the status of the minimum principle in the theoretical

Psychological Bulletin1985, Vol. 97, No. 2, 155-186

Copyright 1985 by the American Psychological Association. Inc.0033-2909/85/$00.75

The Status of the Minimum Principle in the TheoreticalAnalysis of Visual Perception

Gary HatfieldDepartment of PhilosophyJohns Hopkins University

William EpsteinUniversity of Wisconsin—Madison

We examine a number of investigations of perceptual economy or, more specifically,of minimum tendencies and minimum principles in the visual perception ofform, depth, and motion. A minimum tendency is a psychophysical finding thatperception tends toward simplicity, as measured in accordance with a specifiedmetric. A minimum principle is a theoretical construct imputed to the visualsystem to explain minimum tendencies. After examining a number of studies ofperceptual economy, we embark on a systematic analysis of this notion. Weexamine the notion that simple perceptual representations must be defined withinthe "geometric constraints" provided by proximal stimulation. We then take upmetrics of simplicity. Any study of perceptual economy must use a metric ofsimplicity; the choice of metric may be seen as a matter of convention, or it mayhave deep theoretical and empirical implications. We evaluate several answers tothe question of why the visual system might favor economical representations.Finally, we examine several accounts of the process for achieving perceptualeconomy, concluding that those which favor massively parallel processing are themost plausible.

The notions of "simplicity" and "economy"have been used in varied contexts within thesciences (see Sober, 1975). It was a common-place of classical physics and astronomy that"nature acts by the simplest means." Euler,Lagrange, Hamilton, and others have shownthat the central equations of mechanics canbe formulated isoperimetrically (in terms ofmaximum/minimum solutions). In a broadervein, methodologists have proposed that sci-entists proceed in accordance with the prin-ciple of parsimony, which holds that of twotheories with equal empirical adequacy, thesimpler theory should be chosen. A centuryago Mach (1883/1960, 1919) referred thisprinciple to a psychological preference of thescientific investigator for economy of thought.Finally, psychologists have found a tendency

Support for this research was provided by a fellowshipawarded to Gary Hatfield by the American Council ofLearned Societies and by a grant to William Epsteinfrom the Wisconsin Alumni Research Foundation. Theauthors express their gratitude to W. Anderson, J. Cutting,L. J. Daston, H. Egeth, J. E. Hochberg, S. M. Kosslyn,T. J. Sejnowski, and E. Sober for their comments, criticism,and discussion of various drafts.

Requests for reprints should be sent to Gary Hatfield,Department of Philosophy, Johns Hopkins University,Baltimore, Maryland 21218.

in perception toward phenomenal simplicityand regularity.

It is only natural that connections shouldbe made among these diverse concerns withsimplicity by virtue of their common label.However, as useful as it may be heuristicallyto see these diverse concerns as manifestationsof a single principle of economy, one mustkeep in mind that what is meant by "sim-plicity" in a scientific context is closely relatedto how one measures it. A general result ofphilosophical treatments of simplicity is thediscovery that simplicity metrics (e.g., formeasuring the simplicity of theories or otherdescriptions of the world) are highly sensitiveto oftentimes arbitrary terminological con-ventions within a theoretical or other descrip-tive vocabulary; no universally applicablesimplicity metric is close to formulation(Goodman, 1972). Given the current meansfor measuring and comparing instances of"simplicity," it is possible that simplicity asmanifested in, say, mechanics and perceptionare merely analogically related. Sweepingclaims that see in all of the mentioned phe-nomena the operation of a single MinimumPrinciple must be treated with great caution.

Within the psychology of perception, di-verse theoretical approaches have led to dif-

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156 GARY HATFIELD AND WILLIAM EPSTEIN

fering conceptions of perceptual economyitself, which may be grouped into three cat-egories. The first category is the notion thatperceived objects will tend toward phenomenalsimplicity: All else being equal, the object ofthe perceptual experience will have, for ex-ample, the simplest shape possible. This no-tion is connoted by the terminology of "goodform" or "Pragnanz." The second categoryis descriptive economy: Here the idea is thatthe objects of perceptual experience will besuch that they can be described using thefewest predicates in a given language (perhapsEnglish, but more likely a symbol systemdeveloped especially for describing percepts).This conception of simplicity in perceptionis often characterized as "informationaleconomy." Finally, there is the notion ofeconomy of process: All else being equal, theobject of perceptual experience will be theone that results from the most economicalinternal process. This emphasis is reflectedin references to "minimum processing load,"but also in the Gestalt notion of pragnantphysiological processes. Our approach to thetopic of perceptual economy will emphasizesimplicity in the phenomenal organization ofperceived objects and events, although anyapproach to this topic must take up economyof description and economy of process.

Investigations during the past three decades(Hochberg & McAlister, 1953, to Buffart,Leeuwenberg, & Restle, 1983) have shownthat perceived objects exhibit a tendency to-ward minimum diversity and change or to-ward maximum regularity and simplicity (asmeasured by various metrics of simplicity).We label these diverse psychophysical findingsthat perception tends toward economy orsimplicity instances of various minimum ten-dencies. A number of studies have been con-ducted to establish empirically whether per-ception in fact exhibits minimum tendencies,without seeking to establish any particularexplanation of such tendencies (e.g., Hochberg& Brooks, 1960; Hochberg & McAlister,1953).

In addition to studies motivated chiefly byempirical goals, a number of investigatorshave sought to develop an explanation of theobserved minimum tendencies. Two types ofexplanatory problems arise: (a) Why does the

visual system operate in such a way thatperceptual economy is achieved? (b) Howdoes the visual system achieve this economy?The first question concerns the origin ofminimum tendencies, and it seeks an answerwithin a broad theoretical approach to theexplanation of the behavior of the perceptualsystem. The second question concerns theactual processes that result in the manifesta-tion of minimum tendencies, and it requiresan answer in terms of specific processingmechanisms.

Regarding the origin of minimum tenden-cies, some investigators have suggested thatthese tendencies reveal the operation of acardinal principle of perceptual processing.As illustration, Hochberg (1964) suggestedthat a minimum principle could provide thefoundations of a general psychophysics ofspace, and thus yield general explanations ofform and depth perception. This suggestionis reminiscent of the position of the Gestaltpsychologists, who believed that a minimumprinciple could provide a unified explanationof a broad range of empirical findings, in-cluding figure-ground organization, shapeperception, and depth perception (Koffka,1935, chap. 4). Other investigators admitminimum tendencies at the empirical levelbut do not seek their explanation in a generalminimum principle. Hochberg (1974, 1981)and Perkins (1976) attributed minimum ten-dencies to the operation of a likelihood orhypothesis-formation principle. Perkins ar-gued that minimum solutions are favoredbecause they are a "good bet" (a likely hy-pothesis) about the actual properties of objectsin the environment.

The goals of this article are threefold. Thefirst is to gather the various strands of recentempirical investigation into minimum ten-dencies for purposes of comparison and eval-uation, with an eye toward discerning com-mon approaches and common problems. Lo-cal criticisms are presented along the way.Our second goal is to explore the theoreticalstatus of minimum tendencies and proposedminimum principles. Here we engage in sys-tematic critical analysis. Concern with theo-retical formulations leads to our third goal:consideration of the chief types of processmodels proposed to account for minimization.

MINIMUM PRINCIPLES IN THE ANALYSIS OF VISUAL PERCEPTION 157

Investigation of Minimum Tendenciesand Minimum Principles inContemporary Psychology

Any evaluation of minimum tendenciesand the minimum principles in contemporarypsychology must begin with their empiricalstatus. Minimum tendencies have been ex-amined in a variety of empirical settings,including the perception of form, depth, andmotion configurations. The theoretical ques-tions suggested by various attempts to inves-tigate minimum tendencies empirically arenoted as they arise.

W X Y ZFigure 1. The Kopferman cubes. (Pattern W is onlyrarely seen bidimensionally, Pattern 2 more than half thetime. From "A Quantitative Approach to Figural 'Good-ness' " by J. Hochberg and E. McAlister, 1953, Journalof Experimental Psychology, 46. p. 363. Copyright 1953by the American Psychological Association. Reprintedby permission.)

Perception of Form and Depth

Hochberg and McAlister. Hochberg andMcAlister (1953) should be credited withinitiating the efforts to place the notoriouslyqualitative notion of figural "goodness" on aquantitative footing (see also Attneave, 1954).These investigators used line drawings whichcould give rise to alternative perceptual or-ganizations, and they assumed that the "bet-ter" of these organizations would be perceivedmore often or for a longer span of time thanthe alternatives. They predicted that "good-ness," as defined by response frequency, wouldbe correlated with informational economy asdetermined by their postulated metric of"information": "the less the amount of in-formation needed to define a given [percep-tual] organization as compared to other al-ternatives, the more likely that the figure willbe so perceived" (Hochberg & McAlister,1953, p. 361). A test of this formulationrequires an objective specification of the in-formational economy of rival percepts andthe means for ascertaining the likelihood ofthese percepts.

Hochberg and McAlister's (1953) treatmentis best understood by considering the fourpatterns they adopted for study (see Figure1). Each drawing may be seen either as abidimensional patterned hexagon or as a tri-dimensional cube. The authors proposed thatthe four patterns differ with respect to theamount of information needed to specifyeach as a bidimensional pattern. The amountof information was determined by countingline segments, angles, and points of intersec-

tion. For example, to describe Pattern W asa bidimensional pattern the authors listed thelength of 16 line segments, 25 angles, and thelocus of 10 points of intersection. In contrast,the corresponding numbers for Pattern Z areonly 12, 17, and 7. According to this descrip-tion, Pattern W should be less likely to appearbidimensional (more likely to appear tridi-mensional) than Pattern Z (assuming that theamounts of information for the two tridimen-sional cubes are equivalent). Estimates of thelikelihood of the alternative percepts werederived from a two-alternative forced-choicetask. Each stimulus pattern was presented for100 s. Tones were presented at random inter-vals during the presentation period. On theoccasion of each tone, the observer reportedwhether the pattern appeared bidimensionalor tridimensional. Hochberg found that thebidimensional appearance was reported forPattern W for only 1.3% of the probes,whereas Pattern Z elicited reports of bidi-mensionality for 60.0% of the probes. Theseresults, as well as the responses to Patterns Xand Y, are taken as evidence for a minimumtendency in the perception of form.

It should be noted at this juncturethat Hochberg and McAlister's procedureavowedly did not produce a direct test of thetendency toward informational economy (oneform of minimum tendency). This should beapparent from the fact that figural goodnessis measured in terms of response frequency(Hochberg & McAlister, 1953, p. 189). It isassumed that the perceptual system exhibitsa preference for economy. The investigators'task is then to devise a measure of perceptual


economy—in this case, in terms of "amountof information"—that accords with the per-ceptual system's assumed preference forgoodness or simplicity. The investigators thusare actually testing their own measure ofeconomy for its psychophysical adequacy. Analternative strategy would be to assume thatone's measure of economy is adequate (or toargue that it is so on intuitive grounds, or onthe basis of predictive success and generality)and then to use response frequency to assesswhether the visual system actually exhibits apreference for economy. In either case, aprior assumption must be made, either aboutthe visual system's preference for economyor about the adequacy of one's metric ofsimplicity. The lack of a direct test of mini-mum tendencies (and of the minimum prin-ciple) is a common feature of empirical workin this area.

The measure of informational economy iscrucial to any empirical test of the minimumtendency. Although Hochberg and McAlister'sprocedure for assessing informational econ-omy may seem straightforward, its applicationin this study raises questions. Ostensibly, theintention is to measure the informationaleconomy of alternative perceptual represen-tations. In actuality, the procedure is appliedonly to the stimulus patterns. This may seeminconsequential inasmuch as the bidimen-sional appearances of the four patterns maybe assumed to have the same properties asthe patterns and no others. Thus, when theinformation needed to describe the drawingis listed, it is plausible to assume that theinformation needed to describe its appearanceas a bidimensional form has also been spec-ified. However, no application to the drawingcan measure the information when the per-ceptual representation is of a tridimensionalform, because by definition the latter is char-acterized by properties that are absent in thedrawing (e.g., relative orientation of the facesand internal depth). In fact, the procedure,applied to the drawings, in principle cannotuncover any differences among the tridimen-sional representations of the patterns. Inas-much as information is measured simply bycounting lines, angles, and intersections inthe stimulus drawings and because all of thedrawings are projections of the same positedcube, the procedure will necessarily yield the

same measure of information for the respec-tive tridimensional representations of the fourpatterns. Indeed, for the comparison of thefrequency of reports of bidimensionalityamong patterns to be a valid test of theminimum tendency, the authors must assumethat descriptive information is constant forthe patterns perceived as tridimensionalforms. No account is taken of the fact thatthe drawings present four phenomenally dis-tinct views of a tridimensional cube, andtherefore no assessment is made of the relativesimplicity of these distinct percepts.

