Detection of symmetry and anti-symmetry

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tryL.Sherhesteved inceptioetristimuli were only moderately aected by these manipulations. Anti-symmetrical stimuli composed of large black and white checksfacts across time and cultures. The biological signi-of view because it can be used to probe the nature ofspatial coding mechanisms in the human visual of bilateral symmetry about the y axis ifFor images satisfying Denition 2, there is a perfectnegative correlation between the intensities of sym-metrically placed points. Therefore, symmetrical andanti-symmetrical stimuli have the same information con-tent. However, identical information content does not.* Corresponding author. Tel.: +1 514 848 2243; fax: +1 514 848 4545.E-mail address: (R. Gurnsey).Vision Research 45 (20050042-6989/$ - see front matter 2005 Elsevier Ltd. All rights reservedcance of symmetry has been widely discussed and ithas been shown that symmetry inuences the behaviourof many animals, including birds and bees (e.g., Hor-ridge, 1996; Swaddle & Cuthill, 1994). Humans are verysensitive to the presence of symmetry in images (e.g.,Barlow & Reeves, 1979) and many vision scientists havespeculated about the mechanisms underlying symmetrydetection (see Wagemans, 1995 for a recent review).Symmetry is interesting from a signal processing point8x; yIx; y Ix; y: 1For images satisfying Denition 1 it is clear that thereis a perfect positive correlation between the intensities ofpoints that are symmetrically placed across the y axis.From this correlational point of view, it is natural toask whether we are equally sensitive to anti-symmetrydened as8x; yIx; y Ix; y: 2elicited low thresholds. However, anti-symmetry became essentially undetectable at small check sizes. Removing low frequenciesfrom large-check-size, anti-symmetrical stimuli had little eect on thresholds whereas removing high frequencies had a pronouncedeect. Moving the stimuli from xation to 8 eccentricity caused a dramatic increase in thresholds for anti-symmetrical stimuli butnot symmetrical stimuli. When the grey scale range was increased anti-symmetry was undetectable at any check size whereas sym-metry was easily seen at all. We argue that these results and others in the literature suggest that anti-symmetry is only detected underconditions favourable to selective attention. 2005 Elsevier Ltd. All rights reserved.Keywords: Polarity; Symmetry; Eccentricity; Spatial Vision; Second-order channels1. IntroductionBilateral-or mirror-symmetry is extremely salient tohumans and has been a prominent feature of our arte-The denition of bilateral symmetry provides a pointof departure for posing such questions.Points I(x,y) and I(x,y) are symmetrically placedwith respect to the y axis and an image possesses theDetection of symmeSandra Mancini a, Sharona Department of Psychology, Concordia University, 7141b Department of Neurobiology and Anatomy, University of RocReceived 13 August 2003; receiAbstractTo assess the role of second-order channels in symmetry pertent, eccentricity and grey scale range on the detection of symmdoi:10.1016/j.visres.2005.02.004and anti-symmetrySally b, Rick Gurnsey a,*brooke Street West, Montreal, Que., Canada H4B 1R6r, 601 Elmwood Avenue, Box 603, Rochester, NY 14642, USArevised form 18 January 2004n we measured the eects of check size, spatial frequency con-cal and anti-symmetrical patterns. Thresholds for 21452160aged over all axes of symmetry tested (vertical, horizon-symmetry and anti-symmetry, although the eect waseseatal, left- and right-oblique), MA, OPP and RA stimulielicited approximately 74%, 70% and 72% correct detec-tions, respectively. Considering only the vertical axis ofsymmetry, MA, OPP and RA stimuli elicited approxi-mately 80%, 82% and 81% correct detections, respec-tively. These results suggest that there is littlerelationship between discriminability and the degree ofcorrelation across the axis of symmetry. (However, ad 0 analysis computed on the group data suggested amodest disadvantage for the OPP stimuli relative toMA and RA.) Similar results were reported by Saari-nen and Levi (2000) who used symmetrical andanti-symmetrical stimuli comprising black and whitenecessarily confer identical perceptual status. For exam-ple, it is well known that faces are much harder to recog-nize in photographic negatives than in positives eventhough the only dierence is a polarity reversal of inten-sity values. If psychophysical observers are equally sensi-tive to symmetry and anti-symmetry they would be saidto show polarity insensitivity (Tyler & Hardage, 1996).Polarity insensitivity might reveal that interesting non-linear representations contribute to symmetry detectionand yield sensitivity to anti-symmetry in spite of its ab-sence in nature and consequent lack of biologicalsignicance.2. Sensitivity to symmetry and anti-symmetryA number of reports have addressed the relative dis-criminability of symmetry and anti-symmetry (Zhang &Gerbino, 1992; Wenderoth, 1996) using relatively sparsedisplays comprising black and white dots on a greybackground. Wenderoths (1996) stimuli, for example,consisted of 50 dots, each subtending 0.2 degrees visualangle within a display that was about 20 in diameter.The participants task was to discriminate randomlypositioned dots from those with symmetrically posi-tioned dots having dierent degrees of correlation intheir intensities. Three conditions of particular interestwere those referred to as MA, RA and OPP becausethey all contained both black and white dots. In MAstimuli, black dots matched black dots and white dotsmatched white dots across the axis of symmetry. Thusthere was a perfect correlation between the grey-levelsof symmetrically placed dots. In OPP displays therewas a perfect negative correlation between the grey-lev-els of symmetrically placed dots; i.e., black matchedwhite, and vice versa. In our terminology, MA stimuliare symmetrical and OPP stimuli are anti-symmetrical.For RA displays there was zero correlation in the polar-ity of the symmetrically placed dots; i.e., half thematches were same polarity and half were of oppositepolarity. Wenderoths (1996) data show that when aver-2146 S. Mancini et al. / Vision RGaussian blobs.more pronounced in the anti-symmetry displays.Rainville (1999) found similar sensitivities forsymmetrical and anti-symmetrical patches comprisingbandpass, centre-surround micropatterns. Detectionaccuracy was measured as a function of positional jitteradded to the individual micropatterns. Accuracy de-clined with increasing levels of positional jitter but wasgenerally comparable for same and opposite polaritystimuli. An exception to this trend was that detectinganti-symmetry was much more dicult than detectingsymmetry in dense displays. Thus, Rainvilles resultsare similar to those of Tyler and Hardage (1996).3. Mechanisms of symmetry detectionSeveral recent models explain sensitivity to symmetryin terms of simple operations on the outputs of linearspatial lters (e.g., Dakin & Hess, 1997; Dakin & Watt,1994; Gurnsey, Herbert, & Kenemy, 1998; Rainville &Kingdom, 1999, 2000, 2002). We will refer to these gen-erally as ltering models because they involve multiplestages of spatial ltering. The components of these mod-els can be explained by considering the examples of sym-metry and anti-symmetry in Fig. 1(a) and (b). Panels (c)and (d) show the results of convolving (a) and (b) with alter selective for horizontal luminance gradients andpanels (e) and (f) show positive half-wave recticationsof panels (c) and (d). (A positive half-wave recticationsets all negative values in the convolution output to 0.) Itis worth noting that many V1 simple cells are welldescribed as linear lters followed by a half-waverectication (Movshon, Thompson, & Tolhurst, 1978).Although a half-wave rectication is a non-linear oper-Tyler and Hardage (1996) also examined the relativesensitivity to symmetry and anti-symmetry (in theirterms, same- and opposite-polarity symmetry). Theirstimuli comprised black and white Gaussian blobs ar-ranged on a grey background. The blobs were eitherdense or sparse, and were symmetrical or anti-symmetri-cal about the vertical axis. The blobs were presentedwithin two sectors either to the left and right of xation(horizontal