D. VICKERSUniversity of Adelaide, Adelaide, South Australia 5001

Perceptual economy and theimpression of visual depth

Three experiments were carried out to investigate the hypothesis that theimpression of visual depth given by gradient patterns is due to the operation of aprinciple of "perceptual economy." The first experiment showed that thegreater the number of values of the gradient variables in a pattern, the strongerwas the tridimensional impression given by that pattern. The second showed thatthe greater the number of gradient variables, the less the amount of the patternthat had to be exposed before a tridimensional response was elicited. The thirdshowed that judged slants increased as a function of the number of gradientvariables in each pattern. The results suggest that the impression of visual depthgiven by a pattern represents a compromise between the economy gained frommaking a tridimensional specification of the elements in the pattern and thepossible cost of making an error in coding the information in this way.

Gibson (1950a, 1966) has arguedthat an adequate basis for visual depthperception is provided by invariants inthe visual array which is sampled by an0, either as he gazes fixedly at aparticular scene or as he scans ormoves around in his surroundings.Although the higher order variablesexamined by Gibson may in mostcases provide effective stimuli for theperception of depth, however, thereare several instances in which it seemsdifficult to apply the Gibsonian theoryof ecological optics. In some cases,such as the "balloon demonstration"described by Ittelson and Kilpatrick(1951), the perceptual system appearsto respond to a higher order variable,which, though present, is misleading.The result is that two stationaryballoons are perceived as movingbackwards and forwards, though it isonly their brightness and size whichcovary. In other cases the perceptualsystem appears to respondinappropriately to a higher ordervariable, as when a rotating trapezoidalwindow is seen as an oscillatingrectangle (lttelson & Kilpatrick,1951), or when a luminous,two-dimensional Necker cube, viewedin darkness, is seen asthree-dimensional, even to the extentthat the faces appear of different sizes(Gregory, 1966, p. 156). In yet othercases the "interpretation" of higherorder variables by the perceptualsystem remains the same, though thestimulus situation is quite different.For example, Clark, Smith, and Rabe(1955) found that an unslantedtrapezoid, viewed monocularly, wasseen as a slanted rectangle, while arectangle at a slant was seenveridically, although Os were warnedof the possibility of being presentedwith a variety of combinations ofshape and slant. As Gibson (1966,p, 287) points out, since his theory isprimarily a theory of correct

perception, it must explain these andother examples of "incorrect"perception by supplementaryassumptions, e.g., that the 0 makes aprobable guess (p. 304). However, theexamples of incorrect perception onwhich Gibson concentrates are thoseelicited by equivocal or contradictoryinformation rather than thoserepresenting an actual distortion of theinformation in the visual array.Moreover, he does not offer a detailedaccount of what happens when theperceptual system is confronted withtwo or more hypotheses, each ofcomparable plausibility, i.e., underwhat conditions the perceptual systemmight be expected to reject onehypothesis, alternate between the two,or settle for a compromise.

An alternative approach whichmight serve both as a theory ofperception and of "misperception" issimilar to that first explored byHochberg and McAlister (1953) andAttneave (1954), and more recentlyby Garner (1962, 1966, 1970). Thissuggests that stimulus information isperceived as invariant because this is amore economical way of encoding thesensory data. For example, it ispossible to specify Pattern A in Fig. 1as four parallel lines of varyingthickness and interline interval. It isalso possible, however, to specify thearray tridimensionally so thatthickness and interline interval becomeinvariant. This form of codingminimizes the variation hypotheticallyadmissible in the pattern, therebyred ucing the redundancy in itsdescription.

