1 IntroductionAccording to the organisational principle of `uniform connectedness' (Palmer andRock 1994), chromatically homogeneous surfaces should be perceived as unitary. Thisis usually true, but, as a matter of fact, many surfaces split perceptually into two (ormore) partially overlapping figures (Koffka 1935). In such cases, modal illusory con-tours are seen to bind part of the occluding figure, and amodal contours to bind theoccluded part of the farther figure (figure 1a). (We will name the global figure `surface'and the results of splitting `figures' or `objects'.) Modal contours are more salient instereoscopic or moving displays, but they can be observed in pictorial conditions too.

The aim of this work is to show that mirror symmetry opposes splitting in thiskind of surfaces; that is, we propose to demonstrate that mirror symmetrical surfacessplit less easily than asymmetrical ones.

Much work has been devoted to the rules determining the direction of occlusion:which figure is seen in front and which behind (Farne© and Mantuano 1971; Kanizsa1979; Metzger 1975; Petter 1956; Stoner and Albright 1993; Singh et al 1999a; Tommasiet al 1995). It must be stressed, however, that both order solutions can usually beperceived (although with different strength), so that many of these surfaces are bistabledisplays, as noted also by Petter.

Petter (1956) carried out the first systematic research on the factors playing a rolein the depth ordering of the figures. According to him, the object whose perceptuallyinterpolated contours are shortest appears usually in front. This principle has becomeknown as `Petter's rule'. Petter identified other factors: (i) if the two objects are identical

Mirror symmetry opposes splitting of chromaticallyhomogeneous surfaces

Perception, 2002, volume 31, pages 693 ^ 709

Giulia ParovelDipartimento di Psicologia Generale, Universita© di Padova, via Venezia 8, 35131 Padua, Italy;e-mail: parovel@psy.unipd.it

Stefano VezzaniDipartimento di Psicologia, Universita© di Milano-Bicocca, Edificio U6, piazza dell'Ateneo Nuovo 1,20126 Milan, Italy; e-mail: stefano.vezzani@unimib.itReceived 29 March 2001, in revised form 9 January 2002

Abstract. Chromatically homogeneous surfaces can be seen as single figures but also as two ormore overlapping figures. Local factors such as relatability have been proposed in order to explainperception of two or more figures (Kellman and Shipley, 1991 Cognitive Psychology 23 141 ^ 221).However, even when these factors are at work, there are conditions favouring the perception ofa single figure, which have not been explored so far. Here we propose that one such factor isthe mirror symmetry of the surface. Three experiments were designed to test: (a) the mainhypothesis, that mirror symmetry enhances perception of a single figure; (b) the role of orienta-tion; (c) the effect of the number of axes of symmetry. The results show that (i) there is a goodgeneral correlation between mirror symmetry and perception of a single figure; (ii) vertical andhorizontal axes of symmetry are the most effective; and (iii) the more axes of symmetry a surfacehas, the more likely is the perception of a single figure. These results suggest that mirror symmetryis an important factor in the perception of chromatically homogeneous displays. Some explana-tions are discussed, particularly one based on the rejection-of-coincidence principle [Rock, 1983The Logic of Perception (Cambridge, MA: MIT Press)], and a version of the minimum principlein which the strength of the global solution depends on symmetry, whereas the strength of thesplitting solution depends on the strength of local factors. In brief, global and local factorscompete in determining the perceptual outcome in chromatically homogeneous surfaces.

DOI:10.1068/p3218

and one of them is in motion, the one in motion appears in front; (ii) if the objects areof the same shape but of different size, the larger one appears in front.

Stoner and Albright (1993) trace Petter's rule back to the second of these otherfactors, or, more precisely, to the heuristic according to which the larger the retinalimage, the closer the object appears. Tommasi et al (1995) and Singh et al (1999a)provide evidence against this interpretation. Often the largest figure has the shortestinterpolated contours, but it is not necessarily so, and when these two factors aredisentangled it becomes clear that Petter's rule dominates.

At least three other factors influencing the depth order of the figures are known:(i) the figure that is placed lower in the visual field is more often seen in the fore-ground, in accordance with a well known depth cue (Farne© and Mantuano 1971);(ii) if one of the figures has blurred contours, it is more often seen in the background(figure 1b) (Farne© and Mantuano 1971); (iii) the figure that has the highest supportratio is seen in the foreground (Singh et al 1999b). The support ratio is the ratio of thelength of the physically specified contour of the figure to the total length of its con-tour. In other words, a figure has a high support ratio if its perceptually interpolatedcontours are short with respect to its physically specified contours. When the support-ratio rule competes with Petter's rule, the latter dominates (Singh et al 1999a).

The above-mentioned works dealt mainly with the direction of occlusion, that iswith the results of splitting, but the conditions leading to or preventing the splittingitself are also well worth considering. Why should a three-dimensional arrangement ofoverlapping surfaces be sometimes preferred by the visual system to a two-dimensionalsolution with a single surface? When does this happen, and when not? Only a coupleof works address these issues, and we are going to review them.