Although Hochberg and McAlister did notseek to provide an explanation of their findingof a tendency toward perceptual economy,their study of alternating perceptual organi-zations suggests one conceptualization of theprocess by which perceptual economy isachieved. Consider the finding that the reportsfor Pattern Y were almost equally dividedbetween bi- and tridimensional responses.One inference which may be drawn from thisresult is that the perceptual system operatesby generating various perceptual representa-tions compatible with the stimulus, assessingthe informational content of each and thenselecting the representation which passes thetest of economical representation. When, asfor Pattern Y, the alternatives are equallyeconomical, a selection is made at random.This process model may be appropriate onlyfor patterns that are perceptually multistable,or it may be proposed as a general model,applicable whenever more than one perceptualorganization is compatible with proximalstimulation, even if only one of these patternsis normally "selected" for perception.

Buffart, Leeuwenberg, and Restle. Re-cently, Buffart, Leeuwenberg, and Restle(1981) applied the "law of simplicity" or"minimum principle" in the analysis of pat-tern perception. Their approach shows a kin-ship with that of Hochberg and McAlister(1953) in its use of feature counting to mea-sure simplicity. However, Buffart et al. (1981)committed themselves to going beyond thepsychophysical investigation of minimumtendencies to investigate the minimum prin-ciple as a cardinal explanatory principle inthe perception of form.

Buffart, Leeuwenberg, and Restle (1981)examined the perceptual tendency, remarked


earlier by Gestalt psychology, of the percipientto complete a figure, part of which is occludedby another. Buffart et al. designed 25 patterns,each consisting of a square and one or moreadditional bidimensional figures. Four ex-amples are shown in Figure 2. For eachpattern, the subject was asked to trace thecontour of the figure or figures that accom-panied the square and, for those cases inwhich the square overlapped the other figure,to be especially accurate in drawing angles(if any) hidden behind the square. The ques-tions of interest were: (a) When will thesubject report a completion in contrast to anonoverlapping mosaic? (b) What completionwill be made? Buffart et al. argued that theanswer to both questions is provided byLeeuwenberg's (1967, 1971, 1978) codingtheory and the minimum principle.

Coding theorists are committed to the ideathat perception is a matter of interpretation.Each interpretation corresponds to a "prim-itive code," which is a set of symbols describ-ing the form of an object within the language

Figure 2. Four of the 25 figures used in the figuralcompletion studies of Buffart, Leeuwenberg, and Restle(1981). (Information loads as computed by the investi-gators favored a completion for A and C, a mosaic forD, and assigned equal loads to each interpretation for B.A and C were chosen as completions by 30 out of 30adult subjects; D was always given a mosaic interpretation.B was interpreted as a mosaic by 5 out of 30 subjects,receiving a completion by the other 25 subjects. From"Coding Theory of Visual Pattern Completion" by H.Buffart, E. Leeuwenberg, and F. Restle, 1981, Journal ofExperimental Psychology: Human Perception and Perfor-mance, 7, pp. 242-243. Copyright 1981 by the AmericanPsychological Association. Reprinted by permission.)

of coding theory. These symbols are relatedto various perceptual configurations accordingto code semantics, which provide rules forconstructing a primitive code that symbolizesthe relative placement of lines and anglesneeded to specify a given form. The idea ofsimplicity is introduced in accordance withvarious syntactic operations on these primitivecodes. The codes are first "reduced" to sim-plest form via a set of formal operations onthe code symbols (e.g., conventional operatorsto indicate iteration of symmetry). Relativesimplicity among alternative codes for thesame ambiguous display is then calculatedby comparing the number of independentparameters within the code needed to specifyfully each perceptual "interpretation." In thewords of the authors:

Coding theory consists of the idea that a given displaymay result in any of several interpretations or codes, thata primitive code can be reduced to a shorter form, thateach such code has an information load that consists ofthe number of independent parameters it uses, and thehypothesis that the perceptual system tends to use thecode with minimum information load, according to thelaw of simplicity or minimum principle.

The law of simplicity, within coding theory, is this:The perceptual system reduces information load andunder ideal conditions will arrive at the interpretationhaving the lowest information load. This, in a naturalsense, is the interpretation having the simplest code andcan therefore be thought of as the simplest interpretation.(Buffer! et al., 1981, pp. 250-251)

As an example of the operation of codingtheory, consider Figure 3. The upper portionshows two interpretations of one of the ex-perimental figures (our Figure 2A). Figure3A is a "figural completion"; 3B is a "mosaic"interpretation. As defined within code se-mantics, a stands for a right angle, and X isa unit line segment. To derive a primitivecode, one stipulates a beginning point anddirection, and then lists the line segmentsand angles that make up the figure. Theprimitive code for Figure 3A, derived byfollowing the path indicated in the figure, is\aX\a\\a\\a\a\ot\\a\\aX\a\. By count-ing the number of symbols, one determinesthe information load; in this case / = 25. Bytaking advantage of the repeating or iteratedelements, such as XaX, the primitive codemay be reduced to the following expression:4*[X«X]2*[aX]3*[XaX], for which 7 = 1 1(counting only the iteration numerals and the


1 JA B

Figure 3. Two possible interpretations of Figure 2A. (Theupper figures indicate completion (A) and mosaic (B)interpretations. The lower figures show coding paths. Thecircles indicate where coding begins, and the arrowsindicate the direction in which coding proceeds. Theletters a and X indicate a right angle and a unit linesegment. The primitive code is constructed by listing theelements a and X in order as one proceeds along thecoding path. From "Coding Theory of Visual PatternCompletion" by H. Buffart, E. Leeuwenberg, and F.Restle, 1981, Journal of Experimental Psychology: HumanPerception and Performance, 7, p. 246. Copyright 1981by the American Psychological Association. Reprintedby permission.)

elements that stand for angles or line seg-ments; neither the reducing operation itselfnor the "conventional"—nonreferring—symbols are counted in the information loadof a reduced code). Further syntactic opera-tions of a conventional (and intricate) naturereduce the information load to four. ForFigure 3B, the primitive code has / = 26 andmay be reduced to 7 = 6. The completioninterpretation is therefore predicted. (It maybe noted that even though the lower squarein Figure 3A is perceived as being behind theupper square, the code proceeds from squareto square without registering a change indepth, thereby treating the squares as co-planar.)

Buffart et al. (1981) evaluated the infor-mation load of the reduced code for thecompletion and mosaic interpretations. Ofthe 25 figures, 16 had completions with lowerinformation loads than the mosaic interpre-tation, 7 had equal information loads, and 2had more economical mosaic interpretations.For the first set, it was found that 96% of the

subjects produced completions; when the in-formation loads were equal, 45% of the sub-jects produced completions; and for the latterset, only 10% of the subjects produced com-pletions. Comparable results were found withsubjects who were graduate students and re-searchers and with secondary school students.Buffart et al. claimed complete success fortheir combination of coding theory and min-imum principle.

As in the case of Hochberg and McAlister's(1953) study, Buffart et al. strictly speakinghave not made a direct test of the minimumprinciple. They have shown that, on the basisof their coding theory and their posited mea-sure of minimum information load, they canpredict subjects' responses. Yet at least twoconcerns are conflated in this test: (a) whetherthe perceptual system operates in accordancewith a minimum principle, and (b) if it does,whether this principle is mirrored in theformal apparatus of Buffart et al. There can,however, be no independent test of (a). Theformal apparatus mentioned in (b), togetherwith the intuitions of past and present exper-imenters, must serve as the test of (a) at thesame time that (b) itself is tested. This not-uncommon situation in the logic of experi-mentation is by no means fatal, and it maybe met by application of converging opera-tions and by the long-run empirical fruitful-ness of a hypothesis. However, the situationis disconcerting when competing explanationsfor the empirical outcome are available. Buf-fart et al. did in fact compare their explana-tion with other accounts of figural completion,including both the familiarity of the com-pleted figure and a reliance on local cues foroverlap, and they claimed relative superiorityfor their theory. Their case is compelling forthe comparison with "local-cue" theory. Ac-cording to a local-cue approach, a local fea-ture such as a T signals overlap. In agreementwith the earlier findings of Donnerstein andWertheimer (1957), Buffart et al. found thatthe T subpattern may be associated with acompletion or a mosaic depending on theglobal context. (In Figure 2A the Ts areassociated with overlap and hence completion;in Figure 2D they are not.) It is more difficultto evaluate the comparative claim of thefamiliarity account, because no measure offamiliarity nor any actual tests of familiarity


are provided. Appeal is made to intuitions,which clearly can vary, because the authorscite a square with one corner lopped off as acase of an unfamiliar figure (the "completion"interpretation of Figure 2C).

A review of the computations required bythe coding theory to arrive at the informa-tional loads of the alternative constructionsreveals a long serial routine. Although thismay be an apt characterization of the calcu-lations required of the investigator, Buffart etal. intentionally left open the question ofwhether this serial process is carried out bythe perceptual system. In fact, they offeredassurance that "theoretical work on the pro-cess of coding is in progress and does notbuild on the hypothesis that the process isprimarily one of generating a code item byitem" (Buffart et al., 1981, p. 272). Nonethe-less, they are committed to the notion thatthe visual system interprets stimulus displaysin terms of primitive codes that embodyfeatures akin to those found in coding theory,and that these are, by some nonserial processas yet unspecified, reduced to simplest formand compared for informational load.

In a subsequent discussion, BufFart, Leeu-wenberg, and Restle (1983) have comparedthe perceptual process to one of hypothesistesting in the tradition of Gregory (1974).They assumed that a registered pattern ofstimulation induces the projection of severalperceptual hypotheses (potential interpreta-tions), out of which the simplest (as deter-mined by information load) is chosen forverification against "sensory constraints" (i.e.,against a more thorough checking of regis-tered stimulation). During the verificationprocedure additional hypotheses may be gen-erated, among which the simplest will againbe chosen for further verification (Buffart etal., 1983, p. 996). Hence, these coding theo-rists have proposed that perceptual hypothesesare generated, selected for minimum-infor-mation load, and tested for accuracy, allwithin an internal coding system.

Attneave and Frost. Vickers (1971) pro-posed that the impression of depth elicitedby gradients of optical texture is due to theoperation of a principle of informationaleconomy. He contended that economy isachieved by assigning the elements of thetexture gradient to that slanted surface on

which they would form a uniform (ungraded)texture. On the basis of his observations,Vickers concluded that the minimum prin-ciple provides a better account of the opera-tion of texture gradients than either traditionalcue theory or Gibson's (1950, 1966) directtheory.

Like Vickers, Attneave and Frost (1969)advocated the "minimum principle" as anaccount of the monocular perception of tri-dimensional configurations which stands asan alternative to both cue theory and Gibson'stheory. Attneave and Frost's formulation ofthe minimum principle has much in commonwith Hochberg and McAlister's proposal:"[Monocular] depth perception is determinedby tendencies to minimize the variability ofangles, lengths, and slopes" (Attneave & Frost,1969, p. 394), with the qualification that thetridimensional organization is "within the setof permissible tridimensional interpretationsof the [optical] projection" (p. 391). Thenotion of "permissible" interpretations istreated by assuming that the rules of per-spective are implicit in an analog mediumrepresenting physical space (p. 395), withinwhich perceptual representations develop.

Attneave and Frost (1969) tested their ver-sion of the minimum principle by examiningthe extent to which subjects perceive mon-ocularly viewed line drawings of parallelepi-peds as tridimensional when this organizationminimizes the variability among lines, slopes,and angles. Consider two of the three typesof display studied by Attneave and Frost(Figure 4). In both cases, the subject mon-ocularly inspected the drawings and aligneda binocularly viewed rod so that the rodappeared to be a colinear extension of oneof the edges of the perceived tridimensionalparallelepiped; this procedure was repeatedfor each of the three leading edges of the twodrawings in Figure 4, plus a third (interme-diate) drawing. In one case the standarddrawing was an orthogonal projection of aparallelepiped such that in the drawing thelines representing edges were of equal lengthand opposite edges were parallel, that is,equal in slope (Condition 1; actually, ninevariant drawings were used in each condition).In the other case, the standard drawing wasa polar projection of a cube (Condition 3).In this drawing, lines representing edges were


CONDITION 1 CONDITION 3

Figure 4. Illustration of two of the three conditions usedby Attneave and Frost (1969, Fig. 2). (In Condition 2[not pictured], sides were parallel as in Condition 1, butline lengths varied in accordance with perspective. Ninevariant drawings [projections from different points ofview] were used in each condition. From "The Deter-mination of Perceived Tridimensional Orientation byMinimum Criteria" by F. Attneave and R. Frost, 1969,Perception & Psychophysics, 6, p. 392. Copyright 1969by Psychonomic Society, Inc. Reprinted by permission.)

unequal, and the slopes were also unequal,as in a conventional perspective rendering ofa cube. This latter drawing affords a projectionconsistent with viewing a real cube monocu-larly from a selected reference point. Toassess the minimum principle, Attneave andFrost calculated the slant-in-depth of thethree leading edges of the hypothetical par-allelepiped having the minimum variabilityamong the angles, lengths, and slopes of itsedges. They predicted that calculated or hy-pothetical slant-in-depth would be positivelycorrelated with the subjects' judged slant.More important, the relationship should bestronger in the second case than in the firstcase. In the second case, conformity with thehypothetical slant-in-depth will equalize thethree variables. In the first case, two of thethree variables—slope and length—are al-ready equal in the picture plane; taking thefigure in depth would make the angles equalbut render the lines and slopes unequal. Theresults reported by Attneave and Frost (1969,p. 394, Figures 3a and 3c) conformed to bothof these predictions.