It may be objected that theconcepts used by Hochberg andMcAlister, Attneave, and Garner arevery different, and indeed are notalways used consistently by theirrespective authors (Vickers, 1967;Evans, 1967). One solution to themany contradictions has been

proposed by Vickers (1967) andindepe ndently (and much moresystematically) by Evans (1967).Instead of the usual two, it is possibleto distinguish between threeinformation measures: (1) the averageinformation associated with anycombination of the values of each of agiven number of variables when noconstraints are imposed on theirselection; (2) the average informationassociated with any combination whensome constraints are used in itsselection; and (3) the averageinformation associated with any of thecombinations actually presented to orsampled by an O. The differencebetween the first and second measuresdetermines the amount of"constraint" redundancy, i.e., theamount of constraint, patterning, orsimilarity in the patterns actuallyshown or sampled. The differencebetween the second and thirdmeasures determines the amount of"discrimination" redundancy, or theextent to which the observed patternsdiffer among themselves (Evans,1967). On this view, an economyhypothesis would imply that in theusual "free" perceptual situation (Le.,where the emphasis is on organizingthe stimulus information) theperceptual system seeks to arrive at aform of coding which will minimizediscrimination redundancy (ormaximize constraint redundancy).That is, the 0 tries to devise a generaldescription which will include thesampled patterns as a subset but whichwill also restrict the number ofhypothetically admissiblecombinations of the values of thevariables observed in the sampledpatterns. One way of doing this, as inthe case of Pattern A in Fig. 1, is tospecify the information in such a waythat a variable with a number of valuesis coded as an invariant.

Since a tridimensional specificationof Pattern A allows all the correlatedvalues of each variable to be predictedfrom the angle of orientation of thearray, then, the greater the number ofcorrelated values in such an array, thegreater will be the relative economy ofsuch a specification. That is, if angle oforientation is regarded as the onlyeffective variable for a tridimensionalspecification, then adding moreappropriate lines will not increase thenumber of angles required for acomplete description of the array,whereas if thickness and interlineinterval are variables, as in abidimensional specification, then thenumber of combinations of the valuesof these variables would increase witheach additional line. Hence we shouldexpect that a tridimensional responseto Pattern A should become moreprobable if more lines were added to

Perception & Psychophysics, 1971, Vol. 10 (1) Copyright 1971, Psychonomic Journals, Inc., Austin, Texas 23

Fig. 1. The four patterns used in Experiment 1.

A

cit, provided the thickness and interlineinterval could remain invariant in atridimensional specification.

According to Gibson's theory, onthe other hand, it would seem that inthe same situation no change inperception should occur fromPattern A to Pattern B, since "thesame stimulus array coming to the eyewill always afford the same perceptualexperience insofar as it carries thesam e variables of structuralinformation [Gibson, 1966, p. 248]."

EXPERIMENT 1Stimuli

The four patterns used are shown inFig. 1. Each pattern corresponds tothe projection of a horizontal grid ofequidistant straight lines onto a planeinclined at 75 deg to the grid, thenumber of lines increasing from 4 in Aup to 19 in D.

ProcedureTwo patterns were displayed side by

B

oside before 0 and at right angles to hisline of sight. Os were instructed toindicate which looked "morethree-dimensional." The six pairs ofpatterns were presented to each of sixOs in a 6 by 6 Latin square design inwhich the two possible positions ofeach pattern were randomized fromone 0 to the next.

ResultsJudgments were recorded for each

pair of patterns, and a coefficient ofrank order agreement for pairedcomparison data was calculated(Kendall, 1948). For six Os and fourpatterns U '" 0.89, so that agreementbetween Os is highly significant, withp < .001. The results support thehypothesis in showing that the greaterthe number of values of the twovariables which can be reduced bychoosing a tridimensional rather than abidimensional response, the greater isthe tendency to choose the former.

