Visual Perception of Spatial Extent

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Visual Perception of Spatial Extent*

WALTER C. GOGEL

Civil Aeroinedical Research Institute, Oklahoma City, Oklahozma 73101(Received 4 October 1963)

This study was concerned with the manner in which perceived depth and perceived frontoparallel sizevaried with physical distance and hence with each other. An equation expressing the relation between per-ceived size and physical depth was developed and applied to size judgments determined with four observersunder two viewing conditions. By use of that equation and an expression of the size-distance invariancehypothesis, an additional equation was developed which related perceived and physical depth. The addi-tional equation, when applied to judgments of perceived depth from the same observers under the sameviewing conditions, produced results not in agreement with those expected from the size-distance invariancehypothesis. This is interpreted as evidence against the validity of the size-distance invariance hypothesisin its usual form. The data from the apparent depth judgments also were applied to the problem of the dis-crepancies in results that have been found in experiments on the perceptual bisection of depth intervals.

INTRODUCTION

MANY studies have been concerned with the prob-Mlem of the relation between physical and per-ceived space. This problem can be conveniently butarbitrarily divided into (1) the perception of verticalor horizontal extents, i.e., the perception of extent(size) in a frontoparallel plane and (2) the perception ofdepth. Frontoparallel and depth extents have sometimesbeen considered to be perceptually interrelated withan expression of the interrelation given by the size-distance invariance hypothesis.' The purpose of thepresent study is to investigate the manner in whichboth perceived size and perceived depth can vary withobservation distance and hence with each other.

The problem of this study can be illustrated with theaid of Fig. 1. In Fig. 1 three objects (e, f, and g) areplaced normal to the line of sight of the observer, i.e.,in frontoparallel planes, at distances De, Df, and Da, re-spectively, from the observer. The perceived fronto-parallel extents (perceived sizes) associated with thewidths Se, Sf, and S, are called S,', Sf/, and S,', re-spectively, with the respective visual angles labeledOG, Of, and a. The perceived depths (perceived relativedepths) associated by the observer with the relativedepth intervals d0f and df,, are called d/f' and df,0',respectively. The perceived distances (perceived abso-lute distances) which the observer associates with De,Df, and D, are called D,', D/', and D,', respectively.The problem of how the perceived frontoparallel sizeS' varies as a function of D is the problem which hasusually been investigated under the topic of "size con-stancy." The problem of how the perceived absolutedistance D' or the perceived relative depth d' varies asa function of D is the problem which has usually beeninvestigated under the topic of "depth constancy."'

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

1 F. P. Kilpatrick and W. H. Ittelson, Psychol. Rev. 60, 223(1953).

2 For a different statement of the problem of depth constancysee R. Over, Am. J. Psychol. 74, 308 (1961).

Size-Distance Invariance Hypothesis

The specification of the relation between perceivedfrontoparallel size and perceived depth by means of thesize-distance invariance hypothesis is given by

Swhr K= KisDs',

where K, is a constant such that

K1= S0.,'1/D 0

1 = S//GfD/, etc.

(1)

(2)

In Eq. (1), S,' is the perceived size of any object whosevisual (retinal) size is 0, and which is located at anyphysical distance D, from the observer. The term D,'is the perceived absolute distance associated by theobserver with D,,. In Eq. (2), Se' is the perceived sizeof a particular object (Object e) whose visual (retinal)size is G, and which is located at the particular physicaldistance D, from the observer with D,' being the per-ceived absolute distance associated with De. In theequations used in this study, the subscript "v" refers toa variable quantity while the subscripts e, f, and g(see Fig. 1) refer to specific values. As is illustrated inEq. (2), whenever a particular subscript such as e isused, any other particular subscript such as f couldhave been substituted for it throughout the expression.

FIG. 1. A schematic drawing for considering theperception of size and distance.

411

MARCH 1964VOLUME 54, NUMBER 3

4WALTER C. GOGEL

Equation of Size Constancy

An equation permitting the quantitative expression ofperceived frontoparallel size S,' as a function of physi-cal distance D, can be developed from the two followinghypothetical situations.