Although Attneave and Frost treated theminimum principle in terms of minimizingvariability among the angles, lengths, andslopes of the posited object, Attneave (1972)favored a view that interprets the minimumprinciple in conjunction with the idea that

physical space is represented perceptually asan approximately isotropic analog space. Theminimum principle then operates not onlyto simplify the relationships among the partsof each perceived object (as in the previouscase), but also to simplify the relationshipbetween perceived objects and an underlyingreference system in this analog medium. Att-neave tentatively postulated:

a mechanism that reads momentary tridimensional valuesof length, slope, angle, and the like out of the spatialrepresentational medium and computes from them anintegrated measure of 'goodness' or simplicity that is fedback as a 'hot-cold' signal guiding the tridimensionalrepresentation to the simplest state consistent with theconstraints of the input, at which the system wouldachieve stability. (Attneave, 1972, p. 302)

The envisioned mechanism is teleological inthe unobjectionable sense that it is guidedtoward a simple end-state by local decisionsalong the way. This formulation implies aprocess that is unlike the model we suggestedfor Hochberg and McAlister's study. Ourinterpretation of the Hochberg-McAlisterdata suggested a process in which the mini-mum principle acts as a selective rule forchoosing among fully formed alternatives, allof which are generated for assessment. Incontrast, Attneave's formulation suggestedthat the minimum principle guides the con-struction of a single (most economical) rep-resentation, consistent with input and thelaws of projective geometry. (Attneave, 1972,imputed the latter to his postulated spatialrepresentational medium; see p. 302.) It is asif the evolution of the perceptual represen-tation is directed in progress; the variousdecisions required as this evolution progressesare resolved by the minimum principle.

Attneave and Frost (1969) contended thattheir results are more consistent with a versionof the minimum principle than with cuetheory or direct theory. However, as was thecase with Vickers' study, their experimentwas not designed to decide among rival ac-counts, and their assessment of the relativemerit of the minimum principle did notdepend upon differing empirical predictionsderived from the competing accounts. As inthe case of Buffart et al. (1981), the charac-terization of the competing theories is onethat adherents of those theories would ques-tion. Cue theory is understood as involving


an item by item list of the relation betweenproximal lines and angles and possible distalslopes, making no mention of the inferencerules or algorithms common to cue theory.Direct theory is characterized as involving acumbersome cognitive apparatus for comput-ing and solving the higher order variables ofthat theory, totally at variance with Gibson's(1966) notion of "information pick-up." In-deed, in a subsequent evaluation of this ex-periment Attneave (1972) remarked: "Theforegoing comparisons show, if nothing more,the extraordinary difficulty of experimentallyconfirming or discontinuing a Pragnanz the-ory as opposed to its alternatives . . . I doubtat this point that any experiment of thepresent kind is going to settle the issue in adecisive way" (p. 300).

Perkins. Perkins (1976; see also Perkins& Cooper, 1980) investigated two issues re-lating to Pragnanz in perception. First, hesought to show that the imposition of Prag-nanz or "good form" in the interpretation ofvisual stimuli is constrained by (or is inaccordance with) the rules of projective ge-ometry (see also Perkins, 1972). Second, heinvestigated the notion that the attributionof good form to visual stimuli is a "goodbet" (i.e., that a Pragnanz assumption wouldguide the visual system toward accurate per-ception of form).

The notions of Pragnanz and of geometricconstraints used by Perkins (1976) may beillustrated in conjunction with Figure 5. For

the present purpose, restrict attention to An-gles L and R and potential axis of SymmetryS (as illustrated in Figure 5a and applied tothe others), and regard each of the shapes asa tridimensional object. A Pragnanz interpre-tation would consist in attributing right anglesto the corners of this object, or in attributingto the object symmetry about Line S. ForFigure 5b, casual inspection may reveal thatthree alternative tridimensional organizationsare perceptually favored: (a) the left endappears rectangular, with Angle L perceivedas a right angle; (b) the right end appearsrectangular, and R is perceived as a rightangle; and (c) the object appears symmetricalabout Line S, with Angles L and R equal toone another. Each of these interpretations isconsistent with projective geometry; that is,each of these posited tridimensional objectscould project Figure 5b in accordance withconventional projective geometry. By contrast,for Figure 5g geometric constraints allow arectangular interpretation of Angle R, but donot permit this attribution to Angle L, noris symmetry permitted. For any of the figures,once a given angle is decided, projectivegeometry can be applied across the rest ofthe figure to determine all sides and anglesof the posited tridimensional object.

Within this framework, Perkins (1976)tested three predictions: (a) perceptions ofsymmetry or rectangularity will occur morefrequently than chance, (b) such percepts willoccur more frequently when they are consis-

Figure 5. The stimulus shapes used by Perkins (1976, Fig. 1). (Angles L and R are defined on each shapeabout the potential axis of symmetry indicated by the dotted line in View a. From "How Good a Bet isa Good Form?" by D. Perkins, 1976, Perception, 5, p. 394. Copyright 1976 by Pion Ltd. Reprinted bypermission.)


tent with protective geometry, and (c) subjects'estimates will approximate the geometricallycorrect values; that is, when a subject reportsrectangularity or symmetry in a shape, theestimates of the nonright angle or the sym-metric angles will accurately reflect the valuedetermined via projective geometry. Eightsubjects were asked to provide verbal estimatesof Angles L and R (to within 5°) in Figure5b-5h, under instructions to view each draw-ing as a depiction of a tridimensional object.Although the data did not conform to thesepredictions in every instance, overall agree-ment with the predictions was good. Subse-quent work by Perkins (1982) tested perceiv-ers' accordance with geometric constraintson a wider variety of stimuli, and indicatedthat perceivers are flexible, if sloppy, geome-ters.

Perkins viewed the process of generatingindividual percepts as a directive one, involv-ing serial application of the Pragnanz hypoth-esis by the visual system. The Pragnanz as-sumption is applied to a salient feature ofthe stimulus figure and is accepted if it isconsistent with the constraints of projectivegeometry. If the Pragnanz attribution (e.g.,this corner is right angled) is adopted, then"the implications in terms of shape, size,slant, connectivity . . . are propagated toother parts of the scene" (Perkins, 1976, p.403). The process might be repeated severaltimes until a consistent figure is discovered.

A more detailed analysis of a serial, direc-tive process as envisioned by Perkins hasbeen provided by Simon (1967) and may beillustrated by his analysis of the perceptionof the Necker Cube:

A. A scanner moves over the stimulus and detects simpleconfigurations. By "simple configurations' are meant figureslike those the Gestalt psychologists call 'good': straightlines, especially if horizontal or vertical, right angles,squares, circles, closed symmetrical forms.

B. When a simple configuration has been detected, thescanner proceeds from it to the other elements of thestimulus, providing them with simple interpretations inrelation to the initially discovered simple configuration.This process continues until an internal representationhas been constructed for the entire stimulus, or a contra-diction has been encountered.

C. If a contradiction is encountered, the interpretationis rejected and step A is repeated—usually with a differentinitial position of the scanner. (Simon, 1967, p. 4)

A Gestalt theorist would be appalled bySimon's process model. The serial, part bypart examination of features, and the incre-mental trial-and-check build up of the figureis as remote as could be imagined from theGestalt approach. (See Koffka's, 1930, pp.162-168, detailed analysis of the NeckerCube.) Nevertheless, in implying (Point B)that what is taken as a simple interpretationof any one part will condition the interpre-tation of other parts Simon adopted a holisticapproach that the Gestaltist might find agree-able. This point in Simon's exposition re-quires clarification. It implies that interpre-tation of one element constrains the interpre-tation of the other elements, thereby helpingto determine what will be accepted as asimple interpretation. This relation of con-straint would seem to reflect some unspecifiedknowledge structure that must be attributedto the visual system, and that presumablyreflects Perkins's (or Attneave's) geometricconstraints.

The directive process model just sketchedaddresses the how of Pragnanz. Perkins (1976)answered the question of why by viewingPragnanz as a working assumption adoptedby the percipient as a result of commercewith the environment, in which the perceptualsystem learns that symmetry and figuralgoodness are good bets. Assessment of thisclaim about the environment would involveconsiderable "ecological sampling" (Brun-swik, 1956) on the part of the investigator.Second, the perceptual system uses past ex-perience (prior frequency) to decide betweenequally possible Pragnanz interpretations (e.g.,rectangular and equilateral interpretations ofthe projection of a parallelepiped). It shouldbe noted that Pragnanz, in Perkins's usage,departs drastically from the Gestalt formu-lation and operates as a favored perceptualhypothesis or schema in a manner compatiblewith the type of hypothesis-testing theoryfavored by Gregory (1974).

Perkins's treatment of Pragnanz as hypoth-esis or "good bet" raises a troublesome ques-tion regarding the relationship between aminimum principle and what may be calledthe likelihood principle. The likelihood prin-ciple, considered by Helmholtz as the fun-damental rule of perceiving, is the claim that


the perceptual system constructs a represen-tation which is the most likely interpretationgiven the history of exposure to proximal-distal correlations (through learning or evo-lution). Often the most economical perceptseems intuitively also to be the most likelypercept.

As illustration, consider the DistortedRoom phenomena. In one version of thiswell-known Ames demonstration, a roomconstructed with trapezoidal floor, walls, andceiling is viewed monocularly from a positionsuch that the resulting retinal image is (ideally)equivalent to the image that would be asso-ciated with a conventional rectangular roomviewed from that position. Under these con-ditions, the room appears to be rectangular,and the perception persists even when personstake up positions in opposite corners of theroom, with the result that the persons appearto be of grossly different sizes. The rectangularappearance of the room is consistent with aminimum principle inasmuch as more infor-mation would be required to define a percep-tual representation of a Distorted Room (e.g.,the angles formed by each of the four cornersare unique and the sizes of the windowsdiffer). However, the standard explanation ofthe appearance of the Distorted Room makesno mention of a minimum principle. Theconventional explanation of these findings,which stresses the assumptions that the ob-server brings to the situation, amounts to anapplication of a likelihood principle: It ismore likely that the given retinal image wascaused by a rectangular room than by adistorted room. Granting the claims of therival accounts, we have a case of correspon-dence between the simplest and most likelyperceptual representation. Similar remarksapply to a number of Ames' other demon-strations (e.g., the Ames Chair).

The kinetic depth effect, a widely citedinstance of the perception of depth throughmotion (Braunstein, 1976; Ullman, 1979),provides another illustration of the congruencebetween the economical and the likely. Whena rotating wire-frame object is illuminatedby a point source of light behind a screen(Figure 6), the shadow cast upon the screenmight be perceived as a two-dimensionalobject changing its shape or as a stable three-

Figure 6. Successive views of a rotating wire objectprojected onto a screen. (From Introduction to Perception[p. 117] by I. Rock, 1975, New York: Macmillan. Copy-right 1975 by Macmillan. Reprinted by permission.)

dimensional object rotating about a centralaxis. Clearly, a minimum principle wouldpredict that the three-dimensional configu-ration would be perceived, rather than a two-dimensional configuration whose componentline segments are constantly altering theirlengths and angular relations. In fact, theobserver does report perceiving a rigid 3-dimensional configuration rotating in depth.Here again, as with the Distorted Room, onemight have forecast the outcome by drawingon the rule that the perceptual system isdisinclined to posit unlikely objects, in thiscase an object that continually undergoescovariations of length and angle. (Evidencepurporting to support this interpretation hasbeen presented by Rock & Smith, 1981).Once more there seems to be correspondencebetween the most economical percept andthe most likely percept.

Man. Marr (1982) and his co-workershave developed a broad theoretical approachto vision that touches on two aspects ofperceptual economy: (a) In its description ofthe structure of the environment it explicitlyrelies on assumptions that can be character-ized as Pragnanz-like, and (b) it provides aninteresting conception of a minimization pro-cess.