separations) or above and below xation(vertical separations). Detection accuracy was measuredas a function of presentation duration for several view-ing distances. Sensitivity was dened as the reciprocalof the exposure duration yielding d 0 = 0.5. As viewingdistance decreased, stimulus size increased and the twohalves of the stimuli moved to greater eccentricities.For both symmetrical and anti-symmetrical displaysperformance varied little at eccentricities beyond 2from xation. Sensitivity to symmetry and anti-symme-try was similar in low-density displays. On average, sen-sitivity was higher in low density displays for bothrch 45 (2005) 21452160ation, many computations applied to half-wave rectiedbeenResearch 45 (2005) 21452160 2147signals yield the same results as operations on the origi-nal (unrectied) signals. Therefore, we dene the arraysof responses in panels (e) and (f) as quasi-linear channels.A representation of roughly the sort shown in Fig.1(e) forms the basis for symmetry detection in Dakinsmodel (Dakin & Hess, 1997; Dakin & Watt, 1994). Inthat model, the strength of the symmetry signal is re-lated to the mass of the blobs that straddle the axisof symmetry and the degree to which their centres arealigned. The model successfully explains a number ofclassic results in the symmetry literature (Dakin & Watt,1994) and new phenomena (Dakin & Herbert, 1998; Da-kin & Hess, 1997). Anti-symmetry would be invisible tothe model (unless a squaring or full-wave rectication isapplied prior to initial spatial ltering). If one considersthe half-wave rectication in Fig. 1(f) it is clear that noblobs straddle the axis of symmetry and hence the basisfor detecting symmetrical structure is lost. In general,anti-symmetry in signals such as Fig. 1(b) cannot be de-Fig. 1. Symmetrical and anti-symmetrical stimuli [(a) and (b)] that have(f)] or squared [(g) and (h)].S. Mancini et al. / Visiontected by mechanisms that compute the equivalent of across-correlation across an axis of symmetry within ahalf-wave rectied representation such as in Fig. 1(f).Tyler and Hardage (1996) were the rst to point outthat second-order channels may play an important rolein symmetry detection. They argued that sensitivity toanti-symmetry and low density patterns indicates thatsymmetry detection is mediated predominately by sec-ond-order processes. Furthermore, because performancevaried little with eccentricity (beyond 2 from xation),these second-order processes were thought to involveconnections that span the cortex.Second-order channels are commonly created bytaking the absolute value of a linear lters response(full-wave rectication) or by squaring the lters re-sponse (squaring rectication). V1 complex cells appearto be well described by a full-wave rectication that isachieved by summing the positive and negative half-wave rectications of simple cell responses (Wilson,Levi, Maei, Rovamo, & DeValois, 1990). For our pur-poses full-wave and squaring rectications have similarconsequences so we do not distinguish between them.Panels (g) and (h) of Fig. 1 show a squaring recticationof the linear lter responses shown in panels (c) and (d),respectively. Note that the representations in both (g)and (h) are symmetrical and may therefore provide a ba-sis for the detection of anti-symmetry. In general, anti-symmetry in signals such as Fig. 1(b) will be detectedby mechanisms that compute the equivalent of a cross-correlation across an axis of symmetry within a squaringor full-wave rectied representation such as in Fig. 1(h).When full-wave or squaring rectications are appliedto linear lter responses the resulting representations areusually referred to as non-Fourier or second-order chan-nels.1 Second-order channels are widely used in modelsof related spatial tasks such as texture segmentation,motion energy extraction, and subjective contours detec-tion, to name a few. Therefore, second-order channelsltered for horizontal [(c) and (d)] and then half-wave rectied [(e) andmay be used to achieve many computational ends.Gurnsey et al. (1998) suggested that a dierencingoperation applied within quasi-linear channels wouldprovide a basis for symmetry detection. Specically, ifthe absolute response dierence is computed betweenhorizontally separated points within panels (e) or (g)then a column of zeros will form at the locus of theaxis of symmetry. This groove could be detected by1 Rainville and Kingdom (2002) have a somewhat dierent view ofFourier and non-Fourier channels. In their model the Fourier channelsproduce local Fourier energy responses that are computed by summingthe squared responses of bandpass, quadrature pair lters. It should benoted that their Fourier channels would yield equal sensitivity tosymmetry and anti-symmetry as dened above. Their non-Fourierchannel responses were computed in the same way as the Fourierchannel responses but the inputs to the quadrature pair lters were theenergy responses in a Fourier channel rather than the original image.So, their denition distinguishes between energy that is or is notavailable within the pass-band of the initial lters.convolving the resulting representation with a lterselective for vertical. All things being equal, the magni-tude of the lters response will be related to the degreeof correlation across the axis of symmetry within thement of individual elements.2148 S. Mancini et al. / Vision Researepresentation to which it is applied. Gurnsey et al.(1998) showed that a model of this sort degrades grace-fully in the presence of manipulations that are known toreduce the salience of symmetry. The model can be eas-ily shown to yield the same sort of behaviour when pre-sented with anti-symmetry if the dierencing operationis applied in a second-order channel [Fig. 1(h)] but notwhen applied in a quasi-linear channel [Fig. 1(f)]. Rain-ville and Kingdom (2000) described a model of symme-try detection that works in much the same way as thatdescribed by Gurnsey et al. (1998) although the strategyof groove detection was applied to the outputs of anumber of symmetry detection units.The focus of the present paper is on the nature of therepresentations involved in the ltering mechanisms thatunderlie symmetry detection. We assume that if second-order channels participate in symmetry detection thenequal sensitivity to symmetry and anti-symmetry wouldbe expected.We have reviewed evidence that psychophys-ical subjects show roughly similar sensitivity to symmetryand anti-symmetry in sparse but not dense displays(Rainville, 1999; Tyler & Hardage, 1996; Wenderoth,1996). Therefore, one explanation for this eect mightbe that sparse images activate second-order channelsand dense displays activate only quasi-linear channels,leading to a loss of sensitivity to anti-symmetry.Tyler and Hardage (1996) argued that equal sensitiv-ity to symmetry and anti-symmetry points to the ubiq-uity of processes operating on second-order channels.They observed that the depth of response modulationin second-order channels decreases as density increasesand used this fact to explain loss of sensitivity to anti-symmetry in dense displays.2 In other words, anti-sym-metry becomes invisible within second-order channelsas density increases. [However, the simulation resultsof Rainville and Kingdom (2002, see Fig. 12) suggestthat second-order channels should produce high sensi-tivity to anti-symmetry seen in even dense displays.]It may be, however, that equal sensitivity to symme-try and anti-symmetry does not reect the fact that theyelicit similar responses from ltering models involvingsecond-order channels. For example, it might be thatsensitivity to symmetry arises from ltering modelsinvolving quasi-linear channels whereas sensitivity toanti-symmetry arises from attentional mechanisms thatare able to operate only in sparse displays. In sparse dis-2 Tyler and Hardage (1996) also point out that anti-symmetry maybe detected within quasi-linear channels if the matching process cantolerate a certain amount of variability in the positions of matcheditems; see also Barlow and Reeves (1979) and Rainville and Kingdom(2002). We return to this in the ANALYSIS section.4. Experiment 1a: the eect of check sizeIn Experiments 13, stimuli comprised black andwhite checks in which the proportion (p) of polaritymatched checks at