Distortions of Length Perception in Human Vision

Transcription

1 Distortions of Length Perception in Humn Vision Distortions of Length Perception in Humn Vision Algis Bertulis, Aleksnr Bultov Deprtment of Biology, Kuns University of Meicine Key wors: geometricl illusions on length, superposition of illusions, luminnce contrst, sptil filtering, visul fiel nisotropy. Astrct. The phenomen of geometricl illusions on extent n of nisotropy of the visul fiel suggest the geometric proportions of perceive fiel eing ifferent from the proportions of physicl stimulus. The present stuy investigtes the Müller-Lyer n the Oppel- Kunt illusions s functions of sptil prmeters of the figures, n constructs neurophysiologicl moel. The moelling is se on the uncertinty principle, ccoring to which istortions of size reltions of certin prts of the stimulus re etermine y processes of sptil filtering in the visul system. Qulittive n quntittive greement ws otine etween psychophysicl mesurements of the strength vlue of the illusions n the preictions of the moel. When comine in one stimulus n presente simultneously, two illusionry figures, irrespective of their type, prouce one common misrepresenttion ut not two seprte istortions on perceive length. Asence of summtion of illusions inictes unifie neurl origin of similr effects prouce y issimilr shpes. Illusionry istortions on perceive length my rise in the monoculr retino-corticl pthwys seprtely in ech eye. A unifie low-level neurl mechnism of two-imensionl sptil filtering is si to e responsile for certin group of geometricl illusions. The illusions on length n the nisotropy of the visul fiel re to e interprete s inepenent istortions of perception, the effects of which converge to n lgeric summtion in simultneous mnifesttion. 1. Introuction We report here our psychophysicl stuies of two kins of istortions on length perception, generlly recognize s istinct phenomen: visul-geometric illusions n visul fiel nisotropy. We exmine the two effects seprtely n in their simultneous mnifesttion n present our resoning concerning nture of the istortions. It is well known tht geometricl illusions re common to vrious types of perception n re not n exceptionl property of vision. With the Müller-Lyer n the Ponzo ptterns, hptic illusions equivlent to the visul ones were oserve; with the hptic Deloeuf stimuli, the size illusion of the outer circle occurre; with the Zöllner figure, hptic illusion irectionlly opposite to the visul one ws otine [Suzuki, Arschi, 1992]. An uitory Oppel-Kunt illusion hs een lso escrie: the suivie soun intervl ppere lsting longer thn n empty intervl of the sme urtion [Russo, Dellntonio, 1989]. Therefore, the geometricl illusions might e consiere s n immnent physiologicl chrcteristic of the neurl networks of sensory systems. In vision, perception of ny imge is ccompnie y lrger or smller misrepresenttions, the extent of which is etermine y sptil n temporl composition of the imge. For exmple, squre mkes the rcs of circumscrie circle pper puffy, n the errors of the sujects justing the rcs to reuce the misrepresenttion to zero re reltively high. On the other hn, the left-right symmetry in isection proceure is oserve ut the error vlues cn hrly e estimte without sttisticl nlysis Distortions of perception of length in illusionry figures We tke the Müller-Lyer n the Oppel-Kunt visul ptterns s stimuli in our experiments (Fig. 1). The two ifferent figures prouce similr visul effects on perceive size: two hlves of the figures seem to e ifferent in length when they re relly equl. In the Müller-Lyer figure, the hlf with inwr-fcing wings seems to e shorter thn the other hlf with outwr-fcing wings. In the Oppel-Kunt pttern, which hs no crossing lines n no flnking contextul figures, the she hlf seems to e longer thn the nonshe hlf. The Müller-Lyer illusion is one of the est known effects hving een stuie intensively since the en of the 19th century. The Oppel-Kunt illusion hs een given less focus y reserchers, though its mnifesttions hve een well ocumente. A question my e ske whether the sme neurophysiologicl mechnism is responsile for the two types of perceive size istortions n if the sme theoreticl pproch might e pplie to the similr effects prouce y iffering geometricl ptterns. The experimentl t in the literture show rther consistent mgnitue of istortions perceive in the Müller- Lyer figure. The illusion hs een mesure s function of the shft length [Fellows, 1967; Brigell et l., 1977; Restle, Decker, 1977; Gillm, Chmers, 198; Mck et l., 198], the wing length [Heymns, 1896; Lewis, 199; Nkgw, 198; Dewr, 1967; Erlecher, Seculer 1969; Fisher, 197; Restle, Decker, 1977], the wing tilt ngle [Biervliet, 1896; Heymns, 1896; Dewr, 1967; Erlecher, Seculer, 1969; Dvies, Spencer, 1977; Pressey et l., 1977; Restle, Decker, 1977], the gp Corresponence to Algis Bertulis, Prof.hil.r, Bertulis@vision.km.lt. Volume 1, Numer 1, Jnury-June 21 BIOMEDICINE 3

2 Algis Bertulis, Aleksnr Bultov etween the shft n the pices of the wings [Yngisw, 1939; Fellows, 1967; Worrll, Firth, 1974; Pressey et l., 1977; Preeon, 1992], n the frming rtio, i.e. the rtio of the totl figure length to the shft length [Brigell et l., 1977; Brigell, Uhlrik, 1979; Schino, 1986]. The contrsting effect of the inwr-fcing wings n the outwr-fcing wings ws exmine simultneously in oule figure; lso, either effect ws mesure seprtely y presenting one-hlf figure with comprison line tht h no wings [Lewis, 199; Pieron, 1911; Erlecher, Seculer, 1969; Fisher, 197; Restle, Decker, 1977]. The evience ville points to the regulrities such s follows. The illusion strength is proportionl to the shft length, wheres it vries s n inverte U-shpe function of the wing length, reching the mximum when the wings re pproximtely 3-4 per cent of the shft length. Lrge inwrfcing wings hve size reuction effect on the shft; the illusion ecreses with enlrgement of the tilt ngle of wings up to 9. The ngle exerts its effect within smll region ner the tip of the shft, the region extening over out oneseventh of the shft length. The reverse irection of istortions is possile to chieve y moving the wings wy from the ens of the shft, n the gp t which the mximum reverse effect occurs is inversely relte to the tilt ngle. The pek overestimtion of the shft is otine when the frming rtio is 3:2. The illusion ws lso mesure s function of the contrst rtio of vrious prts of the figures. Btes [1923] foun tht the lck wings n the lck shft line prouce n illusion tht ws etween two n six per cent lrger thn one oserve with the grey wings n lck shft. Mukuji [197] lso foun tht the illusion ws most pronounce when the wings n the shft were of the sme colour. Wickelgren [196] otine similr results: the illusion epene on the contrst rtio of the wings n the shft line n it i not epen on the sign of the contrst of the figure. The experimentl t hve een nlyse qulittively n quntittively ecuse it ws expecte tht the oserve istortions in the visul fiel woul enle one to elucite the funmentl properties of visul processing. As result, numerous hypotheses hve een propose n vrious explntions of the illusion hve een suggeste. Originlly, the Müller-Lyer illusion ws expline s comintion of two opposing fctors: confluxion n contrst [Müller-Lyer, 1896, ; Heymns, 1896; Lewis, 199; Pieron, 1911]. Confluxion mens tht two points re seen closer thn the ojective isply woul justify, n contrst mens tht they re seen too fr prt. It is suggeste tht the two fctors re presumly implicte t vrious levels of processing stimulus through lterl inhiition n contour repulsion, which hve fixe rnge of ction [Nkgw, 198; Wgner, 1968; Coren, 197]. Another type of explntion ws given y Gregory [1968, 197]. Accoring to it, primry cues of epth n perspective elicit perceive length istortions. The explntion is se on n ie tht epth perception opertes on size constncy mechnism [Dy, 1972; Gul, 197; Lester, 1977; Wr et l., 1977; Smith, 1978]. Accoring to Gillm [198], size constncy is se on the liner perspective scle n not on epth response to the scle. All these sttements re reflecte, in one wy or nother, in cognitive theory [Gregory, 1972; Rock, Anson, 1979], which ssumes tht perception results from the ttempts the system crries out to fin the most likely explntion for given set of sensory t. Nevertheless, the presence of illusions in figures isplying no pprent epth cues, such s the Blwin s n the ivie line figures [Blwin, 189; Ooni, 194; Dy, 196; Znforlin, 1967; Coren, Girgus, 1972; Brigell, Uhlrik, 197; Brigell et l., 1977], is use s evience ginst the perspective theory. Brigell et l. [1977] cme to conclusion tht the rtio of the totl figure length to the focl shft length etermines the reltive mgnitue of the illusion. The illusion on lines flnke y contextul figures like rectngles, rrows, lines re lso interprete in terms of the pttionl level theory [Green, Hoyle, 1964; Helson, 1964; Restle, Merrymn, 1968; Restle, 1971, 1977], which sttes tht jugments on length re irectly relte to the size of focl stimulus n inversely relte to the pttion level of the oserver. In prllel, Pressey s [1967] ssimiltion theory is pplie to the explntion of the illusions: the focl shft ssimiltes to (or is verge with) the mgnitue of the contextul figures tht re flnking the shft [Pressey, 1971; Pressey, Bross, 1973]. A qulittively similr weighte verging moel ws propose y Anerson [1974]. Other theories, such s the confusion theory [Ching, 1968; Erlecher, Seculer, 1969] or those se on receptive fiel moels [Wlker, 1973], clim tht the size of proximl figures shoul etermine the mgnitue of istortions. The concept of the centre of grvity, though, is in some conflict with these theories. It is se on fee-ck from efferent commns for eye movements [Ju, 19; Festinger et l., 1968; Kufmn, Richrs, 1969; McLughlin et l., 1969; Virsu, 1971; Coren, Hoenig, 1972]. The concept explins the perceptul istortions y fee-ck resulting from inpproprite tenency to fixte the centre of grvity of contextul figures when ttempting to fixte the en-points of the focl shft. These opposing concepts tken together yiel further proposls. For instnce, some mong the receptive fiel moels evelope ccept the ie of oriente line-etectors in the visul cortex. The etectors presumly mesure ny orienttion s the verticl-to-horizontl extent rtio, wht cuses overestimtion or unerestimtion of the outwrfcing or inwr-fcing wings [Celli, 1977]. The concept of oriente receptive fiels cting s sptil frequency filters is lso use to ccount for the length istortions. Kwt [1976] n Ginsurg [1984, 1986] presente evience tht filtering processes cuse y lterl inhiition n proucing certin mount of lurring of the retinl picture cn prouce the Müller-Lyer illusion. The similrities n ifferences existing mong numerous theoreticl pproches ville o not ssign priority to ny one of them. Comprison of the effects preicte y 4 BIOMEDICINE Volume 1, Numer 1, Jnury-June 21

