Perceptual grouping in Gabor lattices: Proximity and alignment

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1 Journl Perception & Psychophysics 2005,?? 67 (?), (8),???-??? Perceptul grouping in Gor lttices: Proximity nd lignment PETER M. CLAESSENS nd JOHAN WAGEMANS University of Leuven, Leuven, Belgium We propose the Gor lttice s new stimulus designed to del with multiple orgniztions in perceptul grouping, llowing oth comprison etween psychophysicl dt nd neurl findings nd systemtic investigtion of grouping sed on severl low-level chrcteristics nd their interctions. A Gor lttice is geometric lttice with Gor ptches, evoking multistle glol orienttion percept. Visul grouping in Gor lttices with elements ligned in glol orienttion ws compred with grouping of nonligned Gor ptches nd of Gussin los. The effect sizes of proximity nd lignment were estimted in logistic regression nlyses. The results confirmed the importnce of proximity nd locl element lignment s fctors in dynmic grouping. We lso found smll ut consistent enhncement of grouping long the glol vector orthogonl to the locl ptch orienttions. In light of these results, we further motivte the relevnce of these stimuli nd the ssocited experimentl prdigm. This reserch ws supported y the Fund for Scientific Reserch, Flnders (Grnt G ). We thnk Krl Delrge for his ssistnce in dt collection, s well s Peter Zenen, Filip Germeys, nd Jn Vnrie for reding nd commenting on the mnuscript. We re lso grteful for the comments of two nonymous reviewers nd of Todd Mddox. Correspondence relting to this rticle my e sent to J. Wgemns, University of Leuven, Lortory of Experimentl Psychology, Tiensestrt 102, B Leuven, Belgium (e-mil: john.wgemns@psy.kuleuven.c.e). From retinl input, the visul system hs to infer the identities nd loctions of ojects with which we might interct in motoric or cognitive wy. To successfully ccomplish this tsk, the morphous informtion mosic needs to e ordered so tht coherent regions nd lines tht will provide cndidtes for oject surfces nd oundries cn e retrieved. Phenomenologicl demonstrtions within the frmework of Gestlt psychology hve een n importnt source of inspirtion for lter psychophysicl nd neurophysiologicl experiments investigting the rules of this process of erly, precognitive prsing. Unlike in the heydy of Gestlt psychology, the mens to unrvel the mechnisms underlying grouping nd segmenttion re now ville. A full explntion of the multitude of phenomen nd experimentl results in this domin requires converging movement of the severl disciplines involved. The present rticle discusses stimulus composed of elements specificlly imed t mtching wellknown properties of the neurl sis of the visul system within psychophysicl experimentl methodology. Grouping prolems occur t mny different sptil scles, from ridging interruptions in the retinl imge cused y lood vessels to wide occlusions of distl stimuli y nery ojects. Therefore, it is not surprising tht the visul system does not rely on one single mechnism to resolve occlusions nd other forms of grouping miguities in its input. Wtt nd Phillips (2000) distinguished etween prespecified nd dynmic groupings. In the prespecified type, grouping is consequence of converging input in the system hierrchy. It is prespecified in tht grouping procedures in this ctegory will lwys converge towrd the sme solution for given input. An exmple of this principle simplified here for rgument s ske cn e found in the erliest stges of imge processing. Retinl gnglion cells project to the lterl geniculte nucleus, from which locl orienttion informtion is extrcted y the pproprite pttern of projections to V1. This hrdwired structure, which results in locl orienttion selectivity known s the receptive field of orienttion-selective neurons, cn e considered n erly implementtion of grouping of locl retinl signls. Current Approches to Grouping Since it ecme ovious tht the visul system does not merely propgte locl mesurements in strictly feedforwrd fshion, the notion of receptive field hs ecome more fuzzy (see Fitzptrick, 2000, for review). A clss of experiments with line segments (see, e.g., Dresp, 1999; Tzevnov & Dresp, 2002) nd Gor ptches (reviewed y Polt, 1999) definitively refuted the ide of the primry projection region s n rry of pssive loclized filters. Polt nd Sgi (1993, 1994) hve shown tht detection of low-contrst Gor ptch in two-lterntive forced choice (2AFC) prdigm is fcilitted or inhiited y flnking Gor ptches well outside the clssicl receptive field of n orienttion-selective cell in V1. The sign nd mgnitude of the modultion is dependent on oth the distnce of the trget to the flnkers nd the glol configurtion of the stimulus. Psychophysicl findings in this line of reserch re supported y neurophysiologicl methods in experiments Copyright 2005 Psychonomic Society, Inc. 1446

2 PERCEPTUAL GROUPING IN GABOR LATTICES 1447 involving visully evoked potentil recordings in humns (Polt & Norci, 1996), single-cell recordings in ct strite cortex (Polt, Mizoe, Pettet, Ksmtsu, & Norci, 1998), nd single-cell recordings of V1 complex cells in mcque monkeys (Kpdi, Ito, Gilert, & Westheimer, 1995; Kpdi, Westheimer, & Gilert, 2000). In generl, the distnce, lterl offset, nd ngulr offset of flnker nd trget weken threshold fcilittion. These nd similr findings hve plyed n importnt role in estlishing the modlities of interctions eyond the extent of the receptive field in oth the psychophysicl nd the neurophysiologicl domins. Although these experiments hve explored how contextul informtion modultes locl fetures such s perceived contrst of n element, there hs lso een gret interest in the role of locl fetures on contour integrtion. In clssicl prdigm, dotted line is emedded in ckground of rndomly plced dots in order to investigte the detection of liner structure for different levels of proximity nd collinerity (Beck, Rosenfeld, & Ivry, 1989; Smits, Vos, & Vn Oeffelen, 1985; Uttl, 1975). Field, Hyes, nd Hess (1993) pplied the sme principle in the construction of their stimuli. A set of Gor ptches ligned long smoothly undulting curve ws emedded in noise ckground of rndomly oriented Gor elements. Suject performnce ws mesured s function of interelement ngle in 2AFC prdigm. This experimentl pproch hs proven very populr in susequent studies (reviewed y Hess & Field, 1999, nd Hess, Hyes, & Field, 2003) in which lrge numer of locl or glol stimulus properties hve een ssessed, such s contour closure (Kovcs & Julesz, 1993), polrity nd luminnce contrst long the pth (Field, Hyes, & Hess, 2000), chromticity nd color contrst (Beudot & Mullen, 2003; McIlhgg & Mullen, 1996), sptil frequency tuning (Dkin & Hess, 1998), eccentricity (Hess & Dkin, 1999; Nugent, Keswni, Woods, & Peli, 2003), temporl properties of the presenttion (Beudot, 2002; Hess, Beudot, & Mullen, 2001), nd sttionry motion (Bex, Simmers, & Dkin, 2001). Most relevnt in the context of this rticle is the consistent finding tht contour sliency increses with element lignment nd proximity. Field et l. (1993) dvocted the existence of n ssocition field s constrint to contour integrtion processes. According to this concept, the fcilittion of nery locl orienttion filters propgtes long pth fvoring smooth contours. The connection weight etween locl orienttion chnnels thus declines with length nd curvture of the (virtul) contour frgment connecting them. The emergence of the ssocition field is thought to reflect sttisticl regulrities in the visul environment, which in turn seem to rise from the good-continution constrint (Elder & Golderg, 2002; Sigmn, Cecchi, Gilert, & Mgnsco, 2001). In generl, the psychophysicl results nd the neurophysiologicl mesurements in oth contrst enhncement nd snke detection experiments re qulittively well explined in modeling efforts (mong others, y Usher, Bonneh, Sgi, & Herrmnn, 1999, nd Pettet, McKee, & Grzywcz, 1998, respectively) motivted y recent discoveries in corticl rchitecture. V1 pyrmidl cells in mcque cortex hve xon collterls tht rech up to eight hypercolumns, integrting informtion over distnces tht re multiples of clssicl receptive-field size (Gilert, 1992; Stettler, Ds, Benett, & Gilert, 2002). In the superficil lyers of primry visul cortex, these horizontl connections re mostly orienttion specific. This long-rnge connectivity hs prominent plce in the discussion out the mchinery of contour detection modulting processes, s found in psychophysicl nd neurophysiologicl experiments (Gilert, Ds, Ito, Kpdi, & Westheimer, 1996). Extrpolting from findings in the visul cortex of the tree shrew, some reserchers hve suggested tht these horizontl projections in V1 might extend preferentilly to corticl sites long the elongtion xis of the receptive fields they rdite from (Bosking, Zhng, Schofield, & Fitzptrick, 1997; Chisum, Mooser, & Fitzptrick, 2003). The Multistle Lttice Prdigm s Complementry Approch This set of psychophysicl, neurophysiologicl, nd neurontomicl results converges into firly unified, consistent view on contextul processing nd the sustrtes for grouping. Yet, we elieve tht n pproch sed on Gor lttices will offer some new methodologicl possiilities nd elicit novel questions. (1) Despite quite it of reserch designed to give experimentl support to Gestlt lws, lot remins to e discovered out how they comine. We feel tht n pproch sed on stimuli in which different orgniztion principles cn led to multiple solutions rther thn one hs gret potentil to solve questions out coopertive nd competitive interctions etween orgniztion principles. In the Figure 1. Exmple of n olique dot lttice (fter Kuovy & Wgemns, 1995). Spontneous visul grouping clusters the dots in lines, with proximity s the crucil fctor determining the odds of orgnizing the lttice in one or the other orienttion.

3 1448 CLAESSENS AND WAGEMANS γ Figure 2. Prmeters of the sic prllelogrm spnning point lttice. d c Gor snke detection prdigm, fcilittion or inhiition of orgniztion cn only e evluted reltive to good continution, simply ecuse tht is wht defines the signl. (2) The sic detection prdigm in which Gor snke is emedded in noise elements is well suited to revel how scene consisting of single contour is nlyzed y the visul system. In nturl scene, however, multiple edges nd lines usully suprthreshold cross nd intersect. This poses nother clss of prolems to the visul system: not whether locl orienttion chnnel elongs to contour or not, ut which contour it elongs to. There is need for prdigm tht reltes orgniztion of Gor stimuli explicitly to the ltter issue. But how could lttice prdigm ddress these two concerns? A two-dimensionl lttice is sptil pttern consisting of elements tht re lid out in very regulr fshion. The lttice elements re spced evenly to e colliner long different orienttions. In response to the first issue rised ove, note tht the lttice structure hs the inherent dvntge tht the reltive weights of up to three or four different fctors cn e pitted ginst ech other in one presenttion frme (e.g., proximity versus similrity in the studies of Ben-Av & Sgi, 1995; Quinln & Wilton, 1998; nd Zucker, Stevens, & Snder, 1983). In the cse of two fctors, the stimulus cn induce coopertivity etween Gestlt principles when oth fvor the sme lttice orienttion. The visul system cn likewise settle in one of two competitive lttice vectors if the underlying grouping forces re in opposition. As to the second issue, the percept of sufficiently entropic lttice is not prespecified; this is known s the multistility property (Kuovy & Wgemns, 1995). Although severl contour cndidtes re simultneously present s the different lttice vectors, the lttice structure elicits single emergent glol orienttion. This glol percept is the mcroscopic, conscious consequence of the identifiction of the reltions etween elements their elonging together rther thn of the mere presence of contour. The issue cn e compred with the difference etween the representtion of curve in itself nd the representtion of which of the curve prts elong to one unit in other words, etween contour representtion nd curve trcing (Roelfsem, Lmme, & Spekreijse, 1998). In ddition to the enefits discussed ove, we think tht the use of lttices hs dditionl vlue. (1) The findings with Gor ptterns we reviewed ove do not redily explin n importnt phenomenologicl spect of lttice stimuli: The suggested process of prllel computtion cross the visul field does not result in independent locl orgniztions. In other words, it is not ovious how loclly operting ssocition fields of the type proposed y Field et l. (1993) could led to single dominnt orienttion tht is consistent cross the entire stimulus. More generlly stted, little is known or modeled out the properties of perceptul orgniztion with Gor stimuli eyond the scope of nerest-neighor ptches, nd we elieve tht Gor lttices cn contriute to filling tht gp. (2) Any explntion tht relies on contrst enhncement to explin the emergence of slient orienttion (such s in the model y Pettet et l., 1998) fils to explin the sliency experienced in Gor lttices. A B Figure 3. Exmples of control conditions. (A) Exmple of Gor lttice without lignment, with rectngulr lttice structure. The orienttion of ech Gor element is rndom; pprent locl collinerities or prllelisms re simply y-products of the sttisticl properties of rndomly generted orienttions. (B) Exmple of Guss lo lttice, with n olique lttice structure.

