Perceptin & Psychphysics 976, Vl. 20 (6), 438-444 Mdel fr three-dimensinl pticl illusin M. E. JERNGAN nd M. EDEN Reserch LbrtryfElectrnics Msschusetts nstituteftechnlgy Cmbridge, Msschusetts 0239 A hmgeneus crdinte system is used t describe the trnsfrmtin frm rel threedimensinl stimulus t n illusry three-dimensinl perceptul bject. The mdel cmprises series f trnsfrmtins f hich ne cts s n illusin pertr. The illusin pertr is specified by single prmeter hse vlue determines hether the rel r the illusry bject is perceived. An experiment t test ne predictin derived frm the mdel s perfrmed. The results cnfirm the predictin. Humn mnculr depth perceptin is cmplex functin f the nture f the stimulus nd the previus experience f the bserver. Vrius cues nd subjective expecttins re emplyed t cnstruct three-dimensinl percept frm the tdimensinl imge hich is the pticl prjectin f the bject n the retin. Cnceptully, the prblem cn be structured s in Figure. nitilly, the bject in three-spce is mpped t t-dimensinl imge by perspective prjectin (H p ). Such trnsfrmtin is mst cnveniently represented in hmgeneus crdinte system, derived frm the ell-knn thery f prjective gemetry, mst recently discussed by Rberts (963) nd Dud nd Hrt (973). Such system emplys furth crdinte s scle fctr. Pints in three-spce re represented by fur-element vectrs: (X,Y,Z) = (x,y.z, ) = V X = x/, y = y/, Z = z/ An bject is represented s set f pints (Vi] nd is trnsfrmed by pstmultiplying by 4 x 4 mtrix H. The perceptul mdel cnsists f sequence f pprprite trnsfrmtins tht mp the rel bject t the perceived bject. The secnd trnsfrmtin represents the influence f mnculr depth cues nd cgnitive expecttin in recnstructing the depth crdinte lst in the three-spce t t-spce mpping. Exmples f mnculr depth cues include reltive size, interpsitin, perspective, texture grdients, etc. Cgnitive expecttin represents the effect f the bserver's pst experience n the frmtin f the percept. Finlly, percept cnstructin pertr (Hpc) frmultes three-dimensinl percept cnsistent ith the t-dimensinl imge nd the reltive depth cnstrints. Three-dimensinl pticl illusins ccur hen the percept differs in its gemetricl descriptin frm the bject under bservtin. n the mdel, the illusin is represented by mdifictin nly f the reltive depth cnstrint pertr, H x Tht is, the gemetricl descriptin is cmpletely cnsistent ith the perspective trnsfrmtin. The frms f the 4 x 4 trnsfrmtin mtrices re best reveled by cnsidering the mpping f pint in rel spce t pint in percept spce. Cnsider the pint (x y z ) in the crdinte system f Figure 2. The perspective prjectin H p mps (x y z) t pint n the y-z plne (the picture plne) frm the prjectin pint D n the x xis (the bserver's eye). The perspective prjectin mtrix is given by: PERSPECTVE PROJECTON 3-D REAL SPACE ----i... Hp (xyzl) 2-D MAGE (yz) MONOCULAR DEPTH CUES COGNTVE EXPECTATON Z-D MAGE RELATVE DEPTH (xyz) PERCEPT CONSTRUCTON Figure. Cnc:eptul mdel fr mnc:ulr depth perc:eptin. 438
3 D OPTCAL LLUSON 439 / // (',Y,Z) es [. y z ] Figure 2. Crdinte system. 6 -/D This trnsfrmtin pplied t (x y z) results in the t-dimensinl imge: [x y z ][Hp] = [x y z ( - x/d)] The crdintes in the picture plne re given by y/(l - x/d) nd z/(l - x/d) nd cn esily be verified gemetriclly. The x crdinte hs n mening t this pint since the reltive depth cnstrint represented by H, hs nt been pplied. H, hs the frm: Applying this trnsfrmtin e btin [x y Z ( - x/d)]. N the x term cntins the reltive depth cnstrint. All tht remins is t cnstruct the percept ith n inverse perspective prjectin. The hmgeneus crdinte vectr fr the pint in percept spce is then: [xy Z l][hp][hx][hp-l] = x y z[ + ( - /D l)x/d] Nte tht, fr =, percept nd bject pints re identicl, implying n ccurte perceptul recnstructin. llusins result frm the interesting cses hen cgnitive expecttins cn prduce n #= hich is nt t incnsistent ith the mnculr depth cues. T clrify the prceeding discussin, cnsider n exmple. Cnstruct three-dimensinl bject frm three identicl flt shpes such s tht in Figure 3. Nte tht ech shpe hs nly ne right-ngle crner. Let the right-ngle crners f the shpes be jined t frm right-ngle crner in three dimensins, ith the btuse ngles prjecting trd the bserver. Figure 3b shs t-dimensinl gemetric prjectin f the bject n plne perpendiculr t the subject's line f sight. Vieing mnculrly, n bserver cn esily chieve n illusry perceptin in hich the bject ppers t invert ith the center crner ppering clsest t him rther thn furthermst. Fr ech bject fce dignl, d, there is unique distnce D here the prjectin f the_ rel cncve bject nd the prjectin f n externl (cnvex) cube re identicl. ndeed, regrdless f the mthemticlthery used t describe the trnsfrmtin, the prjectin f the test bject is cnsistent ith cube nly t the distnce D. Tht this is true cn esily be seen gemetricllyin Figure 4. Fr the O. FACE OF OBJECT 2 ---e--- 3 './ 4 b. PROJECTON ON PLANE PERPENDCULAR TO SUBJECT'S LNE OF SGHT. Figure 3. () Fce f bject. (b) Prjectin n plne perpendiculr t subject's line f sight.
