of

equations visual

if

they

derivative

Empirical and

spatial

vision.

Also,

normal various

wera developed which should hold true

cells

Gaussian

tools for helping to understand

make

themselvas

are

performing

analysis of the retinal

tests of these aquations were

tor

image.

then

·made,

will be presented in the Results section of this

paper.

ll.L.. O\htr Fynctlon1 Various derivatives

functions were

functions for vision.

other

examined

as

than candidate

Legendre,

basis

Functions rejected for various

reasons included the fallowing types: Laguerre,

Gaussian

Chebyshev, 34

Bessel,

Airy,

and sine functions.

Introdu.ct ion

Ga.u.ssia.n Derivative Theory

Yet, to

four functions other than sine waves ware found have

sufficiently close affinities to

fields to warrant further examination. we

could

safely

functions

reject

the

by inspection,

receptive

We found that

first

two

of

and the la.st two

these

required

empirical testing, &s did sine waves.

A... Pv&bol ic cxl ind1r .m:.

I in ear harmonic

osc i 1 lator

fu.nct ions The closely ha.ve

cylinder

a number of interesting features.

shape

which

linear

harmonic

oscillator

.to Harmuth .. Pulses of use the

'best',"

small pa.rt

percentage of their energy." shown

functions,

allied to the Gau.ssi&n derivative functions,

physics, as

parabolic

of

time-

and

of

the

the

they time-

cartai n

a.

Their orthogonality was 4 graphs the first

seven

para.bolic cylinder functions. insert Fig. 4 a.bout here Parabolic

cylinder 35

functions

also

have

the

Gau.ssian Derivative Theory

In tr odu.c ti on

int•restinq prop•rty that th•Y ar• th•ir own transforms

.

Fou.rier

Therefore,

one

0

wou.ld the if

expect inhibitory and excitatory sidebands Fou.rier domain as w•ll as in th• spatial

domain

receptive fields were in fact parabolic

fu.nctions.

Ind•ed,

have

reported

bean

also report two cortical

calls with secondary maxima in the frequency domain. As

mentioned,

idantic&l linear 1978>

239>. the thesa

to

h&rmon i c which

quantu.m

parabolic cylinder fu.nctions are

osc i 11 a tor

simpl~

e iqenfu.nc ti ans

so that no

frequencies

in

maintained video

the

neural

activity

raster

or

could

spike

regular train

due

responses to th• contaminate

to hertz

60

the

high

data.

by

masquerading a.s low frequencies, a.n artifact known a.s aliasing. To rad.u.ce the neura.l data. from the drH'ting gratings into a. meaningful form, th•

post-stimulus

programs

we

time

Fourier analysis of

histograms was

done

wrote to extra.ct th• amplitude

using of

the

fund.a.mental and the mean response. These measurements ware usually ma.de a.ta. number of spatial and

at a number al contrasts.

frequencies

To obtain sensitivity

functions, th• contra.st required to achieve a. certain criterion

response

amplitude 43

from

the

cell

was

Gaussian Derivative Theory

Methods

estimated

•Y••

Tha

;ave

the

by

reciprocal

;raphical curv•-f itting by of

this

contrast

level

sensitivity at that fr•quency.

lL. Hatbtat.ic&I

H1thpd.I

Unfortunat•ly, working

with

th•

math•m&tical

methods

for

Gaussian derivatives hav• not been

as

wall developed as with Fourier analysis. For example, no

computer

program

was

found

which

would

automatically estimate th• co•fficiants of a Gaussian der i vat i va sari es when step-by-step

r

procedur•

was unknown. was

Therefore

d•vised

which

a

worked

sat i sf actor i ly.

ill!. J.n .1bl. I pt. t. l I I Me

d OJI& i D

first wished to fit Gaussian derivatives

rec•ptiv• f ialds measured in the spatial order

to

test

the

validity

of

our

to

domain,

in

methods~

We

therefore us•d data in the literature which are known to

have been measured with a large number of

so

as to ensure reliability.

points

Me obtain•d the

by measuring the published curves

trials data

.

in

The

c e 11

number is given in the column marked "CELL". insert Tabla III about here As

can be saan from Fig.

9 and

Tabla

single Gaussian derivative term produced a close fit to in

onl~

ever~

III~

a

reasonabl~

receptive field measured.

Indeed,

9 cases out of 55 was there a fit in

which

less than 75% of the varianca was explained. The mean "variance cells

explained"

for the entire

sample

of



55 was

85.5% + 1.8% standard error.

At

the time the

anal~ses

57

of Fig.

9 were

done,

Gaussian Derivative Theory

the

G

term

Results

was not tasted for a fit to

the

data

0

because

of the common assumption in

the

literature

that cortical cells always have low spatial frequency fall-off , which G

does

0

not.