4 PERCEPTUAL GROUPING IN GABOR LATTICES γ Lttice type Rectngulr/squre Olique/rhomic Centered rectngulr/hexgonl Figure 4. Smpling of the lttice spce. Vlues of γ were set t 90º (rectngulr lttices, or squre if ), the miniml vlue given prticulr rtio of / (centered rectngulr, or hexgonl if ), nd the vlue midwy etween these (olique structure, or rhomic if ). The vector length rtios vried from to t intervls of Since the locl structure in lttice is y definition invrint over the visul field, contrst enhncement or other effect would pply to every single element nd increse the perceived contrst of the stimulus s whole. The independence of contour sliency nd perceived contrst found y Hess, Dkin, nd Field (1998) nd Meese, Hess, nd Willims (2001) supports this oservtion. Empiricl studies employing Gor lttice structures will help to develop extensions or lterntives to current theories of contour grouping. (3) Finlly, the phenomenologicl multistility property cretes different perspective. It llows for mking link to the erlier work on lttices nd its methods of mesurement nd quntifiction tht goes eyond mere comprison of proportions correct in different conditions. It is lso idelly suited to ddressing questions relting to dynmicl properties of the underlying processes, such s hysteresis nd dpttion s pplied to perceptul grouping. Fixtion 960 msec Fixtion Gor Lttice Tril Blnk 120 msec Gor 120 msec120 msec 240 msec Blnk Guss Guss Lttice Tril Guss Response Unlimited Response Figure 5. Schemtic overview of the course of n experimentl tril. Upper row: Time course for Gor lttice presenttion. Lower row: Time course for tril with Guss lo lttice. Gor Lttices The psychophysicl procedure used here minly drws on work with dot lttices, clss of stimuli consisting of co-oriented rows of dots with regulr spcing (see Figure 1). The spce of dot lttices, the txonomy of which dtes ck to the 19th century nd the crystllogrphic work of Brvis (1866), ws prmeterized y Kuovy (1994). In this clss of stimuli, dots re ligned in prllel lines long severl orienttions. Within ech orienttion the interelement distnce is constnt, ut the exct vlue of this distnce vries cross orienttions, depending on the lttice geometry. According to the Gestlt lw of proximity, the dots tend to e perceptully grouped long the orienttion in which the dots re closest (Schumnn, 1900; Wertheimer, 1923/1955). A modern version of this grouping principle, dpted to ccount for stochstic responses, mtched experimentl results very well in forml model with only one free prmeter (Kuovy, Holcome, & Wgemns, 1998; Kuovy & Wgemns, 1995). The quintessence of this quntittive Gestlt lw cn e summrized s follows: Ech lttice orienttion hs n ssocited grouping strength tht exponentilly decys with interdot distnce (the ttrction function); the proility of orgnizing the dot lttice in specific orienttion is given y this grouping strength reltive to the sum of ttrction forces present in the lttice. With ll reltive distnces tken into ccount, no evidence for n dditionl role of configurtion ws found, nd consequently the distnce grouping reltion tht hs een inferred from experiments involving dot lttices is referred to s the pure distnce lw. In the present study, we mde use of lttice stimuli composed of Gor elements rther thn dots for severl

5 1450 CLAESSENS AND WAGEMANS Sujects Both uthors nd reserch ssistnt were sumitted to 10 sessions ech (i.e., totl of 11,250 trils per suject; see the Procedure section elow), nd 5 nive oservers, ll of them volunteering students, prticipted in 2 sessions ech (i.e., totl of 2,250 trils piece). All sujects hd norml or corrected-to-norml vision. Figure 6. Upper pnel: Exmple of stimulus disply (rectngulr Gor lttice, ligned). Lower pnel: Corresponding response disply. resons: (1) The lwfulness of proximity grouping in dot lttice structures hs een demonstrted nd replicted. The use of Gor elements llowed for mesurement nd quntifiction of dditionl grouping principles, such s good continution through the lignment of locl orienttions. (2) Gor ptches crry locl informtion for severl well-studied dimensions sptil position, orienttion, elongtion, frequency, nd phse tht cn e prmeterized nd mnipulted systemticlly. (3) To id understnding of the implementtion of grouping lws in the humn visul system, it is desirle tht stimulus elements reflect receptive field properties of erly processing stges in visul cortex. Since the receptive field properties of orienttion-selective simple cells in V1 cn e descried y Gor function (see, e.g., Mrcelj, 1980), nd given tht the Gor function is lso justified y forml rguments (e.g., Dugmn, 1985), it seems pproprite to use mtching luminnce profiles in order to hve t lest generlized ide of how the stimulus is initilly encoded in the strite res of the visul cortex. METHOD Stimuli The geometric structure underlying the ptterns used in the experiment is the point lttice. A clssicl point lttice is two-dimensionl periodic structure consisting of points spred over the plne, invrint under specific trnsltions. The pttern cn e chrcterized y the prmeters of the sic prllelogrm tht spns the lttice (see Figure 2). The vectors constituting the sides of the prllelogrm re the shortest distnces etween elements in the lttice nd re denoted nd, with n intervector ngle γ. The vector difference nd sum of nd, which re the digonls of the sic prllelogrm, re denoted c nd d, respectively. Comintion of the / nd γ prmeters defines restricted spce of lttices in which six lttice types re emedded (Kuovy, 1994). The severl lttice types differ in the numer nd nture of the visul symmetries present in the stimulus geometry. The txonomy derives from the most regulr polygon leding to the lttice lyout when tiled: squre, rhomic, or hexgonl when, nd rectngulr, olique, or centered rectngulr when. As in the prmetric dot lttice reserch initited y Kuovy nd Wgemns (1995), lttices in the present study were presented in rndom orienttions, with γ rndomly set either clockwise or counterclockwise with respect