In order to explain the splitting taking place in figure 1c, Koffka (1935) appeals tothe minimum principle, according to which the visual system prefers simple and pra« g-nant solutions to complex ones. Commenting upon the only figure that he considers,figure 1c, showing two abutting shapes (although note the puzzling statement of Koffkaabout one figure `partly upon the other'), he says that such a figure ` may be seen as avery irregular form but also as two identical and symmetrical forms, one partly upon theother. In the second case, lines appear indicated in the seen form to which no changesof stimulation correspond. Therefore the unifying forces which are produced by thehomogeneous stimulation of the whole dark area are overcome by segregating forceswhich arise from the unification of well-shaped figures, each one of the two figuresbeing of a better shape than the one irregular figure with homogeneous colouring''(page 141). This way, Koffka can potentially explain both the splitting and the percep-tion of a unitary surface: in every specific case, the simplest and most pra« gnant sol-

(a) (b) (c)

Figure 1. (a) A chromatically homogeneous surface that splits perceptually into two rings.(b) The square that is placed higher in the visual field and whose edges are blurred is seen inthe background (Farne© and Mantuano 1971). (c) It is possible to see two abutting figures, in spiteof the stimulation being homogeneous (Koffka 1935).

694 G Parovel, S Vezzani

ution will be seen. Such a version of the minimum principle is now known as `globalminimum principle' (eg Boselie 1994).

The most serious attempt at formalising the global minimum principle is thestructural information theory, according to which, in the presence of an ambiguousfigure, the solution that will be seen is the one represented by the code with theminimum information load (see for instance Buffart et al 1981). This theory has beenapplied to several visual phenomena, such as amodal completion. However, it hasserious shortcomings (Hulleman and Boselie 1999), so we will not consider it later inour discussion of the minimum principle.

Kellman and Shipley (1991), contrary to Koffka, explain the splitting in terms ofcompletely local factors, and presumably (although not explicitly) the absence of split-ting by the absence of such local factors. These authors propose an ambitious theoryfor all completion phenomena, including the `self-splitting figures' that we considered.Their main thesis is that modal and amodal completions derive from a common unit-formation process. Self-splitting figures are particularly revealing to this purpose:although the depth order of the component figures usually reverses over time togetherwith the modal/amodal character of their interpolated contours, the shape of the figuresdoes not change. There are two necessary and sufficient conditions determining thesplitting of a chromatically homogeneous surface: (i) there must be discontinuities inthe contour of the surface, and (ii) there must be relatable edges. A discontinuity isan abrupt change in the contour direction (figure 2a). The concept of relatability is anattempt at formalising the Gestalt law of good continuation. Intuitively, two edges arerelatable when their linear extensions connect with no discontinuity in between. Also,the angle of intersection of the extensions of the two edges must be obtuse or 908(figure 2b). Relatability is an all-or-none criterion, not a graded one: for instance, twocollinear edges are as relatable as two edges forming a 1008 angle. Subjects' responses,instead, are graded (Kellman and Shipley 1991). See Singh and Hoffman (1999) for aproposal for introducing gradedness into relatability.

In our opinion, the support-ratio rule could also play a role in determining splitting.Shipley and Kellman (1992) showed that the higher the proportion of physically speci-fied edges, the more salient are the subjective figures. It may well be that the lowerthe proportion of interpolated edges with respect to the physically specified edges, themore likely splitting will be. It is plausible that only relatively salient edges will be seen.A high support ratio very probably implies that salient interpolated edges are generated;such edges will be seen modally and splitting will take place. On the contrary, whenthe support ratio is very low, interpolated edges will not be salient enough to be seenmodally, and splitting will not take place.

We agree with Kellman and Shipley (1991) that splitting is (mainly) due to localfactors, although for a critique of the claim that completions do not occur for anglesless than 908 one must see Boselie and Wouterlood (1992). However, we disagree with

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Figure 2. (a) The points of discontinuity in the contour of the surface are marked by the blackdots. (b) On the left there is no relatability, because the extensions of the two lines meet at anangle smaller than 908. On the right there is relatability, because the extensions of the two linesmeet at an angle greater than 908.

Mirror symmetry opposes splitting of chromatically homogeneous surfaces 695

Kellman and Shipley on their explicit rejection of the global minimum principle.In fact, we have observed that at least one global factor exists which, even whenthe conditions pointed out by Kellman and Shipley (1991) are met, opposes splitting.This factor is mirror symmetry (henceforth simply `symmetry'). We maintain thatsymmetrical chromatically homogeneous surfaces are perceived as unitary more oftenthan asymmetrical ones (Parovel and Vezzani 2000). The effectiveness of symmetry inopposing splitting has already been noted, incidentally and briefly, by Boselie andWouterlood (1992), in a work dedicated to pointing out the shortcomings of the localtheory of Kellman and Shipley (1991). See section 5 for the contribution of Boselieand Wouterlood.

Mirror symmetry, especially about a vertical axis, is probably the most salientinstance of regularity. This is confirmed by several works (eg Corballis and Roldan1975; Driver and Baylis 1996; van der Helm and Leeuwenberg 1996; Mach 1886; Palmerand Hemenway 1978; Rock and Leaman 1963; Royer 1981).