Situation I

An object of constant frontoparallel size S is presentedat various distances D,. Suppose that the perceivedfrontoparallel size of this object S,' remains constant.This is an example of perfect frontoparallel size con-stancy and can be expressed as

Sv = Se'. (3)

Equation (3) states that for a constant value of S, theperceived size S,' at any distance D, is equal to its per-ceived size S,' at the particular distance De. Suppose,however, that the perceived size of S decreases pro-portionately to the distance DA,. This is an example ofzero frontoparallel size constancy and is expressed as

(4)

Equation (4) states that the perceived size S,' of S atany distance D, is equal to its perceived size S,' at theparticular distance D, multiplied by the ratio of the twodistances.

Equations (3) and (4) are special cases of the moregeneral equation'

and K2 is a constant such that

K2=Se'/1OeDe S//OfDfr, etc. (11)

Equation (9) specifies the relationship between theperceived frontoparallel size (S,') of an object and itsphysical distance (D,) for any value of visual angle6,, as a function of the measure of frontoparallel sizeconstancy (n). Again, perfect and zero frontoparallelsize constancy are expressed by n= 1 and n = 0, respec-tively, while values of n between 0 and 1, for example,indicate that some amount of, but less than perfect,frontoparallel size constancy is present.

Equation (9) can also be derived directly from theThouless index.4 The Thouless index expressed in sym-bols which are consistent with Eq. (9) is as follows:

logS'- logoT= Thouless index=

logs-logo(12)

whereSo=Sv//Se', 0=c °,v/6e, and S= S,/Sc. (13)

From Eq. (12),(S/0) T = S'/0. (14)

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

S,'= K 20,,Dv".

Thus, comparing Eqs. (9) and (15),

T=n.

(15)

(16)S,' = Se,' (Dv/De) n-1. (5)

In Eq. (5), when nt= 1, Eq. (3) results and when 11=0,

Eq. (4) results. Thus, ni is a measure of frontoparallelsize constancy present. When ib=0 there is no fronto-parallel size constancy. When nt= 1 frontoparallel sizeconstancy is complete. When nt> 1 the perceived size ofa physically constant S increases as the distance D,increases.

Situation II

An object of constant visutal angle 0 is presented atvarious distances D,,. For perfect frontoparallel sizeconstancy,

Sv= Se'(Dv/De),

while for zero frontoparallel size constancy

or, in general,S5' = Sc',

,= Se' (Dy/Dc') n.

(6)

(7)

Equations (5) and (8) are special cases of the moregeneral equation

where0, in radians=S,/D,, (10)

3 A relation similar to Eq. (5) has been developed for use in anequation concerned with the perception of three-dimensional shape[W. C. Gogel, J. Psychol. 50, 179 (1960)].

Equation of Depth Constancy

Combining Eqs. (1) and (9), the relation between per-ceived absolute distance D' and physical distance D is

D,'= K3 D,-, (17)where

K 3 =D,'/D. =D//Dfn, etc.

Equations (9) and (17) can be written as

logS,=' nt logD,+logK2 0,and

logD,,'= n1 logD,+logK3 ,

respectively. Also, from Eqs. (2), (11), and (18),

K1 = K2/K3 -

(18)

(19)

(20)

(21)

If the size-distance invariance hypothesis as expressedby Eq. (1) is correct, n should be the same in Eqs. (19)and (20), and Eq. (21) should be valid. In the presentstudy, the hypothesized equality of n between Eqs.(19) and (20) was tested by experimentally determiningthe function relating physical distance to both per-ceived size [Eq. (19)] and perceived distance [Eq. (20)]for two different conditions of observation.

I R. H. Thouless, Brit. J. Psychol. 21, 339 (1931).