to. These procedures constitute cler improvement over the methods in the clssic Gestlt demonstrtions nd the older psychophysicl literture, in which only squre nd rectngulr lttices were shown in verticl orienttion. An impression of looking through circulr perture ws evoked so s to reduce the interction with externl orienttion cues (see Figure 1). In order to include orienttion lignment s grouping cue, nd for the resons mentioned in the introduction, Gor ptches were used rther thn dots. A two-dimensionl Gor function is chrcterized y the prmeters of Guss function (mplitude, loction, nd spred) nd y those of the sine function it is multiplied with (period nd phse). The locl orienttion of the Gor elements could e ligned with the,, or c vector of the lttices. Gor ptches could lso e independently rotted to rndom orienttion (i.e., nonligned condition; see Figure 3A). We did not lign the ptches long the orienttion of the d vector, since nlysis of previous results showed tht corresponding responses often result from mistking the d-response lterntive for the or groupings, prticulrly in centered rectngulr lttices (Kuovy et l., 1998). It cn e expected tht element lignment long prticulr orienttion will increse the odds of glol grouping long this orienttion. To mesure the mgnitude of this effect, the Gor lttices were compred with stimuli in which the elements hd no intrinsic orienttion. The lttice elements in this control condition only reflected the Guss function, without the sine component (see Figure 3B). We used mtched vlues for the spred prmeter with Guss los nd Gor ptches. In this study, the prmeters of the loclized elements were fixed. We chose to use rdilly symmetric Gussin profile with spred σ of four pixels nd sine period of eight pixels for the Gor ptches. At viewing distnce of pproximtely 90 cm, the ltter ws equivlent to 9' of visul ngle. Applying sine phse of 90º, we otined Gor ptches centered round luminnce minimum (even symmetric, off center). With identicl vlues for corresponding prmeters, the center of the Gussin dots ws luminnce mximum. 1 With these prmeter vlues, we otined Gor ptches with luminnce rnge etween pproximtely 0.5 nd 80 cd/m 2 nd low-contrst Guss los with mximum of pproximtely 40 cd/m 2, oth presented ginst ckground of 32 cd/m 2. Figure 4 illustrtes how the lttice spce ws systemticlly smpled for five levels of /, from 1 to 1.5, with three levels of the intervector ngle γ per distnce rtio. Scling remined fixed t

6 PERCEPTUAL GROUPING IN GABOR LATTICES 1451 Rectngulr Olique Centered Rectngulr responses responses c responses Blnk responses Figure 7. Response frequencies ggregted over sujects for the Guss lo lttices. The pie chrts re rrnged in the sme order s the positions tht they correspond to in the lttice spce (see Figure 4) tht is, with incresing levels of / from left to right nd with levels of γ incresing to 90º from the ottom to the top. For incresing vlues of, nd thus of /, the proility of orgniztion long the vector increses. Similrly, the likelihood of c-oriented orgniztion decreses for higher vlues of γ, which for geometric resons yield higher vlues of c. equl to 60 pixels, or 1º of visul ngle. The viewing perture hd rdius of 380 pixels, or out 14º dimeter. In sum, we used 15 different lttice configurtions, crossed with 5 stimulus types Gor lttices either ligned long,, or c or nonligned, nd Guss lttices resulting in 75 stimulus conditions. Apprtus Stimuli were presented on 17-in. noninterlced CRT driven y n S3 Vision 968 PCI grphics dpter nd 120-MHz Pentium PC. The experiment ws run, t screen resolution of 1,024 horizontl 768 verticl pixels nd verticl refresh rte of 70 Hz, y n MS- DOS rel-mode ppliction. Procedure The dot lttice prdigm typiclly involves rief lttice exposures, fter which the sujects indicte the grouping orienttion tht emerged from the lttice presenttion. Pilot experiments (Wgemns, Clessens, Delrge, Eycken, & Kuovy, 1999; Wgemns, Eycken, Clessens, & Kuovy, 1999) showed tht with clssicl presenttion durtion of 300 msec, some sujects tended to focus on the orienttion of centrl ptch insted of estlishing glol orgniztion. Apprently evlution of physicl locl orienttion is esier, nd thus requires less effort nd ttention, thn deriving virtul glol orienttion. To decrese the sliency of locl orienttion informtion so s to void tsk confusion mong the nive sujects, we decided to mke the locl orienttion informtion ville only very riefly. After 120 msec, the Gor ptches were replced y Guss los tht remined on screen for 120 msec, thus preserving proximity informtion. In the conditions with Guss lttices, the stimulus remined unchnged for totl durtion of 240 msec. The ctul stimulus presenttion ws preceded y 960-msec presenttion of the circulr window with centrl fixtion dot, followed y lnk disk for 120 msec (see Figure 5). Immeditely fter the presenttion of the lttice, the response disply ws shown (see Figure 6). It contined four circles: three disks with lines representing the most likely grouping lterntives (corresponding to the,, nd c vectors) nd lnk circle for cses in which suject hd clustered the elements long the d orienttion or in ny other wy. The sujects responded y clicking with mouse pointer on the disk in the response menu with the ppropritely oriented isecting line. After suject hd indicted his or her choice with the mouse pointer, new tril cycle egn. Given the possile interference of locl orienttion with response ehvior, the sujects were explicitly sked to keep their ttention spred while fixting on the middle of the screen. During n experimentl session, ech of the 75 conditions (15 lttices 5 lignment mnipultions) ws presented 15 times. Trils were presented in rndom order, with n opportunity for rek every 225 trils. Ech 1,125-tril session lsted pproximtely 1 h. RESULTS We will first present the min findings grphiclly nd support them with logistic regression nlyses. Then, n nlysis of the experimentl mnipultions nd their interctions will e provided. 2 Role of Proximity nd Alignment Given the lrge numer of conditions nd response lterntives, it is not fesile to show the dt for ech of the conditions for ech of the sujects. Figure 7 shows response frequencies per stimulus condition within the set of Guss