Several functional effects of symmetry are known. For instance: (i) an area is morelikely perceived as figure rather than as ground if it is symmetrical (Bahnsen 1928;Driver and Baylis 1996; Peterson and Gibson 1994) (figure 3); (ii) symmetry influencesamodal completion (eg Buffart et al 1981; Sekuler 1994); (iii) symmetry influences visualsearch (Wolfe and Friedman-Hill 1992).

In short, we believe that splitting is due (mainly) to the local factors pointed outby Kellman and Shipley (1991) (but see Boselie and Wouterlood 1992), but also thatsuch factors can be opposed by the global factor of symmetry. If favouring localfactors are lacking, chromatically homogeneous surfaces do not split.

We are not claiming that all symmetrical surfaces will be perceived as unitary.In fact, whether splitting takes place depends also on local factors, and if local factorsare strong enough (that is, if there are many relatable, especially collinear, edges), asymmetrical surface can certainly split. What we state is simply that, other factorsbeing equal, a symmetrical surface will split less often than an asymmetrical one.

We have run three experiments, in order to provide answers to the followingquestions:

(i) Does symmetry indeed oppose splitting?(ii) What is the role of the orientation of the axis of symmetry?(iii) What is the effect of the number of axes of symmetry?

In all experiments we have used surfaces with at least two relatable edges.

Figure 3. Symmetrical areas are usually seen as figures, although the alternative percept is alsopossible (Purghe 1999).

696 G Parovel, S Vezzani

2 Experiment 1The first experiment aimed to establish whether symmetry really opposes splitting.To this end, we compared vertically symmetrical surfaces and increasingly asymmetricalsurfaces. We expected to find that splitting responses increased with the asymmetry ofthe surfaces. In this and the following experiments, all the component figures had atleast one axis of symmetry.

2.1 SubjectsTwenty subjects participated in the experiment. All had normal or corrected-to-normalvisual acuity, and were na|« ve as to the purpose of the experiment.

2.2 Stimuli and procedureSixteen black surfaces were printed at the centre of separate 21 cm629.7 cm sheets ofwhite paper. The sheets were shown vertically oriented and in frontal position at adistance of 80 cm. Each surface was composed of an upper and a lower part. Four ofthe surfaces were symmetrical about the vertical axis. From each symmetrical surfacethree asymmetrical surfaces were obtained by an anticlockwise rotation of the upperpart amounting to, respectively, 88, 168, and 328. All surfaces are shown in figure 4.The surfaces were 42.3 mm to 59 mm wide, and 56.5 mm to 82 mm high. The stimuliwere randomised, and their order was different for each subject.

Series Rotation upper part=8

0 8 16 32

A

B

C

D

Figure 4. Stimuli of experiment 1. There were four series of surfaces. Each series was composedof one symmetrical and three asymmetrical surfaces. The asymmetrical surfaces were obtainedby an anticlockwise rotation of the upper part of the symmetrical surfaces by 88, 168, and 328.

Mirror symmetry opposes splitting of chromatically homogeneous surfaces 697

Each subject read the following written instructions: ` You will be shown uniformlycoloured surfaces. You are to say whether each surface is composed of a single figure orof two partially superimposed figures. The surfaces are all different from one another,although sometimes only slightly, so each of them must be judged separately. Answeraccording to what you see, not according to what you believe or think''. Afterwardsthe subject carried out his task without time limits, putting into words his responses,which the experimenter wrote down. Only one response per surface was required. Theallowed responses were ` one'' and ` two''. From now on, the ` one'' responses will benamed global responses, and the ``two'' responses will be named splitting responses.

2.3 Results and discussionEach series was treated as a repetition of the same experimental stimuli; data werecombined, and a repeated-measures ANOVA was performed on the proportion of globalresponses.

Results are clear: the greater the degree of asymmetry, the larger the proportionof splitting responses (figure 5). The factor of `symmetry' is significant (F3 57 � 13:313,p 5 0:001); moreover, a trend analysis indicates that the linear component visible infigure 5 is significant (F1 19 � 24:587, p 5 0:001). There are no other significant trends.The sole pair of conditions whose difference is not significant is the 88 versus 168.

We applied also the nonparametric one-tailed McNemar test (McNemar 1955) toeach experimental series, with a continuity correction. All the differences betweenthe first and the last stimuli of each series were significant in the expected direction(series A: p 5 0:005; series B: p 5 0:025; series C: p 5 0:005; series D: p 5 0:01). Theother differences were not significant.

Therefore, our hypothesis is corroborated: symmetry does oppose splitting in all fourexperimental series. Moreover, it is confirmed that perceptual symmetry is a matter ofdegree (eg Palmer and Hemenway 1978).Van der Helm and Leeuwenberg (1996) proposea holographic approach to perceptual symmetry, which can account for the graded-ness of perceptual symmetry, contrary to the traditional transformational approach.

