s

Qhaan’

Froune 1. A schematic drawing for considering the perception of size and distance.

When n = 1, perfect frontoparallel size con-
stancy is present. When n > 1, over-
frontoparallel size constancy is present, i.e., the
Perceived size of a physically constant $ in-
creases as the distance D, increases,

Situation II. An object of constant visual
angle 6 is presented at various distances D,. It
follows in a development similar to that used
for Eq. (5) that for perfect frontoparallel size
constancy, '

;
= S! (D./D.), (8)
j

while for zero aniount of frontoparallel size
constancy ny

S/= S$, , (7)
or, in g&neral,
8’ = S/ (D,/D.)*. (8)

Eqs. (5) and (8) are special cases of the
more general equation

S/ = K,6,D, (9)
where
. 6, in radians = e- (10)
and K, ig.a constant such that
= __S
= EDs aD? etc. (11)

Eq. (9) specifies the relationship between
the perceived frontoparallel size (S,’) of an ob-
ject and its physical distance (D,) for any
value of visual angle 9, as a function of the
amount of frontoparallel size constancy (n).

-3-


relative distance (an apparent depth between
objects), The successively perceived relative
depths in the present experiment were all ad-
justed to be equal to the perceived absolute
distance to Object p and hence equal to each
other, It follows that the perceived absolute
distance throughout the visual field can be
expressed in units of the perceived absolute dis-
tance to the nearest object (Object p). Rele-
vant to Eq, (20), the logarithms of the result-
ing perceived absolute distances D’ (with the
perceived absolute distance to p set equal to

unity) and the associated physical distances D
in centimeters required to produce the per-
ceived absolute distances are shown in Fig, 4,
The data points of Fig. 4 are reasonably linear.
Using the method of least squares, a straight
line of best fit was determined and n and K,
were computed for each of the curves of Fig, 4,
The results are shown in Table I. In all cases
but one, the amount of depth constancy (as
shown by the value of n) was between zero
constancy (n= 0) and perfect constancy (n= 1),

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Froune 4. The relation between the
to produce that

thm of perceived depth (expressed in units of

S logarit
the nearest Object p) and the logarithm of the physical distance in centimeters required
ived depth,

perceived
ss

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Ficunr 2. A top view diagram illustrating the physical conditions involved in the size and

The Measurement of Perceived
Frontoparallel Size

To measure the perception of frontoparallel
size along the alley, red rectangular objects
(Objects h, i, j, I, and m, of Fig, 2A) were
placed one at a time at erent distances
along the alley and the observer adjusted the
lateral distance between two white rods (Ob-
jects a and b) untfl this distance seemed the
same as the width of the particular rectangle
at the particular distance. The two white rods
(6 mm in diameter and 10 cm high) were ver-
tically presented on the floor gf the alley at a

constant distance of 117.5 cm from the ob-*

server. The right rod was stationary while the
left rod could be moved laterally by the experi-
menter, A small black barrier (2.5 cnr high)
which extended across the alley at the base of
the rods prevented the observer from seeing the
slot on the alley floor in which the right rod
was movable, The red rectangles subtended
a constant visual angle (1° 47’ 24”) and had
the following sizes and positions in the alley:

judgments.

Object h was 7.6 cm wide by 1.7 cm high at
243.8 cm from the observer.

Object i was 11.4 cm wide by 2.5 cm high
at 365.7 cm from the observer.

Object j was 15.2 cm wide by 3.4 cm high
at 487.6 cm from the observer.

Object 1 was 22.8 cm wide by 5.1 cm high
at 731.4 cm from the observer.

Object m was 30.4 em wide by 6.8 cm high
at 975.2 em from the observer,

The Measurement of Perceived Depth

To measure perceived depth in the alley, Ob-
jects a, b, h, i, j, l, and m were removed and
Objects p, gq, and w were used instead
(Fig. 2B). Object p was placed at a constant
distance from the observer (171.5 cm) and the
observer adjusted a depth interval (usually be-
tween Objects g and w) at different distances
from himself to duplicate the perceived dis-
tance from himself to Object p.

—5—


Again, perfect and zero frontoparallel size con-
stancy are expressed by n=1 and n=O, re-
spectively, while values of n between.0 and 1,
for example, indicate that some amount of, but
not perfect, frontoparallel size constancy is
present.