3 Distortions of Length Perception in Humn Vision vrious moels with the experimentl evience fils to estlish quntittive mtch: Eijkmn et l. [1981] foun tht moel consisting of size-constncy opertor triggere y epth cues preicte effects lrger thn those tht were ctully oserve; filter moel seeme prtly responsile for the oserve illusion, ut sufficient mgnitue of illusion coul not e otine; oriente retectors were even less effective in explining the oserve length illusions. Therefore some investigtors [Dy, 1972; Restle, Decker, 1977; Pressey, Di Lollo, 1978; Eijkmn et l., 1981] suggeste tht simple strightforwr explntions were questionle n there might e more thn one mechnism contriuting to the oserve effect. Some others, e.g. Preeon [1992], though still emphsize the lck of necessity of invoking ifferent cusl mechnisms. One my come to conclusion then, tht stisfctory n relile moel for the Müller-Lyer illusion hs not een foun yet. The sme is true of the Oppel-Kunt illusion. Accoring to Crven n Wtt [1989], the essentil explntion is tht the verge contour ensity mkes the fille hlf of the Oppel-Kunt figure to pper wier thn the empty hlf. The question of wht in n orinry imge woul correspon to contour ensity my e nswere y stting tht zerocross numers within rnge of sptil scles re equte n contriute quntittively to the illusion [Wtt, 199]. Crven [199] mesure the Oppel-Kunt illusion fter pttion of the suject to spce prllel lines, n otine no fter-effect, wht let him to conclue tht the illusion ws not prouct of continuous sptil clirtion mechnism. The Oppel-Kunt illusion, just s the Müller-Lyer illusion, epens on the rtio of the luminnce or colour contrsts of the components of the figure [Dworkin, Bross 1998]. So, wht is the origin of perceptul istortions generte y the two ifferent geometricl ptterns? The overll oy of the experimentl evience on illusions n pproches ltogether suggest tht unerstning of perceptul istortions my e improve y ppliction of the signl theory. The uncertinty principle, efine y terms of the Fourier trnsform, inictes certin reltion etween the extent of signl n its spectrl chrcteristics. Therefore sptil filtering processes unvoily prouce istortions in the size n shpe reltions of vrious prts of n imge. The mgnitue of the istortions epens irectly on the prmeters of filtering system, therey suggesting tht length illusions, such s the Müller-Lyer n the Oppel- Kunt effects, re etermine y filtering processes in the neurl networks of the visul system. This gives grouns to the filter theory to regin its importnce, n for the filter moels to e re-exmine. Therefore, we hve constructe filtering moel [Bultov et l., 1997] tht employs the neurophysiologicl t on the properties of retino-corticl pthwys, the principles of sptil orgniztion of simple n complex corticl receptive fiels, n the scheme of their istriution in the retin - which mkes our moel essentilly ifferent from tht of Ginsurg [197] or Eijkmn et l. [1981]. The vriility of the prmeters of the filters in our moel epens on eccentricity of the visul fiel. To test our moel for the two illusions, we hve performe series of experiments with the sme sujects uner conitions of constnt oservtion. The results were similr to those reporte in the literture, ut they lso e some etils n revele iniviul ifferences etween the sujects. Our moel is le to preict the solute vlues of the mgnitue of the illusions otine in ll our experiments, therey proviing proof of the importnce of filter mechnism in the origin of illusions on extent Distortion of length perception ue to non-homogeneity of the visul fiel Another psychophysicl phenomenon, the ril nisotropy of the visul fiel, which lso yiels istortions in length perception, seems to hve nture ifferent from tht of geometricl illusions. The nisotropy oes not pper to e meite y sptil filtering. It oes not occur ue to visul experience, ut seems to e geneticlly etermine [Leehey et l., 197]. The visul fiel nisotropy is consequence of topogrphicl interreltions etween the retin n the primry cortex, the re V 1. The rtio of the horizontl-to-verticl imeters of the visul fiel is out 1.23 for monoculr n 1.4 for inoculr vision [Berri, Fiorentini, 1991]. The scle of size estimtion vries with the stimulus orienttion, which cuses perceptul istortions of vrious prts of n imge projecte on ifferent regions of the retin [Steven, 198; Brun, 193; Kunnps, 197,, c; Prinzmetl, Gettlemn, 1993]. Orienttion nisotropy ws oserve while mesuring visul cuity [Rovmo et l., 1982], contrst sensitivity [Pointer, Hess, 1989], vernier cuity [Srinen, Levi, 199] s well s curvture [Fhle, 1986] n sptil frequency [Egr, Smith, 199; Berri, Fiorentini, 1991] perception. Orienttion nisotropy my e consiere s olique effect (for reviews see Appele, [1972] Heelye, Buchmn-Smith, [199]). There were suggestions on how to explin nisotropy in terms of the frming effect of the ellipticl shpe of the visul fiel [Kunnps, 197], of perspective [Gregory, 1963], or y mens of low-level neuronl moels, such s neuronl sensitivity [Rose, Blkemore, 1974], neuronl tuning [Anrews, 1967; Thoms, Gille, 1979], neuronl ensity [Rose, Blkemore, 1974; Mnsfel, 1974; Mnsfel, Ronner, 1978; Orn, Kenney, 1981; De Vlois et l., 1982] or non-homogeneity of the fctor of mgnifiction of the retinl representtion on the strite cortex [Bultov et l., 1996]. A possiility of higher level contriution to orienttion nisotropy ws not omitte either [Buchmn-Smith, Hrley, 1993]. To estimte the influence of nisotropy on the illusion, we hve employe oth three-ot stimuli n the Oppel-Kunt figures me of ots n hving vrious orienttions n sizes of their internl ngle. It coul e ssume tht the three-ot stimuli provie estimtion of visul fiel nisotropy lone while the Oppel-Kunt figures ssess comine effect of nisotropy n illusion. We my sk if there is ny interction etween the two psychophysicl phenomen oth hving presumly lowlevel locliztion, n if the sptil filtering moel is pplicle quntittively to pure Oppel-Kunt illusion s mesure t vrious orienttions of the two hlves of the pttern. Volume 1, Numer 1, Jnury-June 21 BIOMEDICINE