In hindsight, it app•ars plausible that sevara.l

of the cells in the G group in Fig. 9 and ·

in the Introduction enable a chart to be drawn 10>

a

which

illustrates

the

translation

between F and BW

negative

partialled

of

( 1962).

bacama

when the variable hald

,

partial correlation given

Technically

statistical it

= -.59>

Cr

out

technique

Results

speaking.,

an

was

via

the

in

McHemar as

a

which means that

stronger

even

tr

is acting

fT

"suppressor" variable,

counteracts

even

dee line

in

that is max Calls with relatively high f hidden in th• data. max should have relatively low tr by common assumption, logarithmic

but

bandwidth

with increasing f

low o 's result empirically in

logarithmic

as

well

relatively

as arithmetic

high

bandwidths

for

Thus the influence of a declining tr as

one

I

calls. to

goas

hidden peak

higher frequencies serves

t~

counteract

empirical very sharp decline in bandwidths as fraquancy

influence

of

constant>.,

a

then

increases. changing

If r

one

removes

is that it is

still unknown how the important cell

classifications

of X-like and Y-like , simple and complex, and

oriented

and non-oriented,

non-color opponent, derivatives.

nomenclatures and

ate.

are

These

associated

classical

have had tremendous

~enetits

characteristics

over

the

color opponent

cellular

predictive

years

in

with

power

explaining

the

that visual neurophysiologists

have

most easily noticed when they initially looked at the properties derivative

of

visual

theory

calls.

has not yet

Although been

Gaussian

considered

in

relation to these classifications, or for that matter to the detailed anatomy of Area 17 know

, there would be uncorrectable chromatic aberration affects that would cause two images -- one rad

and

one

blue

communication>.

,

Center and

Monasterio

mechanisms

F.

M.

of

macaque

1394-1417. surround

of opponent-color X and Y ganglion

cells

of retina of macaques. ;[,,:,. Meurophvs. 41, 1418-1434. De Monasterio properties

F. M. and Gouras

of

can

K.

Functional

ganglion cells of the

retina. ;[,,:,. Phvsiol., Land. De Valois

P.

K.

~,

rhesus

monkey

167-195.

Spatial frequency adaptation

enhance contrast sensitivity.

Vision

Res. ,

.!.Z,

1057-1065. De Valois (1979>

K.

K.,

De Valois

R. L.

and Yund

Responses of striate cortex cells to

and checkerboard patterns.

~

fh~siol

•.

E. W. grating

Land. 291,

483-505. De Valois (1978>

R. L., Albrecht

Cortical

cells:

113

D. C. Bar

and Thorell

detectors

or

L. C. spatial

Gua.ssia.n Derivative Theory

frequency

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illusory

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Guassian Derivative Theory

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... Guassian Derivative Theory

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122

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to

.!:L._

f'raquency

Physiol. 226,

... Gua.ssia.n Derivative Theory

Tyl er

w.

C.

.

independent

observed

agreement

frequency

and

trequ.ency



is

Gaussian

der iva.tive

generally

orderly progression ot symbols from

indicating

lower

number

observed

the

A scan column

peak

between

predicted

good:

The

the

observed down

any

indicates

frequencies

to

a.

those those

indicating higher observed trequ.encies .

147

Gaussian Derivative Thaory ,

Fig.

12.

Fourier

Figura Lag ends

parameters of a large sample

monkey cortical cells,

of

based on data from De Valois,

Albracht, and Thorell , plotted on a chart for

going from the Fourier domain into the

derivative domain, as in Fig. 10. the

solid

circles

Gaussian

The digits next to

represent the

number

of

cells

observed with the peak spatial frequencies and median bandwidths

indicated

respectively. over

cells

the

fall

at

each

deviation>.



bandwidth with increasing which De Valois

.!! A!·

as statistically significant.

open

Note

peak

the

spatial



cu

>

:;:: 0 4J

a::

1.00

o.eo

Ct

0

..J

0.0 -0.50..___________-.....-.---.---..-----.--.--....---..---. -0.14 -0.10 -0.06 -0.02 0.02 0.06 0.10 0.14

20

.,

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10

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a:: 0

•

0

.,"":s

z

-10

20 -1.4

-1.0

-0.6

-0.2

0.2

0.6

Spatial Position (Degrees)

7

1.0

1.4

Fourier Domain

A

Gaussian Domain

A*

BW 1.4

10

10

5

5

3 2

3 2

1

1

r· .3

.5

1

2

5

0 1

10

s* 10

5

5

3 2

3 2

1

1

BW 0.9

B

2

5

10

1

2

5

10

0 1

2

5

10

r-

.3

.5

c

1

2

o

5

c*

BW 0.8

10

10

5 3 2

5 3 2

1

.3

.5

1

2

5

Frequency (cpd)

Derivative no. ( Gn)

8

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~

.,

54

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30

•

42

•

45

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