Marr and his colleagues have enlisted thepowerful resources of artificial intelligencetechniques to create a machine model thatsimulates the human visual system. Theirapproach consciously seeks to constrain themany possibilities for creating machine "vi-sion" programs by taking into account thefindings of neurophysiology and psychophys-ics. The primary focus of the work discussedby Marr (1982) is "early vision," which isregarded as the generation of a perceptualrepresentation prior to such cognitive activi-


ties as object recognition (cf. Marr, 1982, pp.268-269; Marr & Nishihara, 1978). The the-ory proceeds against the explicit backgroundthat the purpose or biological function ofearly vision is to engender a representationof the distal scene, including the shape, ori-entation, texture, and color of surfaces. Atthe core of this approach is the idea that thevisual system has been engineered to embodycertain "assumptions" about the physicalproperties of the environment, which underliethe processes that extract information aboutthe environment from retinal stimulation.

Although Marr (1982) presented his theoryindependently of any explicit discussion ofPragnanz, the assumptions that are assignedto the visual system generally attribute Pra'g-nanz-like qualities to the environment. In thefirst assumption, the Pragnanz quality ofsmoothness is conjoined with elaboratenessof articulation: "The visible world can beregarded as being composed of smooth sur-faces having reflectance functions whose spa-tial structure may be elaborate" (Marr, 1982,p. 44). A later assumption attributes homo-geneity within various scales of this elaboratestructure: "The items generated on a givensurface by a reflectance-generating processacting at a given scale tend to be more similarto one another in their size, local contrast,color, and spatial organization than to otheritems on that surface" (Marr, 1982, p. 47). Afurther assumption seeks object boundarieson the basis of common fate: "If direction ofmotion is ever discontinuous at more thanone point—along a line, for example—thenan object boundary is present" (Marr, 1982,p. 51). Finally, in the case of stereopsis it isassumed that "disparity varies smoothly al-most everywhere" (Marr, 1982, p. 114). Fromthese basic assumptions and others, Marr andhis co-workers have developed accounts ofedge detection (Marr & Hildreth, 1980), ste-reopsis (Grimson, 1981; Marr, 1982; Marr &Poggio, 1976, 1979) and of the perception ofmotion (Ullman, 1979), shape (Grimson,1981; Horn, 1977; Ikeuchi & Horn, 1981),surface texture, brightness, lightness, and color(Marr, 1982). Interestingly, these Pragnanzassumptions are imputed to the visual systemas assumptions about the actual structure ofthe physical environment. If Marr's theory iscorrect, the operation of the visual system

does indeed depend on the assumption thatthe world has Pragnanz qualities (i.e., thatobjects in the typical environment have phys-ical properties that are characterized bysmooth variation, homogeneity, and expansesof rigid structure).

Marr's work has provided an interestingconception of the minimization process. Marrand Poggio (1976), drawing on Julesz (1971,1974), developed a variant of the directiveprocess models previously discussed. Juleszproposed a "cooperative" model of globalstereopsis (i.e., of the process by which sur-faces-in-depth are determined from binoculardisparity among local features). A character-istic of cooperative models is that they dependon numerous local interactions in parallelthat "cooperate" to achieve a result. Marrand Poggio implemented a cooperative algo-rithm that computed surfaces-in-depth fromdisparity, through iterative operations on localdisparities. The operation of the algorithmincorporates the assumptions of continuityor smoothness and uniqueness (i.e., a featurefrom one image is matched with only onefeature from the other image). The algorithmis "computed" by iterative local interactionsamong "cells'" that respond to local dispari-ties, and that are connected to other "cells"in their neighborhood through either excit-atory or inhibitory connections (by adjustingthe excitatory and inhibitory connections,one can realize various distinct algorithms).

An iterative, cooperative algorithm, realizedby a net of "cells," provides an example of adirective, bottom-up process that iterativelyconverges on surfaces-in-depth possessing thePragnanz quality of smoothness. AlthoughMarr and Poggio (1979) subsequently aban-doned this algorithm for one that does notdepend on cooperativeness in its fundamentaloperation (it rather compares spatial fre-quency filterings of monocular input fromeach eye), their new model does use a coop-erative process for resolving ambiguities(Marr, 1982). In general, cooperative processesprovide a means for envisioning a directiveprocess that converges on smooth formsworking bottom-up from many local inputs.

Perception of MovementRestle. We conclude our presentation of

illustrative applications with three studies of


the perception of motion. The first is providedby Restle's (1979) reanalysis of Johansson's(1950) widely cited experiments.

Restle applied a line of analysis similar tothat of BufFart et al. (1981) previously dis-cussed and draws on Leeuwenberg's (1978)coding theory. As with BufFart et al., percep-tion is viewed as the genesis of an interpre-tation, in this case of the motion of dots onan oscilloscope. The motion patterns can giverise to various groupings of the dots, depend-ing upon how common and relative motionsare allotted (cf. Duncker, 1929). Each patterncan give rise to two or more interpretations(see Figure 7). These are described in codeswhich are reduced to simplest terms andcompared for information load. In Restle'swords:Different interpretations, when fully reduced, may endup with different information loads. The theory statesthat the observer then will perceive the simplest interpre-tation, that is, the interpretation with the minimuminformation load (Hochberg & McAlister, 1953). If twoor more interpretations have equal information load,then the display is ambiguous in practice, and either orboth interpretations may be seen. (Restle, 1979, p. 2)

As noted earlier, the general approach issimilar to the one introduced by Hochbergand McAlister (1953) in that the codes areessentially lists of the features that have to bespecified to describe a particular perceptualrepresentation. Over a large and diverse setof motion configurations, Restle found a highdegree of agreement between the interpreta-tions that were actually perceived and theranking of the various possible interpretationswith respect to their informational loads ascomputed by the investigators.

Restle mentioned briefly two mechanismsby which a perceptual system could arrive atthe minimum information load. One is bygenerating interpretations in nonsystematicorder and settling for the intepretation thatis most economical. This is an example ofthe "selective" process discussed in conjunc-tion with Buffart et al. (1981) and Hochbergand McAlister (1953). Restle was wary of thishypothesized process because of the cumber-some number of serial computations thatmust be performed and the difficulty of de-vising a means for determining the order inwhich the candidate interpretations will begenerated and tested. As an alternative, Restle

o o

o o

a b e d

Figure 7. A motion pattern (top) and three interpretations.(Dots a and b move upward, while Dots c and d movedownward; velocity is equal for all dots and varies as asine function [with highest velocity at the center of themotion path]. In Interpretation A, the four motions areperceived as independent. In Interpretation B, the dotsare perceived as two independent pairs. In InterpretationC, the whole system is perceived as moving upward, witha subunit [Dots c and d] moving downward relative tothe other subunit [Dots a and b]. Interpretation C couldbe reversed, so that the whole system moves downwardand Dots a and b move upward relative to Dots c and d.The information loads for the three interpretations, ascalculated by Restle (1979), decrease from A to B to C.Of Johansson's 14 subjects, 13 definitely reported seeingtwo pairs of dots, and 11 of these reported the motionsas a single system with two subunits. From "CodingTheory of the Perception of Motion Configurations" byF. Restle, 1979, Psychological Review, 86, p. 7. Copyright1979 by the American Psychological Association. Re-printed by permission.)

cautiously toyed with the idea of a mechanismin the spirit of that favored by the Gestalttheorists (e.g., Kohler, 1920):

168 GARY HATF1ELD AND WILLIAM EPSTEIN

It is even possible that the process of finding the minimuminformation load may resemble physical processes de-scribed by thermodynamics—though physical systemsreliably find minimum energy states, any calculations areperformed not by the system under study but by thescientist. This may be just as true in the analysis ofperception. (Resile, 1979, p. 23)

If this analogy with thermodynamics is fol-lowed closely, here is an example of a "direc-tive" process in which the minimal "solution"is found by the momentary interaction of alarge number of microprocesses. An econom-ical percept is realized as a stable state of theperceptual system, which is achieved at aminimum energy level. According to thisconception, the features of the perceptualrepresentation are presumed to be directlyrelated to the characteristics of a distributedprocess in the nervous system (e.g., an eco-nomical motion path is "computed" by adistributed neural process achieving equilib-rium at a minimum energy level). Restleregarded it as an advantage of this tentative,qualitative account that it might achieve theresults described by coding theory withoutengaging in the cumbersome serial calcula-tions that must be performed by the investi-gator in applying the codes.

A third reading of Restle's analysis is thateconomy of representation resides exclusivelyin the scientist's description of the percept inaccordance with a simplicity metric, ratherthan in the physico-chemical or psychologicalprocesses that generate the percept. This for-mulation is suggested by the fact that incommon with earlier mentioned investigators,Restle made no effort to consider the pro-cessing load implied by the various codes.For example, the common motion (directionand velocity) of a set of points is said to havea smaller informational load than the samenumber of points interpreted as independentlymoving. This is indisputably true if the claimis confined to the descriptive codes requiredby the two alternatives taken as givens. How-ever, the economy is achieved by imputingto the visual system what may not be availableto it as information except as the product ofconsiderable processing (i.e., that the pointshave a common fate). Thus, when the pro-cessing demands of the rival perceptual inter-pretations are compared, they may not differdespite the fact that investigators can offer

descriptive codes of the final percept whichdiffer in informational content. Economy ofform as measured by a formal metric maynot correspond to economy of process. Insuch a case, any tendency of the perceptualsystem toward the perception of simple con-figurations would have to be explained ongrounds other than economy of informationload as determined within coding theory.

Ullman. Working in the tradition of Marr(1982), Ullman (1979) developed a model ofmotion perception that relies on a "minimalmapping" principle. Ullman divided theproblem of motion perception into two parts:the correspondence problem and the three-dimensional interpretation problem. The firstis the problem of matching the various ele-ments within temporarily separate "frames"of retinal stimulation (either separate timeslices of continuous motion or successivepresentations in apparent motion). On thebasis of empirical study of apparent motion,Ullman concluded that correspondence occursamong "elements" or "tokens" within theimage (rather than among entire object con-figurations, although the entire configurationmay influence the matching). Local corre-spondences are governed by "affinity" amongtokens, which is determined by a similaritymetric that includes the spatial proximity,brightness, orientation, and length of thetokens (see Figure 8). The problem is thento derive a globally consistent set of matches

A B

Figure 8. The effect of distance on the affinity betweenline segments. (The solid lines indicate the first frames;the dashed lines, the second; the actual stimulus consistedof solid line segments in both cases [presentation time120 ms and ISI 40 ms; monocular viewing]. For PatternA, in which the second frame contains two line segmentsequally distant from the first and the same in otherrespects, the correspondence function yields a one-manymapping [two concurrent motions are perceived]. Whenone of the two distances is now increased as in PatternB, the likelihood of only one motion increases, until adistance is reached at which only the one motion isperceived. From The Interpretation of Visual Motion [p.36] by S. Ullman, 1979, Cambridge, MA: MIT. Copyright1979 by MIT. Reprinted by permission.)


based on the competing affinities among to-kens in successive "frames" or "snapshots,"in such a way that every element in eachframe is matched with at least one elementin the other frame. Ullman contended thatthis matching process is a low-level, autono-mous process that generally proceeds inde-pendently of semantic interpretation.

Ullman proposed that matching is achievedby computing the "minimal mapping" amongelements. In effect, the minimal mapping isthe one that minimizes the sum of the dis-tances traveled between frames among thematched elements. Ullman suggested thatthis minimizing computation can be achievedby a large number of local, simple processingunits operating in parallel and interacting.These provide a computational architecturefor solving a minimizing function throughlocal, iterative interactions from the bottomup. (Ullman used a modified Lagrangianfunction, a relative of the equations used todescribe some of the thermodynamic pro-cesses to which Restle alluded.) Although

Ullman did not develop a fully implementedmodel, his characterization of the computa-tional process provided an interesting con-ception of a bottom-up directive process andis akin to a growing number of proposedparallel computational architectures (Feldman& Ballard, 1982; Hinton & Anderson, 1981;Uhr, 1982).