440 JERNGAN AND EDEN LLUSORY x = D. PERSPECTVE PROJECTON OBJECT C D The cscde [Hp][Hx][Hp- ], representing the illusry perceptin pertr, hs thus been specified. EXPERMENT The simplest test f the mdel is the determintin f the distnce t hich the illusin ppers t be cube. The mdel predicts distnce, D, s functin f d, the fce dignl, nd e, the fce edge, given by _ VTd/e - vm De - VT - die b. CROSS-SECTON. x-z PLANE Figure 4. () Perspective prjectin. (b)crss-sectin, x-z plne. rel bject nd the illusry cube t be cnsistent the pint V4' the lest pint f the rel bject: nd the crrespnding pintfthe illusry cube must be cliner. The line determined by the t pints inter~ects the x xis t nly ne pint, specifying the distnce D. The bject cn be represented by n pprprite se~ f pints, such s the seven cmers (see Figure 4), ith ech pint r in n bject mtrix (7 x 4 in this cse). Fr this clss f bjects, then, the reltive depth cnstrint reflecting n illusry perceptin cn be derived by requiring tht the percept crrespnding t the bject [0] be cube ith mutully perpendiculr nd equl length edges. Since the rel bject cntins cncve right-ngle crner t V hich is. mpped under the illusin int cnvex right-ngle crner, the edge fthe perceived cube ill be ssumed t be equl in length t e, the length f the t edges hich mke up the right ngle f the three shpes hich frm the stimulus. This ssumptin is equivlent t ssuming tht the three crners, V., Vl' nd v 3, crrespnding t pints, 2, nd 3 f Figure 3b re invrint under the illusin trnsfrmtin. These three pints lie in the y-z plne f Figure 4, hich is ssumed t be the picture plne. Pints physiclly n the picture plne must be invrint under the illusin trnsfrmtin if the perspective prjectins re t be cnsistent. Applying the perspective prjectin, illusin pertr nd inverse perspective prjectin t the stimulus [0] must then prduce the cube [C), f edge 3. We hve: Figure is plt f D vs. d in units f e. Six Bstn re grdute students ere pid t ct s subjects fr the experiment. All ere in their mid-20s nd hd' nrml r fully crrected visin. Nine bjects, vrying in the length f the dignl f the three plnr shpes (d in Figure 3) ere cnstructed (d = 3.3, 3.4,..., 4. in.; e = 3.0 in.). Object fces ere sectins f Frmic, hite ith light green unstructured pttern. The bjects ere munted n stnd hich culd be mved lng n pticl bench by mens f string, pulley, nd crnk rrngement. By turning the crnk, the subject culd psitin the bject nyhere in rnge frm lmst tuching his hed t 48 in. y. Ech subject s presented ech bject times. The rder f presenttin f the bjects s rndmized. The subject's tsk s t btin the illusin nd psitin the bject t the pint (distnce lng the visul xis) here it lked mst like cube. The results re pltted in Figure 6, shing men "- '" 3 2 20 [C) is 7 x 4 mtrix hse rs crrespnd t the seven crners f n externl cube. This cnstrint yields -d =----- 2d -..;-'e..67.233 3 33 2.267.333 de Figure. Die s functin f die.