EXPERIMENT 2The four patterns used are shown in

Fig. 2. When specifiedbidimensionally, Pattern I has onevariable, interline interval, with severaldifferent values; II has an additionalvariable, namely, line thickness; whileIII has three and IV has four variables,each with several values. If. eachpattern is specified tridimensionally,however, the number of differentvalues for each variable can be reducedto one. The economy hypothesisimplies that the greater the number ofvariables that can be made invariant byadopting a tridimensionalspecification, the greater should be theprobability of making such a response.Hence, we should expect a greatertendency for a tridimensional responseto IV than to III, to III than to II, andto II than to I. In this case, predictionfrom Gibson's theory is more difficult.As Gibson (1950b) points out, theassertion that a gradient in texture is astimulus for perceived slant wouldassume, in its simplest form, that theperception was a response to amathematical property of the texture.In his later statement of the theory ofecological optics, this seems to be hisown preferred interpretation (Gibson,1966). Since the gradient variables inPatterns I-IV all specify a surface atthe same slant, it would seem tofollow that the depth impression fromeach should be the same. Toparaphrase Gibson's own (1950b)analogy: the melody is the same, eventhough it is played on several differentinstruments in unison.

StimuliPattern I corresponds to the

projection of a horizontal grid ofequidistant parallel lines onto a planeat 75 deg to the grid. The lines,however, are relatively thin and do notdecrease in thickness from bottom totop in their projection. In Pattern IIthe lines are of appreciable thicknessand decrease in thickness at the samerate as the interline intervals.Pattern III is similar to I, except that itis composed of broken lines. Thelengths of the lines and the distancesbetween them decrease at the samerate as the interline intervals. Finally,Pattern IV is similar to Pattern III,except that the broken lines are of thesame thickness as those in Pattern IIand decrease similarly in thicknessfrom bottom to top. The patternswere constructed by photographicallyreducing 43 x 36 ern graphicalprojections, and mounting 15 x 10 emmatt prints of each reduction in blackcardboard frames.

ProcedureEach 0 was given one pattern at a

time just before him on a table. After

24 Perception & Psychophysics, 1971, Vol. 10 (1)

estimation should become moreconsisten t as increased "cues,""information," or "redundancy" areadded to a visual display. Thealternative view suggested by thisobservation is that Os becomeprogressively "bolder" in theirestimations of slant as redundancy isincreased, i.e., that their estimatesbecome greater rather than simply lessvariable. The third experiment wasconducted to test this observation, andto gain some measure of the possiblerange of quantitative differences inresponse.

I IIStimuli

The four patterns used were thesame as those of Experiment 2, shownin Fig. 2.

p < .001p < .001P < .001

Level

Significance of Differences

I-IIII-IIIIII-IV

Patterns

ProcedureThe apparatus is shown in Fig. 3.

Each 0 was shown one of Patterns I,II, III, or IV, viewed monocularlythrough a window in a screen at adistance of about 2 m, and a fifth,finely-textured pattern (X), viewedbinocularly through another windowin the same screen. At the beginning ofeach trial, Pattern X was set to aroughly perpendicular or horizontalposition. Os were asked to increase ordecrease the slant of X to correspondas closely as possible with the apparentslant of the gradient pattern. Osperformed one type of adjustment forall four patterns before performing theopposite adjustment. The order of the"increase" and "decrease" adjustmentswas balanced, and all 24 permutationsof the order of the four patterns wereused once. Each of the 12 student Ostook about 15 min to complete alleight adjustments.

------------_ .....-_---~

Number of Mean NumberGradient of Lines Exposed

Pattern Variables for 3-D Response

I 1 10.7II 2 8.4III 3 5.1IV 4 3.0

ResultsThe mean angle of all settings for

each pattern was calculated for each0, the overall mean settings in degreesfrom the vertical being 48.67, 50.67,

EXPERIMENT 3 64.50, and 66.67 for I, II, III, and IV,In the course of Experiment 2 respectively. A test of average rank

several Os reported that the more correlation (Jonckheere, 1954) withvividly tridimensional patterns the number of gradient variables inappeared to slope away at a more each pattern gave a value ofr= .639,extreme angle than the others, so that which is highly significant (p < .001,the top of IV, for example, seemed one-tailed). Since the previous"further away" than that of III. This experiment has shown that thewould seem to throw doubt on a amount of a pattern which has to becommon assumption that distance exposed before a tridimensional

Table 1Number of Gradient Variables in Each Pattern in Experiment 2, and the Number of

Lines That Have to Be Exposed Before a Tridimensional Response is Obtained

number of variables which can bemade invariant by choosing atridimensional rather than abidimensional response, the greater isthe tendency to choose the former.