symmetrically placed positions variedfrom 1 to 0. As p increases from 0.5 to 1 the stimuli be-come increasingly symmetrical and as p decreases from0.5 to 0 the stimuli become increasingly anti-symmetri-cal. Fig. 2 panels (a), (b) and (c) show stimuli for whichp = 1.0, 0.75 and 0.5, respectively, and panels (d), (e) and(f) show stimuli for which p = .0, 0.25 and 0.5, respec-tively. Thresholds were dened as the proportion ofpolarity matched checks (pt) required for a display tobe discriminable from a random pattern (i.e., p = 0.5).To compare sensitivities to symmetrical and anti-sym-metrical stimuli, thresholds are put on the same scaleand reported as jpt 0.5j + 0.5.In Experiment 1a thresholds were measured for sym-metrical and anti-symmetrical stimuli having a range ofcheck sizes. As check size increases, the stimuli becomeless dense in the sense that responses in high-frequencychannels become sparser. To the extent that check sizemay be seen as an analogue of density in the Rainville(1999) and Tyler and Hardage (1996) studies, one mightexpect thresholds to decrease as a function of check sizefor anti-symmetrical stimuli. On the other hand, if sec-plays there are large regions of zero contrast (homoge-neous grey) and the occasional points of non-zerocontrast. An attentional process might note the symmet-rical placement of isolated tokens that dier in arbitraryways. Attention might be the only route to detectinganti-symmetry and a second route to detecting symme-try.Throughout this investigation we employ stimuli de-signed to eliminate position matching as a strategy fordetecting anti-symmetry. These stimuli dier from thoseused by Rainville (1999), Tyler and Hardage (1996), andWenderoth (1996) and more closely resemble those usedby Jenkins (1983) and Tyler (1999). Recall that the RAstimuli used by Wenderoth (1996) involved isolated,symmetrically placed dots whose colours (black orwhite) were uncorrelated across the axis of symmetry,yet observers were able to detect the symmetrical struc-ture in these displays. In contrast, all experiments re-ported below involve stimuli comprising denselypacked checks; i.e., checks were never isolated on a greybackground but completely covered the stimulus area;e.g., Fig. 1(a) and (b). In such a situation RA textureswould be devoid of symmetrical structure. Thus, in thepresent study position alone cannot be used to conveysymmetry so judgements about symmetry or anti-sym-metry cannot be based solely on the symmetrical place-rch 45 (2005) 21452160ond-order channels contribute to the coding of symme-entscondical (ertureReseatry, then symmetrical and anti-symmetrical stimuli mayelicit similar thresholds at all check sizes.Fig. 2. Examples of stimuli with dierent proportions of matching elembottom: p = 1.0, 0.75 and 0.5 in panels (a), (b) and (c), respectively; serespectively. Examples of symmetrical (third column) and anti-symmetr0.296 and 0.594 degrees of visual angle windowed within a circular apS. Mancini et al. / Vision4.1. Method4.1.1. ParticipantsThere were ve participants, three of which hadextensive experience in other psychophysical tasks andthe other two were novices. All had normal vision orwore the appropriate corrective lenses during the trials.4.1.2. ApparatusThe experiments were conducted using a MacintoshG4. Stimuli were presented on a 21-inch multiscan col-our monitor with display resolution set at 1024 768pixels. Pixel width was 0.37 mm and the screen refreshrate was 85 Hz. The gamma correction software avail-able in the Psychtoolbox (Brainard, 1997) was used tolinearize the screen luminance and a Minolta CS-100photometer was used to nd the absolute luminance lev-els. Stimuli were created and experiments were run in theMATLAB (Mathworks Ltd.) environment using func-tions in the Psychtoolbox (Brainard, 1997) that providehigh level access to the routines of the VideoToolbox(Pelli, 1997).4.1.3. StimuliStimuli comprised black and white checks havingwidths of 2, 4, 8 and 16 pixels which, from a viewing dis-tance of 57 cm, corresponded to 0.074, 0.148, 0.296 and0.594 degrees of visual angle. The stimuli were win-dowed within a circular aperture of 9.5 degrees in diam-eter. The third column of Fig. 2 shows examples ofacross the vertical axis of symmetry. Pattern in the rst column, top tocolumn, top to bottom: p = .0, 0.25 and 0.5 in panels (d), (e) and (f),fourth column) stimuli. From top to bottom, the check sizes are 0.148,of 9.5 degrees in diameter.rch 45 (2005) 21452160 2149symmetrical stimuli (p = 1.0) having check sizes of 4, 8and 16 pixels [panels (g), (h) and (i), respectively]. Thefourth column of Fig. 2 shows examples of anti-symmet-rical stimuli (p = 0.0) having check sizes of 4, 8 and 16pixels [panels (j), (k) and (l), respectively]. The maximumand minimum stimulus luminances were 84.2 and0.06 cd/m2, respectively.4.1.4. ProcedureParticipants were seated 57 cm from the monitor andasked to xate a black dot at its centre. On each trialtwo stimuli were presented in succession. One stimuluswas completely random and the other had some degreeof correlation across the axis of symmetry (p50.5). Thetask was to determine which interval contained the non-random stimulus. Stimuli were presented for 300 ms andseparated by an inter-stimulus interval (ISI) of 300 ms.Therefore, the task was a two-interval forced choice(2IFC) and observers responded by clicking the mouseonce or twice to indicate their choice. Visual feedbackafter each trial was given in the form of a + or to indicate correct and incorrect responses. An adaptiveprocedure (Pentland, 1980) using a Weibull function wasused to nd thresholds corresponding to 82% correctdetections. At least six thresholds were recorded for eachobserver for each of the eight conditions of the experi-ment. At least one threshold measurement was made0. 1areshold (|p -0.5|+0.5) 1bSymmetry2150 S. Mancini et al. / Vision Reseafor each subject, for each experimental condition beforedata collection began. Thresholds for symmetry andanti-symmetry were obtained independently; i.e., sym-metry and anti-symmetry trials were not interleaved.4.2. ResultsThe left panel of Fig. 3 summarizes the results ofExperiment 1a. We note that in many cases we were un-able to obtain thresholds less than one for the anti-sym-metrical stimuli. Therefore, for sessions in which PESTdid not converge on a value (jpt 0.5j + 0.5) less than1, the recorded threshold was set to 1.3 The thresholddata were submitted to a 2 (polarities) by 4 (check sizes)ANOVA. The ANOVA revealed a main eect of size[F(3,12) = 7.5, p < 0.05], a main eect of polarity[F(1,4) = 110.2, p < 0.05] and a signicant interaction2 4 8 160.5Check Size (pixels)Th100 101 102 1030.5Aperture Size (pixels)Anti-Symmetry(a) (b)Fig. 3. (a) Threshold (SEM) as a function of check size in pixels forsymmetrical stimuli (unlled circles) and anti-symmetrical stimuli(lled circles) (n = 5). The check sizes were 0.148, 0.296 and 0.594degrees of visual angle windowed within a circular aperture of 9.5degrees in diameter. (b) Threshold (SEM) as a function of aperturesize in pixels for symmetrical stimuli (unlled circles) and anti-symmetrical stimuli (lled circle) (n = 5).