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Figure 5. Results of experiment 1. Percentage of global responses as a function of degree ofasymmetry of the surfaces. `Global response' means that a single figure was seen. 08 asymmetrycorresponds to perfectly symmetrical surfaces; other degrees of asymmetry correspond to asym-metrical surfaces differing from the symmetrical ones for an anticlockwise rotation of their upperparts respectively by 88, 168, and 328. Each data point represents the responses given by twentysubjects to four different surfaces with the same degree of asymmetry. Each surface received a singleresponse by each subject.

698 G Parovel, S Vezzani

3 Experiment 2The salience of an axis of symmetry depends on its orientation (eg Mach 1886).Usually it is found that the vertical orientation is the most salient (easily detected),followed by the horizontal, and finally by the diagonal (Barlow and Reeves 1979; Palmerand Hemenway 1978; Pashler 1990; Rock and Leaman 1963; Royer 1981; Wagemans et al1992;Wenderoth 1994, 1995, 1996a, 1996b, 1997, 2000;Wenderoth and Welsh 1998a, 1998b).A few other works report a different ordering, with diagonal symmetry more salientthan horizontal symmetry (Corballis and Roldan 1975; Sekuler 1994; Evans et al 2000),although Wenderoth (2000) failed to replicate the Corballis and Roldan (1975) results.So, it is not easy to make precise predictions about the relationship between theorientation of the axis of symmetry and the salience of symmetry. Moreover, the kindsof patterns mostly used in literature, like dot patterns or outlined figures, are quitedifferent from our figures. As a matter of fact, the visual system seems to process dotpatterns quite differently from unitary figures (Liu, Kersten, and Knill 1995, in Sekulerand Swimmer 2000). We believe that it would be interesting to know how the percep-tion of chromatically homogeneous figures depends on the orientation of their axis ofsymmetry. This was the aim of experiment 2.

3.1 SubjectsTwenty subjects, different from those in experiment 1, participated in the experiment.All had normal or corrected-to-normal visual acuity, and were na|« ve as to the purposeof the experiment.

3.2 Stimuli and procedureThe surfaces were 53.3 mm to 57.9 mm wide, and 57.5 mm to 76.9 mm high. Each oneof them was symmetrical about a single axis. Each surface was shown in five differentorientations: 08 (horizontal), 22.58, 458, 67.58, and 908 (vertical). So there were 15 stimuliin all (figure 6). The other conditions were identical to those of experiment 1.

Series Orientation of the surface=8

0 22.5 45 67.5 90

A

B

C

Figure 6. Stimuli of experiment 2. There were three series of five surfaces. Within each series,the surfaces differed only in orientation; between series, the surfaces differed in shape, and theorientations were the same.

Mirror symmetry opposes splitting of chromatically homogeneous surfaces 699

3.3 Results and discussionResults are shown in figure 7. Each series was considered as a repetition of thesame experimental stimuli; data were combined, and a repeated-measures ANOVA wasperformed on the proportion of global responses.

The factor `orientation' is strongly significant (F4 76 � 5:869, p 5 0:001). The solesignificant trend is quadratic (F1 19 � 15:58, p 5 0:001). The differences 08 versus 22:58,08 versus 458, 458 versus 67:58, and 458 versus 908 are significant, whereas the importantdifference 08 versus 908 is not significant (F1 19 � 1:342, p � 0:261).

Many of the differences show themselves only when the three series are combined.A one-tailed McNemar test on each series, in fact, revealed only three significantdifferences: the difference between the conditions 458 and 908 in series B ( w 2 � 5:1,p 5 0:025), and the differences between the conditions 08 and 458 ( w 2 � 8, p 5 0:005),and the conditions 458 and 908 ( w 2 � 5:1, p 5 0:025) in series C.

Our hypothesis is corroborated only partially: a vertical or horizontal axis is moresalient than a diagonal one, but there is no significant difference between a verticaland a horizontal axis. This point will be dealt with in section 5.

The surfaces with 22.58 and 67.58 orientations produced intermediate resultsbetween those for the 458 orientation and the horizontal and vertical orientations.This agrees with what Wenderoth (1994) found: axes of symmetry inclined 108 or 208with respect to the horizontal or the vertical are more salient than diagonal axes.

The experiment does not allow us to say whether the diagonal axis had an effecton the results, because a condition without any axis of symmetry was lacking. How-ever, many works suggest that even a diagonal axis of symmetry has a certain salience,although reduced in extent (Wagemans 1995).

4 Experiment 3The third experiment had two aims: first, to replicate partially experiment 1, and,second, to verify whether the probability of global responses increases with the numberof axes of symmetry. It is well known, in fact, that the salience of symmetry is posi-tively correlated with the number of axes of symmetry (Palmer and Hemenway 1978;Wenderoth 1997; Wenderoth and Welsh 1998b), possibly because perceptual symmetry

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Figure 7. Results of experiment 2. Percentage of global responses as a function of the orientationof the surfaces. 08 orientation corresponds to surfaces with a vertical axis of symmetry; 908 orienta-tion to surfaces with a horizontal axis of symmetry. The in-between orientations correspond toanticlockwise-rotated surfaces, starting from the vertical orientation. Each data point representsthe responses given by twenty subjects to three different surfaces (one for each series) with thesame orientation. Each surface received a single response by each subject.