][Fig. 2. The four patterns used in Experiment 2.

~;4'~~8~~!!~~~~~'~

~#.=E~2~~~~-:::':"~~------------­.._---------_.---------------------.

ResultsThe mean number of lines (or rows

of broken lines) at which Os reporteda change were calculated and areshown in Table 1, together with theresults of one-tailed sign tests of thesignificance of the differences betweenthe means for I and II, II and III, andIII and IV. The results confirm thehypothesis in that the greater the

Os had given some spontaneousindication that they obtained atridimensional effect from thecomplete pattern, the pattern wascovered by an opaque slide. Os werethen asked to uncover the patternsslowly, line by line, until they receivedan impression of depth. Immediatelyafterwards, the pattern was completelyexposed, and Os were asked to bringthe slide down from the top until theimpression of depth disappeared. Eachof the 24 student Os took about 7 minto complete all eight adjustments, andall 24 permutations of the four cardswere used.

Perception & Psychophysics, 1971, Vol. 10 (1) 25

Fig. 3. The apparatus used in Experiment 3.

GRADIENT PATTERNVTEWED MONOCULARLYSCREEN FOR

LEFT EYE

CH'NNEST:1

response is elicited is an inversefunction of the number of gradientvariables in the pattern, we mayinterpret this trend as showing that themore readily the gradient pattern isseen as tridimensional, the greater theangle from 0 at which it appears toslope away.

DISCUSSIONIt must be admitted that

Experiment 1 has a certain unnervingsimplicity and that a sceptic mightargue that Os were judging the greaternumber of lines rather thantridimensionality. Although it wouldrequire a separate experiment to refutethis point, an examination of thepatterns in Fig. 1 would suggest that itis subjectively very implausible. Inaddition, the present interpretationagrees well with informal observationof a child's early attempts to masterperspective drawing; when he isdissatisfied with the depth impressiongiven by the railway tracks he hasdrawn as receding into the distance. afavorite strategy employed by a child isto double the number of sleepers orthe number of fenceposts beside thetrack (an endeavor often frustrated byhis inability to carry this out whilepreserving the same gradient).Analogous effects of textural grain ondistance estimation have recently beenreported by Corsini and Pick (1969).

Again, it may be argued, inconnection with Experiment 2, thatPa t t e r n s I and III containcontradictory gradients, since thethicknesses of the lines do notdecrease. If the elements are seen asoverlapping, however, it is possible toperceive the patterns in such a waythat this contradiction is minimized.For example, Pattern I may be seen asthe side of a clinker-built boat, orPattern III as a tiled roof. In both

jTEXT URED PATTERNVIEWED BINOCULARLY

REVERSING MOTOR

cases the shaded parts wouldconstitute the lower edges of theelements, and these do notautomatically decrease in projectivesize as a function of distance from theO. In any case, this objection wouldnot account for the significantdifferences in response between I andIII and between II and IV.

Experiments 1 and 2 were intendedto test the hypothesis that perceptionis a process of encoding sensory datain as economical a way as possible.The experiments illustrate twotechniques of studying the framing ofperceptual hypotheses by 0: the firstconsists of holding the number ofvariables constant, while increasing thenumber of values on each variable; thesecond consists of holding the numberof values constant and increasing thenumber of variables. Similarexperimental situations for studyingthe framing of perceptual hypothesesreadily suggest themselves. Forexample, certain configurations ofconverging circles, drawn on a disk in aMichotte apparatus (Michotte, 1963)and seen through the viewing slot,appear at slow speeds to expand andcontract in a bidimensional way but,at high speeds, to rotate about an axis,i.e., to display movement in threedimensions. Experiment 1 suggeststhat the greater the number ofelements in a pattern which can bemade invariant in a tridimensionalresponse, the more readily will such aresponse be elicited. In a similar way,we may expect that the greater thenumber of appropriate elements insuch moving patterns, the more readilywill a tridimensional response beelicited, e.g., the slower the speed atwhich the disk must spin before thepattern is seen to rotate. Vickers(1967) has obtained results in asituation of this kind which show that

the speed at which the disks have torotate before a tridimensional responseis elicited is an inverse function of thenumber of converging circles drawn onthe disk, and which lend support tothe view that redundancy in the formof spatial repetition of the elements ina pattern may in some way beanalogous to the repetition of thepattern in time. Similar results, withregard to the speed of rotation, havealso been reported by Langdon(1951).