[F(3,12) = 29.6, p < 0.05]. The main eects of size andpolarity and the size polarity interaction explained,respectively, 65%, 96% and 88% of the variability amongthe means. The results show clearly dierent dependen-cies on check size for the symmetrical and anti-symmet-rical stimuli. Symmetrical stimuli elicit thresholds thatare relatively unaected by check size although theydo rise moderately as check size increases. Thresholdselicited by the anti-symmetrical stimuli are extremelyhigh for the smallest check size (indeed, they are essen-tially unmeasurable) but drop as check size increasesThe ndings are depicted in the right panel of Fig. 3.3 This might be considered a questionable practice but it is clearly thelesser of two evils; taking the average of only those thresholds thatconverged to values less than 1 would clearly overestimate sensitivityand lead to far less representative measures of performance. Thispractice also leads to certain complications in the ANOVA because thesphericity assumption will be violated when the analysis includesconditions in which thresholds are uniformly high. To deal with thistechnical problem all reported F-tests have been subjected to theGreenhouseGeisser correction procedure.It is clear that decreasing the aperture size for anti-sym-metrical stimuli (lled circle) with the smallest check sizedid not improve performance; thresholds remained veryclose to 1 on average. It is also clear that increasing theaperture size for the large-check-size, symmetrical stim-to a level almost identical to that elicited by the symmet-rical stimuli. These results are generally consistent withthose of Rainville (1999) and Tyler and Hardage(1996), if one assumes a connection between check sizeand density.5. Experiment 1b: the eect of aperture sizeIn Experiment 1a, the size of the aperture in whichstimuli were presented remained constant throughoutthe dierent check size conditions. Because this meansthat there were more checks within the window for thesmallest sizes than for the largest, we wondered if the in-crease in thresholds for symmetrical stimuli was a conse-quence of a reduced number of checks within theaperture. Therefore, we ran further trials using symmet-rical stimuli having the largest checks from Experiment1a at a range of aperture sizes.It is conceivable that a complimentary limitation ar-ose in the case of the anti-symmetrical stimuli for whichperformance improved as check size increased. If detec-tion of anti-symmetry actually relies on selective atten-tion to individual checks, then this process might beoverwhelmed by the large number of checks in thesmall-check-size conditions. To investigate this possibil-ity, thresholds were obtained for anti-symmetrical stim-uli at the smallest check size within an aperture thatcontained the same number of elements as for the largestcheck sizes in Experiment 1a.5.1. Method5.1.1. ParticipantsThe participants were the same as in Experiment 1a.5.1.2. Apparatus/procedure/stimuliThe apparatus and procedure were the same as inExperiment 1a with the following exceptions. All sym-metrical stimuli comprised checks that were 16 pixelson a side and presented within apertures of 9.5, 11.7,18.5, and 27.5 degrees of visual angle. The anti-symmet-rical stimuli comprised checks that were two pixels wideand presented in an aperture that was 1.18 degrees of vi-sual angle. Five thresholds for each of the ve condi-tions were obtained for each of the ve observers.5.2. Resultsrch 45 (2005) 21452160uli did not improve performance. It is reasonable toconclude that no aperture eect was operating duringExperiment 1a and that changes in performance levelbetween symmetrical and anti-symmetrical stimuli de-pended on check size only.6. Experiment 2a: high-pass lteringThe results of Experiment 1a are consistent with theprevious ndings of Rainville (1999) and Tyler and Har-dage (1996); viz., thresholds for symmetry and anti-sym-metry diverge as check size decreases. An explanationfor this nding may reside in the relationship betweenthe frequency content of the stimuli and the frequencyselectivities of the mechanisms that encode symmetryand anti-symmetry. Stimuli comprising small checkshave at energy spectra, whereas those with large checkshave spectra that resemble sinc functions; as check sizeincreases, energy is increasingly concentrated in thelow frequencies. Low thresholds for large-check-sizestimuli might reveal a full-wave or squaring recticationapplied to the responses of low-frequency selective spa-tial lters. Such a rectication would render the internalrepresentation of an anti-symmetrical stimulus symmet-rical [e.g., Fig. 1(h)]. If low thresholds for large-check-size anti-symmetrical stimuli are a consequence of aquencies are removed. To evaluate this prediction weused high-pass lters to eliminate low frequencies froma subset of the stimuli used in Experiment 1a.6.1. Method6.1.1. ParticipantsFive individuals familiar with the task participated inExperiment 2a and four participated in Experiment 2b.Participants had normal vision or wore the appropriatecorrective lenses during the trials.6.1.2. Apparatus/procedure/stimuliWe measured thresholds for symmetrical and anti-symmetrical stimuli of check size 16 as well as symmet-rical stimuli of check size 4. All stimuli were passedthrough a lter dened asH 11 c f n ; 3where f is frequency expressed in cycles per patch (cpp),c is the nominal cuto frequency and n was set to 5. Cut-o frequencies of 0, 2, 4, 8, 16, and 24 cpp were used.Examples of high-passed symmetrical and anti-symmet-rical stimuli are shown in the rst two columns of Fig. 4.The left column shows symmetrical stimuli ltered withcuto frequencies of 4, 8 and 16 cpp [panels (a), (b)anti-syles/pan) stimS. Mancini et al. / Vision Research 45 (2005) 21452160 2151Fig. 4. Examples of high-passed ltered symmetrical (rst column) andangle. From top to bottom, the cuto frequencies are 4, 8 and 16 cycltered symmetrical (third column) and anti-symmetrical (fourth columsquaring rectication in low frequency channels thenthresholds should increase substantially when low fre-16, 8 and 4 cycles/patch in panels (g)(i) and (j)(l), respectively.and (c), respectively]. The second column showsmmetrical (second column) stimuli of check size 0.594 degrees of visualtch in panels (a)(c) and (d)(f), respectively. Examples of low-passeduli of check size 0.594. From top to bottom, the cuto frequencies areanti-symmetrical stimuli ltered with cuto frequenciesof 4, 8 and 16 cpp [panels (d), (e) and (f), respectively].Filtering produces an artefact in the anti-symmetricalstimuli that makes them easily distinguishable from l-tered random noise. Specically, ltering produces a col-umn of zero-crossings along the axis of anti-symmetry;e.g., Fig. 1(h). To eliminate this dierential cue to thepresence of anti-symmetry we reduced contrast alongall axes of symmetry (i.e., both symmetric, anti-symmet-ric and all null stimuli) by multiplying the signal by aninverse Gaussian weighting functionG 1 expd=r; 4where d is distance from the axis of symmetry and r = 8pixels. Once again, because all symmetric, anti-symmet-ric and null stimuli have reduced contrast along the axisof symmetry this cannot serve as a dierential cue to thepresence of anti-symmetry. Thresholds were measuredas before and three replications of each condition werewas reduced. In fact, there was no interaction between2152 S. Mancini et al. / Vision Research 45 (2005) 21452160obtained from each participant.6.2. ResultsThe results of Experiment 2a are summarized in theleft panel of Fig. 5. The data were submitted to a 3 (pat-tern types) by 6 (lters) within subjects ANOVA. TheANOVA revealed a main eect of cuto frequency[F(5,20) = 9.8, p < 0.05] which explained 71% of the var-iability in the means. This indicates a general increase inthresholds as more low frequency energy is removedfrom the display. Although statistically signicant, theincrease in thresholds from least to most ltering wasat most 6%, which is quite modest in comparison tothe eect of check size for anti-symmetrical stimuli, inExperiment 1a. There was also a main eect of patterntype [F(2,8) = 18.3, p < 0.05] which explained 82% of2 4 8 16 Pass FilteringCutoff frequency (cpp)Threshold (|p -0.5| + 0.5)Symmetry (size 4)Symmetry (size 16)Anti-Symmetry (size 16)2 4 8 16 32 640. Pass FilteringCutoff frequency (cpp)Fig. 5. (left) Threshold (SEM) as a function of frequency cuto forsymmetrical checks of 0.148 and 0.594 degrees of visual angle and foranti-symmetrical checks of 0.594 degrees of visual angle (n = 5).Stimuli were passed through a high-pass lter with frequency cutos of0, 2, 4, 8, 16, and 24 cycles/patch. (right) Threshold (SEM) as afunction of frequency cuto for symmetrical and anti-symmetricalchecks of 0.594 degrees of visual angle (n = 4). Stimuli were passedthrough a low-pass lter with frequency cutos of 2, 4, 8, 16, 32 and 64cycles/patch.ltering and pattern type. This suggests that partici-pants did not rely on the low-frequency content of theanti-symmetrical stimuli in Experiment 1a. Given the ex-tremely modest increase in thresholds with increasinglysevere high-pass ltering, it is more likely that partici-pants relied on the high frequency content present inthe anti-symmetrical displays in Experiment 1a. Thispossibility was addressed in Experiment 2b.7. Experiment 2b: low-pass lteringThis experiment is identical to Experiment 2a exceptfor the following. The stimuli were passed through alow-pass lter dened asH 1 11 c=f n ; 5where f is frequency expressed in cpp, c is the nominalcuto frequency and n was set to 5. Cuto frequenciesof 2, 4, 8, 16, 32 and 64 cpp were used.Examples of low-passed symmetrical and anti-sym-metrical stimuli are shown in the right two columns ofFig. 