700 G Parovel, S Vezzani

increases with the number of axes of symmetry (another possibility is that perceptualregularity increases, rather than perceptual symmetry). So, our thesis about the influenceof symmetry would be corroborated if we found such an effect.

4.1 SubjectsTwenty subjects, different from those in experiments 1 and 2, participated in the experi-ment. All had normal or corrected-to-normal visual acuity, and were na|« ve as to thepurpose of the experiment.

4.2 Stimuli and procedureAll the experimental surfaces are shown in figure 8. There were four surfaces withrespectively 4, 2, 2, and 8 axes of symmetry. Starting from each of them, we obtainedsurfaces with a decreasing number of axes of symmetry by changing the relative posi-tion of the component figures; the latter were obtained by extending the relatableedges. So, four series of surfaces were used, for a total of thirteen surfaces. It can beseen in figure 8 that the surfaces of series A had respectively 0, 1, and 4 axes ofsymmetry, those of series B and C had 0, 1, and 2, and those of series D had 0, 1, 4,and 8. There was always a vertical axis of symmetry, provided, of course, that therewas at least 1 axis of symmetry. We used also figures with no axes of symmetry in orderto replicate partially experiment 1 with different surfaces. The surfaces were 57.5 mm to78.3 mm wide, and 52.2 mm to 69.5 mm high. The other conditions were identical to thoseof experiment 1.

Series Number of axes of symmetry

0 1 2 4 8

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Figure 8. Stimuli of experiment 3. There were four series of surfaces. Within each series, thesurfaces were obtained by arranging in different ways the same component figures; each surfacehad a different number of axes of symmetry, and a surface with no axis of symmetry wasalways present. The numbers in the small boxes are those to which we refer when speaking inthe text of `surface D1', `surface B3', etc.

Mirror symmetry opposes splitting of chromatically homogeneous surfaces 701

4.3 Results and discussionResults are shown in figure 9. It was not possible to combine all data, because therewas a variable number of axes of symmetry between series. Therefore we combinedthe data of series B and C, and, separately, of series A and D, ignoring the conditionwith 8 axes of symmetry in series D. Two distinct repeated-measures ANOVAs wereperformed on the data combined in this way. The difference between 1 and 2 axes ofsymmetry was significant in the expected direction (F1 19 � 5:630, p � 0:028), as wasthe difference between 1 and 4 axes (F1 19 � 8:941, p � 0:008). We also combined datafrom all the series to check the difference between 0 and 1 axes of symmetry, whichwas significant in the expected direction (F1 19 � 6:047, p � 0:024).

A one-tailed McNemar test revealed significant differences between the first andthe last stimuli of series A, B, and D, but not of series C (series A: w 2 � 9, p 5 0:005 ;series B: w 2 � 9, p 5 0:005 ; series C: w 2 � 0:9, ns; series D: w 2 � 5:1, p 5 0:025). Con-sidering, within each series, only comparisons between surfaces with at least 1 axis ofsymmetry, the sole significant difference was between the surfaces with 1 and 8 axesof symmetry of series D ( w 2 � 5:1, p 5 0:025). No other comparison was significant.

So experiment 1 was successfully replicated (partially): asymmetrical surfaces splitmore often than symmetrical ones. Moreover, the number of axes of symmetry waseffective in the expected direction: the proportion of global responses increased with it.

Series A and C are very similar: they differ only because the two componentrectangles are identical in series A, whereas they are different in series C. Surface A3has 4 axes of symmetry, whereas surface C3 has only 2: we obtained 80% of globalresponses in the first case and 40% in the second case ( w 2 � 6:125, p 5 0:01), whichagrees with our hypothesis.

Surface D4 has 8 axes of symmetry. Therefore we could expect that it was an idealcandidate for receiving global responses. Yet, it received only 40% of global responses.In our opinion, however, the explanation for this result is rather simple, althoughimportant: a very strong factor (that is, high degree of symmetry) testifies to theunitariness of this surface, but very strong factors testify to its composedness too,because the surface has no less than sixteen relatable edges. Factors are at work, then,that have opposite effects. All surfaces of series D have sixteen relatable edges; as aresult, this series received the lowest mean percentage of global responses (18.75%),in spite of its having the highest mean number of axes of symmetry.

The same explanation applies to another result. Surfaces B3 and D4 are composedof the same figures, that is two squares. These surfaces have respectively 2 and 8axes of symmetry, so we could expect that D4 had more global responses, whereas the

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Figure 9. Results of experiment 3. Percentage of global responses as a function of the numberof axes of symmetry of the surfaces.

702 G Parovel, S Vezzani

opposite took place: B3 had 80% of global responses versus 40% for D4 ( w 2 � 4:9,p 5 0:025). Very likely, this result is due to D4 having many more relatable edgesthan B3. Moreover, the edges of surface D4 are collinear in pairs, whereas those ofsurface B3 form angles of 908; in all probability, this factor has an effect too.