Cutting and Proffitt. Wheel-generated mo-tions have been the focus of the theoreticalanalysis of organizational principles in motionperception (e.g., Borjessen & von Hofsten,1975; Cutting & Proffitt, 1982; Duncker,1929; Hochberg, 1957; Johansson, 1973;Wallach, 1965). These motions consist of twoor more luminous points placed anywherefrom the center to the rim of a rolling wheel(see Figure 9). The trajectory of each of theseelements through a fixed set of spatial coor-dinates defines its absolute motion. The ab-solute motions of a given set of elementsusually are analyzed perceptually into thecommon motion of the whole configuration(e.g., the horizontal motion of a wheel rolling

absolutemotion (a)

rolling wheel

relative motion

common motion

rolling wheel B*~

relative motion common motion

tumbling stick

Figure 9. Two stimuli used as prototypes to describe wheel-generated motion, (a. A two-light stimuluswith lights mounted 180° opposite from one another on an unseen wheel rim. The absolute motion pathsare two cycloids, 180° out of phase. The relative motion paths, in contrast, are circular and 180° out ofphase around their midpoint. Common motion is the path of this midpoint, which is linear, b. A two-light stimulus with one light on the perimeter, and one at the center. Absolute motion paths are a straightline and a cycloid. The relative and common motion paths, however, depend on the particular version ofthe object seen either as a rolling wheel or as a tumbling stick. In the rolling-wheel version, relative motionoccurs only for Light A, rotating about Light B. Common motion occurs for both and is linear. In thetumbling-stick version, both lights have relative motions, rotating 180° out of phase around their midpoint,and they both have common motion, describing a prolate cycloid. From "The Minimum Principle andthe Perception of Absolute, Common, and Relative Motions" by J. Cutting and D. R. Proffitt, 1982,Cognitive Psychology, 14, p. 221. Copyright 1982 by Academic Press, Inc. Reprinted by permission.)


on a flat surface) and the relative motion ofeach element to other configural elements(e.g., a point on the rim revolving about apoint at the center of the wheel). On theassumption that "geometric constraints" asdiscussed previously are in place, the absolutemotions of any set of elements may be ana-lyzed in principle into any of a potentiallyinfinite set of combinations of common andrelative motions. It is the widespread viewthat the visual system settles on a particularcombination of common and relative motionsthrough the operation of a minimum prin-ciple.

Two separate minimum principles havebeen proposed to account for the selection ofone among the many possible common/rel-ative motion pairs. The first operates in con-junction with the assumption that commonmotion is the first element to be abstractedfrom the absolute motions of the stimulusconfiguration, leaving relative motion to bespecified as the residual. A minimum prin-ciple is operative insofar as common motionis abstracted in such a way that "the perceptentailing the least number of changes is ob-tained" (Hochberg, 1957, p. 82). In the caseof a wheel with a luminous point at thecenter and on the perimeter rolling along aflat surface, this principle predicts the per-ception of a horizontally moving center witha satellite element revolving about it, ratherthan various other allotments of commonand relative motion in which the center ofrotation bobs up and down. The minimumprinciple operates to minimize common mo-tion as a straight line.

According to the rival account, relativemotion is abstracted first in such a way thatthe sum of the momentary relative motionsequals zero. This entails that the points rotateabout their centroid. In the case of a wheelwith one light at the center and one on therim, this principle implies that the two lightsare perceived as opposite ends of a tumblingstick whose center of rotation describes aprolate cycloid (see Figure 9). Although pre-vious authors had urged either that commonmotion or that relative motion is abstractedfirst, calling upon one or the other of theminimum principles, Cutting and Proffittproposed a third view: that minimizationprocesses operate on both common and rel-

ative motions simultaneously and that theprocess which is completed first determinesthe percept. The first motion component tobe minimized is extracted from the event,and the second motion is treated as theresidual.

The work of Cutting and Proffitt (1982)raises an important question concerning thenotion that perception is broadly guided bya minimum principle. These authors referredto their work as one example of a generalrule that perception tends toward simplicity,not only in motion perception but in patternperception and the perception of ambiguousfigures as well. At the same time, these authorswere careful in the case of motion perceptionto distinguish between the two different min-imum principles mentioned previously. Theseare cast as distinct minimum principles pre-sumably because they involve the minimiza-tion of quite distinct quantities or attributes.In what sense may they both be said toinstantiate the operation of a single MinimumPrinciple? Although one may reply on intu-itive grounds that these principles and theothers we have discussed are linked under thecommon notion of economy or simplicity, inthe absence of a commonly applicable metricof simplicity it is difficult to make a case thatthey are all manifestations of a single, for-mulable principle of perceptual processing.

This question as well as others pertainingto the idea that the minimum principle canbe used as a broad explanatory principle inperceptual theory are addressed in the follow-ing section.

Evaluation of the Status of MinimumTendencies and Minimum Principles in

Theories of Perception

Our selection of experimental illustrationsreveals that minimum tendencies are notmerely another exhibit in the arcade of per-ceptual curiosities. Nor have the suggestedapplications of the minimum principle beenrestricted to a narrow band of phenomena.It has been regarded by some theorists (Att-neave & Frost, 1969; Hochberg, 1964) as acore explanatory principle in perception; oth-ers (e.g., Hochberg, 1974; Perkins, 1976) haveexplained the minimum tendency via likeli-hood.


In preparation for evaluating the minimumprinciple and other proposed explanationsfor minimum tendencies, we address twoquestions regarding the idea of perceptualeconomy. The first pertains to the relationbetween the presumed preference for perceptsthat satisfy a minimum principle and theseeming obligation of the percept to adhereto the "geometric constraints" of opticalstimulation. The second question pertains tothe theoretical implications of adopting oneor another metric of simplicity in evaluatingthe "minimal" properties of stimulus andpercept.

Minimum Principle and GeometricConstraints

Whatever the ultimate status of the mini-mum principle, clearly such a principle mustoperate within the constraints imposed byproximal stimulation. The need to take intoaccount constraints imposed by stimulationis obvious, for otherwise the minimum prin-ciple could not be expected to culminate inperceptual representations that are signifi-cantly correlated with the environment. (Inthe absence of constraints, the principlemerely predicts a relative preponderance ofpercepts classified as "simple" by some metric;indeed, in the absence of constraints, areasimaged next to one another on the retinaneed not be contiguous in the experiencedvisual field.) However, the notion of con-straints provided by proximal stimulationitself needs further clarification.

A number of the authors discussed in thepreceding section explicitly construed theminimum principle as operating within theconstraints of projective geometry. It is amatter of empirical fact that the perceptualrepresentations generated in response to agiven pattern of proximal stimulation fall (atleast roughly) within such constraints: A cir-cular pattern on the retina gives rise to therepresentation of any of a family of ellipses(depending upon the perceived slant), but notto the representation of a square. Yet thelaws of projection relating distal configura-tions and their retinal projections are notthemselves given in the pattern of stimulation.Indeed, the laws of geometric optics thatapply to the light stimulating the retinas are

one thing; the physiological and psychologicalprocesses that are initiated by stimulation ofthe retinas are another. The means by whichthe perceptual system generates representa-tions that are more or less in accordance withprojective geometry must be sought withinthe perceptual system itself, as psychologicalrules or mechanisms. A clear distinction mustbe maintained between projective geometryper se, and "geometric constraints" regardedas rules of the perceptual process. Such con-straints include the familiar relation in whichprojective shape limits the permissible rangeof perceived shape-at-a-slant, and in whichvisual angle limits the permissible range ofperceived size-at-a-distance.

Given that geometric constraints are inplace, what would be left in perception forthe minimum principle to accomplish? Theminimum principle would operate withinthese constraints by determining whichmember of the range of perceptual represen-tations compatible with stimulation will beexperienced. More specifically, if a minimumprinciple is to operate, two conditions mustbe met. First, there must be a range ofrepresentations compatible with stimulationfor the minimum principle to decide among.Second, the members of this range must differaccording to a common metric of simplicityin such a way that there can be defined asimplest, or locally simplest, representation.

For the first condition to be met, registeredstimulation, together with the psychologicallyreal geometric constraints, must be insufficientto determine a percept. Whether this is thecase may be a subtle question, because itmay be difficult to draw an absolute boundarybetween those mechanisms that enforce geo-metric constraints and further processingmechanisms. A Gibsonian might contendthat in most cases the constraints placedupon permissible perceptual representationsby proximal stimulation are sufficient to de-termine a percept, so that the minimumprinciple must be regarded as a default ruleoperating in those few cases of impoverishedstimulation. However, it should be noted thatif our argument about geometric constraintsis correct, this Gibsonian contention cannotbe justified solely on the basis of ecologicaloptics (Gibson, 1966, 1979). (Our argumentdoes not depend on adopting the conventional


premise that stimulation is inherently equiv-ocal from a geometrical point of view.) Evenin a Gibsonian account of perception, mech-anisms are required to "pick up" or "detect"the information available proximally. It seemsunlikely that an a priori argument could beformulated to preclude the operation of aminimum principle within these detectionmechanisms. Even if proximal stimulation isunequivocal with respect to distal propertiesfrom the point of view of ecological optics,this does not entail that the mechanisms thatdetermine the percept need fall under ourcategory of geometric constraints; the psy-chologically real detection mechanism mightuse geometric constraints to determine arange of percepts, with a minimum principletaking up the slack. In any event, whetherour first condition obtains or not is a questionthat can be answered only by an investigationof the actual functioning of the visual system.

The second condition states that the rangeof representations permissible within the geo-metric constraints must be comparable on asingle metric of simplicity. We may askwhether there are any classes of situations forwhich this clearly is or is not the case. Aclear set of cases meeting our second conditionis that in which the proximal stimulus con-strains the percept in all but a single dimen-sion or attribute, along which a simplestrepresentation may be defined. We have dis-cussed several cases of this type. In the workof Hochberg and McAlister (1953), Buffartet al. (1981), and Perkins and Cooper (1980)the dimension of variation was "good form."Although a unique minimal representationcould not always be defined for the stimuluspattern, the competing economical represen-tations generally were comparable and soresolvable within the individual metrics ofsimplicity (e.g., by perceptual alternationamong two or more representations that areequally or nearly equally "simple" or "good").Similar remarks apply to Restle's (1979)analysis of motion perception in terms ofcoding theory.

In contrast, a set of cases in which nosingle minimal solution (or no set of obviouslyequivalent minima) presents itself is that inwhich the proximal stimulus allows for atrade-off along two or more dimensions. Cut-ting and Proffitt's (1982) study of wheel-

generated motions is a case in point. Suchmotions can involve minimization of com-mon or relative motion. As we mentioned,the metrics provided by Cutting and Proffittyield no basis for comparing these competingdimensions of minimization with one another,even though the measurement of, say, mini-mal relative motion in terms of the sum ofmomentary relative motions provides an un-equivocal measure of economy within thatdimension. Another example involves theperceptual trade-off between size and distance.Continuous increase or decrease in visualangle is compatible with either an expandingand contracting object at a constant distanceor an object of constant size approaching andreceding. What would a minimum principlepredict in such a situation? Change in eithersize or distance might be minimized, or acompromise might be reached. When consid-ering a single case in isolation, there seemsto be no reason a priori for finding thatminimization of change in one attribute issimpler than minimization of change in theother. Resolution of the trade-off in this casewould have to be determined empirically.Further explication of the empirically deter-mined operation of the minimum tendencywould then involve other principles of per-ception.

Beyond these two conditions, there is anadditional question that arises in applyingthe notion of geometric constraints, andtherefore, in deciding what minimum config-urations may be available within such con-straints. How much of the environmentshould be included in setting the constraintswithin which a minimum configuration maybe defined? One might include the spatiallayout as sampled by a freely moving organ-ism over a period of a few minutes, thecurrent visual field, or some portion of thelatter (these possibilities are not exhaustive).If a minimum tendency or minimum prin-ciple is to operate as envisioned by theseauthors, the segment of the environmentmust not be too small. For example, theapplication of a coding analysis to the fourdots in Figure 7 depends upon all of thembeing registered by the perceptual system.

The authors discussed previously fall intotwo groups on this matter. Those treating theperception of form and depth used static


figures; here it is expected that the singledrawing, perhaps as filling the visual field atany given moment, is the effective stimulus,and its proximal registration sets the geomet-ric constraints within which a minimum (orrelative minimum) configuration will be de-termined. By contrast, the motion studies ofRestle (1979), Ullman (1979), and Cuttingand Proffitt (1982) obviously demand that aminimum motion configuration be deter-mined over time. The interval between framesdetermined the time limit in Ullman's studies.Cutting and Profiitt's wheel-like motions wereon the screen for 2.5 s; Restle's dots cycledin 1 to 4 s. Neither of these authors discussedthe relation between exposure time and theperception of an "economical" motion con-figuration.

Hochberg (1974, 1982) has claimed thatthe difficulty in setting bounds on the extentof the visual scene that must be included ina minimizing calculation poses a perhapsinsurmountable obstacle to any account ofperception based upon a minimum principle(and, by implication, to the possibility ofmeasuring a minimum tendency). Hochbergobserved that "impossible" figures, such asthose in Figure 10, present a difficulty forthe application of a minimum principle tothe perception of form. For example, anassumption of "figural goodness" leads toviewing the left and right sides of Figure 10Aas composed of solid rectangular prisms, puttogether as in a picture frame. Yet the figureas a whole cannot be consistently organizedas a picture frame—the middle line on thetop bar must be seen as bounding both thetop and bottom edge of the front plane ofthe upper bar! According to Hochberg, theminimum principle should predict that this"contradiction" would yield organization ofFigure 10A as a flat line drawing, whereas ittends to be seen as a figure in depth. Hochberg(1982) attributed the depth result to thedistance separating the left and right sides ofFigure 10A, which allows each end to beorganized in depth without forcing one totake in the contradiction. He contended thatthis interpretation is supported by the factthat Figure 10B tends to be organized bidi-mensionally, a finding that does not resultmerely from its dimensions, because FigureIOC elicits the perception of depth.