7 1452 CLAESSENS AND WAGEMANS J.W. K.D. P.C. Nive responses responses c responses Blnk responses Figure 8. Effect of proximity for different oservers, ggregted nd verged over γ. The response frequencies presented in the nive lock re ggregted cross the 4 nive sujects in order to otin totl numer of trils, nd hence sttisticl power, of the sme order of mgnitude s for the expert sujects. lo lttices reltive to the position in the lttice spce (i.e., prticulr / nd γ). In this summrizing grph, response frequencies re ggregted cross sujects. It is cler tht lttice geometry influences the response pttern. Over lttice types, lrger / vlues (columns in Figure 7) yield fewer responses. This reflects the effect of the distnce mnipultion. Similrly, for lrger vlues of γ (rows in Figure 7), corresponding to lrger c / rtios, the numer of c responses decreses. These oservtions re congruent with previous results (Kuovy & Wgemns, 1995) nd with the principle of grouping y proximity (Kuovy et l., 1998). The results otined with nonligned Gor lttices (not shown) re similr. Figure 8 (results ggregted cross ll lignment conditions nd ll γ vlues) shows tht the weight of proximity s grouping fctor vries somewht cross oservers, lthough the pttern of results is identicl. In comprison with the Guss lttices nd the nonligned Gor lttices, the ligned Gor lttices elicit more responses in -ligned lttices, more responses in -ligned lttices, nd more c responses in c-ligned lttices (Figure 9). This result shows tht locl orienttion lignment cts s grouping cue. In lttices with smll differences etween the - nd -vector lengths, orienttion lignment long could override proximity s grouping cue in limited rnge. Below, we will estimte the trnsition point t which orienttion lignment strts dominting the grouping response. The sttisticl nlyses re focused on the nd responses. Since the corresponding vectors re shortest, they generte more responses thn did the c (nd d) vectors; consequently, these dt re less prone to sttisticl floor effects. Moreover, judgment errors in selection for the response disply throughout the experiment most frequently confusion of the, c, nd d orienttions for lttices with high / rtio (Kuovy et l., 1998) hve less influence on response ctegories with higher frequencies. The pure distnce lw proposed y Kuovy nd Wgemns (1995) cn e simplified to clssicl logistic model in which the generlized logit ln[p()/p()] is linerly dependent on / 1 (Kuovy et l., 1998). Figure 10 shows the oserved log-odds s function of the corresponding distnce rtio; in Figure 11, the sme dt re plotted for ech suject individully, with ggregted control conditions. Since responses constitute the numertor, higher position of point in the grph corresponds to higher likelihood of choosing. As descried previously, lignment yields more responses, which is why there is positive intercept for the -ligned conditions with respect to the log-odds for the other lttices. Conversely, lignment leds to more responses, nd therefore to negtive intercept for the -ligned conditions. The lttices ligned long the c vector hve spline with course coinciding with those for lttices with Guss los nd nonligned Gor ptches. This indictes tht, lthough c lignment evidently yields more c responses, the reltive

8 PERCEPTUAL GROUPING IN GABOR LATTICES 1453 ligned ligned c ligned Guss Gor Nonligned J.W. K.D. P.C. Nive responses responses c responses Blnk responses Figure 9. Influence of the lignment mnipultion for the different sujects, ggregted over distnce conditions nd lttice types. The first three columns show the response frequencies for Gor lttices tht were ligned with the,, nd c vectors, respectively. The fourth nd fifth columns contin the pie chrts for the control conditions: the lttices consisting of Guss los ( Guss ) nd the Gor lttices in which ech ptch ws ssigned rndom orienttion ( Nonligned Gor ). proportions of nd responses re still lrgely equl. This fct is reminiscent of n xiomtic ide of choice theory generlly referred to s independence of irrelevnt lterntives. It is stted s proilistic lemm in Luce (1959): The proility rtio of two choices should not depend on the set of lterntives tht is under considertion. Our dt seem to stisfy this condition. There is n ovious prllelism etween the splines for the conditions in which Gor ptches were used, except for the c-ligned lttices, for which ecuse of the lower numer of nd responses vriility is the lrgest (compre full lines in Figure 10). For ll other conditions involving Gor ptches, whether -ligned, -ligned, or nonligned, the proximity effect levels off. The grph suggests more liner trend for the stimuli with Guss los in comprison with those from the Gor lttice conditions, in line with the nlysis for dot lttices y Kuovy et l. (1998). At the individul level (Figure 11), this curviliner trend shows up in 1 nonnive nd 3 nive sujects. The level t which the logit symptotes, in prticulr for the -ligned Gor lttices, is quite fr from the extremes in severl sujects. This mkes n explntion in terms of sttisticl floor effect or rndom errtic responses unlikely. It seems to indicte insted tht mechnisms for grouping diverge for Gor elements nd other stimuli when extreme distnce rtios come into ply. Note, however, tht the curviliner trends in different lignment conditions do not necessrily correspond to one nd the sme cuse. Logistic Weights of Proximity nd Alignment The nd response frequencies were processed in logistic regression nlyses in order to ssess sttisticl significnce of the visul oservtions ove. The simplest sttisticl model for the logistic link function ln[p()/p()] tht cptures the sic chrcteristics of the set of splines for ll of the lignment conditions contined / 1 s numericl vrile, its squre ( / 1) 2 to cpture curvilinerity, nd lignment s clss-level vrile. Both this model nd those in susequent nlyses were formulted s inry logistic models nd fitted y mximumlikelihood estimtion sed on n itertive Fisher scoring optimiztion (SAS sttisticl softwre, PROC LOGISTIC procedure). 