4 Algis Bertulis, Aleksnr Bultov A B w α w α w α c w α e f c C Û Û e y y 1 mt -Û Y ; x c Fig.1. Fcsimiles of the stimuli. A: the Müller-Lyer (-), n Oppel-Kunt (e, f) figures;, height of the verticl line;,, lengths of the reference n test prts of the figures, respectively; w, wing length; α, wing tilt ngle. B: superposition of two ifferent ptterns. C: fcsimiles of the Oppel-Kunt figure (), the three-ot stimulus () n the symmetry justment illustrtion (c), where ϕ, θ n, re the tilt ngle n the length of the test n the reference prts of the stimulus, respectively; n X tst n X ref re projections of the test n reference prts on the xis of scisss, X. For the rest of the Xtst Xref x 1 6 BIOMEDICINE Volume 1, Numer 1, Jnury-June 21

5 Distortions of Length Perception in Humn Vision 2. Methos 2.1. Stimuli The experiments were conucte uner computer control, with softwre of our originl esign rrnging the orer of stimuli, presenting them on the monitor, introucing ltertions ccoring to the suject s commn, recoring the suject s responses, n hnling the results. The experiments were crrie out in rk room, so tht the isply frme coul not e iscerne. The sujects viewe the stimuli monoculrly. The right eye ws usully teste irrespective of whether or not it ws the leing eye ut, in some series of experiments, the left eye ws lso exmine. The viewing istnce ws 18 or 4 cm. An rtificil pupil (3mm in imeter) ws use. A chin holer limite movements of the suject s he. In one set of experiments, two prts of stimulus were presente to ifferent eyes of the sme suject. Ech prt ws expose with ifferent frmes, t the rte of 12 Hz, so tht the suject woul see the figures without flickering. To mesure the perceive length istortions, the Müller-Lyer n the Oppel-Kunt figures or just threeot stimuli were generte ginst rk or grey ovlshpe ckgroun y the Cmrige Reserch Systems VSG 2/3 on the monitor EIZO T62 with gmm correction. The grey ckgroun ws of 2 y 2 min rc size. The stimuli ptterns were oriente horizontlly with two prts of the figures lying sie y sie (Fig. 1A) or one elow the other (Fig. 1B). Superposition of the two illusionry figures ws lso performe (Fig. 1B, 1B). For comprison, verticl figures were teste. One prt of the Müller-Lyer figure h inwr-fcing wings while the other h the outwr-fcing wings. The Oppel-Kunt figures h fille n n empty prts. The Oppel-Kunt figures were forme either of stripes (Fig. 1Ae, 1Af, 1B, 1Bc) or ots (1C). Originlly, the Oppel-Kunt figure ws forme of stripes. We hve moifie the figure replcing the stripes y ots. The ppernce of ot oes not vry with rottion: tht is, single element of the stimulus remins unchnge if the orienttion of the two prts of the figure chnges. In our experiments, vrious orienttions of ech prt of the Oppel-Kunt figure n of the three-ot stimulus were use. The with of the lines within the stimuli ws out.3 or.8 min rc, n the imeter of the ots ws out 2 min rc. In the experiments, the length of the reference prt of the stimuli,, n the length of the wings of the Müller-Lyer figure, w, vrie from.1º to 1.º; the tilt ngle of the wings, α, from 1 to 17 ; the numer of the stripes within the fille prt of the Oppel-Kunt figure, n, from 2 to ; the orienttion of the test prt of the Oppel-Kunt figure or the three-ot stimulus, ϕ, from to 36. The reference prt of the stimuli, θ, h four possile orienttions, 9, 18, n 27. The contrst of the vrile memer of the two superpose illusionry figures vrie from 1. to.68; the contrst of the sttic memer ws fixe t 1. or.68. The contrst ws clculte ccoring to the Michelson formul: c = (f ) / (f + ), where f is the stimulus luminnce, n is the ckgroun luminnce. Five grtions of the ckgroun luminnce were present:, 6, 9, 1, n 21 c/m Proceure The sujects estimte the perceive length of the test prt of the figure y justing it to e equl to the length of the reference prt. No instructions concerning gze fixtion point were given. The sujects were provie with three uttons, n instructe to press utton 1 if they wnte to mke the test prt of the figure longer, n utton 2 if they wnte it shorter. A single press vrie the size y one pixel, which correspone to.8 or.3 min rc in our experiments. Auitory fee-ck ws provie. Oservtion time ws unlimite n the sujects mnipulte the figure with uttons 1 n 2 until the esire length equlity ws chieve. Then the suject presse utton 3 to trnsfer the response into computer. At the susequent presenttion, certin prmeter of the reference prt of the figure ws chnge, n the proceure ws repete. During the presenttions, the sujects were ske to isregr the rnom chnges in the reference vlue. The length of the test prt of the figure ws lso rnomize, n the sujects i not know in vnce whether the computer woul mke it longer or shorter n how much ifferent it might e s compre with the length of the reference prt. Two hunre n fifty presenttions were inclue in single experiment, i.e. vlues of ech prmeter were repete five times. The experiment ws performe repetely two or three times uring session. In vrious sessions, ifferent prmeters of the figure were teste Sujects Dt were collecte from 16 sujects (femles n mles). None of the sujects h history of visul isorer. Their visul cuity ws 1. One of the sujects h his vision correcte to norml y 1. D lenses. The otine results were qulittively ienticl for ll the sujects wheres quntittive iniviul ifferences were estlishe. 3. Results In pilot experiments, we introuce isection proceure to test the suject s ility for sptil intervl iscrimintion n to estimte the left-right symmetry vlue. Three verticl lines, ech.8 min rc wie n 28 min rc high, were generte on the monitor. Sujects were ske to equlize the intervls etween the left-hn line n the centre line n the right-hn line n the centre line y pushing one of the flnking lines to the left or to the right. The ifference etween the test n the reference intervls ws plotte ginst the reference intervl length (Fig. 2). Asolute error vlue of the perceive equlity increse with the reference intervl n coul rech 6-8 min rc t the size of 1 min rc. Some of the sujects showe preference for one sie, while the others i not. Volume 1, Numer 1, Jnury-June 21 BIOMEDICINE 7

6 Algis Bertulis, Aleksnr Bultov Reference prt length, min rc Reference prt length, min rc Fig. 2. Bisection s function of the length of the reference prt for two sujects: ER (fille circles) n UL (open circles). The height of verticl lines ws fixe t 28 min rc. Asciss, reference prt length, in min rc; orinte, solute error, in min rc. Continuous lines, moel preictions with vrious gze fixtion points: on the centre of the left prt of the stimulus (upper curve), on the centre of the right prt (lower curve), on the centre of the whole stimulus (mile curve). Fig.. The Müller-Lyer illusion s function of length of the reference prt. Two pirs of wings were present, s in Figure 1A; ws fixe t 7 min rc. Further etils re s in Figure Reference prt length, min rc Reference prt length, min rc Fig. 3. The Müller-Lyer illusion (orinte) s function of length of the reference prt (sciss). Three pirs of wings n the shft line were present s in Fig. 1Ac. w ws fixe t 28 min rc, n α t 4. One cn notice tht the experimentl t for suject ER (fille circles) fit well the upper curve of the moel s preictions, n those for suject UL (open circles) fit the lower curve. This les to the ssumption tht suject ER keeps looking t the left prt of the stimulus while estimting equlity of size, wheres suject UL concentrtes on the right prt. Fig. 6. The Müller-Lyer illusion s function of length of the reference prt. Only one pir of wings ws present, s in Figure1A. Further etils re s in Figures 3 n Reference prt length, min rc Wings tilt ngle, eg Fig. 4. The Müller-Lyer illusion s function of length of the reference prt. The shft line ws sent in the stimulus, s in Figure1A. Further etils re s in Figure 3. Fig. 7. The Müller-Lyer illusion s function of wing tilt ngle. The shft line ws present; ws fixe t 7 min rc, w t 28 min rc. 8 BIOMEDICINE Volume 1, Numer 1, Jnury-June 21