Eq. (9) can also be derived directly from the
Thouless Index.‘ The Thouless Index expressed
in symbols which are consistant with Eq. (9)
is as follows:

_ — logS’~log@
T= Thouless Index Tog § = log 6 (12)

where
S’=S,'/S,, 6 = 6,/0, and S = S,/S,. (13)
From Eq. (12),
(S/8)* = $'/0, (14)

Using Eqs. (10), (11), and (13), Eq. (14)

S/=K,6,D,. (15)
Thus, comparing Eqs. (9) and (15),
T=n. (18)
An Equation of Depth Constancy

Combining Eqs. (1) and (9), the relation
between perceived absolute distance D’ and
physical distance D is

D,=K.D," » (17)
where
_D! _D’
Ke pro prt (18)

+
Eqs. (9) and (17) can be written as
log S,’ = n log D, + log K.6,. ° (19)
.

“RH. Thouless, Brit. J. Psychol. 21, 339 (1931).

and

log D,’ = n log D, + log K,, (20)
oe. Also from Eqs, (2), (11), and
(18),

K, = K,/K.. (21)

__ If the size-distance invariance hypothesis as ex-

pressed by Eq. (1) is correct, n should be the
same in Eqs. (19) and (20), and Eq. (21)
should be valid. In the present study, the hy-
pothesized equality of n between Eqs. (19)
and (20) was tested by experimentally deter-
mining the function relating physical distance
to both perceived size (Eq. 19) and perceived
distance (Eq. 20) for two different conditions
of observation.

APPARATUS
The Visual Alley

Perceived frontoparallel size and perceived
depth were measured in an alley 1020 cm long
and 128 cm wide. The floor of the alley con-
sisted of either (1) a uniform black cloth, or
(2) a checkerboard pattern of light and dark
Gray rectangles (each 41 cm long and 23 cm
wide). The right wall of the alley was covered
with tan cloth. Tan curtains formed the left
wall, and the end of the alley was covered by
Mack velveteen. Schematic drawings of the

y are shown in Fig. 2. A baffle, 28.5 cm
high, covered with tan clgth, extended along
the left side of the alley, 33 cm from the left
wall. Ten white {9 cm) squares (not shown in
Fig. 2) vertically presented in frontoparallel
planes were distributed in depth on the floor
of the alley, with five of these along the baffle
ani five along the right side of the alley. The
center of the alley was clear of white squares
permitting the placement of the lettered ob-
jects shown in Fig. 2, The alley was illumi-
nated by overhead fluorescent lights. The alley
and every object in the alley was viewed by
the observer with his right eye only (the left
eye-piece at the viewing position was oc-
cluded). The eye of the observer at the view-
ing position was 23.8 cm above the floor of the
alley.

—4-

had PRT FLY



C

q
Cc

U
Cc
B
Cc
]

TABLE I

Values of n, K;, and K, as Determined from the Slopes and Intersection
Values" of the Straight Lines of Best Fit for the Data of Fig. 3 and Fig. 4.

Checkerboard Floor

Uniform Black Floor

Judgoents_| “jdgnere | “| felgmens | ignore |S
Obs. n K, n K,| K. n K /n K] K
A 0.50 21.25 | 0.80 2.07} 10.27 | 0.47 25.16| 0.66 4.32] 5,82
B 0.67 «689/091 0.95) 7.25047 33.28 081 1,74} 19.13
c 0.61 12.83] 0.91 1.03] 12.46 | 0.48 29.80] 1.05 0.47] 63.40
D 0.86 1.96}0.93 0.91} 2.15/0.72 4.62] 0.90 1.07] 4.32

* For convenience, values of K, in this table have been arbitrarily multiplied by 100.