4.7.1. ResultsThe results are summarized in the right panel of Fig.5. The data were submitted to a 2 (polarities) 6 (cutofrequencies) ANOVA. The ANOVA revealed a main ef-fect of cuto frequency [F(5,15) = 10.3, p < 0.05] indi-cating a general increase in thresholds as more highthe variability among means. There was no interaction[F(10,40) = 0.54, p > 0.05] between pattern type andltering.At all levels of ltering, the symmetrical stimuli hav-ing check size 4 produced lower thresholds than sym-metrical stimuli having check size 16. This is consistentwith the results of Experiment 1a in which it was shownthat thresholds for symmetrical stimuli decreased withdecreases in check size. In contrast to Experiment 1a,thresholds were higher for anti-symmetrical stimuli hav-ing check size 16 than for symmetrical stimuli havingcheck size 16 in the case of no ltering (see the leftmostdata-points in the left panel of Fig. 5). This might sug-gest that participants in Experiment 1a relied on infor-mation along the vertical midline to discriminate theanti-symmetrical from random stimuli, and that reduc-ing contrast along the vertical midline for null stimulias well may have eliminated this cue.The most important result of Experiment 2a is thateliminating low frequencies from large-check-size, anti-symmetrical stimuli did not lead to the dramatic increasein thresholds found in Experiment 1a when check sizefrequency energy is removed, a main eect of pattern0.8|p -0. 0.8Fig. 7. Threshold (SEM) as a function of eccentricity in degrees ofS. Mancini et al. / Vision Reseatype [F(1,3) = 92.2, p < 0.05] with lower thresholds over-all for symmetrical stimuli, and a signicant interaction[F(5,15) = 3.8, p < 0.05]. The latter result reects a moresubstantial increase in thresholds for anti-symmetricalthan symmetrical stimuli as more high frequencies wereeliminated from the displays. Cuto frequency, polarity,and cuto frequency polarity explained 77%, 97% and56% of the variability in the treatment conditions,respectively.To facilitate a comparison of the eects of removinglow and high frequencies from the displays, we have re-plotted in Fig. 6 the check size 16 data from the left andright panels of Fig. 5 as a function of percent retained en-ergy; i.e., total stimulus energy after ltering relative tototal stimulus energy before ltering. The gure showsthat for symmetrical stimuli, detection performance ismoderately aected by energy reductions and there isvery little dierence in the performance changes forreductions of high and low frequencies. For anti-sym-metrical stimuli, however, performance is more seriouslyimpaired by the loss of high frequencies than by the loss100 101 1020.50.60.7Energy Retained (%)Threshold (100 101 1020.50.60.7Energy Retained (%)SymmetryAnti-SymmetryFig. 6. Threshold (SEM) for symmetrical and anti-symmetricalstimuli as a function of energy retained after ltering for check size0.594 degrees of visual angle. Left (n = 5) and right (n = 4) panelsrepresent high-, and low-pass ltered stimuli, respectively.0.91.0High Pass Filtering5| + 0.5)0.91.0Low Pass Filteringof low frequencies.The most important result of Experiment 2b is thestatistically signicant interaction between polarity andcuto frequency. This indicates that high frequenciesare more important for the detection of anti-symmetrythan symmetry. If the loss of high frequencies impairsthe detection of anti-symmetry but not symmetry, wewould expect that when symmetrical and anti-symmetri-cal stimuli are moved into the periphery thresholdsshould increase more for anti-symmetrical than symmet-rical stimuli.8. Experiment 3: eccentricityIn this experiment observers were presented withlarge-check-size (16 pixels) symmetrical and anti-sym-metrical patterns. Thresholds were obtained at eccen-tricities of 0, 1, 2, 4 and 8 degrees of visual angle inthe right visual eld. The participants, apparatus andmethodology were the same as those of Experiment1a. Viewing was binocular.8.1. ResultsThe results of Experiment 3 are summarized in Fig. 7.The data were submitted to a 2 5 within subjects AN-OVA. The ANOVA revealed a main eect of eccentric-ity [F(4,12) = 48.8, p < 0.05], a main eect of polarity[F(1,3) = 23.3, p < 0.05], and a trend toward a signi-cant interaction [F(3,12) = 2.58, p = 0.09]. The treat-ment eects for eccentricity, polarity and eccentricity polarity, respectively explained 94%, 89% and 46% ofthe variability in the data. Fig. 7 shows that thresholdsfor symmetrical and anti-symmetrical stimuli weresimilar at xation and clearly diverged as eccentricity in-creased. At 8 participants were unable to detect struc-ture in the anti-symmetrical stimuli whereas thresholdsfor the symmetrical stimuli reached a plateau atvisual angle for symmetrical stimuli (unlled circles) and anti-symmetrical stimuli (lled circles) (n = 5). The check size was 0.594degrees of visual angle windowed within a circular aperture of 9.5degrees in diameter.0 1 2 4 ()Threshold (|p -0.5|+0.5)SymmetryAnti-Symmetryrch 45 (2005) 21452160 2153p 0.86. There is a clear parallel between Fig. 7 andthe right panel of Fig. 6; in both cases as high frequen-cies are removed thresholds for anti-symmetry rise dra-matically and those for symmetry are only modestlyaected.9. Experiment 4a: greyscale stimuliThe stimuli used in the rst three experiments wereintended to defeat an explicit position matching strategyfor detecting anti-symmetry. It might be argued, how-ever, that as check size increases (and check density de-creases) it would be easier for subjects to explicitlycompare individual checks at symmetrically placed loca-tions to assess the probability that the grey-level corre-lation is zero. This strategy is quite dierent fromthe idea that sensitivity to anti-symmetry arises fromltering models involving second-order channels. Itseems reasonable to assume that such an attentional pro-cess would also be defeated (or more severely chal-lenged) by increasing the number of grey-levels in thedisplay. However, it can be easily shown that lteringmodels involving second-order channels would not beaected by the number of grey-levels in the display.Therefore, some insight into the origins of sensitivityto anti-symmetry in large-check-size, anti-symmetrical,such as used in Experiments 13, might be gained byexamining the eects of increasing the number of grey-levels in the stimuli.To pursue this point we constructed symmetrical andanti-symmetrical stimuli from samples of Gaussiannoise; all values greater than 3 standard deviations fromthe mean were clipped. A copy of each sample was re-ected about the y axis and joined to the original. Foranti-symmetrical stimuli, symmetrically placed elementswere sign reversed, yielding a perfect negative correla-tion across the axis of symmetry. Performance was lim-ited by adding random noise (drawn from the samedistribution) to the symmetrical or anti-symmetrical sig-nals. The contrasts of the signal and noise were set asfollows:s cp signal 1 cp noise 6tively. The signals were then quantized to 254 greylevels for presentation on the computer monitor.9.1. Method9.1.1. ParticipantsFive individuals participated in all conditions. Allwere experienced psychophysical observers and all hadnormal vision or wore the appropriate corrective lensesduring the trials.9.1.2. Apparatus/procedure/stimuliSymmetrical and anti-symmetrical displays were cre-ated with check-sizes of 2, 4, 8 and 16 pixels. Examplesof the stimuli are presented in Fig. 8. The left column[(a)(c)] shows symmetrical stimuli with c = 1. The sec-ond column [(d)(f)] shows symmetrical stimuli withc = 0.8. The third column [(g)(i)] shows anti-symmetri-cal stimuli with c = 1 and the fourth column [(j)(l)]shows anti-symmetrical stimuli with c = 0.8. Althoughthe patterns in columns 1 and 3 are set to full contrast(c = 1) and those in columns 2 and 4 have c = 0.8, tomost observers all patterns in columns 1 and 2 appearsymmetrical whereas those in columns 3 and 4 appearrandom.] greyottom2154 S. Mancini et al. / Vision Research 45 (2005) 21452160where c is contrast (which ranges for 0 to 1) and signaland noise represent the symmetrical (or anti-symmetri-cal) image and noise component of the display, respec-Fig. 8. Examples of symmetrical [(a)(f)] and anti-symmetrical [(g)(l)second and fourth columns were stimuli with c = 0.8. From top to bwindowed within a circular aperture of 9.5 degrees in diameter.The apparatus and procedure were exactly as inExperiment 1a, with the following exceptions. The PESTprocedure was replaced by the QUEST procedure(Watson & Pelli, 1983) and stimulus contrast (c) wasscale stimuli. The rst and third columns were stimuli with c = 1 and, the check sizes are 0.148, 0.296 and 0.594 degrees of visual angleThree thresholds were obtained in each of the eightscale stimuli, the QUEST procedure, varied stimulusand noise contrast to limit performance, and reducedthe contrast along the axis of symmetry (as in Experi-ment 2). Experiment 4b was conducted to assess the rel-ative salience of binary and greyscale stimuli under moresimilar conditions.The greyscale stimuli were essentially as shown inFig. 