Figure 10. Two "impossible" figures (A and B); one"possible" figure (C). (Pattern A tends to elicit a strongerdepth response than Pattern B, which cannot be duesolely to Pattern B's small size, because Pattern C elicitsa depth response. From "How Big is a Stimulus?" by J.Hochberg. In Organization and Representation in Percep-tion [p. 192] edited by J. Beck, 1982, Hillsdale, NJ:Erlbaum. Copyright 1982 by Erlbaum. Published withpermission of Lawrence Erlbaum Associates.)

Hochberg's (1974, 1982) arguments basedupon Figure 10 and other line drawings un-derscore the need to address the question ofhow great a segment of the display must beincluded for purposes of calculating simplicity(e.g., whether this should be set as a fixedextent within the visual field or determinedon the basis of segmentation of the environ-ment into objects). However, it is difficult tosee how such figures confute explanations ofform perception based on a minimum prin-ciple. Indeed, Simon (1967), in presentingthe process model that we reviewed, explicitlyconsidered the analysis of "impossible" fig-ures. He suggested that the depth response tosuch figures arises because they are locallyinterpretable as consisting of tridimensionalstructures with rectangular sides. Once the"contradiction" is detected (Step C in hismodel), the depth response is maintained byrestricting the "scanning" portion of the pro-cess to the noncontradictory portions of thestimulus. More recently, Perkins (1982) sug-gested that the depth response in "impossible"figures results from a "partial determination"of perceptual organization by the stimulusfeatures. The depth response is elicited despiteits "contradictoriness" because the perceptualsystem only takes a part of the stimulus intoaccount at a given time. Thus, althoughHochberg's arguments regarding "impossible"figures do not show that application of aminimum principle within perceptual theorywould be "misguided" (1982, p. 195), theydo emphasize the need to take into accountthe effective size of the stimulus in determin-ing the relevant "geometric constraints"within which an economical configurationmay be defined.


Choice of Simplicity Metric

The idea that the perceptual system oper-ates according to a minimum principle isopen to a number of interpretations, depend-ing on how one construes the notions ofsimplicity and regularity, and on what onethinks is minimized by the perceptual system.We have seen that our authors generallyinterpreted their minimum tendencies orminimum principles as applying to limitedextents of the spatial layout, or to alterationsin spatial configuration over a brief period oftime. In focusing on this interpretation, weleave aside possible construals of the mini-mum principle as a broad principle of cog-nitive economy, that is, as a principle forachieving the simplest overall description ofthe environment (say, by achieving the sim-plest taxonomy of the objects in the environ-ment).

Yet the idea of perceiving a minimumconfiguration may itself require reference toa description of the configuration and itsalternatives. At least from the investigator'spoint of view, in order to characterize and tocompare the simplicity of various configura-tions some metric of simplicity must be ap-plied, and its application must involve atleast a rudimentary description of the targetobjects. We have seen that a commonlyadopted approach is to count and comparethe number of primitive features that variousconfigurations comprise (the fewer features,the simpler the configuration). This countingpresupposes a taxonomy of features and,hence, a descriptive language. Because a givenconfiguration may be described in more thanone way, depending on what one chooses tocall a feature, the theorist is faced with aproblem of choosing the appropriate simplic-ity metric.

An example may help. Consider the fourprojections of a cube in Figure 1. Hochbergand Brooks (1960) devised 17 different mea-sures of simplicity to be applied to suchdrawings. The relative simplicity of the bidi-mensional shapes to one another and therelation of the measured simplicity of theseshapes to that of the tridimensional cubevary depending upon the measure adopted.In one of the measures, simplicity was deter-mined by counting the number of continuous

(but possibly transected) rectilinear segmentsin the figures; in another, by counting thenumber of unbroken line segments. (In eachcase, the lower the score, the simpler thefigure.) The results for the four patterns are12, 12, 11, 9, and 16, 16, 13, 12 for therespective measures. A tridimensional cubegets a score of 12 in either case. Notice thataccording to the first measure, the strongbidimensionality of Pattern Z and slightlyless strong bidimensionality of Pattern Y isnicely set off from the other two patterns.According to the metric, the bidimensionalorganizations of these drawings clearly aresimpler than those of the other two drawings,and perceivers are in agreement. Notice alsothat Patterns W and X have simplicity scoresof 12, which is equal to the score of thetridimensional organization. This leads oneto expect an equivocation between the bi-and tridimensional organizations of thesepatterns, an expectation that is not borne outempirically. (The drawings tend to be orga-nized as cubes.) A similar situation obtainswith the second test. The simplicity measuresuggests equivocation on Pattern Z, whichin fact tends to be seen bidimensionally.Hochberg and Brooks did not make compar-isons between scores for bi- and tridimen-sional organizations. They used the scoressolely for comparisons of simplicity withinsets of bidimensional shapes such as those inFigure 1. Our remarks are not meant toimpugn this procedure, but rather to exem-plify the fact that specific comparisons ofsimplicity are dependent upon the metric ofsimplicity that is chosen.

Although Hochberg and Brooks (1960)conveniently make our point for us by sup-plying alternative metrics, the point can bemade with regard to any metric of simplicity.Thus, in the recent application of codingtheory by Buffart et al. (1981) in assessingthe simplicity of figural completions, it isclear that metrics other than those used bythe authors could be proposed. This shouldbe apparent from the fact that Buffart et al.applied their metric of simplicity in an en-tirely formal manner to the elemental symbolsof the code itself. Thus, measured simplicitydepends on what gets counted as a separateelement of the code. For example, Buffart etal. (1981) did not include some elements of


their code among the features to be countedto determine simplicity (e.g., operators thatdo not refer directly to elements of the targetconfiguration). Their rationale was that thesefeatures "have only a notational meaning"(Buffart et al., 1981, p. 261). This decision isplausible only if one emphasizes simplicityof perceived form rather than economy ofprocess. Further, they included no elementin their code whatsoever for specifying thatthe completed figure is perceived as lyingbehind the occluding figure. Hence, the ele-ment of depth which is lacking in the mosaicinterpretation received no weight in the com-pletion interpretation. Nor is it obvious howmuch weight should be assigned to depthrelative to other features, such as a linesegment or an angle.

The studies that we have examined embodya variety of metrics. These include the follow-ing proposals for what gets minimized inperception: (a) the number of distinct featuresand relations among features, counted in oneway or another (Buffart et al., 1981; Hochberg& McAlister, 1953; Restle, 1979); (b) thevariability among angles, line lengths, andline slopes (Attneave & Frost, 1969); (c) thesum of distances traveled between successiveframes among elements matched through ap-parent motion (Ullman, 1979); (d) the num-ber of changes in the "common motion"component of a complex motion (Cutting &Proffitt, 1982); and (e) the sum of elementalmotions in the "relative motion" componentof a complex motion (Cutting & Proffitt,1982). It was also proposed that perceptiontends toward qualitatively "good" figures,which were specified as figures with rightangles or with symmetrical forms (Perkins,1976). In addition, we have noted that Marr(1982), without connecting his work explicitlywith the notion of a minimum principle,attributed to the visual system the assump-tions that the environment contains smoothsurfaces and relatively homogeneous opticaltextures. Consideration of this diverse groupof simplicity metrics and qualitative specifi-cations of Pragnanz reveals that the studieswe have reviewed do not constitute a unifiedbody of work investigating a common mini-mum principle or minimum tendency. Rather,these studies are only loosely related by theircommon use of such terms as "simplicity,"

"economy," "minimal," and "Pragnanz," to-gether with the intuitive notions of simplicityor economy that lie behind the various pro-posed measures of such attributes.

Given the multiplicity of metrics, whatshould be the attitude of the investigatortoward the choice of a metric of simplicity?Of course, the attitude may vary, dependingon one's particular research strategies andgoals. Hochberg and Brooks (1960), for ex-ample, were seeking to develop a purelypsychophysical law that would predict sub-jects' responses. This led them to seek toestablish empirically which of the 17 metricsof simplicity they examined best fit the be-havior of subjects. The metric of simplicitythey established would then act as a predictor,but need not reveal anything regarding theprocesses that lead to subjects' responses. Itremains an open question whether the per-ceptual system registers as primitive thosefeatures that Hochberg and Brooks counted.With this approach, the only constraint onthe choice of a metric of simplicity is theempirical adequacy of the psychophysical lawsbased on the metric.

Without denying the usefulness of a psy-chophysical approach, we wish to examineits limitations. The situation is parallel to thecase of geometric constraints. Although theperceptual system may operate in such a waythat it conforms to geometric optics, onlyfacts about the processing mechanisms of thevisual system can determine the extent towhich perception actually matches geometricoptics. Similarly, simplicity is not a directdatum available in proximal stimulation, noris what is measured as "simplest" on a givensimplicity metric necessarily that which aneconomizing perceptual system would deemsimplest. A coding theory may predict theresponse of the visual system, but this canonly be because the processing mechanismsof the visual system are what they are. The"simplest" construal of a given proximal pat-tern depends upon both the geometric con-straints and the metric of simplicity implicitin the visual system's processing mechanisms,if there is one. Put another way, even thoughit is well established that arbitrarily manysimplicity metrics can be constructed forassessing the simplicity of a range of phenom-ena, this fact does not imply that an econ-


omizing visual system has access to a multi-plicity of metrics. An answer to the questionof whether the visual system embodies aunique simplicity metric depends upon factsabout the system itself.

An arbitrarily chosen metric of simplicitymight be brought by a program of empiricalresearch into closer and closer match withthe behavior of the visual system and ulti-mately might achieve predictive generality. Astronger program of research would be toregard the choice of a metric of simplicity aspart of a hypothesis about the process mech-anisms of the visual system. This wouldentail imputing a specific set of mechanismsfor realizing geometric constraints and a spe-cific simplicity metric to the visual system.This makes for a stronger hypothesis becauseit yields more routes of empirical testability.Any such hypothesis of course should be asaccurate and as general a predictor as are thesimplicity metrics of the psychophysical ap-proach. In addition, particular structures areattributed to the visual system that can beexpected to yield testable results along di-mensions other than simplicity (e.g., reactiontime or error curves). Moreover, if one actuallyimputes a coding system to the visual system,then it is natural to seek independent confir-mation that the visual system registers theprimitive features of the code.

Competing Explanatory Foundations for theMinimum Tendency

Recall that in the introduction we indicatedthat minimum tendencies have been estab-lished in the experimental literature indepen-dently of the idea that a minimum principlemust provide the ultimate explanation forsuch tendencies. In fact, as an examinationof contemporary studies reveals, a numberof distinct explanatory foundations have beenproposed for various minimum tendencies.We wish to examine the relation of theempirically observable minimum tendenciesto these various explanatory frameworks. Ouraim is not to find the one true explanatoryframework. Certainly, the prime considerationis to determine which, if any, of the candidatesare desirable, but this aim is complicated bythe fact that the various frameworks are notuniversally incompatible with one another,

nor do they all bear the same explanatoryrelation to minimum tendencies.

The types of explanations can be dividedinto two broad categories, corresponding totwo distinct explanatory questions. The firstgroup attempts to explain the presence of (orto "ground") specific minimum tendencies;these attempts differ according to how theyanswer the question of why the visual systemhas minimizing tendencies. The second groupincludes accounts of how perceptual economyis achieved; this is a question of process.

Grounding. Rival answers to the questionof grounding divide into functional and non-functional accounts. Functional accountsstress the adaptive advantage of minimumtendencies, of which two have been proposed:(a) the adoption of a minimum principleyields veridical percepts because simplicity isa "good bet" (likelihood), and (b) economicalrepresentations make fewer demands on lim-ited processing capacities (economy of pro-cess). On either of these proposals, the originof a minimum tendency could in principlebe attributed to either evolution or learning(or some combination of the two). As ithappens, Perkins (1976), one adherent of thelikelihood account, has assumed that likeli-hood is adopted as a good hypothesis on thebasis of learning rather than through evolu-tion. Conversely, economy of process as afundamental strategy in perception is gener-ally, but tacitly, regarded as arising throughevolution (Mach, 1919, was explicit), althoughVickers (1971, p. 27) has suggested that thestrategy is learned.