3 Prt of the weight estimtes for the link function re given in Tle 1. Contrst comprisons etween nd levels versus the other lignment conditions were ll highly significnt (two-tiled, p.0001), nd ll pirwise comprisons within the other lignment conditions hd high p vlues. The estimtes ssocited with nd lignments were not significntly different in their solute vlues. Disregrding the smller interction effects, one could stte tht locl lignment enhnced the odds y fctor of lmost three in the direction of the ptch lignment orienttion. Estimtes for second-order polynomil prmeteriztion of the -ligned condition seprtely, without ny other effects, yielded estimtes corresponding to the model ln[p()/p()] ( / 1) ( / 1) 2. Clcultion of n equilirium point,

9 1454 CLAESSENS AND WAGEMANS ln p() p() p() p() p() p() Alignment condition ligned ligned c ligned Guss Nonligned Figure 10. Plot of the log-odds of to responses s function of / 1 for the different lignment conditions. Schemtic views of stimuli re connected with their corresponding dt points; the line segments in the smllest circles indicte the locl orienttion of Gor ptches. Note tht dt points ove 0 represent conditions in which more responses were given thn responses. Per level of / 1, there re three dt points per lignment condition, ecuse of the different lttice structures involved. The splines run through the verge over the three lttice types within ech comintion of the other mnipultions nd connect verges cross the distnce rtios per level of the lignment mnipultion. where the ttrction forces of lignment nd proximity rtio re eqully strong, is possile y solving the qudrtic eqution given ove for p() p(), or ln[p()/ p()] 0. Applying this principle to the estimtes for -ligned Gor ptches, the interpolted solution is / Some of the prmeter estimtes vried considerly over sujects; see Tle 2 for n overview. The individul psychometric functions cn e inspected in Figure 12. When the individul dt sets for the nive sujects were included, the prmeter estimtes spnned rnge of to , to , nd to for the intercept, liner distnce, nd qudrtic distnce, respectively. The inset to Figure 12 provides sctterplot of the intercept nd the slope of qudrtic pproximtion to the logit, with ech plot symol representing 1 suject. Our discussion of the issue of interindividul differences will e resumed lter. Stepwise Logistic Modeling of Effects A further investigtion of the role of the experimentl mnipultions nd their interctions consisted of stepwise forwrd-selection procedure. Alignment, distnce, nd lttice geometry, s well s their interctions, were ssessed in terms of their dvntge for logistic model fit nd were dded sequentilly. Distnce ws treted s clss-level vrile so s to keep the prmetric shpe of the proximity mnipultion open. The significnce of the score sttistic relted to ech extension served s criterion for priority of inclusion. The logistic regression prmeters were reestimted fter ech itertion. This itertive procedure ws ended when dding term did not yield significnt (i.e., p.05) increse of the score sttistic. The finl model otined with this procedure contined ll min effects s well s their two-wy interctions, except for the interction etween distnce nd lignment. The set of model vriles s whole ws highly significnt ( p vlue well elow.0001) in comprison with n intercept-only model (score χ 2 10,905, likelihood rtio χ 2 (26) 10,842); in fct, the model lredy reched p.0001 significnce s soon s it included the distnce rtio vrile. The Wld χ 2 vlues for the effects re informtive s to their respective weights nd importnce in the finl model fit (see Tle 3). The interction etween lttice type nd lignment, which ws dded s the lst predictor to the model, hd p vlue of out.022, which is only mrginlly significnt. However, this interction ws quite consistent cross oservers, nd it hs n interesting interprettion (see elow). There ws smll ut sttisticlly significnt role of lttice type in the results: Its min effect nd significnt interction with distnce, contriuted y the responses from only 1 oserver (Suject P.C.), were entirely due to the centered rectngulr lttice geometries. This prticulr suject gve more responses in these conditions for intermedite distnce rtios in ll lignment conditions. The forementioned vrile selection procedure ws lso pplied for ech suject seprtely (i.e., for the 3 in-

10 PERCEPTUAL GROUPING IN GABOR LATTICES Alignment condition 2 Alignment condition ligned ligned 1 J.W. ligned 1 K.D. ligned Control Control Alignment condition 2 Alignment condition ligned ligned 1 P.C. ligned 1 Nive ligned Control Control Figure 11. Plot of the log-odds of to responses per suject. We grouped dt over the control conditions for the ske of grph redility. formed oservers nd the comined sessions of the nive sujects, or 4 sujects in totl). The otined models were not entirely consistent with respect to included interction terms; however, these terms were lwys the lst to join the model, ecuse of their smll effect size in comprison with the significnt min effects. On the other hnd, Wld χ 2 sttistics showed tht oth the effects of distnce rtio nd lignment were highly significnt within ech suject ( p.0001). These effects were lwys the most powerful fctors, with proximity lwys the strongest. This order of importnce vlidtes the roustness of the model otined with the full dt set s it pplies to ll sujects. Figure 13 depicts the evolution of the devince ( 2 log-likelihood) when dding fctors to the model. Although the differences in slopes indicte tht the weights for the different model terms vry sustntilly cross sujects, the reltive rnk order of the effects remined constnt. One interction, lignment lttice type, ppered in the finl forwrd-selection model for ll sujects except P.C., lthough with smll effect size. Within the different lttice types, comprtively more responses were given for rectngulr lttices tht were ligned long the vector, nd fewer for rectngulr lttices ligned long. This represents reltive grouping dvntge for orienttions orthogonl to the direction of lignment. In sum, the most conspicuous results discussed here re the lrge effect sizes of oth distnce rtio nd lignment Tle 1 Prmeter Estimtes for Min Effects in the Qudrtic Proximity Model Model Term for Prmeter Estimte Stndrd Error Wld χ 2 (1) Alignment: ligned *** Alignment: ligned *** Alignment: c ligned n.s. Alignment: Rndomly ligned n.s. / *** ( / 1) n.s. Note The condition with Guss lo lttices is used s the reference level within the lignment clss-level vrile. *** p n.s., not significnt, p.05.