Vol. 54412

S, = Set (D,1D,) -

M 1 VISUAL PERCEPTION OF SPATIAL EXTENT 413

APPARATUS

Visual Alley

Perceived frontoparallel size and perceived depth weremeasured in an alley 1020 cm long and 128 cm wide. Thefloor of the alley consisted of either (1) a uniform blackcloth, or (2) a checkerboard pattern of light and darkgray 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 thealley was covered by black velveteen. Schematic draw-ings of the alley are shown in Fig. 2. A baffle, 28.5 cmhigh, covered with tan cloth, extended along the leftside of the alley, 33 cm from the left wall. Ten white(9-cm) squares (not shown in Fig. 2) vertically pre-sented in planes normal to the line of sight were dis-tributed in depth on the floor of the alley, with five ofthese along the baffle and five along the right side of thealley. The center of the alley was clear of white squares,permitting the placement of the lettered objects shownin Fig. 2. The alley was illuminated by overhead fluo-rescent lights. The alley and every object in the alleywere viewed by the observer with his right eye only (theleft eye-piece at the viewing position was occluded).The eye of the observer at the viewing position was 23.8cm above the floor of the alley.

Measurement of Perceived Frontoparallel Size

To measure the perception of frontoparallel sizealong the alley, red rectangular objects [Objectsh, i, j, 1, and m, of Fig. 2(A)] were placed one at a timeat different distances along the alley and the observeradjusted the lateral distance between two white rods(Objects a and b) until this distance seemed the sameas the width of the particular rectangle at the particulardistance. The two white rods (6 mm in diameter and10 cm high) were vertically presented on the floor of thealley at a constant distance of 117.5 cm from the ob-server. The right rod was stationary while the left rodcould be moved laterally by the experimenter by meansof an attachment which permitted the experimenter to

baffle-,

0abs.

I I Ih i i

A

B

FIG. 2. Top view diagrams illustrating the physical conditionsinvolved in the size and depth judgments.

remain invisible to the observer. A small black barrier(2.5 cm high) which extended across the alley at thebase of the rods prevented the observer from seeing theslot on the alley floor in which the left rod was movable.The red rectangles subtended a constant visual angle(10 47' 24") and had the following sizes and positionsin the alley:

Object h was 7.6 cm widefrom the observer.

Object i was 11.4 cm widefrom the observer.

Object j was 15.2 cm widefrom the observer.

Object 1 was 22.8 cm widefrom the observer.

by 1.7 cm high at 243.8 cm




Object m was 30.4 cm wide by 6.8 cm high at 975.2 cmfrom the observer.

Measurement of Perceived Depth

To measure perceived depth in the alley, Objectsa, b, h, i, j, 1, and m were removed and Objects p, q,and w were used instead [Fig. 2(B)]. Object p wasplaced at a constant distance from the observer (171.5cm) and the observer adjusted a depth interval (usuallybetween Objects q and w) at different distances fromhimself to duplicate the perceived distance from himselfto Object p.

Objects p and q were red rectangles 15 cm high by 5cm wide which were vertically positioned on the floorof the alley. Object w was a red 7.6-cm square placedin a frontoparallel position with the bottom of the squarealways resting on the floor of the alley. Object w wasattached by a glass rod to a cart (invisible to the ob-server) which moved on a track behind the baffle. Byturning a knob at the viewing position, the observercould move the red square directly toward or away fromhimself in depth.

PROCEDURE

Frontoparallel Size Adjustments

The observer signaled to the experimenter to moveObject a [Fig. 2(A)] laterally to the right or left untilthe distance between the inner edges of the rods visuallyappeared equal to the width of Rectangles h, i, j, 1,or m. The adjustment was repeated four times, afterwhich a different rectangle at a different distance waspresented. The distances in centimeters adjusted be-tween the rods measured the perceived width S' of theparticular rectangle. The order of presenting the rec-tangles was randomly determined for each observer.

Depth Adjustments

The observer moved Object w (the red square) backand forth in depth until the distance of w behind p vis-ually seemed equal to the distance of p from himself.

413March 1964

WALTER C. GOGEL

This was done three times. The average of the threeadjusted distances was calculated by the experimenterand q was placed at this distance. The observer thenmoved w behind q until the visually perceived depthbetween w and q (dw') seemed to duplicate the visuallyperceived distance of p from himself. Again, this ad-justment was made three times. The new average posi-tion of w was found, q was moved to this new distance,and the process was repeated. This process resulted ina series of depth intervals (dq') each of which appearedto the observer to be equal to the distance of p fromhimself and thus equal to each other.