40G PERCEIVED SIZE

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LOG PHYSICAL DISTANCE

Ficunr 3, The relation between the logarithm of the perceived size in centimeters of the
rectangles and the logarithm of their physical distance in erntinerne

-7-


63-20

as

AD 482 587 7

THE VISUAL PERCEPTION
OF
> SPATIAL EXTENT

FEDERAL AVIATION AGENCY
CIVIL AEROMEDICAL RESEARCH INSTITUTE
‘ AERONAUTICAL CENTER
OKLAHOMA CITY, OKLAHOMA

SEPTEMBER 1963

‘fmePRODUCED BY

NATIONAL TECHNICAL
INFORMATION’ SERVICE

US. DEPARTMENT OF COMMERCE ~
‘SPRINGHELD, VA." 22161

a


distance. These studies together with previous
evidence indicate that the size-distance invari-
ance hypothesis in its usual form is not always
valid.

Fig. 5 is useful in analyzing errors which can
occur in the perception of distance. The first
point on all the graphs of Fig. 5 has an ordinate
value of 171.5 cm which is the physical dis-
tance of Object p from the observer. The ob-
server's perception of the distance of p is a
perception of absolute distance. All other
points on the graphs represent the results from
judgments of relative distances, i.e. results
from perceptions of the depth between two ob-
jects. The ordinate value of the second point
for three out of four observers is less than

171.5 cm. This means that for these observers
the perceived depth per unit of physical depth
was usually greater in the first relative depth
judgment than in the judgment of the absolute
distance of p. For the following two relative
depth judgments, however, the perceived depth
per unit of physical depth usually decreased
with the final relative judgment showing a de-
crease in this ratio for the uniform black floor
but not for the checkerboard floor, From these
data, it is possible to infer the results which
would have occurred if the observer had been

*See (a) F. P. Kilpatrick and W. H. Ittelson, Psychol.
Rev. 60, 223 (1955); (b) W. Epstein, J. Park, and A.
Casey, Psychol. Bull. 58, 491 (1961); and (c) W. C.
Gogel, Vision Research 3, 106 (1963).

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PHYSICAL DISTANCE OF INTERVAL (CM)
Lo

Ficure 5. The relation between the Physical depth interval required to produce re
tually equal depth intervals and the Physical distance of the hegianing of the interval | from
the observer,

-10-—

*


The average results for each observer from
the depth adjustments, using the checkerboard
pattern and the uniform black floor, is given
in Fig. 5. The average adjusted depth interval
in centimeters which appeared to the observer
to be equal to the distance from himself to
Object p is given on the ordinate with the
physical distance in centimeters from the ob-
server to the least distant part of this interval
shown on the abscissa.

DISCUSSION

From the essentially linear graphs of Fig, 3,
it follows that a single value of n for a par-
ticular observer and a particular floor type
(checkerboard or uniform black floor) appro-
priately describes the frontoparallel adjust-
ments. A similar conclusion applies to the
depth adjustments shown in Fig. 4, But, in
every instance (see Table I) the value of n
from the same observer and same floor type is
greater with the depth than with the fronto-
parallel size adjustments. According to the
size-distance invariance hypothesis, however,
n should be the same for the depth adjust-
ments as for the frontoparallel size adjustments.
The differences in n for the two types of judg-
ments in Table I therefore provides evidence
against the validity of the size-distance invar-
iance hypothesis. ”

Using Eq. (21), K, was determined from K,
and K, with the results shown in Table I. From
Eqs. (2), (11), (18),gand (21), differences
between the K, values calculated from the re-
sults from the checkerboard floor and from the
uniform black floor would be expectéd to occur
as a consequence of differences inn. Further-
more, since n differed for the two floor types,
K, in Table I would be expected to vary as a
function of the unit used in the specification
of D. The differences in the calculated valties
of K, in Table I reflect both the differences in
n and the%ize of the unit used in the specifica-
tion of D. Differences in K, in this experiment
therefore do not constitute a direct or inde-
pendent evaluation of the validity of the size-
distance invariance hypothesis.

|

Eq. (9) is convenient for describing size con-
stancy data, If, as this study indicates, n is
generally a constant for the entire visual field
(or for a large portion of the visual field) for a
particular observer, the usefulness of Eq. (9)
will be increased. Eq. (17) provides an equally
convenient description of depth constancy data.
When n = 1 (perfect depth constancy) per-
ceived and physical depth are proportional.
When n = 0 (zero depth constancy) perceived
depth does not increase even though physical
depth increases. From Fig. 1, it is clear that

D/=d7 +DY (22)

and therefore, Eq. (17) can be expressed in
terms of d’ instead of D’ as follows:

d.’ = K,(D, — D,"), ete, (23)

In some studies, it might be more convenient to
use Eq. (23) rather than Eq. (17) in sum-
marizing data from depth judgments.