8 except that contrast was not reduced along theaxis of symmetry. Binary stimuli were created exactlyas were the greyscale stimuli except that in an added stepthe grey-levels were made binary by setting intensitiesgreater than the mean to white and those less than themean to black. This manipulation produces a continu-ous variation in the strength of the symmetry andanti-symmetry signals; i.e., it accomplishes essentiallythe same thing as the proportion matching manipula- (contrast)S. Mancini et al. / Vision Research 45 (2005) 21452160 2155conditions for four of the ve participants, and onereplication was obtained for the remaining participant.9.2. ResultsThe results are summarized in Fig. 9. The data weresubmitted to a 2 (polarities) 4 (check size) within par-ticipants ANOVA. The ANOVA revealed a main eectof polarity [F(1,4) = 142.7, p < 0.05] but no main eectof check size [F(3,12) = 0.71, p > 0.05] and no interac-tion [F(3,12) = 1.26, p > 0.05]. Polarity explained97.2% of the variability in the data.The results in Fig. 9 are very dierent than those inthe left panel of Fig. 3. The principal dierence is thatthresholds for anti-symmetrical stimuli do not drop forlarge check sizes in Fig. 9 as they do in the left panelof Fig. 3. These results suggest that the ease with whichlarge check size, anti-symmetrical stimuli were detectedin Experiment 1a has something to do with the limitednumber of grey levels rather than with a general sec-ond-order rectication process.10. Experiment 4b: binary and greyscale stimulivaried rather than the proportion of matching elements.42 4 8 Size (pixels)ThreshoSymmetryAnti-SymmetryFig. 9. Contrast thresholds (SEM) as a function of check size inpixels for symmetrical stimuli (unlled circles) and anti-symmetricalstimuli (lled circles) (n = 5).Although a comparison of Experiments 4a and 1asuggests that sensitivity to anti-symmetry is very dier-ent in the two conditions, it could be objected that thereare many dierences between the two experiments thatmight contribute to the observed dierences. For exam-ple, Experiment 1a used binary stimuli, the PEST proce-dure and varied proportion of matching checks tocontrol performance, whereas Experiment 4a used grey-4 When dealing with binary stimuli it makes sense to talk aboutproportion of matching elements whereas it is not as obvious whatshould constitute a match given that each check may take on one of254 dierent grey-levels.tion used in Experiments 13, but in a slightly dierentway. In both cases thresholds were obtained using theQUEST procedure.10.1. Method10.1.1. ParticipantsFive individuals participated in all conditions. Allwere experienced psychophysical observers and all hadnormal vision or wore the appropriate corrective lensesduring the trials.10.1.2. Apparatus/procedure/stimuliExcept for the changes in stimuli described above, allaspects of the procedure were as described in Experi-ment 4a. Two to four thresholds were obtained in eachcondition for each participant.10.2. ResultsThe results are summarized in Fig. 10. The mainpoints can be made by noting that the left panel ofFig. 10 is essentially identical to the left panel of Fig.2 4 8 StimuliCheck Size (pixels)Contrast2 4 8 StimuliCheck Size (pixels)SymmetryAnti-SymmetryFig. 10. Contrast thresholds (SEM) as a function of check size inpixels for symmetrical stimuli (unlled circles) and anti-symmetricalstimuli (lled circles) for both binary (left, n = 5) and greyscale (right,n = 5) conditions.3 indicating that the interaction between polarity andcheck size does not depend on the method used to de-grade symmetry and anti-symmetry. As well, the rightpanel of Fig. 10 is essentially identical to Fig. 9, indicat-ing that the results are little aected by the presence orabsence of the contrast reduction along the axis ofoccurring and result in reduced sensitivity. Increasing2156 S. Mancini et al. / Vision Research 45 (2005) 21452160the number of grey-levels in a display might also disruptan imprecise matching strategy. The results of Experi-5 For simplicity, in this section we do not consider specic models ofsymmetry detection. Rather we consider the information availablewithin quasi-linear and second-order channels and their variants. Weuse cross-correlation to assess this information. The precise- andimprecise matching terminology arises from this correlationalsymmetry.11. AnalysisThe results of Experiments 2 and 3 suggest thatdetecting anti-symmetry requires the presence of highfrequencies and the results of Experiments 1a and 2asuggest that the responses of high frequency selectivelters must be relatively sparse. These results cannotbe explained by a squaring rectication (or energycomputation) in high frequency channels followed by aprecise matching procedure.5 Although this kind of pro-cedure would produce good performance in large checksize, anti-symmetrical displays, it can be easily shownthat it would produce even better performance in smallcheck size, anti-symmetrical displays and this would beinconsistent with the results of Experiment 1a. A half-wave rectication followed by a precise matching proce-dure cannot explain the pattern of results. Again, it canbe easily shown that such a procedure would producepoor performance for all anti-symmetrical displays andthis would be inconsistent with the results of Experi-ments 1a, 2a and 3. The low thresholds for large-check-size, anti-symmetrical stimuli in Experiments 1a,2a and 3 might be explained by imprecise matching ina quasi-linear channel.Passing anti-symmetrical displays through an isotro-pic, zero-mean, bandpass lter produces responses thatare phase reversed on either side of zero crossings. InFig. 4f, for example, symmetrically placed lter re-sponses are sparse and sign reversed on either side ofzero-crossings. If this representation (Fig. 4f) werehalf-wave rectied, then an imprecise matching proce-dure might note the high correlation in activity inroughly symmetrical positions. Furthermore, as sug-gested by Tyler and Hardage (1996) such a processmight be defeated as check size decreases because therewould be many more candidate matches within any re-gion. This would increase the chances of false matchesperspective.ments 4a and b might be seen to support this qualitativeaccount. Therefore, it is possible that all the data in Fig.10 could be explained by a ltering model that somehowembodies imprecise matching within quasi-linearchannels.To assess this hypothesis we implemented a simplecomputational model of symmetry detection. The modelconsisted of ltering with a $2G lter (Marr & Hildreth,1980), a half-wave rectication followed by Gaussianblurring which in turn was followed by cross-correlationto assess symmetry. The $2G lter is dened asr2Gr; r 1=p4 1 r2r2 er2=2r2 7where r is the distance from the centre of the windowand s, the spread of the Gaussian component of the l-ter, was set to 0.75 pixels, which made it sensitive toabout 60 cpp (corresponding to about 6.3 cpd in termsof the displays used in the experiments). We chose a l-ter tuned to high spatial frequencies for two reasons.First, the results of Experiment 2 suggested that detec-tion