7 Distortions of Length Perception in Humn Vision 3.1. Vritions of sptil prmeters of illusionry figures The t collecte y employment of the Müller-Lyer figures showe errors consierly exceeing those chieve y ppliction of the isection test. Illusion strength vlue might rech 16 to 2 or even 3 min rc t the size of 1 min rc (Fig. 3). Iniviul ifferences were oserve, the illusion strength vlue iffering y fctor of 2 or even of 3 etween the sujects. Removl of the shft line showe some ltertions in the function (Fig. 4): the ifference in illusion strength vlue etween oservers ecrese (compre Fig. 3 n Fig. 4). Removl of the wings wekene the illusion, e.g. figures with two pirs of wings showe smller errors (Fig. ), n those with one pir still smller (Fig. 6). On the whole, ech pir of wings h its own influence on the strength of illusion, n some itivity ws stte. Illusion s function of the tilt ngle α showe curve with positive n negtive vlues (Fig. 7). The curve ws not quite symmetricl ner the zero point. Firstly, for some sujects, the zero point ws shifte from 9 º. Seconly, the solute vlues of negtive n positive t points were slightly ifferent. Removl of the shft line chnge the chrcter of the α function (Fig. 8). The ifference etween the t of the sujects ecrese, n some extreme vlues ppere on the curves. Vritions in wing length, w, mnifeste themselves s reltively flt curves when the shft line ws present (Fig. 9). A tenency to isply pek t lower w vlues (1-3 min rc) coul e suspecte from the evience otine. The peks ecme clerly visile on the w curves when the shft line ws sent (Fig.1). Verticl orienttion of the Müller-Lyer figure prouce illusion of the sme strength s horizontl orienttion (Fig. 11). So, in summry, the Müller-Lyer illusion extreme points chnge their position ccoring to the vlues of, w, n α. The Oppel-Kunt illusion lso increse grully with length of the figure reference prt, (Fig. 12). The Oppel- Kunt illusion s function of the numer of stripes, n, h locl mximum somewhere in etween 7 n 13 stripes (Fig. 13) Superposition of illusionry figures n contrst vritions In these experiments, two figures of the comine pttern h the sme length n coincie precisely, with their ens mtche. Two superpose Müller-Lyer figures iffere in orienttion (y 18 ) n contrst (Fig. 1B). In the experiments, the illusion strength vlue vrie with ltertions of the luminnce mplitue (f ) of the vrile figure in oth situtions s follows: either the sttic figure ws rk, the contrst eing 1 or it ws light, the contrst eing fixe t.4 (Fig. 14). The function h nerly symmetricl shpe with cler-cut mximum t the luminnce mplitue roun c/m 2 of the vrile figure, t which it ctully coul not e seen. The mximum vlue of the length istortion, in this cse, ws out 2 per cent of the reference prt length, thus mtching the verge of the illusion s Wings tilt ngle, eg Fig. 8. The Müller-Lyer illusion s function of wing tilt ngle. The shft line ws sent. Further etils re s in Figure Wing length, min rc Fig. 9. The Müller-Lyer illusion s function of wing length. The shft line ws present; ws fixe t 7 min rc, α t Wing length, min rc Fig. 1. The Müller-Lyer illusion s function of wing length. The shft line ws sent; ws fixe t 7 min rc, α t 4. strength in the experiments with single figures (Fig. 3). If the luminnce mplitue of the vrile figure ws 9 c/m 2 or 9 c/m 2, the illusion ws sent. Consequently, the illusion fe if the solute vlues of the luminnce mplitues of the two figures were the sme, n oth figures were rk, or light, or one of the figures ws s rk s the other ws light. Volume 1, Numer 1, Jnury-June 21 BIOMEDICINE 9

8 Algis Bertulis, Aleksnr Bultov 3 28 A Reference prt length, min rc Numer of stripes Fig.13. The Oppel-Kunt illusion s function of the numer of stripes in the she prt of the figure. ws fixe t 7 min rc, α t 28 min rc. B A Wings tilt ngle, min rc Luminnce mplitue, c/m² Fig. 11. The Müller-Lyer illusion s function of length of the reference prt (A) n wing tilt ngle (B), with the Müller- Lyer figure oriente verticlly. In A, w ws fixe t 28 min rc, α t 4 ; in B, ws fixe t 7 min rc, w t 28 min rc. 1 B Luminnce mplitue, c/m² Reference prt length, min rc Fig. 12. The Oppel-Kunt illusion s function of length of the reference prt. The height of verticl lines forming the stimulus ws fixe t 28 min rc. The numer of stripes in the she prt of the figure ws 1. Fig. 14. The illusion on length s function of the luminnce mplitue of the vrile Müller-Lyer figure. The luminnce mplitue of the sttic Müller-Lyer figure ws 9 c/m 2 n its contrst 1. (A); or 21 c/m 2, n.4 (B). The ckgroun luminnce ws 9 c/m 2. The length of the reference prt of the stimulus ws 6 min rc, the wing length ws 1 min rc, n the tilt ngle of the wings 4º. The sujects were TR (fille symols) n UL (open symols). If the luminnce mplitue ws greter thn 9 c/m 2, the illusion ws reltively wek n h the opposite sign (Fig. 14). In the experiments with superposition of the Oppel-Kunt n the Müller-Lyer figures (Fig. 1B), similr effects were oserve: the experimentl curve ws symmetricl roun the zero vlue of the luminnce mplitue of the vrile Oppel-Kunt figure, n the illusion ecrese to zero if the solute vlues of the vrile Oppel-Kunt figure n the sttic Müller-Lyer figure were equl (Fig. 1). The nlogy etween the t from superposition of figures of vrious types ecme more evient if the illusions 1 BIOMEDICINE Volume 1, Numer 1, Jnury-June 21

9 Distortions of Length Perception in Humn Vision Fig. 1. The illusion on length s function of the luminnce mplitue of the vrile Oppel-Kunt figure. The luminnce mplitue of the sttic Müller-Lyer figure ws 1 c/m 2 n its contrst 1.; the ckgroun luminnce ws 1 c/m 2. Further etils s in Figure Luminnce mplitue, c/m² A Contrst B Contrst Fig. 16. The illusion on length s function of contrst of the vrile Müller-Lyer figure shown ginst the ckgroun of 21 c/m 2 (tringles), 1 c/m 2 (squres), 9 c/m 2 (circles); lso, the illusion s function of contrst of the vrile Oppel- Kunt figure ginst the 1 c/m 2 ckgroun (continuous curve) for oservers TR (A) n UL (B). were exmine s function of the contrst of the vrile figures seen ginst vrious ckgrouns (Fig. 16). The curves were similr in shpe n lmost overlppe wht inictes tht the istortions on length perception o not epen on the contrst of the sttic figure, n o not vry with the level of pttion. Eviently, the Oppel-Kunt n the Müller-Lyer illusions my hve the sme neurophysiologicl sis. In the experiments with superposition of the Oppel-Kunt figure n non-illusionry isection stimulus me of three verticl stripes (Fig. 1Bc), the first figure ws vrile n the ltter ws sttic. The three-stripe figure my e consiere s Müller-Lyer pttern with 9 ngle etween the wings n the shft. In the experiments, the illusion vrie with ltertion of the luminnce mplitue of the illusionry figure n isppere s the mplitue pproche zero (Fig. 17) ut, contrry to the previous t, the length istortions were stronger when oth figures h the sme sign of luminnce mplitues - e.g. ll stripes were light (the right sie of the curve) - n were weker when the figures iffere in luminnce mplitue signs (the left sie of the curve). Proly, the sitution with ifferent signs of the contrst of the stripes refers to the cse with vrile numer of stripes in the fille prt of the Oppel-Kunt figure (Fig. 13) n to the cse in which the contrst rtio for vrious segments of the figure ws chnging [Dworkin, Bross, 1998] The experiments with single Müller-Lyer n Oppel- Kunt figures showe invrince of the illusion strength vlue t stimulus contrsts rging from 1. to.68 (Fig. 18). The results run in prllel with the reporte t [Wickelgren, 196; Long, Murtgh, 1984; Pollck, Jger, 1991; Li, Guo, 1993, 199; Dworkin, Bross, 1998; Spehr, Gillm, 1998]. At superposition of two Müller-Lyer figures, ltertions of length of the wings of the vrile figure yiele the following vritions of the illusion s strength vlue: the illusion ws sent if the ifference of length of the wings etween the two figures equle zero; n the illusion ugmente if the ifference increse (Fig. 19). The curves were lmost symmetricl roun the zero point. At the zero point, the stimulus converte itself into composition of three crosses rrnge in line t equl intervls. Altertion of the wings tilt ngle gve symmetricl curves (Fig. 2). When oth the tilt ngle in the sttic figure, α 1, n the tilt ngle in the vrile figure, α 2, were less or more thn 9, the illusion increse with enlrgement of α 2. When the ifference etween α 2 n α 1 ws equl to 9, the illusion increse with enlrgement of α 2. When α 2 α 1 = 9, the illusion isppere ecuse the sujects sw just three-cross stimulus Perception of illusionry figures through ifferent eyes y the sme suject In seprte presenttions of two illusionry figures (Fig. 1B) to ifferent eyes of the sme person, the sujects reporte the figure with inwr-fcing wings s eing shorter Volume 1, Numer 1, Jnury-June 21 BIOMEDICINE 11