40 3 2 '" "- C 20 COMBNED DATA -SX SUBJECTS PLOT MEAN ± 3-D OPTCAL LLUSON 44 smller thn expected stndrd devitin fr lrge vlues f D my be prtly ttributed t limittins in the experimentl setup. Subjects ere re f the mximum distnce setting nd their judgments my hve been effected by such knledge. n serching fr the crrect psitin, subjects felt the bject cme up ginst the end f the pticl bench. A lnger bench uld hve lled greter vritin. EXPERMENT 2 A secnd clss f bjects, similr t the first clss, but differing in t spects, hs been subjected t the sme experimentl test. n this cse, the fces hve dignl d > V2e, ith ech 2 67 233 3 367 33 2.267.333 die Figure 6. Dt. psitin nd indicting stndrd devitin fr ech bject. An lternte presenttin f the results is given in Figure 7, here the experimentl vlues re pltted ginst the predicted vlues f the distnce D. A liner regressin nlysis results in the equtin fr the line f best fit: Y =.9837 X +.436. The crreltin cefficient is.9899. The results re in generl greement ith the mdel. Vritins frm the predictin re primrily due t the difficulty f the tsk. The judgment f cubeness is itself subject t vritin, prticulrly t greter distnces frm the subject here reltively lrge chnge in D results in reltively smll chnge in the prjectin. Fr smller d (3.3 t 3. in.), the bject is firly clse t the subject nd the illusin is difficult t mintin. One expects tht the subject's bility t judge gemetricl reltinships beteen bject fce edges uld be relted t the vritin in his judgment f cubeness s reflected by the stndrd devitin f his estimte f D. A strightfrrd gemetricl clcultin yields the ffset (reltive t D) necessry t btin Yz 0 chnge in the ngle f the edge prjectin. Figure 8 shs the result ith d representing the ffset. Superimpsed re dt pints indicting stndrd devitin. There ppers t be sme evidence tht the subjective criterin f cubeness my be relted t judgment f prllelism fr the fce edges. The..J «!z 20 ::!i ii: ~ <, '" C 2 8 8 6 4 2 20 2 De PREDCTED Figure 7. Plt f experimentl vs, predicted vlue f Die fr Experiment. The line f best fit is given by Y =.9837 X +.436, nd the crreltin cefficient is r =.9899. O' GROUP STANDARD DEVATON "------'---'---_--'- "--_--'---_--'-_-----.JL-_-'--- 20 2 3 D Figure 8. d, the distnce frm the pint f cube prjectin (D) tht the bject must be mved t btin ljz chnge in the ngle f the right-side-edge prjectin in the picture plne. d': trd bserver; d-: y frm bserver. 8-8+
442 JERNGAN AND EDEN mpping is mny-t-ne; hy shuld ne prticulr percept result frm illusry perceptin? We hve derived n illusin pertr tht seems t pply t the pint here the bject prjects under perspective s cube. Des the sme pertr pply t ther distnces frm the bserver? At the pint f cube prjectin, ne might rgue tht cgnitiveexpecttin-cubes being verlerned bjects-cnstrins the perceptul inversin f the bject t cnstruct 4 40... e Figure 9. n this cse, tbe bject s frmed frm tbree identicl fces S befre; bever, sides nd 2 re cmmn t ecb f t fces, nd A is tbe center vertex cmmn t ll tbree fces. externl fce ngle being right ngle. n dditin, the fces re jined ith the cute ngle meeting t the internl vertex, resulting in the fces nt being perpendiculr t ech ther s s the cse fr the first bjects. Figure 9 shs the bject fce. The cmputed distnce f cube prjectin is: d/3e + [(V2die - )2-3)/2 Die = -" die - V2 ith reltive depth cnstrint f: -d/v'3e [(v'2die - )2-3)/2 Figure shs dt frm three subjects tken in mnner similr t tht fr the first clss f bjects. The cmputed distnce f cube prjectin is ls pltted. A plt f experimentl vs. predicted vlue f the distnce D is shn in Figure. The line f best fit is given by Y =.823 X +.4, nd the crreltin cefficient is.9893. As in the first cse, the dt grees resnbly ell ith the cmputed vlues f the pint f cube prjectin. The dded inclintin beteen bject fces fr the secnd grup my cntribute t the difficulty f mintining the illusin. This uld be reflected by less relible cube judgments nd my ccunt fr the greter vritin in the results f the secnd grup. The effect f the limited length.f the pticl bench is gin evident in the bject hving lrge vlue fd. As prelude t discussing further experiments, cnsider fe bservtins n the nture f these prticulr illusins. The perspective prjectin.. " 3 2 20..J ~ 20 ~ il: ~ ~ 2.437.483.4.46.4.84 de Figure. Dt..627.73~.677 20 2 3 Ole PREDCTED Figure. Plt f experimentl vs, predicted vlue f Ole fr Experiment 2. Tbe line f best fit is given by Y =.823 X +.4, nd tbe crreltincefficient is r =.9893.