OBSERVERS AND ORDERS

Four men were used in the experiment as observers.Previously these observers had experience in makingsize and distance judgments in experimental situations.Each observer had a visual acuity in his right eye (cor-rected if necessary) of at least 20/22.

The depth and frontoparallel size adjustments alwayswere completed first with the checkerboard patternand following this with the uniform black covering onthe floor of the alley.

RESULTS

Frontoparallel Size Adjustments

The data from the experiment relevant to Eq. (19)are shown in Fig. 3. Since the red rectangles subtendeda constant visual angle, 0 in Eqs. (9) and (19) is con-stant. The ordinate of Fig. 3 gives the logarithms of theaverage results in centimeters from each observer forthe frontoparallel size adjustments, for both the checker-board pattern and the uniform black floor. The abscissais the logarithm of the physical distance in centimeters

1.4

1. 2- -= o

1.0/ 0I .0o- *oZ

Ac)0V)

0

C)

.8

o--o CHECKERBOARD FLOOR*-* UNIFORM BLACK FLOOR

1.4 -

.2 - ' -

007

1.0 5

C

2.3 2.5 2.7 2.9 2.3 2.5

Log Physical Distance2.7 2.9

FIG. 3. The relation between the logarithm of the perceived sizein centimeters of the rectangles and the logarithm of their physicaldistance in centimeters.

of the frontoparallel targets from the observer. The datapoints of Fig. 3 are reasonably linear. Using the methodof least squares, a straight line of best fit was computedfor each of the distributions of Fig. 3. From Eq. (19),the slope of the straight line of best fit is n with logK20the ordinate intercept. The resulting values of n and ofK 2 (with K 2 determined for the constant value of 0expressed in radians) are given in Table I. As might beexpected, there was greater frontoparallel size constancywhen the checkerboard pattern rather than the blackcloth floor was used. In all cases the measure of fronto-parallel size constancy was between no constancy (n= 0)and perfect constancy (it= 1).

Depth Adjustments

In the previous equations, D' is a perceived absolutedistance (an apparent distance from the observer tothe object). This is to be distinguished from d' which isa perception of relative distance (an apparent depthbetween objects). The successively perceived relativedepths in the present experiment were all adjusted to be

TABLE I. Values of it, K2, and K3 as determined from the slopesand intersection values, of the straight lines of best fit for thedata of Figs. 3 and 4.

Checkerboard floor Uniform black floorFrom size From depth From size From depthjudgments judgments K2/K3 judgments judgments K2/K3

Obs. At K2 it K3 K, i K2 a1 K3 Ki

A 0.50 21.25 0.80 2.07 10.27 0.47 25.16 0.66 4.32 5.82B 0.67 6.89 0.91 0.95 7.25 0.47 33.28 0.81 1.74 19.13C 0.61 12.83 0.91 1.03 12.46 0.48 29.80 1.05 0.47 63.40D 0.86 1.96 0.93 0.91 2.15 0.72 4.62 0.90 1.07 4.32

a For convenience, values of K3 in this table have been arbitrarily multi-plied by 100.

equal to the perceived absolute distance to Object pand hence equal to each other. It follows that the per-ceived absolute distance throughout the visual field canbe expressed in units of the perceived absolute distanceto the nearest object (Object p). Relevant to Eq. (20),the logarithms of the resulting perceived absolute dis-tances D' (with the perceived absolute distance to pset equal to unity) and the associated physical distancesD in centimeters required to produce the perceivedabsolute distances are shown in Fig. 4. The data pointsof Fig. 4 are reasonably linear. Using the method of leastsquares, a straight line of best fit was determined andit and K13 were computed for each of the curves of Fig. 4.The results are shown in Table I. In all cases but one,the measure of depth constancy (as shown by the valueof it) was between no constancy (n=0) and perfectconstancy (1= 1).

The average results for each observer from the depthadjustments, using the checkerboard pattern and theuniform black floor, are given in Fig. 5. The averageadjusted depth interval in centimeters which appearedto the observer to be equal to the distance from himselfto Object p is given on the ordinate with the physical

414 Vol. 54

VISUAL PERCEPTION OF SPATIAL EXTENT

distance in centimeters from the observer to the leastdistant part of this interval shown on the abscissa.