In a recent study testing the size-distance in-
variance hypothesis, Ueno’ has developed equa-
tions similar to Eqs. (9) and (17) of the pres-
ent study from Stevens’ power law’ -which
states that psychological magnitude is a power
function of stimulus magnitude. Using the
psychological method of transposition of
Oyama’, Ueno has compared the value of the
exponerft in the power function involving per-
ceived size with that involving perceived dis-
tance. Exponents computed from the two types
of judgments were not always similar with
large differences sometimes occurring for mo-
nocular and reduced conditions of observation.
Both the study by Ueno and the present study
indicate, contrary to the size-distance invari-
ance hypothesis, that the power function re-
lating perceived size and physical distance does
not always involve the same value of exponent
as that relating perceived distance and physical

*T. Ueno, Japan. Psychol. Research 4, 99 (1962).
*5. S. Stevens, Am. Psychologist 17, 20 (1962).
* T. Oyama, Psychol. Bull. 56, 74 (1959).

-9-


Objects p and q were red rectangles 15 cm
high by 5 cm wide which were vertically po-
sitioned on the floor of the alley. Object w
was a red 7.6 cm square placed in a fronto-
parallel position with the bottom of the square
always resting on the floor of the alley, Ob-
ject w was attached by a glass rod to a cart
(invisible to the observer) which moved on a
track behind the baffle. By tuming a knob at
the viewing position, the observer could move
the red square directly toward or away from
himself in depth.

PROCEDURE

The Frontoparallel Size Adjustments

The observer signaled to the experimenter to
move Object a (Fig. 2A) laterally to the right
or left until the distance between the inner
edges of the rods visually appeared equal to
the width of Rectangles h, i, 4,4 orm. The
adjustment was repeated four times after which
a different rectangle at a different distance was
presented. The distances in’ centimeters ad-
justed between the rods measured the per-
ceived width S’ of the particular rectangle,
The order of presenting the rectangles was
randomly determined for each observer.

The Depth Adjustments

The observer moved Object w (the red
square) back and forth in depth until ‘the dis-
tance of w behind p visually seemed equal to
the distance of p from himself, This was done
three times. The average of the three adjusted
distances was calculated by’ the experimenter
and q was placed at this distance, 'The ob-
server then moved w behind gq until the vis-
ually perceived depth between w and q (d.’)
seemed to duplicate the visually perceived dis-
tance of p from himself. Again, this adjust-
ment was wade three times. The new average
Position of w was found, @was moved to this
new distance and the process was repeated,
This process resulted in a series of depth in-
tervals (d.) each of which appeared to the
observer to be equal to the distance of p from
himself and thus equal to each other.

Observers and Orders

Four men were used in the experiment as ob-
servers. Previously these observers had experi-
ence in making size and distance judgments in
experimental situations, Each observer had a
visual acuity in his right eye (corrected if nec-
essary) of at least 20/22.

The depth ‘and frontoparallel size adjust-
ments always were completed first with the
checkerboard pattern and following this with
the uniform black covering on the floor of the
alley.

RESULTS
The Frontoparallel Size Adjustments

The data from the experiment relevant to
Eq. (19) are shown in Fig. 3. Since the red
rectangles subtended a constant visual angle,
9 in Eqs. (9) and (19) is constant. The or-
dinate of Fig. 3 gives the logarithms of the
average results in centimeters from each ob-
server for the frontoparallel size adjustments
for both the checkerboard pattem and the uni-
form black floor. The abscissa is the logarithm
of the physical distance in centimeters of the
frontoparallel targets from the observer, The
data points of Fig. 3 are reasonably linear.
Using the method of least squares, a straight
line of best fit was computed for each of the

jdistributions of Fig, 3. From Eq. (19), the
slope of the straight line of best fit is n with
log K,9 the ordinate intércept. The resulting
values of n and of K, (with K, determined for
the constant value of 8 expressed in radians)
are given in Table I. As might be expected,
d was greater frontopaPaflel size constancy
whee the checkerboard pattern rather than the
black cloth floor was used. In all cases the
amount of frontoparallel size constancy was
between zero constancy (n=0) and perfect
constancy (n= 1),