11 1456 CLAESSENS AND WAGEMANS Suject Tle 2 Qudrtic Approximtion of the Logit for the -Aligned Gor Lttices Intercept Estimte Liner Estimte nd the consistency of their role s min effects cross sujects, s shown y the dt-driven vrile selection procedure. The nonliner nture of the influence of distnce rtio on grouping of Gor ptches nd the lrge interindividul differences in the weights of the vrious experimentl mnipultions on perceptul grouping were lso ovious throughout the different nlyses. The order of reltive importnce of the min effects ws consistent cross oservers. DISCUSSION Qudrtic Estimte Equilirium Point J.W *** *** n.s K.D *** *** n.s P.C *** *** * Nive oservers *** *** *** Note The intercept, liner, nd qudrtic estimtes should not e compred with ech other directly; their vlues weight different vrile rnges. The lst column shows the equilirium point tht is, the vlue of / for which there is lnce etween proximity nd lignment groupings, nd hence where p() p() for the -ligned lttices. * p.01. *** p n.s., not significnt, p.05. Irrespective of lignment condition, the log-odds of choosing the lterntive re liner function of interelement distnce over lrge rnge of the distnce mnipultion. This finding is in line with results from previous reserch (Kuovy et l., 1998), nd thus supports the vlidity of the pure distnce lw (Kuovy & Wgemns, 1995) for Gor lttices. In two respects, however, the dt of the present study devite from the pure distnce lw. First, for lrger vlues, the reltion etween distnce nd perceptul grouping levels off for most Gor lttice conditions, wheres locl element lignment remins eqully significnt for ll ligned lttice conditions. Second, the explortory nlysis reveled wek, lthough firly consistent, interction etween lttice geometry nd lignment. The reltively lrge numers of responses for -ligned rectngulr lttices nd of responses for -ligned rectngulr lttices supports the existence of fcilittion long grouping xis orthogonl to the locl element orienttion. This result prllels those with sidey-side Gor ptches in the snke detection prdigm, s otined y Field et l. (1993) nd others (e.g., Bex et l., 2001), nd is reminiscent of the contrst enhncement for orthogonl configurtions reported y Polt nd Sgi (1993). For 1 suject (P.C.), the finl forwrd-selection model hd slightly different composition thn it did for the other sujects, in tht it contined n effect of lttice type nd n lignment lttice type interction. This is the result of higher numer of responses for centered rectngulr geometries with intermedite-distnce rtios, where the -grouping lterntive normlly domintes. Of ll geometries, centered rectngulr lttices hve the highest grouping entropy (Kuovy & Wgemns, 1995). A possile explntion of the devint pttern of results is more rndomness in the dt due to either high threshold for use of the lnk-response lterntive or lrge influence of prior orienttion ises in conditions of high perceptul uncertinty for centered rectngulr lttices. The results otined with the Gor lttice prdigm llow for precise evlution of the reltive importnce of the fctors involved in grouping, nd they corroorte the conclusions from previous experiments on contour integrtion. At the sme time, perceptul orgniztion in Gor lttices lso sheds new light on theories of contour integrtion tht hve trditionlly een developed to model contrst enhncement dt or snke detection performnce. The lttice method genertes dt tht exhiit lwful reltionship etween stimulus chrcteristics nd multistle, phenomenlly coherent orgniztion. This finding leds to numer of questions out how glol orgniztion reltes to existing models. First, it is not ovious whether one-lyered network with integrtion mechnisms sed solely on ssocition field type connectivity (Pettet et l., 1998; Yen & Finkel, 1998) could explin why Gor lttice orgniztion is coherent nd not prtitioned tht is, why different dominnt orienttions do not occur in different prts of the lttice. This coherence feture of glol orgniztion requires integrtion of informtion spnning more thn the clssiclly ssumed extent of n ssocition field. Second, the populr explntion tht reltes locl contrst enhncement to contour sliency (see, e.g., Usher et l., 1999) is not pplicle to grouping in lttices. In the cse of lttices, the sptil reltions etween elements re y definition invrint throughout the stimulus. Therefore, contrst enhncement or similr mechnisms would pply to the lttice s whole nd would not generte pop-out of one specific orienttion. This does not men tht contrst modultion hs no role to ply in grouping phenomen. The sitution with ner-threshold contrst imges or stimuli with low signl-to-noise rtio might e different from perceptul orgniztion in suprthreshold regime. Although the primry tsk of the visul system in the former cse is to identify whether or not ny contour cndidte is present, in the ltter sitution this system must decide etween severl competitive interprettions. In other words, the mechnisms nd sustrtes for grouping might e sudivided in two distinct ut interoperting systems: one Tle 3 Evlution of Model Terms Effect df Wld χ 2 Distnce 4 5, *** Alignment 4 1, *** Distnce lttice type *** Lttice type *** Alignment lttice type ** * p.01. ** p.001. *** p p.022.