of large-check-size, anti-symmetrical stimuli re-quires the presence of high frequencies. Second,symmetry detection at all check sizes was not severelyimpaired by the elimination of low frequencies. Thehalf-wave rectier set all negative values in the convolu-tion output to 0, thus yielding a quasi-linear channel asdened in Section 1.The imprecise matching was accomplished by blur-ring the rectied response with an isotropic Gaussian l-ter. The use of Gaussian blurring to model imprecisematching follows from the observations of Barlow andReeves (1979) that slight perturbations of dot positionsin random dot, symmetrical displays had little eect onhuman performance but completely defeated an idealobserver that computed correlations at symmetricalpositions. Barlow and Reeves pointed out that if the im-age were blurred before correlations were computed theperformance of the ideal observed could be substantiallyimproved. In a similar manner, an ideal observer thatuses precise positional information would not be ableto detect anti-symmetry in any half-wave rectied repre-sentation. However, if the half-wave rectied representa-tion is blurred, then the correlation in activity atsymmetrically positioned points will increase.The question now is whether the pattern of results inFig. 10 can be captured by the model. Three versions ofthe model were evaluated, each diering only in thespread of the Gaussian blurring applied to the quasi-lin-ear channel. The three Gaussian kernels had rs of 1, 2and 4 pixels. The stimuli were identical in all respectsto those used in Experiment 4b except that they werescaled to 50% of the original size. The conditions testedwere exactly the 16 cells of Experiment 4b; i.e., 4 checksizes (1, 2, 4 and 8) 2 display types (symmetry vs. anti-symmetry) 2 grey-scales ranges (2 vs. 254) = 16.On each simulated trial a stimulus display havingsome non-zero contrast (c) and a null stimulus (c = 0)were submitted to the model. For both displays thecross-correlation across the axis of symmetry was com-puted and the one that produced the largest cross-corre-lation was chosen as symmetric. Thresholds were foundusing the QUEST procedure exactly as in Experiment4a. Ten thresholds were obtained for each of the 16 cells,for each of the three models.The simulation results are shown in Fig. 11. The leftand right panels shows results for binary and greyscalestimuli, respectively. Unlled symbols indicate symmet-rical displays and lled symbols indicate anti-symmetri-cal displays. The three degrees of post-recticationblurring are indicated by circles, squares and trianglestries in terms of second-order channels (see also Rain-ville, 1999). Experiment 1a generally replicated theS. Mancini et al. / Vision Reseafor rs 1, 2 and 4, respectively. There are three importantobservations to be made. First, as blurring increases,thresholds decrease for anti-symmetrical stimuli and in-crease for symmetrical stimuli. This apparently paradox-ical result is easily explained by the eects that blurringhas on the target displays (symmetrical and anti-sym-metrical) and null displays. Reduced thresholds foranti-symmetrical displays are attributable to the in-creased correlation across the axis of symmetry thatarises from blurring; i.e., blurring produces the expectedresult. On the other hand, blurring also increases thecorrelation across the axis of symmetry for null displays.Consequently, contrast in the symmetrical image mustbe increased to exceed the blur induced correlations inthe null displays.Second there is little evidence that thresholds decreasefor anti-symmetrical stimuli as check size increases forany of the levels of blurring. To the contrary, for r =2 and 4 thresholds for anti-symmetrical stimuli increasewith check size. It might be objected, however, that ourmodel of symmetry detection is not realistic in that deci-sions are based on the magnitudes of cross-correlations(e.g., Dakin & Hess, 1997; Dakin & Watt, 1994) ratherthan on a biologically plausible mechanism of symmetrydetection (e.g., Gurnsey et al., 1998; Rainville &2 4 8 Size (pixels)Threshold (contrast)Binary2 4 8 Size (pixels)Threshold (contrast)Grey ScaleFig. 11. Simulated thresholds for binary (left) and greyscale stimuli(right), symmetrical (unlled symbols) and anti-symmetrical (lledsymbols) stimuli. Three degrees of post-rectication blurring (rs = 1, 2and 4 pixels) are shown as circles, squares and triangles, respectively.results of Tyler and Hardage; thresholds for symmetryand anti-symmetry were similar in binary displays com-prising large checks but diverged as check size de-creased. This result might be seen as consistent with arole for second-order channels in symmetry perception.However, Experiment 4 showed that anti-symmetry wasundetectable even in displays comprising large checkswhen the grey scale range of the checks was increased.This result is inconsistent with the idea that second-order channels generally play an important role in sym-metry detection.Our experiments show that thresholds for symmetri-cal stimuli were only modestly aected by manipulationsof check size, spatial frequency content, eccentricity andgreyscale range. It seems reasonable, therefore, to positthe existence of linear or quasi-linear channels that sup-port sensitivity to symmetry. Such channels might oper-ate as described by Gurnsey et al. (1998) or RainvilleKingdom, 1999, 2000, 2002). If such models were stud-ied in conjunction with the assumption that perfor-mance is limited by internal noise then it might be thatan appropriate level of internal noise would yield thepattern of results seen in the left panels of Figs. 3 and 10.However, our nal point is that the simulation resultsin the left and right panels of Fig. 11 are identical. Thatis, even if introducing internal noise into a biologicallyplausible model were sucient to capture the patternof results in the left panels of Figs. 3 and 10 (i.e., binarystimuli) exactly the same pattern of results would beproduced for the greyscale stimuli (right panel of Fig.10). We conclude that an imprecise matching procedureapplied within a high-frequency, quasi-linear channelcan improve sensitivity to anti-symmetry. However,such a strategy would produce similar thresholds forbinary and greyscale stimuli and this result is con-tradicted by the psychophysical data.12. General discussion12.1. Symmetry vs. anti-symmetryA review of the literature showed that symmetricaland anti-symmetrical stimuli elicit comparable perfor-mance when composed of isolated tokens separated byregions of intermediate grey (Rainville, 1999; Saarinen& Levi, 2000; Tyler & Hardage, 1996; Wenderoth,1996) and when tokens are sampled from binary distri-butions such as black and white dots (Wenderoth,1996), black and white Gaussian blobs (Saarinen &Levi, 2000; Tyler & Hardage, 1996) or centre-surroundelements of opposite polarity (Rainville, 1999). Tylerand Hardage explained sensitivity to such anti-symme-rch 45 (2005) 21452160 2157and Kingdom (2000) and take a half-wave rectiedand Watt (1994), Gurnsey et al. (1998), Rainville andKingdom (1999, 2000, 2002) or Tyler and Hardageesearch 45 (2005) 21452160signal as input. Assuming the entire mechanism inte-grates information across scales (Rainville & Kingdom,1999) it is easy to understand why eliminating some spa-tial frequency information through explicit ltering(Experiment 2) or by moving the stimulus into theperiphery (Experiment 3) would lead to modest in-creases in thresholds. It can also be shown that increas-ing check size reduces information density (Rainville &Kingdom, 2000) which leads to modest threshold in-creases (Experiments 1 and 4).Assuming the existence of a second-order channel toaccommodate a subset of the present results predictssensitivities that are inconsistent with other results re-ported here. For example, positing the availability oflow frequency, second-order channels to explain lowthresholds for large-check-size, anti-symmetrical stimuliin Experiment 1a is inconsistent with the eects of lter-ing in Experiment 2 and the complete insensitivity tolarge-check-size, anti-symmetrical greyscale stimuli inExperiment 4.Our results do not imply that second-order channelsnever contribute the the detection of symmetry andanti-symmetry. For example, there are many