The results from Experiment 3show that the order of magnitude ofthe corresponding slant setting followsthe order of tendency to evoke atridimensional effect, as measured inExperiment 2. In other words, thefewer the number of variables, orvalues of a given variable, that have tobe exposed before a given patternlooks tridimensional, the greater theslant will appear to be when the wholepattern is exposed, even though theslant setting necessary to achieve themaximum invariance remains thesame.

It may be that the phenomenon ispeculiar to the particular set ofpatterns used, although somepreliminary investigation with actualplane surfaces suggested this was notso; indeed, this result resembles thewell-known continuum of judgmentsof phenomenal regression in relationto the amount of information aboutdistance (Holway & Boring, 1941;Thouless, 1931). Similar results havefrequently been reported in theliterature (e.g., Clark, Smith, & Rabe,1956; Attneave, 1969). Usually, theseare interpreted as showing thatperception of depth or slant becomes"more accurate" as increased "cues"are added to any array (Dember, 1965,p. 180; Fergus, 1966, p, 213). Theambiguity in the use of the term"accurate," however, disguises thepoint that the most direct way ofapplying the Gibsonian hypothesis tothis situation would be to suppose thatas more gradient data become availableto 0, the less variable his estimates ofslant based on the data shouldbecome. In a similar way, a simpleeconomy hypothesis would postulatethat Os adopt a perceptual responsethat will make all gradient variablesinvariant. However, there is strictlyonly one angle at which the figurescould be specified so as to make thevalues of the gradient variables equal,and all other slant responses wouldseem on this view to be moreredundant.

The present results, on the otherhand, suggest that 0 attempts to havehis cake and eat it at the same time,i.e., he perceives the variables in thepatterns as invariants, but adjusts theslope at which the hypothetical plane

26 Perception & Psychophysics, 1971, Vol. 10 (1)

containing them must lie, according tothe amount of confirming evidence,even though this means distorting thisevidence. Such an explanation couldalso account for the perception of arotating trapezoidal window as anoscillating rectangle and would predictthat simpler structures would be lesslikely to evoke the impression ofmovement in depth of a rigid body(e.g., Wallach & O'Connell, 1953). Itseems worthwhile, therefore, toconsider the possibility that theperceptual system does not simplytend to reduce redundancy in thecoding of visual data to a minimum,but seeks to balance the advantagegained by this against the possible costof making an error, or introducingsome distortion into the data, by moreefficien t recoding. A moderatered uction in redundancy mightwarrant a moderate distance or slantestimate, while a high reduction mightjustify a bolder or increased estimate.

These effects must be studied inmore detail before proposing theoriesof greater precision. It would seemuseful, however, to try to develop theeconomy hypothesis, modified alongthe lines suggested above. It appears tobe a hypothesis of possibly greatergenerality than the Gibsonianapproach, while at the same timedepending upon the same rich varietyof stimulus information to which thetheory of ecological optics drawsattention. Besides suggesting anaccount of incorrect as well as correctperception, it could deal with theperception of two-dimensionalstimulus patterns or "noninformative"stimuli (Gibson, 1966, p. 245) in thesame way as that of arrays specifyingreal objects in depth. Finally, itsapplication to perceptual learningwould follow very similar lines tothose proposed by Gibson (1966) andE. J. Gibson (1969). As Bower (1966)

suggests, infants may begin byperceiving invariants in stimuli andsequences of stimuli rather thanlearning how to perceive byprogressing from simple percepts tocomplex ones, in the way we learnelementary geometry at school. On thepresent view, the perception ofinvariants would be an economynecessitated by the enormousinformation potential of the visualarray and the limited cognitivecapacity of the infant. Perceptuallearning would accordingly take theform of developing perceptualhypotheses that combined economywith the discrimination redundancynecessary for particular situations.Variations in the achievement of sucha compromise would seem tocorrespond naturally with differencesin perceptual or cognitive style such asthose distinguished by Barron (1952).