12 PERCEPTUAL GROUPING IN GABOR LATTICES 1457 ln 2 1 p() 0 p() 1 Suject J.W. K.D. P.C. Nive Weight of Alignment Weight of Proximity Figure 12. Plot of the log-odds of to responses for the -ligned condition, dt ggregted over lttice type. The different dshed lines show the qudrtic logistic model for the different sujects (see legend), nd the solid line is the fit for the dt merged cross sujects. The inset sctterplot shows the weight of lignment (the intercept in the qudrtic model, horizontl xis) nd the corresponding liner weight for proximity (verticl xis), s reported in Tle 2, per suject. The estimtion derived from dt merged over the nive sujects is shown s filled upwrd-pointing tringle in the inset, nd ech of the individul nive sujects is represented s n unfilled tringle. Horizontl nd verticl lines correspond to one stndrd error of estimte elow nd ove the weight estimtes. 2 log-likelihood 9,500 9,000 8,500 8,000 7,500 7,000 6,500 6,000 5,500 5,000 4,500 J.W. K.D. P.C. Nive 4,000 Intercept Distnce Alignment Suject model Full model Figure 13. Interlock differences etween effect devince gins. Devince ( 2 log-likelihood) is sttistic mesuring dness of fit; dding more model terms increses goodness of fit, nd thus decreses the devince. The downwrd slope cn e considered to e the weight of the dded term in explining remining vriility in the log-odds of versus responses. From left to right on the sciss of this grph, distnce rtio, lignment, further terms from the suject-specific forwrd-selection procedure ( Suject model ), nd ll other terms ( Full model ) re dded. The resulting devince is plotted s function of these model expnsions. The devince decrese otined y dding the first term nd, prticulrly, the second term vries cross sujects.

13 1458 CLAESSENS AND WAGEMANS tht identifies the presence or sence of contour in given loction nd orienttion, nd nother tht selects which contour elements re prt of the sme contour, in effect dismiguting multistle orgniztion into one solution. These systems seem to correspond lrgely to the frmeworks of prespecified nd dynmic grouping, respectively (see Wtt & Phillips, 2000). Neurl models tht concentrte on one of these spects of grouping re ound to hve different fetures or to exhiit different ehviors. Models tht strt y using correlted firing of neurl units s hypothesis for the emergence of dynmic inding (Choe & Miikkulinen, 2004; Li, 1998; Yen & Finkel, 1998) might hve more potentil s n explntion of trnsient glol grouping (ut note the criticl evlution y Shdlen & Movshon, 1999, nd the recent dt on curve trcing y Roelfsem, Lmme, & Spekreijse, 2004). In their specifictions of neurl rchitecture, the uthors of these models often rely on the now well-estlished finding tht long-rnge connections etween pyrmidl neurons in re V1 tend to connect groups of cells with similr orienttion tuning (Gilert, 1992; Gilert et l., 1996; Stettler et l., 2002). In the present study, ll Gor ptches hd the sme orienttion. Therefore, the stronger grouping long the vector with the sme orienttion s the locl lignment necessittes the dditionl constrint tht connections re more hevily weighted in orienttion-selective cells tht lie long the sme glol xis. Bosking et l. (1997; see lso Chisum et l., 2003) hve found similr rchitecture of horizontl connections in tree shrew strite cortex, ut it is currently uncler whether such n rchitecture is ville in the strite cortex of other mmmls. The reltive weights of feedck versus locl interctions in the phenomen discussed here re still n importnt unresolved issue. Exctly the modultory input from regions with lrger receptive fields in higher lyers might explin the sptil coherence in lttice stimuli. The finding tht some sujects show n incresed grouping tendency for prllel Gor elements points to the need for reserch into the functionl significnce of psychophysiclly nd neurophysiologiclly estlished fcilittion long prllel contour elements tht re not likely to elong to the sme contour. The cses in which the logicl solutions of contour integrtion nd the oserved phenomenl grouping diverge could potentilly yield vlule informtion out the reltive shre of prespecified nd dynmic spects of grouping in visul experience. In summry, we elieve tht the use of Gor lttices in rigorous psychophysicl experiments nd nlysis yields importnt prospects for the quntifiction of severl grouping fctors nd their interctions nd for further evlution of current models of sptil integrtion. 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