10 Algis Bertulis, Aleksnr Bultov Luminnce mplitue, c/m² Wing length, min rc Fig. 17. The illusion on length s function of the luminnce mplitue in the vrile Oppel-Kunt figure. The luminnce mplitue of the sttic three-stripe figure ws 3 c/m 2 n its contrst.33; the ckgroun luminnce ws 1 c/m 2. Further etils s in Figure 14. Fig. 19. The illusion s function of wing length in the vrile Müller-Lyer figure. Contrst for oth vrile n sttic figures ws 1.; the ckgroun luminnce, 1 c/m 2. Further etils s in Figure 14. A 1 A Luminnce mplitue, c/m² Wing tilt ngle, eg B B Luminnce mplitue, c/m² Wing tilt ngle, eg Fig. 18. The illusion on length s function of the luminnce mplitue of single Müller-Lyer (A) or Oppel-Kunt (B) figure ginst the ckgroun of 6 c/m 2. Further etils s in Figure 14. Fig. 2. The illusion on length s function of wing tilt ngle in the vrile Müller-Lyer figure. The tilt ngle of the sttic figure ws 4º (A) or 13º (B). The contrst of the figures ws 1.; the ckgroun luminnce, 1 c/m 2. thn tht with the outwr-fcing wings. The illusion s strength ws foun to e s lrge s 2 per cent of the reference figure length. It i not chnge with ltertion of ifference of the contrsts of the figures (Fig. 21), s if single figure were present within the stimulus. The strength of the illusion i not vry either with incresing istnce etween the figures (Fig. 1Be), from to 2 min rc (Fig. 22). On the other hn, the illusion s strength grew up with increse of the istnce when these figures were presente to the sme eye (Fig. 22). 12 BIOMEDICINE Volume 1, Numer 1, Jnury-June 21

11 Distortions of Length Perception in Humn Vision Luminnce mplitue, c/m 2 Fig. 21. The illusion on length s function of the luminnce mplitue in the vrile Müller-Lyer figure.vrile n sttic figures were presente to ifferent eyes of the sme oserver. The luminnce mplitue of the sttic figure ws c/m 2 ; the contrst, - 1.; the ckgroun luminnce ws 6 c/m c Distnce, min rc Fig. 22. The illusion on length s function of the istnce etween two Müller-Lyer figures presente together to one eye (circles) or to ifferent eyes of the sme oserver seprtely (tringles). The luminnce mplitue of the figures ws c/m 2 n 2 c/m 2 ; the ckgroun luminnce ws 6 c/m Visul fiel nisotropy n geometricl illusions We ssume tht the mesurements me with the threeot stimulus (Fig. 1C) showe the contriution of the visul fiel nisotropy to perceive length istortions (Fig. 23). Sptil filtering i not interfere significntly in these experiments ecuse the reference n the test prts of the stimulus h ienticl sptil structure. Since four sic orienttions (, 9, 18, n 27 ) of the reference prt of the stimulus were use, four chrcteristics of meriin scles on perceive length were otine. They might e interprete s the ifferentil chrcteristics representing error vlue ltertions epenent on vrition of the tilt ngle of two meriins compre. All the four functions of Figure 23 were clerly ifferent in terms of shpe n mplitues. They illustrte the vriility of meriin scles s function of Fig. 23. Length comprison errors (orinte) s function of the orienttion of the test prt in three-ot stimulus (sciss). Orienttion of the reference prt ws in, 9 in, 18 in c, n 27 in. The length of the reference prt ws 14 min rc. The sujects were TR (open circles) n UL (fille circles). Error vlues were otine y sutrction of the length of the test prt from tht of the reference prt of the stimulus fter the perceive equlity h een estlishe y the oserver. Volume 1, Numer 1, Jnury-June 21 BIOMEDICINE 13

12 Algis Bertulis, Aleksnr Bultov the orienttion of the reference meriin. Thus, for instnce, if the scle of meriin ws tken s the reference (Fig. 23), the scles of the other meriins seeme to e more or less extene. The egree of extension increse with the orienttion to 1 n 18 to 21, ut ecrese with the orienttion 13 to 18 n 22 to 36. Therefore, two regions of mximum error vlues were oserve on the curve (Fig. 23). The regions were efine pproximtely s 1 to 13, n 21 to 22 wie. Their solute error vlues were not equl, thus giving us evience of the meriin scle vriety within the upper n lower prts of the visul fiel. With the scle of 9 meriin serving s the reference one, the scles of the other meriins seeme to e compresse (Fig. 23). Most of the errors h negtive vlues. The error vlue pproche zero within the regions of 9, n 27. This suggests tht the length comprison errors ecme negligile when the internl ngle of the stimulus pproche 18, not only. Quite ifferent functions were oserve if the 18, n 27 orienttions were use for the reference prt of the stimulus (Figs. 23c, ). Their shpes were ifferent, the loction of positive n negtive wves vrie from curve to curve, ut the regions of minimum error vlue correspone to 18 internl ngle. This supporte n ssumption tht the nisotropy effect is miniml in cses when the three-ot stimulus is rrnge long stright line. The experiments with the Oppel-Kunt figure (Fig. 1C) revele comine effect of stimulus orienttion n structure. The two prts of the figure iffere in sptil structure. The chrcteristics of perceive length istortions for the sme sic orienttions of the reference prt were mesure, n four functions were otine (Fig. 24). They iffere from the corresponing functions in Fig. 23 n showe quite specific chrcter of their own. No fetures of symmetry of the functions were foun. There ws some evience tht the internl ngle of 18 etermine the mximl error vlues. To compre the scles of horizontl n verticl meriins of the visul fiel, supplementry series of experiments with the Oppel-Kunt figure of vrying size n open t 18 ngle were crrie out. The four orienttions of the stimulus yiele rther monotonous t (Fig. 2). The length comprison errors increse grully with length of the reference prt of the stimulus. The slopes of the functions n the corresponing error vlues were close to ech other (Fig. 2). The strength of illusion seeme to e nerly the sme with ny of the four orienttions. 4. Discussion The phenomen of geometricl illusions n visul fiel nisotropy inicte tht the scle of the perceptul fiel is ifferent from the scle of the physicl stimulus. The t reporte ove suggest tht the perceive size of n oject is result of compoun interction of oth sptil structure of the oject, n its position n orienttion in the visul fiel. Ech prt of the stimulus unvoily plys Fig. 24. Length comprison errors (orinte) s function of the orienttion of the test prt of the Oppel-Kunt figure (sciss). Further etils s in Figure 22. role in creting istortions of perception. Consequently, ny shpe is enriche y more or less noticele misrepresenttions. c 14 BIOMEDICINE Volume 1, Numer 1, Jnury-June 21