5 General discussionOur main result is that symmetry opposes splitting in chromatically homogeneoussurfaces. Moreover, in agreement with previous works on symmetry, we have foundthat, the larger the number of axes of symmetry, the more effective are the symmetryresults, and that vertical and horizontal symmetries are more salient than diagonalsymmetry.

First of all, there is one point we want to make clear. We believe that most or allof our surfaces were bistable; that is, at least in most cases, both the global and thesplitting solutions could be seen. In fact, some subjects reported spontaneously thatthey had seen both solutions when presented with some of the surfaces (in such a casethe experimenter told them to report the prevailing percept, which the subjects didwithout any problem). We ourselves are able to see both solutions in at least some ofour surfaces, and the solutions alternate, although one of them usually prevails.According to us, the `strength' ratio of the two solutions determines how long eachsolution is seen. Of course, this idea about competing percepts is not new (see forinstance van Lier et al 1995; Sekuler 1994).

Our data do not allow us to tell whether the visual system generates the twosolutions sequentially or in parallel. A microgenetic analysis (Sekuler and Palmer 1992)is called for answering such a question. Palmer and Rock (1994) proposed that, inchromatically homogeneous surfaces, the global figure is seen first and only later,possibly, the component figures; according to such hypothesis, then, the global figureshould always be seen, although maybe only for a short time. We do not have a soleexplanation for our data. We think, rather, that a few explanations are feasible.

The first explanation was proposed by Boselie and Wouterlood (1992, page 282):` The easily perceived vertical symmetry of figure 1cJ [which was a cross almost identi-cal to our surface A3 (experiment 3)] implies a decomposition of the pattern in a leftand right part, with collinearity still playing a role within each of these halves. Thisdecomposition prevents a split up along collinear lines only''. This hypothesis is basedon the sensible consideration that the visual system must split the surface into twoparts in order to detect symmetry; the hypothesis implies that perception of symmetryprecedes perception of the component figures.

However, this hypothesis must face an important objection. In fact, it is doubtfulwhether the decomposition in mirror-symmetrical parts `prevents' the splitting incomponent figures. There are two reasons to think, instead, that the former type ofsplitting competes with the latter: (i) we said previously that at least in most casesboth the global and the splitting solutions can be seen, although they have different`strength'; (ii) above all, there are symmetrical surfaces which produce a high percent-age of splitting responses. Symmetry does not always win. Surface D4 in experiment 3is especially significant. This surface has 8 axes of symmetry, and this should blockvery well its splitting into component figures. Yet, twelve subjects out of twenty saw thecomponent figures.

Moreover, our data suggest that the surface is decomposed along not only thevertical axis but also the horizontal axis, and other axes of symmetry could possiblybe taken into consideration, because many works suggest that even symmetry abouta diagonal axis is somewhat salient. Admittedly, however, this does not appear to be acritical point in Boselie and Wouterlood's proposal.

Mirror symmetry opposes splitting of chromatically homogeneous surfaces 703

Our results admit at least three other possible explanations: the first is foundedon Rock's (1983) `rejection-of-coincidence principle', and the other two are founded onthe minimum principle. These explanations are not necessarily mutually exclusive(Rock 1983).

5.1 The rejection-of-coincidence principleThe rejection-of-coincidence principle states that the visual system tends to avoidunexplained coincidences in the interpretation of the retinal image. According to thisprinciple, each one of our symmetrical surfaces could be obtained by superimposingtwo (symmetrical) component figures only thanks to the coincidental perfect alignmentof the axis (or of one of the axes) of symmetry of each component figure. So, whenthe global surface is symmetrical, the visual system tends to avoid the splitting solutionbecause it involves a coincidence.

The effect of the number of axes of symmetry can be explained by the same style ofreasoning. Consider, for instance, series A of experiment 3: it would be a coincidence ifthe two rectangles were placed in such a way as to produce a surface with one axis ofsymmetry (as in surface A1), but it would be a greater coincidence if they were placedin such a way as to produce a surface with four axes of symmetry (as in surface A3).Only in the latter case, in fact, their centres would coincide. Therefore, there will be astronger avoidance of coincidence in the latter case.

The rejection-of-coincidence principle can explain also why the component figuresare perceived, at least when there are collinear edges: in fact, it would be an unexplainedcoincidence if two edges were perfectly collinear without actually being parts of thesame contour (Rock 1983).

5.2 The minimum principleTwo possible explanations appeal to the minimum principle. The minimum principle hasbeen used to explain several visual phenomena, such as perceptual transparency (Gerbino1994) and amodal completion (Buffart et al 1981), and, by the Gestaltists, to explainperception generally (for a review, see Hatfield and Epstein 1985). If the visual systemworks according to a minimum principle, it will prefer simple to complex solutions,and symmetry has always been considered an instance of simplicity. Simplicity is avery important part of the concepts of `Pra« gnanz' and `goodness', which are synonymous.