The Depth Adjustments

In the previous: equations, D’ is a perceived
absolute distance (an apparent distance from
the observer to the object), This is to be
distinguished from d’ which is a perception of

-6-

ss



The perceived distances (perceived absolute dis-
tances) which the observer associates with D.,
D,, and D, will be called D,’, D/, and Dy, re-
spectively, The problem of how the perceived
frontoparallel size S’ varies as a function of D
is the problem which has usually been investi-
gated under the topic of “size constancy.” The
problem of how the perceived absolute distance
D’ or the perceived relative depth d’ varies as
a function of D is the problem which has usu-
ally been investigated under the topic of “depth
constancy.”*

The Size-Distance Invariance
Hypothesis

The specification of the relation between per-
ceived frontoparallel size and perceived depth
by means of the size-distance invariance hy-
pothesis is given by

S/=K,0,D, (1)

where K, is a constant such that

n=

6D. ~ apy

(2)

In Eq. (1), S/’ is the perceived size of any ob-
ject whose visual (retinal) size is 9, and which
is located at any physical distance D, | the
observer. The term D,’ is the perceived abso-
lute distance associated by the observer, with
D,. In Eq. (2), S,’ is the perceived size of a
particular object (Object e) whose visual (ret-

inal) size is 8, and which is located at the par- ‘

ticular physical distance D, from the observer
with D,’ being the perceived absolute distance
associated with D,. In the equations used in
this study, tl subscript “v” will refer to a var-
iable quantity while the subperipts e, f, and g
(see Fig. 1) will refer to specific values, As
is illustrated in Eq. (2), whenever a particular
subscript such as e is used, any other particular
subscript such as f could have been substituted
for it throughout the expression.

An Equation of Size Constancy

An equation permitting the quantitative ex-
pression of perceived frontoparallel size S,’ as
a function of physical distance D, can be de-
veloped from the two following hypothetical
situations.

Situation I. An object of constant fronto-
parallel size S is presented at various distances
D,. Suppose that the perceived frontoparallel
size of this object S,’ remains constant. This is
an example of perfect frontoparallel size con-
stancy and can be expressed as

S’= Sy. (3)
Eq, (3) states that for a constant value of S,
the perceived size S,’ at any distance D, is
equal to its perceived size S,’ at the particular
distance D,. Suppose, however, that the per-
ceived size of S decreases proportionately to
the distance D,. This is an example of zero
amount of frontoparallel size constancy and is
expressed as

S/ = S.(D./D,). (4)
Eq. (4) states that the perceived size S,’ of S
at any distance D, is equal to its perceived size
SJ at the particular distance D, multiplied by
the ratio of the two distances.

Eqs. (3) and (4) are special cases of the
more general equation’

‘

S/=S/ (D,/D.)"*. (5)

ty
. é . (5), when n=1, Eq. ,(3) results and
when n=0, Eq. (4) results, Thus, n is an
expression of the amount of frontoparallel size
constancy present. When n=0, zero amount
of frontoparallel size constancy is present.

*For a different statement of the problem of con-
stancy see R. Over, Am. J. Psychol. 74, 308 (1961).

*A relation similar to Eq. (5) has been developed for use
in an equation concerned with the perception of three-
dimenonal shape” (W. ©. Cogs, FP Peychol. 50, 179

1960)1.

~-2—

. *



THE VISUAL PERCEPTION OF SPATIAL EXTENT®

WALTER C. GOGEL

“Sn =
study was concemed with the manner in which and
frontoparallel si variod with physical distance and hence Giir'ses ake An

ing the relation between perceived
Sod to frontoparallel

size

under two viewing conditions. U: the saine “aquation and. an’ exprocine
an equation was developed which related per-

ceived. and physical depth. The additional equation, when
depth from same observers under the same viewing conditions, produced results
in ‘agreement with those expected from the size-distance invariance hypothesis.
evidence against the validity of the size-distance invariance by
its usual form. The results from the apparent

problem of the nonveridicality of the perceptual

distance invariance hypothesis, i

interpreted as

INTRODUCTION

Many studies have been concerned with the
problem of the relation between physical and
perceived space. This problem can be con-
veniently but arbitrarily divided into (1) the
perception of vertical gr horizontal extents (the
perception of frontopgrallel size) and (2) the
perception of depth extents. Frontoparallel ands
depth extents have sometimes been considered
to be perceptually interrelated with an expres-
sion of the interrelation given by the size-
distance invariance hypothesis.*;/ The purpose
of the present study is to investigate the man-
ner in which both perceived frontoparallel size