DISCUSSION

From the essentially linear graphs of Fig. 3, it followsthat a single value of n for a particular observer and aparticular floor type (checkerboard or uniform blackfloor) appropriately describes the frontoparallel adjust-ments. A similar conclusion applies to the depth adjust-ments shown in Fig. 4. But, in every instance (seeTable I) the value of n for the same observer and samefloor type is greater with the depth than with the fronto-parallel size adjustments. According to the size-distanceinvariance hypothesis n should be the same for the depthadjustments as for the frontoparallel size adjustments.The differences in n for the two types of judgments in

400-

300-

200-

>

.2

Q~

400-

300-

200- -o

I00

0 200 400 6(

A iBmI I I ,

°--0 CHECKERBOARD FLOOR*-e UNIFORM BLACK FLOOR

C 8 D

oo Soo 0 200 400 600 800

.75-

0 0

.60- 0 0 Z00

0 0 0.45 /

// /

.30 C~ | e

o--o CHECKERBOARD FLOOR

*-* UNIFORM BLACK FLOOR

.75 -

.60 00

.G- ///

.45 /

C/

2.4 2.6 2.8 . ') 2.,

Log Physical Distance

/*//

// .0:

//A/

D2 .6 I 2 3.0

q 2.6 2.8 3.0

FiG. 4. The relation between the logarithm of perceived depth(expressed in units of the nearest Object p) and the logarithm ofthe physical distance in centimeters required to produce that per-ceived depth.

Table I therefore provide evidence against the validityof the size-distance invariance hypothesis.

Using Eq. (21), K1 was determined from K2 and K3with the results shown in Table I. From Eqs. (2), (11),(18), and (21), differences between the K1 values cal-culated from the results from the checkerboard floorand from the uniform black floor would be expected tooccur as a consequence of differences in n. Furthermore,since n differed for the two floor types, K1 in Table Iwould be expected to vary as a function of the unit usedin the specification of D. The differences in the calcu-lated values of K1 in Table I reflect both the differencesin n and the size of the unit used in the specification of D.Differences in K1 in this experiment, therefore, do notconstitute a direct or independent evaluation of thevalidity of the size-distance invariance hypothesis.

Equation (9) is convenient for describing size con-

Physical Distance of Interval (cm)

FIG. 5. The relation between the physical depth interval re-quired to produce perceptually equal depth intervals and thephysical distance of the beginning of the interval from the observer.

stancy data. If, as this study indicated, n is generally aconstant for the entire visual field (or for a large portionof the visual field) for a particular observer, the use-fulness of Eq. (9) will be increased. Equation (17) pro-vides an equally convenient description of depth con-stancy data. When n= (perfect depth constancy),perceived and physical depth are proportional. Whenn=O (zero depth constancy), perceived depth does notincrease even though physical depth increases. FromFig. 1, it is clear that

Df'= de±/+De' (22)

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

def=K3(Dfn-D6 8), etc. (23)

In some studies, it might be more convenient to useEq. (23) than Eq. (17) in summarizing data from depthjudgments.

In a recent study testing the size-distance invariancehypothesis, Ueno5 has developed equations similar toEqs. (9) and (17) 'of the present study from Stevens'power law' which states that psychological magnitudeis a power function of stimulus magnitude. Using thepsychological method of transposition of Oyama,7 Uenohas compared the value of the exponent in the powerfunction involving perceived size with that involvingperceived distance. Exponents computed from the twotypes of judgments were not always similar with largedifferences sometimes occurring for monocular and re-duced conditions of observation. Both the study byUeno and the present study indicate, contrary to the

S T. Ueno, Japan. Psvchol. Res. 4, 99 (1962).6 S. S. Stevens, Am. Psychologist 17, 29 (1962).7 T. Oyama, Psychol. Bull. 56, 74 (1959).

a

.-

66

04ASo

March 1964 415

WALTER C. GOGEL

size-distance invariance hypothesis, that the powerfunction relating perceived size and physical distancedoes not always involve the same value of exponent asthat relating perceived distance and physical distance.These studies together with previous evidence' indicatethat the size-distance invariance hypothesis in its usualform is not always valid.