13 Distortions of Length Perception in Humn Vision Length of the reference prt, min rc 4.1. The neurophysiologicl moel of two-imensionl sptil filtering Recent t on shpe perception threshols hve inicte ominnt role of filtering processes of sptil frequencies in perceptul performnce [Ginsurg, 1984, 1986; Fogel, Sgi, 1989; Sgi, 1991; Chen et l., 1993]. Consequently, psychophysicl n neurophysiologicl evience hs een ccumulte tht the visul system in ll the cortex my e crrying out sptil frequency filtering of visul informtion [Kulikowski, Bishop, 1981; De Vlois et l., 1982; Glezer et l., 199]. Neurophysiologicl t suggest tht simple corticl cells my e escrie y prouct of the Gussin envelope n perioic functions [Mrcelj, 198; Kulikowski, Bishop, 1981; Kulikowski et l., 1982]. Therefore, they might e consiere s two-imensionl sptil frequency filters (Gor elements). The weighting function of the Gor element is compute from the eqution 2 1 w(x,y) = Acos(ω x + θ) exp ( - x² / 2σ x ² - y²/2σ y ² ), (1) Length of the reference prt, min rc c Length of the reference prt, min rc Length of the reference prt, min rc where ω is the optiml frequency; θ is the phse shift; σ x,σ y re prmeters etermining chrcteristics of the Gussin envelope in the preferre orienttion n orthogonl to it. In ition, σ y = 2σ x. A is the coefficient of contrst sensitivity which oes not epen on pek frequency [De Vlois et l., 1982]. Accoring to the reporte t [Pollen, Felon, 1979; Glezer et l., 199], the pek frequencies of corticl simple units re rther ifferent n prouce geometricl progression with step of. of n octve. The highest frequency is foun to e out 2 cycles per egree (c/eg). The nwith of the Gor element equls octves [De Vlois et l., 1982] n oes not epen on pek frequency. Therefore the weighting functions of vrious corticl neurons hve out the sme shpe. A single Gor element is tune oth to frequency n orienttion. Their complete set shows continuum of sptil frequency peks n orienttions over reltively wie rnge. Also, the non-linerity of neuronl output chrcteristics shoul not e overlooke in the moelling. In series of stuies of simple n complex corticl units [Movshon et l., 1978; De Vlois et l., 1982; Henry, 198], specific estimtion metho of neuronl response to rifting sine grtings ws pplie. The uthors mesure mplitue n men vlue rtios. If simple cell output ws consiere s hlf-wve rectifier with liner chrcteristics, the rtio shoul e f 1 /f = π/ sin(x)x = π/2 1.7 (2) Fig. 2. Length comprison errors (orinte) s function of the length of the reference prt (sciss) in the Oppel-Kunt figure with internl ngle 18. Orienttion of the reference prt ws in, 9 in, 18 in c, n 27 in. Soli lines re preictions of the filter moel (Bultov et l., 1997) with two gze fixtion positions: t the centre of the reference prt (upper), n t the centre of the test prt of the stimulus (lower). We ssume tht simple neurons hve qurtic rther thn liner output chrcteristics: f 1 /f = π/ sin²(x)x = 2 (3) Volume 1, Numer 1, Jnury-June 21 BIOMEDICINE 1

14 Algis Bertulis, Aleksnr Bultov This ssumption provies proper escription of neuronl output non-linerity n goo corresponence etween the clculte vlues n the experimentl t [Movshon et l., 1978] (Fig. 26). There is nother ssumption in our moel concerning the signl phse. The phse shift in the perioic function of Gor element my prouce significnt phse ltertions of the output signl. To voi it, the nervous system possesses certin mechnisms of conjugte units. The simple cells which re locte close to ech other within the sme corticl lyer n hve overlpping receptive fiels of the sme size, orienttion, n pek frequency re joine in pirs. The weighting functions, w 1 (x) n w 1 (x), of the two memers in couple re conjugte y the Hilert trnsform: w 1 (x) = A(x) cos(ω x + θ), w 2 (x) = A(x) sin(ω x + θ), (4) where A(x) escries the Gussin envelope, ω is optiml frequency, θ is phse shift. The outputs of the two memers converge on the next unit. Chrcteristics of the output unit resemle those of the complex cells: they re tune to sptil frequency n orienttion, n re not selective to stimulus position in the receptive fiel [Glezer et l., 1973, 198]. Therefore the response of the complex unit oes not epen on phse shift in the perioic component of the Gor element: R 12 (x) = [ f(x-ξ) A(ξ)cos(ω ξ + θ)ξ] 2 = [cos(θ)h c (x) - - sin(θ)h s (x)] 2, R 22 (x) = [sin(θ)h c (x) - cos(θ)h s (x)] 2, r(x) = [r 12 (x) + r 22 (x)]. = [h c2 (x) + h s2 (x)]., () where h c (x) = f(x-ξ) A(ξ)cos(ω ξ )ξ, h s (x)= f(x-ξ) A(ξ)sin(ω ξ)ξ (6) The complex units re optimize for extrction of the envelope of signl n weighting function convolution. Inirect evience of this cn e foun in literture [Kulikowski, Bishop,1981]. Filters conjugte y the Hilert trnsform might e constructe y excittory n inhiitory pthwys of neurl network. An lterntive mechnism for elimintion of phse ltertions might e suggeste. A lrge numer of simple corticl units with overlpping receptive fiels of the sme size n orienttion my converge on the output unit. Ech of them is consiere to hve iniviul phse shift of the perioic component. If one suggeste even phse istriution within the rnge to 2π n qurtic output chrcteristic, then the output of the ssemly woul not epen on phse shift of the perioic component of Gor elements: Reltive moultion, f1/f Sptil frequency, c/eg Fig. 26. Reltive moultion (f 1 /f ) in responses of two simple cells to rifting sinusoil grting plotte ginst the sptil frequency of the grting (c/eg). Squres, responses from close-to-liner cell; circles, responses from non-liner cell. (From Movshon et l., 1978). R 2 (x)= θ[ f(x-ξ) A(ξ)cos(ω ξ+ θ)ξ]² = π [ h c2 (x)+ h s2 (x)] (7) One more ssumption of our moel els with contrst elevtion in the corticl excittion ptterns. To escrie this process in the excittion pttern of complex units, we use the two-imensionl Lplcin opertor: 2 f(x,y) = 2 f(x,y)/ x f(x,y)/ y 2 (8) The Lplcin opertor ws pplie to the moelling of retinl excittion ptterns y Kovszny n Joseph [193] n Kelly [1974]. The lst ssumption is concerne with receptive fiel istriution in the retin. In spite of the mgnifiction fctor [Dniel, Witterige, 1961; Gul, Bertulis, 1976; Schwrtz, 198; Dow et l., 1981], the numer of cells per squre unit is constnt ll over the strite cortex [Huel, Wiesel, 1974]. The fovel projection contins mximum numer of units tune to higher frequencies. With eccentricity, the high-frequency representtion iminishes while tht for lower frequencies increses, s oes receptive fiel size. We mke n ssumption tht istriution of fiels of the sme size within the retin my e etermine y lw of χ² type: p i (ξ) = k i (ξ+) n exp { - (ξ+) 2 / 2σ i2 }, (9) where ξ is eccentricity; p i (ξ) is istriution ensity of the fiels with given size; k i is the normliztion coefficient; is the prmeter etermine y the smllest fiel size; n is the power exponent; n σ i = 2.2i. We ssume n = 2 e- 16 BIOMEDICINE Volume 1, Numer 1, Jnury-June 21