REFERENCESATTNEAVE, F. Some informational aspects

of visual perception. PsychologicalReview, 1954, 61, 183-193.

ATTNEAVE, F. The determination ofperceived tridimensional orientation byminimum criteria. Perception &Psychophysics, 1969, 6, 391·396.

BAR RON, F. Personality style andperceptual choice. Journal of Personality,1952,20, 385-40l.

BOWER, T. G. R. The visual world ofinfants. Scientific American, 1966, 215,80-92.

CLARK, W. C., SMITH, A. H., & RABE, A.Retinal gradient of outline as a stimulusfor slant. Canadian Journal ofPsychology, 1955,9, 247-253.

CLARK, W. C., SMITH, A. H" & RABE, A.Retinal gradients of outline distortionand binocular disparity as stimuli forslant. Canadian Journal of Psychology,1956, 10, 77-8l.

CORSINI, D. A., & PICK, H. L. The effectof texture on tactually perceived length.Perception & Psychophysics, 1969, 5,353-356.

DEMBER, W. N. Psychology of perception.New York: Holt, Rinehart, & Winston,1965.

EV ANS, S. H. Redundancy as a variable inpattern perception. PsychologicalBulletin, 1967, 75, 18-26.

FORGUS, R. H. Perception. New York:McGraw-Hill, 1966.

GARNER, W. R. Uncertainty and structureas psychological concepts. New York:Wiley, 1962.

GARNER, W. R. To perceive is to know.American Psychologist, 1966, 21, 11-19.

GARNER, W. R. Good patterns have fewalternatives. American Scientist, 1970,58,34-42.

GIBSON, J, J. The perception of the visualworld, Boston: Houghton-Mifflin, 1950a.

GIBSON, J. J. The perception of visualsurfaces. American Journal ofPsychology, 1950b, 63, 367-384.

GIBSON, J. J. The senses considered asperceptual systems. London: GeorgeAllen & Unwin, 1966.

GIBSON, E. J. Principles of perceptuallearning and development, New York:Appleton-Century-Crofts, 1969.

GREGORY, R. L. Eye and brain. London:World University Library, 1966.

HOCHBERG, J. E., & McALISTER, E. AQuantitative approach to figural"goodness." Journal of ExperimentalPsychology, 1953,46,361-364.

HOLWAY, A. H., & BORING, E. G.Determinants of apparent visual size withdistance variant. American Journal ofPsychology, 1941, 54, 21·37.

ITTELSON, W. H., & KILPATRICK, F. P.Experiments in perception. ScientificAmerican, 1951, 185, 50·55.

JONCKHEERE, A. R. A test of significancefor the relation between m rankings and kranked categories. British Journal ofStatistical Psychology, 1954, 7, 93-100.

KENDALL, M. G. Rank correlationmethods. London: Griffin, 1948.

LANGDON, J. The perception of a changingshape. Quarterly Journal of ExperimentalPsychology. 1951,3, 157-165.

MICHOTTE, A. The perception of causality.London: Methuen, 1963.

THOULESS, R. H. Phenomenal aggressionto the real object. II. British Journal ofPsychology, 1931, 22, 1·30.

VICKERS, D. Theories and experiments onvisual discrimination and the perceptionof visual depth. Unpublished dissertationsubmitted for the degree of PhD in theUniversity of Cambridge, 1967.

WALLACH, H., & O'CONNELL, D. N. Thekinetic depth effect. Journal ofExperimental Psychology, 1953, 45,205-217.

(Accepted for publication December 12,1970.)

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