. and perceived depth can vary with observation

distance and hence with each other. ,

Tiedion we ”

el

ganged

|
‘
af]

the
applied to judgments of percei

g
eee

&

The problem of this study can be illustrated
with the aid of Fig. 1. In Fig, 1, the fronto-
parallel extents S,, S,, and S,, are placed at
distances D., D,, and D,, respectively, from the
observer, The perceived frontoparallel extents
(perceived sizes) associated with the widths
S., Si, and S, will be called S,’, S’, and S,’, re-
spectively, with the respective visual angles la-
beled 6,, 6,, and 8. The perceived depths (per-
ceived relative depths) associated by the
observer, with the relative depth intervals dy
and d, be called d.’ and d,,’, respectively.

* The data of this study were collected at the U. S. Army
Medical Research Laboratory, Fort Knox, Kentucky.

‘K.P, Kilpatcick and W. H. Ittelson, Psychol. Rev. 60,
223 (1953).

» -1l-


#

b

fo
XN

4

ol j THE VISUAL PERCEPTION OF SPATIAL EXTENT >

ag
Poe,
f fe ; WALTER C. GOGEL, Pee:
~ Chief, Spatial Perception Section
Psychology Branch
_ Tm,
init l¢ if : j 63}20
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/ '

, Vv
FEDERAL AVIATION AGENCY

CIVIL AEROMEDICAL RESEARCH INSTITUTE /
AERONAUTICAL CENTER

W aigteon CITY, OKLAHOMA
EPTEMBERAOCS

1


given the task of adjusting a movable object to
perceptually bisect the distance from himself to
a stationary object. This task would produce
two perceptually equal intervals with the
nearer interval being a judgment of absolute
distance (a judgment of the distance from the
observer to the movable object) and the far-
ther depth interval being a judgment of relative
distance (a judgment of the depth between the
movable and the stationary object), From the
present study, for near portions of the alley,
the observer would usually underestimate the
nearer (absolute) distance with respect to the
farther (relative) distance. However, if a depth
between two visible stationary objects (a rela-
tive depth) were perceptually bisected by a
movable third object, the reverse sometimes
would occur. The physical depth between the
nearest object and the movable object some-
times would be overestimated relative to the
physical depth between the movable object and
the farthest object. These inferences-from the
data of the present study might help explain
some of the discrepancies in the results some-
times encountered in studies involving the per-
ceptual bisection of a depth interval.” “

* Reference 8b, pp. 495-496.

an

The reason for the increase in perceived
depth per unit of physical depth from the
checkerboard but not from the uniform black
floor at the far distances is not clear, Perhaps
some complex effects are to be expected from
judgments involving depth near the terminating
wall of the alley.

In many respects it is unfortunate that the
size-distance invariance hypothesis of Eq. (1)
probably is not valid, If Eq. (1) had been
valid, for a constant value of n, Eqs. (1), (9),
and (17) would have together resulted in a
simple predictive system of considerable par-
simony, A few measures of perceived extent
for a particular observer would have permitted
the prediction of perceived extent at any orien-
tation (frontoparallel, depth, or a combination
of both) throughout the visual field. A size-
distance invariance hypothesis can be deter-
mined from Eqs. (9) and (17) without as-
suming that n is the same in the two equations.
However, the resulting equation would be more
complicated and less parsimonious than Eq. (1).

ACKNOWLEDGMENTS

The author wishes to thank James M, H.
Gregg for his assistance in collecting the data
and Frank L, Agee, Jr., and James C. Hester
for their assistance in its analysis.

-ue-

#U. 8 GOVERNMENT PRINTING OFFICE : 1964 0 - 720-901