15 Distortions of Length Perception in Humn Vision cuse it gives the est correltion etween compute vlues n experimentl finings [De Vlois et l., 1982; Dow et l., 1981] (Fig. 27). The prmeters of the istriution function vry with size of the receptive fiel. Consequently, the set of receptive fiels ifferent in size prouce system of overlpping concentric rings roun the fove. For this reson, we come to the conclusion tht, in sptil frequency terms, the visul fiel is not homogenous system, n tht the prmeters of sptil filtrtion, oth in the visul system n in our moel, epen on () sptil properties of the imge n () its position in the visul fiel. Therefore, vrious prts of the retinl imge re perceive with ifferent ccurcy. Due to ttention n eye movements, the gze fixtion point jumps, n the sptil filtrtion prmeters chnge in vrious prts of the visul scene, which etermines the iniviul evitions of illusion strength vlue, especilly when ig ojects re oserve. One cn imgine two lterntive situtions in the experiments: (i) the left hlf of the figure is locte in the fove n the right one in the prfovel prt, n (ii) the left hlf of figure is situte in the prfove n the right one in the fove. The experiments with geometricl illusions provie cler inictions tht the oserver keeps looking involuntrily either t the left or t the right hlf of the figure (Figs. 3, 4). Therefore non-homogeneity of sptil filtrtion prmeters in the visul fiel my influence illusion strength n its iniviul vritions, s well s my give rise to left-right symmetry. Errors of symmetry re much smller thn those of illusion. The former re cuse y properties of the visul Normlize istriution Pek sptil frequency, c/eg Fig. 27. Distriution of simple cells in the fovel projection s function of pek frequency (c/eg). Squres, t from De Vlois et l., 1982; circles, vlues preicte y our moel (eqution 9). fiel only, wheres the ltter re lso conitione y ifferences in sptil orgniztion of the ojects. To test the hypothesis tht sptil filtering cuses geometricl illusions, we put the ptterns of Müller-Lyer n Oppel- Kunt figures through the moel (Fig. 28). There were five min stges of imge processing in the moel: 1 clcultions of the two-imensionl spectrum of n imge, F(ω x,ω y ); W 1 (ω x,ω y ) = F(ω x,ω y ) H 1 (ω x,ω y ) g 1 (x,y) = s 1 (x,y) P 1 (x,y) s 1 (x,y) = Φ 1 W 1 F(ω x,ω y ) i f ( x, y) = gi ( x, y) i 2 f (x,y) s n (x,y) = Φ 1 W n W n (ω x,ω y ) = F(ω x,ω y ) H n (ω x,ω y ) g n (x,y) = s n (x,y) P n (x,y) Fig. 28. The min scheme of the moel. F(ω x,ω y ) is the input signl spectrum; H i (ω x,ω y ) re the spectrl chrcteristics of simple receptive fiels of i-th size; Φ -1 is complex inverse Fourier trnsform; P i (x,y) is istriution function for the fiels of i-th size; n 2 is Lplcin opertor. Volume 1, Numer 1, Jnury-June 21 BIOMEDICINE 17

16 Algis Bertulis, Aleksnr Bultov 2 clcultions of proucts of the two-imensionl spectrum, F(ω x,ω ), n the spectrl chrcteristics, H (ω,ω ), of y i x y the filters of vrious size conjugte y the Hilert trnsform; 3 inverse Fourier trnsform, Φ -1, of the proucts, F(ω x,ω ) y H i (ω x,ω ); y 4 summtion of the moules of the results of Φ -1 using weighting functions, P i (x,y); ppliction of the two-imensionl Lplce opertor, 2. In stge 1, the two-imensionl spectrum of the Müller- Lyer figure ws clculte y the following eqution: ML(Ω,Ψ,α,w) = K(Ω,Ψ,α,w)exp[iΩcos(Ψ)] + + K(Ω,Ψ,18 - α,w) + K(Ω,Ψ,α,w) exp[ - iωcos(ψ)], (1) where K(Ω,Ψ,α,w) is the spectrum of single rrow of the figure with wing tilt ngle α n wing length w; Ω n Ψ re polr co-orintes; n re the lengths of the reference n the test hlves of the figure, respectively. 2 sin[ωcos(ψ- α)w/2] K(Ω,Ψ,α,w) = exp[- iωcos(ψ - α)w/2] + Ωcos(Ψ α) 2 sin[ωcos(ψ+α)w/2] + exp[- iωcos(ψ+α)w/2] (11) Ωcos(Ψ+α) is The two-imensionl spectrum of the Oppel-Kunt figure 2sin[Ωsin(Ψ)/2] OK(Ω,Ψ, n)=r(ω,ψ, n)+ exp[- iωcos(ψ)(+/2)], Ωsin(Ψ) (12) where R(Ω,Ψ, n) is the spectrum of the grting locte t the co-orinte system centre n consisting of n stripes; is the height of the stripes; is the length of the she prt, n is the length of the test prt. 2sin[Ωsin(Ψ)/2]sin[Ωcos(Ψ)n(n - 1 ) -1 ] R(Ω,Ψ, n)= Ω sin(ψ)sin[ωcos(ψ)(n-1) -1 ] (13) At stge 2, the spectrl chrcteristics of conjugte filters re expresse vi two-imensionl Gussins of certin orienttions shifte with respect to the optiml frequencies. The optiml frequency n the prmeters for ny of the Gussins re etermine y the size of the simple corticl cells. At stge 3, the inverse Fourier trnsform is crrie out. The output ptterns of stges 4 n of the moel were isplye on the monitor. Some of the ptterns re shown in Figure 29. In ll the cses, the output ptterns contine istortions equivlent to the illusions oserve in the experiments (Fig. 2 to Figs. 1, 12, 13). In other wors, the compute size istortions were foun to e in goo greement with the experimentl t. Iniviul vritions etween the sujects were reprouce y the moel s well Fctors etermining illusions Our experimentl n moelling t furnish grouns to consier the physiologicl processes of two-imensionl sptil filtering s the primry fctor in so-clle visul illusions of extent. While eling with the tsk of jugement on length, the retinl pttern of n oject is processe s whole, n the perceive size is result of comine interction mongst sptil prmeters of ll prts of the oject. From this point of view, there is no conflict etween the filtering hypothesis n the theories n concepts, such s confluxion n contrst, ssimiltion, pttion, frming rtio, centre of grvity or verging of weights, size constncy, n orientte line etectors. The theories n concepts cope with the filter moel n flow together in the moel, which yiels n explntion of the illusions in neurophysiologicl n mthemticl terms. The present finings o not suggest tht it might e superfluous to invoke some other cusl mechnisms of the illusions. On the contrry, our experimentl n moelling evience o show tht it is eye movements tht etermine the etween-sujects vriility of illusion strength vlue, especilly when lrge ojects re oserve. Apprently, pttion properties of the sptil filters, visul fiel nisotropy, ttention, n memory might ct itionl fctors or corrections in perceive istortions. Even lurring of the retinl imge ue to opticl errtions of the eye exerts quntittive influence on certin visul illusions [Ching, 1968; Coren, 1969]. But reuction of the errtions in these experiments oes not ecrese the Poggenorff illusion t nticipte proportion. In generl, two types of fctors my e in effect: physiologicl properties of low-level visul pthwys n psychologicl events t higher centres of the nervous system. The numer n weight of the fctors my vry from illusion to illusion. The very existence of n evience on so mny types of illusions [Gregory, 197, 199] inicte tht ny level of the visul system my contriute to perceptul istortions. We hve pplie the filtering moel to the two illusionry figures, n the iscussion on misperceptions of other type, such s illusion of orienttion, curvture or re, is eyon the scope of the present communiction. On the other hn, there is evience tht filtering processes re involve in these cses s well. For instnce, the mgnitue of vrious illusions of size n re, such s the Deloeuf [Ike, Ooni, 19] n the Titchner (Zigler, 196) illusions, epens on the rtio of contextul size to focl stimulus size. Inverte U-shpe functions hve een foun when 18 BIOMEDICINE Volume 1, Numer 1, Jnury-June 21

17 Distortions of Length Perception in Humn Vision A C X X X X X X X X min rc min rc B D X X X X X X X X min rc Fig. 29. The output ptterns of the moel t stges 4 (top) n (mile), n the normlize trnsverse sections of the ptterns on xes X-X (ottom) of the Müller-Lyer figures (A, B, min rc C), n the Oppel-Kunt figures (D, E, F) with vrious gze fixtion points: on the centre of the left prt of the figure (A, D), on the centre of the figure (B, E), n on the centre of the right prt Volume 1, Numer 1, Jnury-June 21 BIOMEDICINE 19