dierencesbetween our experimental methodology and that of Ty-ler and Hardage (1996), which yielded equivalent sensi-tivity to symmetry and anti-symmetry. Therefore,determining the conditions under which second-orderchannels make a contribution to symmetry detectionrequires further study. However, it seems that second-order channels make little if any contribution to sym-metry detection under the conditions of the presentexperiments. In fact, the results suggest two kinds ofprocesses available for the detection of symmetry andanti-symmetry. One type of process has access to rst-order channels and is able to match information acrossthe axis of symmetry; such a process might have thecharacter of the mechanisms described by Dakin andWatt (1994), Gurnsey et al. (1998) or Rainville andKingdom (2002). When rst-order information isunavailable this process fails. For example, in high den-sity displays subjects are very sensitive to symmetry con-veyed by rst-order information but if the rst-orderinformation is removed, by making the display anti-symmetrical, then detection becomes impossible, at leastunder the conditions of our experiments.A second process is revealed by the fact that anti-symmetry is detected when items are sparsely distrib-uted on a neutral background (Rainville, 1999; Tyler& Hardage, 1996; Wenderoth, 1996). We suggest thata second route to the detection of symmetry andanti-symmetry involves an explicit assessment of tokenposition (independently of their colour) when density islow (Rainville, 1999; Tyler & Hardage, 1996; Wende-roth, 1996). When density increases this strategy be-comes ineective because there are too many2158 S. Mancini et al. / Vision Rpositions to consider in the time available. Further-(1996).There are a number of paradigms that could be usedto assess whether detection of anti-symmetry diersfrom detection of symmetry in its reliance of attentionalresources. One might be to assess the susceptibility ofsymmetry and anti-symmetry detection to attentionalmanipulations. For example, participants could beasked to perform a second, resource demanding taskwhile trying to detect symmetry or anti-symmetry(Braun & Julesz, 1998). The dual route proposal sug-gests that anti-symmetry detection might be more im-paired by this manipulation than symmetry detection.The eects of invalid, exogenous cuing might be as-sessed in a similar way. The appearance of an invalidexogenous cue prior to the presentation of a symmetri-cal or anti-symmetrical display might draw certain re-sources away from the patterns and the consequencemight be greater disruption of anti-symmetry detectionthan symmetry detection.12.2. Frequency contentThe eects of frequency manipulation in Experiment2 may be compared to the results of Rainville and King-dom (1999). They created symmetrical patterns fromwhite noise to which was added uncorrelated white noise(much as in Experiment 4). The resulting patterns(which had at spectra) were then ltered to have 1/fbspectra for b ranging from 2 to 5. When b = 0 stimulihave at spectra, when b < 0 there is high frequencyemphasis and when b > 0 there is low frequency empha-sis. Natural images are found to have spectral slopes inthe range 1.2 6 b 6 3.2. Rainville and Kingdom (1999)wondered if there might be an advantage for symmetrydetection for slopes in this range. To answer this theyvaried the signal-to-noise ratio to nd threshold (81%more, in displays such as ours detection of anti-sym-metry (and symmetry for that matter) might beachieved by combining information about positionand colour when there are few items in the display[e.g., Fig. 2(l)]. Specically, it may be possible to com-pare symmetrical locations, note their colours and as-sess the probability that there is zero correlationbetween colour and position across the axis of symme-try. However, such a process would be defeated ifeither the number of locations to compare is too great(i.e., small-check-size displays), or if ne discrimina-tions between grey-levels are required (e.g., the grey-scale stimuli used in Experiment 4). Because thissecond strategy seems to operate in conditions that fa-vour selective attention, we assume that it is not low-level in the sense of the models described by Dakincorrect responses in a 2AFC task). As expected, thresh-visual eld (Barrett et al., 1999; Sally & Gurnsey,2001). Our results suggest that much steeper scalingthe present experimental conditions sensitivity to sym-metry and anti-symmetry does not generally arise fromResearch Grants to Rick Gurnsey. Portions of this pa-per were presented at the ARVO annual meeting,Research 45 (2005) 21452160 2159olds show a U shaped dependence on b, with lowestthresholds in the range associated with natural images.We found a similar result. When combining the re-sults of Experiments 2a and b, for large-check-size sym-metrical stimuli, we nd a U shaped dependence onspatial frequency content (i.e., concatenate the unlledcircles in the right panel of Fig. 5 with the unlled circlesin the left panel of Fig. 5). The unltered stimuli in thiscase have spectra that resemble sinc functions with en-ergy concentrated around the origin. Thus, the spectraof our unltered, large-check-size stimuli are more sim-ilar to b > 0 spectra than b < 0 spectra. Removing highor low frequencies increased thresholds, although as sta-ted the increases were rather modest. It is dicult tocompare directly the results of Rainville and Kingdoms(1999) study with ours because the psychometric vari-ables were dierent. They varied signal-to-noise ratio,which ranged from 0 to 1 and we varied percent match-ing, which ranged from 0.5 to 1. When the thresholdsfrom the two studies are expressed as a proportion ofthe range of the variable, the largest average dierencein their data was about 0.24 and in ours about 0.13.Therefore, the consequences of spectral manipulationswere generally greater in their experiment than in ours.These comparisons should be treated with caution, how-ever, because of the signicant dierences in the waythat performance was measured.The results of Experiments 2a and b suggest that sym-metry is computed simultaneously at several scales; i.e.,within several spatial frequency channels. For symmetri-cal stimuli we found comparable thresholds for stimulihaving radically dierent spectra (i.e., low passed andhigh passed stimuli with cut-os diering by more than3.5 octaves). These results are consistent with those ofRainville and Kingdom (1999, Experiment 2) who pro-vide additional evidence for the idea that symmetry calcu-lations are performed in parallel at several dierent scales.12.3. EccentricitySeveral recent studies have examined the eects ofeccentricity on symmetry detection (Barrett, Whitaker,McGraw, & Herbert, 1999; Gurnsey et al., 1998; Sally& Gurnsey, 2001; Tyler, 1999; Tyler & Hardage,1996). Three of these studies show reduced sensitivityto symmetrical stimuli of xed size as they are movedinto the periphery (Barrett et al., 1999; Gurnsey et al.,1998; Sally & Gurnsey, 2001). Although the experimentsdier in detail they converge on the conclusion thatstimuli must be magnied substantially with eccentricityto achieve equivalent-to-foveal performance.On the other hand, Tyler (1999) reported that peaksensitivity to stimuli of xed size does not change sub-stantially with eccentricity. Again, sensitivity was de-ned by the exposure duration required to elicit a d 0 ofS. Mancini et al. / Vision0.5. 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Mancini et al. / Vision Research 45 (2005) 21452160Detection of symmetry and anti-symmetryIntroductionSensitivity to symmetry and anti-symmetryMechanisms of symmetry detectionExperiment 1a: the effect of check sizeMethodParticipantsApparatusStimuliProcedureResultsExperiment 1b: the effect of aperture sizeMethodParticipantsApparatus/procedure/stimuliResultsExperiment 2a: high-pass filteringMethodParticipantsApparatus/procedure/stimuliResultsExperiment 2b: low-pass filteringResultsExperiment 3: eccentricityResultsExperiment 4a: greyscale stimuliMethodParticipantsApparatus/procedure/stimuliResultsExperiment 4b: binary and greyscale stimuliMethodParticipantsApparatus/procedure/stimuliResultsAnalysisGeneral discussionSymmetry vs. anti-symmetryFrequency contentEccentricityConclusionsAcknowledgmentReferences