The notion that a minimum tendencywould be adaptive because of the high likeli-hood of simple configurations in the environ-ment is difficult to assess. Earlier we notedthat for a variety of well-known perceptualphenomena there is an intuitive match be-tween the simplest and the most likely. Directempirical evaluation of this seeming congru-ence would require the construction of precisemeasures of simplicity and likelihood and thedevelopment of suitable techniques for envi-ronmental sampling (Brunswick, 1956). Theemphasis on likelihood may draw encourage-ment from the fact that a number of successfulmachine programs for modeling vision in-corporate Pragnanz-like assumptions intotheir versions of the visual process. We have


mentioned a number of Pragnanz assump-tions in the work of Marr et al. In a similarvein, Barrow and Tenenbaum's (1978) modelof the perception of surfaces, edges, anddepth assumes the smoothness of surfaces.In these various cases, the investigators havebuilt into the operation of their machinemodels assumptions to the effect that Prag-nanz (smooth variation, homogeneous struc-tures) is a good bet. The machine modelswere set to work on objects drawn from thesame environment that confronts the humanvisual system (including conventional stimuliand ordinary objects). The fact that theseprograms yield veridical "percepts" arguesfor the functional validity of the Pragnanzassumptions in that environment. Hence,these assumptions may be regarded as func-tionally valid within the environment facedby the human visual system, although thesuccessful operation of the machine programsdoes not establish that the human visualsystem actually embodies these assumptions.

An intriguing conception of the relationbetween likelihood and simplicity is suggestedby the philosophical work of Goodman (1972,chap. 7) and Sober (1975). It has been shownthat inductive sampling of the type that mustunderlie a judgment that the simpler is themore likely itself relies upon an implicitmetric of simplicity. That is, extrapolation ofa probability or likelihood judgment from asample depends on a preference for fittingsimple curves to the data from the sample.(Such a conception is at the core of Attneave's,1954, discussion of redundancy and simplic-ity.) This might suggest that simplicity andlikelihood are two faces of the same coin,and one need not treat the minimum prin-ciple and the likelihood principle as realalternatives. However, although the relationbetween inductive likelihood and simplicityis well established, it does not entail a blendingof the concepts for present purposes. Foralthough inductive sampling must indeed relyupon an implicit metric of simplicity (e.g.,in curve fitting), the various metrics of sim-plicity that may be imputed to the visualsystem need not directly reflect the implicitsimplicity metric (or metrics) behind theempirical practices of scientific investigation.It is empirically conceivable that environ-mental sampling as suggested by Brunswik

(1956) would show that the simpler (as definedon a given metric) is not the most likely.Here, the metric of simplicity that underliesthe inductive sampling (curve fitting) can bequite independent of that which is used tomeasure the simplicity of environmentalforms. The question of whether the visualsystem might tend toward simplicity on thebasis of likelihood, or whether it does so onother grounds, retains its meaning.

Economy of process provides an alternativeconception of the grounds for the minimumprinciple. There is intuitive appeal to thenotion that cognitive systems are built so asto carry out cognitive processes with relativeeconomy. This assumption was explicitly dis-cussed by Mach (1906, 1919), who advocateda "biological-economical" orientation to thepsychology of perception and cognition onevolutionary grounds. Mach argued thateconomy of thought would be of clear survivalvalue to an organism faced with a bewilderingarray of sensory inputs, and he suggested thatprinciples of economy operate from the lowestlevels of sensory organization right up to theintellectual processes of the scientific inves-tigator. The actual means for economizingare not given by the evolutionary approachas such; they depend upon the evolutionaryhistory of the organism and must be investi-gated separately for different types of sensorysystems. What the evolutionary approachdoes as a whole is provide a schema foranswering the question of why the visualsystem economizes, thereby providing a con-ception of the functional role of minimumtendencies. The specific adaptive features ofminimum tendencies also are not specifiedby an evolutionary approach as such, andthey must be investigated for each type ofsensory system in relation to its environment.Similar remarks apply to the programmaticsuggestion that the minimum principle is anacquired strategy of cognitive economy(Vickers, 1971).

Finally, the chief exemplar of a nonfunc-tional grounding of the minimum tendencyis that of the Gestalt psychologists, who de-rived the economy of perceptual configura-tions from the physico-chemical structure ofprocesses in the brain. As conceived especiallyby Kohler (1920, 1947), these brain processeswere regarded as occurring in gradient media


or ionic "fields" in which the tendency towardminimal configurations is governed by thevariational principles of classical mechanics(Planck, 1915, Lees. 2, 7; Yourgrau & Man-delstam, 1968). The appeal to spontaneousorganizational forces in a cortical mediummight seem implausible given what we knowabout the highly articulated architecture ofthe visual cortex. Kohler (1969, chap. 2)presented arguments to the contrary. He ob-served that even in highly articulated tissue,certain processes (e.g., steady-state physico-chemical processes in the intercellular media)are not explained histologically. He contendedthat the evolution of articulated structurecannot explain facts about the physiology oforganisms that result from the basic laws ofphysics and chemistry applied to the mediasurrounding these structures.

Kohler's contention cannot be dismissedout of hand. It accords with the notion,emphasized by evolutionary thinkers, that"mechanically necessary" aspects of an or-ganism (such as the mass of a flying fish,which "functions" to bring it back to thewater) are not to be treated as evolutionaryadaptations (Williams, 1966, chap. 1). More-over, appeal to "fields" or gradient media inanimal tissue has long formed part of thetheoretical landscape in developmental em-bryology (Kohler, 1927; Weiss, 1939, PartIII, pp. 289-294), and it has been suggestedthat such fields exhibit simple, universal lawsacross a broad range of animal species, in-cluding hydra, sea urchins, fruit flies, newts,and chickens (Gierer, 1977; KaufTman,Shymko, & Trabert, 1978; Wolpert, 1970).Schwartz (1977) extended this line of thinkingto the development of receptotopic structuresin the brains of such diverse species as themonkey and the goldfish. He proposed a setof "minimal developmental rules" that treatthe developing neural structures as conform-ing to variational equations of the sort familiarwithin classical mechanics, and he com-mented that these rules "allow the final de-tailed structure of the receptotopic mappingto be determined by general physico-mathe-matical principles rather than via biologicalencoding of detailed positional information"(Schwartz, 1977, pp. 670-671). Consequently,it is a well-established pattern of thoughtwithin biology to seek to explain some phys-

iological processes in terms of general physico-chemical properties of intercellular media.Of course, because all known processes inorganisms are in conformity with the laws ofphysics and chemistry, an evolutionary ac-count may still be required to explain whycertain processes following these laws arefound in the brain and not others. Kohler'spoint that some of the processes underlyingperception may reflect general physical con-ditions of animal tissue is not thereby negated,but its alleged independence from evolution-ary considerations is weakened. The presenceof specific kinds of ionic fields in developingor in mature animal tissue is not like havingmass (mass being a property of all materialthings above the atomic level). Thus, Kohler's"grounding" is perhaps not nonfunctionalafter all, but his type of explanatory accountretains its distinctiveness on other grounds,as we discuss in our consideration of processaccounts.

Process. The two chief types of accountof how perceptual economy is achieved are:(a) the so-called "soap bubble" accounts (Att-neave, 1982), which explain perceptualeconomy in terms of physico-physiologicalprocesses, and (b) accounts that treat percep-tion as the formation of internal descriptions(encoded propositionally) and regard percep-tual economy as resulting from operationsthat yield economy of description. This di-vision marks a distinction between two moregeneral strategies in giving process accountsfor perception: appeal to the brute propertiesof neurophysiological processes (as is oftendone to explain color vision and phenomenasuch as Mach bands), and appeal to thenotion that perceptual processing occurs inan internal symbol system (or language ofthought; Fodor, 1975), which serves as thevehicle for generating descriptions of the en-vironment in accordance with encoded rulesand strategies.

The Gestalt psychologists forthrightly pos-ited physico-chemical mechanisms for achiev-ing the minimal "solutions" to various per-ceptual "problems" (Koffka, 1919; Kohler,1920, 1969; Wertheimer, 1912). According toGestalt theory, the process underlying sim-plicity in perception is unmotivated and non-computational. Any computation of simplic-ity is carried out by the scientist who wishes


to predict the percept or assess the fit betweenthe obtained and theoretically expected per-cept. Just as various physical systems achieveequilibrium at minimum states as a result ofpurely physical interactions, so too the ner-vous system achieves such solutions to per-ceptual problems by physico-chemical pro-cesses. This account has the advantage thatit appeals to well-known processes for achiev-ing minimal states or configurations.

The particular model of brain activity ad-vocated by the Gestalt psychologists has longsince been discarded, and a new account ofthe minimum process based on contemporaryknowledge of the brain has not been formu-lated. Nevertheless, a number of contempo-rary investigators have concluded that anexcessively atomizing approach to physiolog-ical psychology is bound to fail, and that theproblems of neuropsychology require "con-fronting the theoretical and experimental per-spectives demanded by the global, statistical,or Gestalt aspects of the nervous system"(John & Schwartz, 1978, p. 25). Withinneurophysiology, Schmitt, Dev, and Smith(1976) reviewed a large body of anatomicaland electrophysiological data in support ofthe view that "graded electrotonic potentials,rather than regenerative spikes, may be thelanguage of much of the central nervoussystem" (p. 116); they emphasized the roleof "the extracellular electric field and ionicenvironment" (p. 117) in brain activity, whilenot denying that regenerative spikes retaintheir importance. From the side of mathe-matical biology, Cowan and Ermentrout(1978) developed mathematical models thattreat the neural nets underlying perception"in terms of the properties of nonlinear fieldsor continua, which we introduce as a suitablerepresentation of the activity seen in neuralnets comprising large numbers of denselyinterconnected cells" (p. 69). The field con-cepts used by these authors must be sharplydistinguished from extracellular fields, forthey comprise interactions among numerousdiscrete neurons connected into nets. Thesevarious developments, taken together, suggestthat the type of process account envisionedby Gestalt psychology cannot be dismissed.

Attneave (1982) recently undertook an in-teresting speculative exercise to reexaminethe prospects of developing a conceptualiza-

tion of perception in the spirit of the soap-bubble metaphor favored by Gestalt theory.The soap bubble is an exemplar of systems"that progress to equilibrium states by wayof events in interconnected and recursivecausal sequences so numerous that their ef-fects must be considered in the aggregaterather than individually" (Attneave, 1982, p.12). Attneave's model is designed to explainmonocular depth perception by positing in-ternal regulatory tendencies which operate ina neuronal manifold representing externalspace. Perceptual organization is explainedby appeal to roughly parallel organizationalproperties in the neural medium. In this case,phenomenal simplicity and regularity dependdirectly upon the simplicity and regularity ofthe structure of physiological processes.Economy of process accords with economicalperceptual organization.

The chief alternative to the soap-bubbleaccount is based on the notion that perceptioninvolves generating a description of the en-vironment, and that economy of perceptionarises from economy of description. The no-tion of perception as description suggests theexistence of a representational medium inwhich various alternative descriptions can begenerated and evaluated for simplicity. Aswas discussed earlier, there are two basicversions of this process of generation andtesting: selective and directive. The selectivemodel suggests that the perceptual systemexamines a number of the permissible per-ceptual representations compatible with theoptical constraints and selects from amongthese representations the alternative thatpasses a computational test for maximumeconomy. In contrast, the directive modelsuggests that the minimum principle guidesthe microgenesis of the perceptual represen-tation to ensure the construction of the mosteconomical perceptual representation.

Elaboration of a selective model requiresconsideration of a variety of subprocesses orcomponent operations relating to the gener-ation, assessment, and selection of alternatives.The decisions concerning these operationswill shape the model. The following briefaccount is intended as illustration. In theinitial stage, a number of the representationscompatible with optical stimulation are gen-erated simultaneously. Then the information


load of each representation is computed con-currently according to a metric of simplicitythat is imputed to the perceptual system. Thecomputed values are scanned, and all repre-sentations but the one having the lowestinformational load are cleared from workingmemory. This representation then serves asthe representation of the distal configuration.

The initial stage of this or any other selec-tive model is the most troublesome. Howmany candidate representations are to begenerated, and how are they chosen? If therange of possibilities is continuous, clearlythey cannot all be generated. The systemmight be set to calculate all possibilitiesseparated by an arbitrarily chosen unit ofsimplicity, but even this suggests an over-whelming task of calculation. A similar prob-lem is familiar to designers of artificial intel-ligence systems. As McArthur (1982) pointedout, a general problem faced by computermodels of vision is the "combinatorial explo-sion" that arises when numerous possibilitiesare defined by a given visual input. Variousstrategies have been developed for narrowingdown the alternatives. Minsky's (1975) frametheory is one such alternative. A selectivemodel might be supplemented by the use ofknowledge structures or "frames" that takeinto account the immediately prior perceptualsituation to limit the generation of alternativesfor simplicity testing. In this manner, percep-tually untenable or unlikely representationsare weeded out early. In any event, there area number of empirical implications of thisselective model which, if substantiated, wouldcontribute to its plausibility. Given that themodel specifies that the alternatives are sub-jected to evaluation, the percipient might beexpected to have access to information aboutthe rejected (less economical) representations.Second, if the observer is forced to makeanother choice, the perceptual representationranked next in informational load shouldemerge.