18 Algis Bertulis, Aleksnr Bultov X X X min rc E F X X X contextul size ws vrie for the Einghus [Zigler, 196] n Deloeuf [Ike, Ooni 19] figures s like s in the Müller-Lyer [Levis, 198], Ponzo [Fisher, 1969; Pressey et l., 1971] n the ivie line [Ooni, 194] illusions of extent. It hs een reporte recently [Gutusks et l., 2] tht preictions se on the two-imensionl filtering moel fit well the experimentl t on illusions of curvture. In ition, the Knizs tringle ws teste y mens of two-imensionl sptil filtering [Ginsurg,197]. The results otine suggest tht ttenution of the low sptil frequencies in the visul system contriute to the formtion of the illusory tringle. However, Ginsurg usee the iel lowpss filter with 16 sptil frequencies. The vriility of filtering chrcteristics of the visul pthwys ws not consiere, wht might hve cuse the filure to explin other sujective contour ptterns y filtering. A possile neurl mechnism for illusionry contour perception ws suggeste [Peterhns, von er Heyt, 1991] employing the en-stoppe cells. It seems to e resonle then to evelop moel of sptil filtering which woul possess the properties of the en-stoppe cells. Furthermore, short ltency of the illusory contour responses [von er Heyt, Peterhns, 1989], orienttion-specific pttion, n isppernce of nomlous contours t equiluminnce [Brigner, Gllgher, 1974; Gregory, 1987] seem to confirm the low-level mechnism. The low-level locliztion of the mechnism is lso supporte y the t presente in Figures 14, 21 n 22 which show tht the strength of illusion vries with ltertions of () the ifference in luminnce contrsts, n () the istnce etween the two figures when oth re presente to one eye; ut the strength of illusion remins constnt when figures re shown to ifferent eyes of the sme suject. Such results suggest tht the illusionry effect might rise in the monoculr retino-corticl pthwys in ech eye seprtely Liner summtion of the effects of illusion n nisotropy The functions in Figure 2 might e recognize s pure chrcteristics of sptil filtering. At first pproximtion, they might e expresse s: X X M = k (14) min rc of the figure (C, F). The tilt ngle of the wings in the Müller-Lyer figure ws 6, n the length of the wings ws 24 min rc. The numer of stripes in the fille prt of the Oppel-Kunt figure ws 8. where M is error vlue, is length of the reference prt, n k is coefficient etermine y the position of gze fixtion point. If the point is t the centre of the reference prt, k = k ref ~.24, n if it is t the centre of the stimulus, k = k test ~.13. The k vlues epen on the prmeters of the neurophysiologicl moel of sptil filtering esigne to stuy the t on the Müller-Lyer n the Oppel-Kunt illusions. When the orienttion n the size of the internl ngle of the Oppel-Kunt stimulus vry in the experiments (Fig. 24), pure chrcteristics of sptil filtering cnnot e oserve irectly in the mesurement t ecuse the effect of orienttion nisotropy s to the effect of the illusion. Even though the unerlying mechnism of their interction is not 2 BIOMEDICINE Volume 1, Numer 1, Jnury-June 21

19 Distortions of Length Perception in Humn Vision known yet, one cn ssume tht the two effects re inepenent from ech other n represent seprte neurl processes in the retinl-corticl pthwys. We checke this ssumption. The t on the three-ot stimulus (Fig. 23) were sutrcte from those on the Oppel-Kunt figure (Fig. 24). The ifference (Fig. 3) might e consiere s the chrcteristics of the illusion if oth phenomen form n itive function n no complicte eformtions of the processes occur when they mnifest together. At first sight, the four chrcteristics in Figure 3 my seem to e quite ifferent. Uner closer scrutiny, however, they o iffer in the phse shift only. If the phse shift is eliminte, the chrcteristics coincie, n the curves correspon to ech other. Therefore, just one single function remins (Fig. 31). It exhiits symmetry n hs the mximum error vlue t the mi-point, which correspons to 18 internl ngle. The error vlue ecreses to zero on oth sies of the mximum, with the internl ngle incresing or ecresing. The function in Fig. 31 resemles sinusoi in shpe. All these t le to conclusion tht the Figure 31 curve represents pure chrcteristics of the Oppel-Kunt illusion t vrious orienttions n mgnitues of the internl ngle of the stimulus. One cn see tht the illusion strength vries with ngle ut not with orienttion of the min xis (isector) of the stimulus. Such result strongly supports the ie of two inepenent itive processes in istortions of visul perception. The ifferences etween the t in Figure 23 n in Figure 24 might e represente s follows: m e [ f ( ϕ, θ ) g( ϕ, )] ( ϕ, θ ) = θ, (1) c where ϕ n θ re tilt ngles of the test n the reference prts respectively; f(ϕ,θ) re the experimentl t with the Oppel-Kunt figure; g(ϕ,θ) re the t with the three-ot stimulus; m e (ϕ,θ) is the chrcteristic of the illusion per se n its contriution to the length istortions. In orer to check the ientity of m e (ϕ,θ) s pure chrcteristic of sptil filtering, two moels were pplie to the t of Figures 3 n 31: the filter moel n n empiricl one. The length istortions preicte y the filter moel for vrious orienttions n mgnitues of the internl ngle of the Oppel-Kunt figure were foun to e quite close to those otine experimentlly (Figs. 3, 31). A smple of the output ptterns of the computtionl moel is given in Figure 32. The empiricl moel ws se on n ssumption tht the tsk of length comprison is performe y the sujects through estimtion of the symmetry of the stimulus. To evelop the moel, we introuce n imginry system of orthogonl coorintes. The system ws interrelte with the stimulus in such wy tht y-xis woul coincie with the isector of the stimulus (Fig. 1Cc). The projections of the two prts of the stimulus on the coorintes were prouce. The projections of the reference n test prts on the x-xis were esignte s X ref n X tst, n those on the y-xis s Y. Oppel-Kunt illusion strength, min rc Fig. 3. Differences etween the t in Fig. 23 n the t in Fig. 24 for oservers UL (fille circles) n TR (open circles). It is ssume tht these errors re only etermine y pure Oppel-Kunt illusion. Dshe curves n crosses re pproximtions y the lestsqures metho for sujects UL n TR, respectively. Soli curves re preictions of the filter moel with upper curve stning for gze fixtion point on the reference prt (k ref ) n lower curve, for gze fixtion point on the test prt (k test ) of the stimulus. Further etils s in Figure 23. Volume 1, Numer 1, Jnury-June 21 BIOMEDICINE 21

20 Algis Bertulis, Aleksnr Bultov Oppel-Kunt illusion strength, min rc c Fig. 32. The output ptterns of the filter moel for the Oppel- Kunt figure with vrious orienttions of the test prt: 3º in, 8º in, 13º in c n 18º in. There were 1 ots in the fille prt of the Oppel-Kunt figure. Fig. 31. Length comprison errors s function of orienttion of the test prt in the Oppel-Kunt figure. Dt in were otine y elimintion of phse shifts present in Figure 3. Orienttion of the reference prt ws (circles), 9 (squres), 18 (tringles), n 27 (imons). Open symols, t for suject TR; fille symols, t for suject UL. Averge t re shown in. Dshe curve n crosses re pproximtions y the lest-squres metho for sujects UL n TR, respectively. Soli lines re preictions of the moel. X ref n X tst were seprte in spce, wheres the two Y projections coincie. Therefore, the symmetry of X projections coul e estimte, n the coincience of Y projections coul e checke. The symmetry n coincience were present when the two prts of the stimulus were equl in length. Furthermore, nisotropy n filtering effects olish the length equlity n ernge the equilirium of the projections. Let us suppose tht the effect of nisotropy is eliminte, wheres the filtering effect is mintine. In such cse, the symmetry of the X projections is ernge y istortion on the reference sie ( X) ue to illusionry elongtion of the reference prt n increse of the internl ngle of the stimulus. To reset the symmetry of X projections, the length of the test prt hs to e increse y m t, which les imlnce mongst Y projections. To restore the lnce of Y, the coorintes hve to e rotte into position X n Y. Mnipultions with m t n the rottion ngle of the coorintes re-estlish the symmetry of the illusionry stimulus n give rise to length equlity errors. This prticulr moel of error vlues m t s function of the internl ngle of the stimulus yiels the curves very similr to those preicte y the filter moel (Figs. 3, 31). The curves simply coincie. Such similrity is preictle since the clcultions re se on moifiction of eqution (14), the ltter eing consiere chrcteristic of sptil filtering. The moifiction eqution is s follows: ( X + X ) l = l cos ( ϕ θ / 2) + ( 1+ k) sin ( ϕ / 2) 1, 2 mt ( ϕ, θ ) = Y + ref θ (16) where X = kx ref. A comprison of theoreticl pproximtions, m t (ϕ,θ), n the experimentl t, m e (ϕ,θ), ws fulfille y estimting two prmeters, the cross-correltion coefficient, r, n the mplitue coefficient, k ppr, which is erive from the pproximtion to m e (ϕ,θ) y the lest-squres metho. The results of estimtion otine y two sujects (UL n TR) show tht r excees the vlue.8 significntly, n k ppr stisfies isprity k test <k ppr <k ref (Tle 1). 22 BIOMEDICINE Volume 1, Numer 1, Jnury-June 21