It has been asserted that the perceived extent S,' aswell as D,' in Eq. (1) is a perceived absolute extent.9

One possible criterion for the measurement of perceivedabsolute extents is that the obtained measures are sensi-tive to changes in the over-all perceptual magnification(or minification) of extents in the visual field. Neitherthe method of measuring S,' nor the method of measur-ing D,' in the present study meets this criterion. Itwill be noted, however, that this criterion, while of con-sequence in the determination of both K2 and K3 in thepresent study is of no importance in determining iz. Thevalue of it in any of the equations of the present studyis unaffected by multiplicative constants applied toeither or both S,' or D,' and, therefore, is independentof over-all perceptual magnification.

Figure 5 is useful in analyzing errors which can occurin the perception of distance. The first point on all thegraphs of Fig. 5 has an ordinate value of 171.5 cm whichis the physical distance of Object p from the observer.The observer's perception of the distance of p is a per-ception of absolute distance. All other points on thegraphs represent the results from judgments of relativedistance, i.e., results from perceptions of the depth be-tween two objects. The ordinate value of the secondpoint for three out of four observers is less than 171.5cm. This means that for these observers the perceiveddepth per unit of physical depth was usually greater inthe first relative depth judgment than in the judgmentof the absolute distance of p. For the following two rela-tive depth judgments, however, the perceived depthper unit of physical depth usually decreased. For thefinal relative depth judgment the decrease in this ratiocontinued for the uniform black floor but not for thecheckerboard floor. From these data, it is possible toinfer the results which would have occurred if the ob-server had been given the task of adjusting a movableobject to perceptually bisect the distance from himself

8 See (a) F. P. Kilpatrick and W. H. Ittelson, Psychol. Rev. 60,223 (1953); (b) W. Epstein, J. Park, and A. Casey, Psychol.Bull. 58, 491 (1961); and (c) W. C. Gogel, Vision Res. 3, 106(1963).

9 W. C. Gogel, E. R. Wist, and G. S. Harker, Am. J. Phychol.76, 537 (1963).

to a stationary object. The task would produce two per-ceptually equal intervals with the nearer interval beinga judgment of absolute distance (a judgment of thedistance from the observer to the movable object) andthe farther depth interval being a judgment of relativedistance (a judgment of the depth between the movableand the stationary object). From the present study, fornear positions of the alley, the observer would usuallyunderestimate the nearer (absolute) distance with re-spect to the farther (relative) distance. However, if adepth between two visible stationary objects (a relativedepth) were perceptually bisected by a movable thirdobject, the reverse sometimes would occur. The physicaldepth between the nearest object and the movable ob-ject sometimes would be overestimated relative to thephysical depth between the movable object and thefarthest object. These inferences from the data of thepresent study might help explain some of the dis-crepancies in the results sometimes encountered instudies involving the perceptual bisection of a depthinterval.'0

The reason for the increase in the perceived depth perunit of physical depth from the checkerboard but notfrom the uniform black floor at the far distances is notclear. Perhaps some complex effects are to be expectedfrom judgments involving depth near the terminatingwall of the alley.

In many respects it is unfortunate that the size-distance invariance hypothesis of Eq. (1) probably isnot valid. If Eq. (1) had been valid, for a constant valueof i, Eqs. (1), (9), and (17) would have together re-sulted in a simple predictive system of considerableparsimony. A few measures of perceived extent for aparticular observer would have permitted the predictionof perceived extent at any orientation (frontoparallel,depth, or a combination of both) throughout the visualfield. A size-distance invariance hypothesis can be de-termined from Eqs. (9) and (17) without assuming thatn is the same in the two equations. However, the re-sulting equation would be more complicated and lessparsimonious than Eq. (1).

ACKNOWLEDGMENTS

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

10 Reference 8(b), pp. 495-496.

416 Vol. 5i4

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Visual Perception of Spatial Extent