We now turn to models of the directivevariety. Several variants of the directive modelmay be formulated. One variant is suggestedby Attneave's (1972) reference to a feedbackor "hill-climbing" system, although the fol-lowing account should not be attributed toAttneave. It is presumed that the perceptualsystem has evolved to favor the most eco-

nomical representation compatible with theinput. Upon reception of optical input, theperceptual system generates a first approxi-mation perceptual representation from amongthe representations consistent with the input.The first approximation is followed immedi-ately by transformation of the representationin an analog spatial representational medium.If the initial transformation generates a moreeconomical representation according to aspecified test, the process of transformationcontinues until it ceases to yield greater econ-omy. It is assumed that once a gain ineconomy is realized the transformation pro-ceeds in the same direction, so that there isno return to a less economical representation.If the initial transformation generates a lesseconomical representation, the original rep-resentation is reinstituted. This serial processof testing ends when a specified number ofsuccessive transformations fails to yield asimpler representation.

Detailed development of a model alongthese lines will have to explicate several mat-ters. As with the selective model, a decisionrule is needed to specify the nature of thefirst approximation. This or a further rulemust also be specified to effect the transfor-mation process. Second, it will be necessaryto arrive at a generalized metric of economyby which the gain or loss of economy isassessed. Because the assessment is executedby a human visual system, the metric shouldbe one that may plausibly be attributed tothe system. Third, the model must considerthe comparison operation by which it isdetermined that a change in economy hasoccurred, and by which the process is ter-minated. As these steps are made explicit,the postulated process takes on an implausiblycumbersome visage.

An alternative to this hill-climbing ap-proach is a directive model of the sort re-viewed previously in connection with thework of Perkins (1976) and Simon (1967).The central idea is that the system seeks a fitamong local features that concurs with Pra'g-nanz (e.g., symmetry) and then propogatesthis interpretation throughout the scene, untilan inconsistency is discovered or a coherentpercept is obtained. As with models of aselective variety, a difficulty with this ap-proach is "combinatorial explosion": a large


number of initially promising but ultimatelyimpossible interpretations may keep the sys-tem long at work. One way to avoid thisproblem is to build in especially strong con-straints on the initial interpretation (Waltz,1975). A mechanism for directly generatingthe simplest representation consistent withthese constraints might then be provided by"filtering" or "constraint relaxation" tech-niques (McArthur, 1982). These operate inparallel to check the permissible combinationsamong local features according to the char-acteristic of the filter, which in this case seeksminimum configurations.

Significant increases in power are achievedby iteration of the operations (Barrow &Tenenbaum, 1978; Rosenfeld, 1978). Indeed,by iterating the interactions among unitsreceiving local inputs, global properties canbe computed, just as a soap film achieves theglobal property of minimum surface areathrough numerous local interactions. Localinteractions propagated across a network tocompute a global property have been appro-priately characterized as "pseudo-local" op-erators (Ikeuchi & Horn, 1981, p. 181). In-asmuch as such mechanisms depend upon alarge number of local interactions that con-verge on a minimal solution, they may beviewed as the computational analogue of theisoperimetric processes upon which Gestaltmodels are based (cf. Barrow & Tenenbaum,1978, p. 15). In fact, Grimson (1981), workingin the tradition represented by Marr (1982),developed a model of stereopsis in surfaceperception that draws upon variational prin-ciples to compute various "minimum" values(of change in surface orientation) in theprocess of arriving at a representation of adistal surface.

Adherents of the notion of perception asdescription, whether working with a selectiveor a directive model, often implicitly assumethat there is a direct relation between econ-omy of description and economy of process(e.g., Buffart et al., 1981). The seeming plau-sibility of this assumption stems from theidea that simpler figures or event configura-tions have fewer distinct elements (e.g., linesor angles), and hence have briefer descriptionsthat require less processing capacity. Yet theassumption cannot be accepted without ques-tion, as we have seen. The question of whether

the two types of economy coincide cannotbe decided simply by examining the descrip-tions or codes that describe the perceivedfigure (the end-product of perceptual pro-cessing). To evaluate economy of process,one must examine the full account of theprocesses required to arrive at the simplestperceptual description. In the case of codingtheory, this account might include detectingand encoding redundancy, generating alter-native codes for a single stimulus, reducingthem to their simplest form, and comparingthem. This does not have the appearance ofan economical process, even if it does succeedin detecting economical forms.

Our review of process accounts suggeststhat the basic approach of the Gestalt psy-chologists retains its attractiveness. The basicattractiveness of the position is that it providesa way of conceiving how economy can beachieved through the direct interaction of alarge number of mutually independent events,such as might be conceived to occur in thevisual cortex through lateral connections(Gilbert & Wiesel, 1983; Rockland & Lund,1983). Yet although this approach has intu-itive appeal, it does not have the specificityof the economy-of-description approach, inwhich explicit, serial procedures (howevercumbersome) for generating representationsand applying a simplicity metric to them canbe worked out and perhaps even realized inthe form of a computer program. However,investigators in biology have been developingmathematical models of brain activity thatretain the local interactionism of the Gestaltapproach, are plausibly considered to modelactual brain processes, and bring the precisionof mathematical modeling to this domainwithout invoking a computational metaphor.Such models have been applied to the activityof neural nets in general (Cowan & Ermen-trout, 1978), to the Gestalt rules of perceptualorganization (Cowan, 1980, pp. 51-52), andto visual hallucinations (Ermentrout &Cowan, 1979).

The related notion of massively paral-lel computational architectures provides anadditional means for modeling processesthat "compute" mathematically precise func-tions. Discussions of parallel computationalarchitectures (Anderson, 1983; Anderson &Hinton, 1981; Feldman, 1981; Hinton &

182 GARY HATF1ELD AND WILLIAM EPSTEIN

Sejnowski, 1983) stress the idea that "com-putations" may be conceived as direct inter-ations among a large number of local unitsin the brain. This conception of computationmay be viewed as a formalized version of twotraditionally distinct traditions of thought: Itis in the spirit of the qualitative Gestalt ideasabout brain function as introduced by Kohlerand his colleagues (Kofflca, 1919, 1935; Koh-ler, 1920, 1947, 1969) and alluded to byRestle (1979), and it may be seen as a des-cendent of the connectionist approach rep-resented by Hebb (1949). By envisioningmassively parallel interactions, this approachcan use conceptions of processing that ap-proach the continuous variation of soap film.

There is great promise in massively parallelcomputational architectures for modelingcognitive functions (on vision see Ballard,Hinton, & Sejnowski, 1983). Several varietiesof parallel computational architectures areunder investigation (Fahlman, Hinton, &Sejnowski, 1983). Among the simpler of theseis the "value-passing" architecture, in whichcomputation is performed by numerous"units" or "nodes" through their mutuallinks. Such links can be excitatory or inhib-itory and can vary in "gain." Input sets up apattern of activity across a given (perhapsproportionally large) set of units. "Compu-tation" occurs as these units mutually interact.The "output," or computed "representation,"is constituted by the pattern of activation inthe units once they have "settled down" toan equilibrium state. The subsequent use ofthis "representation" by other distinct pro-cessing systems would depend upon the in-teraction of this computational net with theseother systems via connections originating innumerous local units.

As an example, consider a recent compu-tational model of the process of deriving theshape of a surface from its projected imagealone. The model was developed by Brown,Ballard, and Kimball (1982) and is in thespirit of Marr's (1982) approach as previouslydiscussed (see also Horn, 1977). Analysisbegins with reception of the shaded image ofan object. Knowledge of the reflectance func-tion of the surface is provided a priori. Thesystem then computes a representation ofshape from the image through massively par-allel, cooperative processes that compute

shape from numerous local inputs while in-teracting with a second computational systemthat computes the direction of the light source.This basic strategy was found to be robustover several variations in the details of thecomputational design.

Computational models that operate throughthe relaxation of parallel networks typicallyare minimizers. Ballard et al. (1983) describedthe computational task of deriving "shapefrom shading" as a "massive best-fit search."Insofar as this computational task is conceivedas a best-fit problem, it is conceived as em-bodying a simplicity preference. Indeed, mas-sively parallel computational architecturesthat compute through relaxation are com-monly thought of as seeking a minimumenergy state (Hinton & Sejnowski, 1983). Ifthe visual system is regarded as having sucha computational architecture, then it mustbe regarded as an economizer. However, itneed not be viewed as seeking "good forms"in the traditional sense of circles, squares,and rectangles. Rather, it seeks a good fit tonumerous data points, where the best fit maybe an irregular, asymmetrical form. The no-tion of "good form" in this case pertains tochanges over small regions of the surface ofan object (not to its global Gestalt properties)and may amount to no more than a contin-uous second derivative for the function de-scribing local changes in surface orientationacross an object. Thus, although the notionthat visual processing draws upon a parallelcomputational architecture may lend supportto the idea that visual processing incorporatesa minimum principle, it does not necessarilysupport the notions of Pragnanz discussed bycoding theorists or found in the qualitativeapproach of Perkins (1976, 1982). In thiscase, economy of process does not necessarilyyield phenomenal simplicity.

We have treated the soap-bubble and therelated massively parallel-architecture ac-counts as opposed to descriptionalist for-mulations. However, adherents of parallel ar-chitecture accounts sometimes characterizethemselves as working within a description-alist framework (e.g., Marr, 1982), althoughothers reject the notions of internal symbols(e.g., Feldman, 1981) and internal rules (e.g.,Anderson & Hinton, 1981). It may be arguedthat one strength of parallel architectures is


the direct implementation of computationalalgorithms in the engineering of the system,without the need for internally representedrules or assumptions. If this point is accepted,the adoption of a soap-bubble or parallel-architecture account need not preclude theidea that perception ultimately involves de-scribing or categorizing objects in the envi-ronment. In fact, such a position leaves openthe possibility of a hybrid conception of therelation between sensory perception and suchcognitive operations as recognition. In thishybrid account any minimum tendencies inphenomenal experience would result frombottom-up processes working in relative in-dependence of attention and semantic mem-ory (as in Marr's, 1982, conception of earlyvision). These processes would yield repre-sentations of the spatial configurations of theenvironment that provide the basis for rec-ognition, identification, and classification ofobjects.

ConclusionWhen considered as a psychophysical law,

the minimum tendency is well established forthe perception of form, depth, and motionunder certain conditions. Or perhaps oneshould say that diverse minimum tendencieshave been established, because it is by nomeans clear that the various metrics of sim-plicity that have been applied to measuresimplicity or Pragnanz in fact measure asingle dimension or attribute. There may beas many minimum tendencies in perceptionas there are metrics of simplicity.

Granting, nonetheless, that there are ten-dencies toward perceptual economy, theirtheoretical implication remains uncertain.Our investigation leads us to believe that aglobal Minimum Principle, which acts as acardinal principle of perception, will not beobtained. Nonetheless, the notion that a min-imum principle (or principles) is operative inperception does raise interesting questions.Does it reflect a fundamental tendency ofcognitive and perceptual systems to prefersimplicity? Is a preference for simplicityadaptive? Does it result from the fact thatthe simpler is the more likely? These questionsremain open.

The idea that a tendency toward perceptualeconomy would be cognitively advantageous

is appealing. Part of its appeal is derivedfrom the notion that simplicity is adaptivebecause simpler representations take up fewercognitive resources. This seems plausible forrepresentations that are already in place.However, perception is a matter of generatingrepresentations as a consequence of stimula-tion. We have seen that if the representationgenerated is regarded as occurring within asymbolic medium as in coding theory, therecurrently is no basis for supposing that econ-omy of description implies economy of pro-cess. In contrast, if phenomenal simplicity isexplained within a soap-bubble account, sim-plicity of the structure of the process iscorrelated with perceptual economy. Withsuch qualitative accounts, it is difficult to seethe implications of the relationship betweenprocess and perception for overall cognitiveeconomy, for it is unclear what to make ofthe relation between pragnant physiologicalstructures and subsequent cognitive processing(e.g., recognition or memory). However, recentwork that extends parallel computationalmodels to higher domains (Anderson, 1983;Feldman & Ballard, 1982) may provide ameans for effecting this connection. Yet thesetypes of models, which provide a precisecharacterization of economy of process, donot seem to favor economy of perceived form(global Gestalt properties), which weakensthe proposed link between cognitive economyand phenomenal simplicity. One can onlyawait further investigations of perceptualeconomy in which the metrics of simplicityand attendant process models are fully spec-ified. Such investigations could provide amore definite specification of the ways inwhich perception tends toward the simple.

In the meantime, one is left with one'scognitive inclination toward perceptual econ-omy. And that is just what wants explaining.

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the status of the minimum principle in the theoretical

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