We can think of at least two ways of applying the minimum principle; it candetermine: (i) the salience of both the global and the splitting solutions, or (ii) only thesalience of the global solution. Koffka (1935) adopted the first interpretation, as it isclear from our quotation in section 1: the visual system compares the Pra« gnanz of thewhole and that of the component figures, and the most pra« gnant solution is chosen.

According to this interpretation of the minimum principle, in our experiments 1and 3 we obtained a higher percentage of splitting responses for the asymmetricalsurfaces because the Pra« gnanz ratio between the whole and the component figures waslower for the asymmetrical surfaces than it was for the symmetrical ones; in otherwords, in the competition between whole and component figures, the latter, beingalways symmetrical, won more often when the whole was asymmetrical than when itwas symmetrical.

The second way of applying the minimum principle is the following. Only thegoodness of the global surface is relevant, and it is not compared with the goodnessof the component figures. The more the global surface is pra« gnant, the greater theprobability of the global solution being seen, but the Pra« gnanz of the whole is notcompared with the Pra« gnanz of the component figures. What is compared is thestrength of the two solutions, which depends on different factors in the two cases.

704 G Parovel, S Vezzani

We have a competition between, on the one hand, a global surface whose strengthdepends on its Pra« gnanz (symmetry), and, on the other hand, component figures whosestrength does not depend on Pra« gnanz but on the strength of local factors.

We think that we can rule out a strong version of the first account, according towhich the visual system simply compares the goodness of the global surface and thegoodness of the component figures. In fact, a comparison between surfaces D4 and B3in experiment 3 proves that the strength of the splitting solution depends at leastpartially on local factors. Both surfaces are composed of two squares, so the goodnessof the component figures is the same in both cases. Moreover, surface D4 has noless than 8 axes of symmetry, whereas surface B3 has only 2. Therefore, one couldexpect surface D4 to produce more global responses, whereas the opposite took place:there were twelve splitting responses for surface D4, and only four for surface B3( w 2 � 4:9 , p 5 0:025). Plausibly this happens because in surface D4 there are muchstronger local clues favouring the splitting solution: there are sixteen relatable edges inthis surface versus the four in surface B3. This implies that splitting depends at leastin part on local factors.

This weakens the first account, so the choice is between a model where thegoodness of the component figures plays a role, but not an exclusive role and maybenot even a predominant one, and a model where it does not play any role. Our datado not allow us to decide between these two possibilities, because we manipulated thesymmetry of the surfaces but not the symmetry of the component figures (the latterwere always symmetrical, being simply rotated in experiments 1 and 2): it is an openquestion whether the component figures are the more easily perceived when theyare symmetrical. However, we are inclined toward the second possibility, because split-ting takes place also with totally irregular component figures, even when a globallysimpler (that is symmetrical) solution is available, as in figure 10.

Whichever solution is chosen, however, we want to stress that, in our opinion, aglobal minimum principle cannot explain the formation of the component figures. Infact, it seems that the visual system must generate all the infinite possible componentfigures compatible with the physically specified edges, in order to choose the simplest,and this is practically impossible (Kellman and Shipley 1991). Moreover, as can beseen in figure 10, the component figures can be completely irregular, which is incom-patible with a global minimum principle. In our opinion, the splitting is entirely dueto local factors, as proposed by Kellman and Shipley (1991). We think that the visualsystem always generates both the splitting and the global solutions, and then comparesthe two, choosing the strongest. The splitting solution is generated by exploiting localfactors, and the global solution by applying the organisational principle of uniformconnectedness (Palmer and Rock 1994).

However, there is a local version of the minimum principle that can explainwhy the component figures are completed the way they are: in fact, the completionbased on the continuation of the physical contour is the locally simplest solution

Figure 10. This surface has a vertical axis of symmetry. Nevertheless,two component irregular figures are perceived. The global surfaceis symmetrical, the component figures are not, but the latter perceptprevails. This suggests that there is no competition between thePra« gnanz of the global surface and the Pra« gnanz of the componentfigures; rather, the component figures emerge thanks to strong localfactors, namely relatable edges. Possibly, another factor might play arole: the shortness of the amodal and modal contours that must beinterpolated in order to obtain the component figures.

Mirror symmetry opposes splitting of chromatically homogeneous surfaces 705

(eg Wouterlood and Boselie 1992). Maybe there is a competition between the globallysimplest solution, which favours the perception of a single figure, and the locally simplestsolution, which favours the perception of the component figures.

Figure 10 suggests that another factor might affect the strength of the globalsolutionöcompactness of the surface: possibly the global solution does not prevail inthis figure because the surface, although symmetrical, lacks in compactness. FollowingHulleman and Boselie (1999), we define compactness as the square of the perimeterdivided by the area. So it is possible that the goodness of the global solution does notdepend so much on its symmetry as on its perceived regularity: the latter is relatednot only to symmetry but also to compactness (Hulleman and Boselie 1999).(1)

The minimum principle can explain the effect of the number of axes of symmetryif one assumes that perceived symmetry increases with the number of axes of sym-metry: the more axes of symmetry a surface has, the more it will look symmetricaland simple, and the stronger will be the opposition to its splitting into componentfigures.