3-D OPTCALLLUSON 443 cube. At ll ther distnces, the illusry ppernce hs n such precise cgnitive expecttin t supprt it. n the mdel, the fct tht unique illusry percept is btined is reflected by the unique vlue f, the reltive depth cnstrint, fr given test bject. At distnces ther thn D, the vlue f crrespnds t cnstrint f minimum perceptul distrtin f reltive depth. Of ll pssible threedimensinl bjects cnsistent ith the perspective prjectin, the ne perceived is tht clsest t the expected cube. The uniqueness f in the pertr cn be tested s flls. The illusin ppers t be edge dminnt, i.e., independent f surfce fetures f the three fces. By inscribing qudrilterl n surfce tht the illusin pertr predicts ill pper rectngulr t sme distnce ther thn D, the cube pint, nd presenting the tsk t n bserver, the illusin pertr cn be verified. An imprtnt chrcteristic f the mdel is the plne f inversin. This cn be tested by sking the subject t estimte the distnce f vrius pints n the bject. n keeping ith ur minimum perceptul distrtin cnstrint, e expect pints vt v2, V3, the three crners hich lie in the picture plne (see Figure 4), t be invrint under the trnsfrmtin, nd the subject t be nerly crrect in plcing mrker in the plne f v tt V2, nd V3. Unfrtuntely, this test hs nt yet been perfrmed in ny rigrus fshin, lthugh infrml tests re encurging. Similr experiments ill determine the vlue f, the reltive depth cnstrint s functin f the bslute distnce D f the bject. The mdel yields n xp crdinte fr percept spce f: Slving fr e btin: x x p = + ( - ) x/d = x p (D - x) x(d - xp) This my be empiriclly checked by sking the bserver fr depth estimtes ith suitble indictr rrngement. x-z D -x p -Z p x p Z p - tn 8 p tn 8 Figure 2. nterprettin f the prmeter. A cmprisn f the reltive depth cnstrints fr the t clsses f bjects cnstructed frm the shpes f Figures 3 nd 8 revels simple gemetricl reltinship cnnecting rel bject nd illusry bject thrugh. Cnsider Figure 2 nd the bsic trnsfrmtin n percept spce: nd Nte nd (x yz )... [x y z (l + ( - ) x/d)] x x -------- p - + ( - ') x/d z zp = + (0- - xp/zp = x/z ) x/d xp/zp tn 9p 0'=--=-- x/z tn 9 n the lst expressin, the ngles 9 nd 9p re the ngles beteen the vectr defining the pints in rel OBJECT - H p H x H p " COGNTVE EXPECTATON Figure 3. Blck digrm f the perceptul prcess f the three-dimensinl illusin. PERCEPT
444 JERNGAN AND EDEN spce nd perceptul spce, respectively, nd the picture plne. The perceptul inversin tht results in the illusin crrespnds t flding thrugh the plne f inversin defined by the picture plne, here the prmeter determines the reltinship beteen the ngle in percept spce nd tht in rel spce. The dditin f feedbck pth t Figure representing the effects f expecttin is shn in Figure 2. Tht cgnitive expecttin is perful fctr in determining H x, the reltive depth pertr, is cler frm nting tht the illusin cn be held in spite f cntrdictry mnculr depth cues. Mre significntly, mtin prllx cn be ignred, ith the bserver ccepting highly unusul bject mvement in respnse t his n hed mvement hile retining the illusry percept. An interesting explrtin f the cgnitive expecttin cncept uld invlve reversing bject nd percept. Tht is, present the bserver ith perfect externl cube nd sk him t btin cncve nncube illusin. One might cmpre ese f inversin nd frequency f illusry percept s mesures f the reltive cgnitive expecttin f cnvex cubes nd cncve nncubes. The effect f cncve vs. cnvex culd be explred by presenting cnvex nncube nd ttempting t elicit cncve cube, illusry percept. n summry, e hve mde sme specultins cncerning the trnsfrmtin frm rel-rld stimulus t n illusry percept. The mthemtics f hmgeneus crdintes is useful frmerk fr describing the perspective prjectin nd prvides n elegnt descriptin f the illusin pertr specified by single prmeter. One predictin hich is esily derived is the distnce t hich rel bject, cnstructed frm nnsqure shpes, shuld pper s three-dimensinl cube under the illusin trnsfrmtin. The experimentl dt tend t cnfirm the predictin. REFERENCES DUDA, R. 0., & HART, P. E. Pttern clssifictin nd scene nlysis. Ne Yrk: Wiley, 973. ROBERTS, L. Mchine perceptin f three dimensinl slids. n J. T. Tippett et. (Eds.), Opticl nd electr-pticl infrmtin prcessing. Cmbridge, Mss: M.LT. Press, 963. (Received fr publictin September 9, 97; revisin ccepted August 2, 976.)