An explanation based on past experience must also be considered, because severalof our surfaces and all of the component figures were familiar. We cannot rule outthat this factor had an influence, but we can rule out that it fully explains our results.For instance, it seems to us that the surfaces of series D in experiment 3 were moreor less equally familiar, and yet surface D4, which had 8 axes of symmetry, receivedmore global responses than the asymmetrical surface D1 (w 2 � 5:1, p 5 0:025).

We failed to find the usual superiority of vertical over horizontal symmetry. Thisresult must be discussed. First, at least another work failed to find it (Fisher andBornstein 1982). Second, the result could simply be due to the weak power of ourstatistical test. Third, there might be a confound. As we said in section 1, Farne© andMantuano (1971) showed that the relative vertical position of the component figuresplays a role in determining which figure is seen in the foreground. It might be that,when the component figures have different vertical positions, the strength of the split-ting solution is increased. In our experiment, when the orientation of each surface isvaried, the relative vertical position of its component figures is varied too. It might bedue to this confound that the surfaces with a vertical axis of symmetry and thosewith a horizontal one produce similar results. In fact, the component figures of thevertically symmetrical surfaces are at different levels, whereas the component figures ofthe horizontally symmetrical surfaces are at the same level: in the first case, then,there might be a stronger tendency toward perceiving splitting, which could counter-balance an opposite tendency due to the verticality of the axis of symmetry. The samereasoning applies to the comparison between horizontally symmetrical surfaces anddiagonally symmetrical surfaces: only in the latter case the component figures are atdifferent levels, and it could be due at least in part to this factor that there are moresplitting responses when there is a diagonal axis of symmetry than when there is ahorizontal axis of symmetry.

An explanation of the effects of orientation is beyond the aims of this paper.However, none of the theories considered above can explain such effects, nor can thestructural information theory and the holographic approach by van der Helm andLeeuwenberg (1996). It has been proposed several times that the superiority of verticalsymmetry is due to biological objects being usually vertically symmetrical, but thesuperiority of the horizontal over the diagonal orientation cannot be explained thisway.

(1) However, there is another possible explanation for what is seen in figure 10: the edges that thevisual system must interpolate in order to obtain two figures are very short with respect to thetotal length of the contour (see section 1 for the likely role of this factor).

706 G Parovel, S Vezzani

Finally, we want to comment upon Kellman and Shipley's (1991) theory. We acceptthat local factors are necessary for splitting to be perceived. However, these authorsreject the idea that global factors contribute to the perceptual outcome in chromaticallyhomogeneous surfaces, and our data suggest that they are wrong in this respect. More-over, we think that relatability theory does not go far enough in explaining splitting.Consider, for instance, figure 11; here two rectangles are seen, one in front of the other,and the depth order of the figures reverses over time. Two edges complete modally, andtwo amodally. But why do all the edges not complete modally at the same time? Moreover,why do always the same couplings of modally completed edges take place? In princi-ple, there could be modal completion of only one edge, or of one edge belonging toone rectangle together with one edge belonging to the other rectangle. Of course, insuch cases figures other than rectangles would be seen. Why does this not happen?Relatability theory cannot answer these simple questions.

5.3 Global and local effects in chromatically homogeneous surfaces and amodally completedfiguresWe are well aware that the problem of the relation between local and global factorshas been widely studied in connection with amodal completion (eg van Lier et al 1994;Sekuler 1994), and the phenomenon we are considering is clearly related to amodalcompletion. The doubt could arise, then, that the literature on amodal completion couldhelp elucidate the relation between global and local factors in chromatically homoge-neous figures. In our opinion, this doubt is unfounded. In amodal completion, globalfactors are deemed to influence the shape of the completing figures. In our case,instead, global factors prevent splitting and hence amodal completion. In our opinion,then, the analogy between amodal completion and chromatically homogeneous figuresis rather superficial and does not deserve to be elaborated. However, it must bestressed that our work agrees with others (eg Boselie and Wouterlood 1992; Sekuler1994) in showing the shortcomings of relatability theory in explaining phenomena ofamodal and modal completion.

6 ConclusionsWe have provided evidence that mirror symmetry (or possibly regularity) does opposesplitting of chromatically homogeneous surfaces. This result must be added to the listof functional effects of symmetry. There is a competition between the global factor ofsymmetry, which opposes splitting, and local factors such as relatability, which favoursplitting. The strength of the global solution depends on the symmetry, or regularity,of the global surface, whereas the strength of the splitting solution depends on thestrength of local factors. Moreover, we have proposed a few plausible explanations.Further work is needed to decide amongst them, but, according to us, the two mostplausible ones are: (i) the explanation based on the rejection-of-coincidence principle,and (ii) the application of the minimum principle where only the Pra« gnanz of thewhole is taken into account.

Figure 11. The depth order of the two rectangles changesover time. Why do all the edges not complete modally atthe same time? And why do always the same couplings ofmodally completed edges take place?

Mirror symmetry opposes splitting of chromatically homogeneous surfaces 707

Acknowledgment.We are grateful to Giovanni B Vicario for his insightful suggestions.

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