Multiresolution wavelet framework models brightness induction effects

Transcription

1 Aville online t Vision Reserch xxx (28) xxx xxx Multiresolution wvelet frmework models rightness induction effects Xvier Otzu *, Mri Vnrell, C. Alejndro Párrg Computer Vision Center/Computer Science Deprtment, Universitt Autònom de Brcelon, Cmpus UAB, Cerdnyol del Vllès, 8193 Brcelon, Spin Received 1 My 27; received in revised form 4 Decemer 27 Astrct A new multiresolution wvelet model is presented here, which ccounts for rightness ssimiltion nd contrst effects in unified frmework, nd includes known psychophysicl nd physiologicl ttriutes of the primte visul system (such s sptil frequency chnnels, oriented receptive fields, contrst sensitivity function, contrst non-linerities, nd unified set of prmeters). Like other low-level models, such s the ODOG model [Blkeslee, B., & McCourt, M. E. (1999). A multiscle sptil filtering ccount of the white effect, simultneous rightness contrst nd grting induction. Vision Reserch, 39, ], this formultion reproduces visul effects such s simultneous contrst, the White effect, grting induction, the Todorović effect, Mch nds, the Chevreul effect nd the Adelson Logvinenko tile effects, ut it lso reproduces other previously unexplined effects such s the dungeon illusion, ll using single set of prmeters. Ó 27 Elsevier Ltd. All rights reserved. Keywords: Visul system; Brightness induction; Wvelet trnsform 1. Introduction In visul perception, the term rightness often refers to non-quntittive perception of light elicited y the luminnce of visul trget (see Gilchrist, 26, p. 6). This rightness depends not only on the light reching the retin from the visul trget ut lso on the sptil distriution of light on its surroundings. Brightness induction refers to this chnge of ppernce due to the surrounding light nd its effects re clssified ccording to the perceptul direction of the chnge. When the chnge in rightness of the visul trget goes wy from the surrounding rightness, it is clled rightness contrst (Heinemnn, 1955) nd when the chnge goes towrds tht of the surrounding rightness, it is clled rightness ssimiltion (Helson, 1963). In the following section, we review severl rightness induction effects which re widely studied in the literture. * Corresponding uthor. E-mil ddress: xotzu@cvc.u.es (X. Otzu) Brightness induction effects One of the oldest known exmples of rightness induction is the simultneous rightness contrst (SBC) effect (Heinemnn, 1955; Wllch, 1948). This effect decreses for incresing test field size, ut is still strong for test fields s lrge s 1 deg (Yund & Armington, 1975). Since this is fr lrger thn receptive fields of retinl nd lterl geniculte nucleus (LGN) neurons in monkey (De Vlois & De Vlois, 1988), it suggests tht other types of neurons my e involved in the process. Such neurons with smll excittory centres nd lrge inhiitory surrounds (which my e suited for the tsk of shifting rightness towrds or wy from lrge test field) were found in re V4 of the primte visul cortex (Schein & Desimone, 199; Spillmnn & Werner, 1996). A second known exmple of rightness induction is the so clled grting induction (GI) effect (McCourt, 1982). The perceived contrst of the induced grting gin decreses with incresing test field, ut lso decreses with the sptil frequency (s.f.) of the inducing grting (Foley & /$ - see front mtter Ó 27 Elsevier Ltd. All rights reserved. doi:1.116/j.visres

2 2 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx McCourt, 1985). The induced grting is still perceived in test ptches s lrge s 6 deg (Blkeslee & McCourt, 1997). It hs een rgued tht these two phenomen (SBC nd GI) re just mnifesttions of the sme underlying mechnisms (Blkeslee & McCourt, 1997) nd their physiologicl sis re relted to the discovery of corticl neurons in ct (Rossi, Rittenhouse, & Prdiso, 1996) nd monkey (Gilert, Ds, Ito, Kpdi, & Westheimer, 1996) tht integrte over reltively lrge distnces. Another well known rightness effect is the White effect (White, 1979), where grey test ptches of the sme luminnce pper to hve different rightness when plced on top of the lck nd white rs of squre grting. Here, the rightness shift is independent of the spect rtio of the test ptch (i.e. it does not depend on the mount of white or lck order ner or in contct with the test ptch). Wht mkes this effect even more interesting is tht the contrst etween the grey ptch nd its orders (or surrounding re) seems to e less importnt thn the contrst with the r upon which it is situted. A similr effect ws descried y Todorović (1997) where the rightness shift seems to e independent of the mount of lck or white ckground in contct with the test ptch. Severl explntions oth t the receptive-corticl level nd t higher perceptul levels hve een ttempted to explin the White effect (see elow). However, it is cler tht the most plusile explntion for this effect t receptive level needs oth elongted corticl filters (Foley & McCourt, 1985; White, 1981) nd the opertion of sptilly extensive neuronl mechnisms, s opposed to isotropic receptive fields nd shorter rnge sptil interctions such s those found in the retin. Mch nds re rightness mxim nd minim perceived t the eginning nd end of luminnce grdients, respectively (Mch, 1865). They hve een interpreted in terms of lterl inhiition of retinl gnglion cells (Goldstein, 22) nd more recently s consequence of the physicl properties of rel world luminnce grdients (Lotto, Willims, & Purves, 1999). The Chevreul illusion (Chevreul, 189) is the nme given to the rightness minim nd mxim, respectively, perceived t the foot nd tip of ech step in luminnce stircse. There hve een ttempts to explin this illusion in terms of single chnnel nd the contrst sensitivity function (Cornsweet, 197) ut this explntion hs een ndoned in fvour of multi-chnnel models nd locl fetures within the steps (Morrone, Burr, & Ross, 1994; Perom & Lurinen, 24). However, there re lso lterntive explntions of this effect in terms of filling-in process triggered y edges t the different sptil scles (Pesso, Mingoll, & Neumnn, 1995). The Adelson tile illusion (Adelson, 1993), ppers when wll mde from homogeneous locks is sptilly modulted y horizontl drk stripe in such wy tht some of the dimonds tht form the top of the locks fll within the righter prt of the wve nd some fll within the drker prt. The top of the locks (horizontl dimond shpes) re constructed to e physiclly the sme (i.e. they reflect the sme mounts of light), ut the dimonds tht fll in the light strip look drker thn the dimonds in the drk strip. By rerrnging the pttern to mke the effect dispper while keeping the locl contrst round dimonds the sme, Adelson (1993) demonstrted tht explntions need to incorporte long-rnging receptive field interctions. A modifiction of the Adelson tile illusion ws introduced y Logvinenko (1999), who lurred the contrst edges of the horizontl strips, thus removing ny pprent trnsprency (nd verifying tht the illusion still holds). There re wide vriety of explntions for these Adelson illusions, rnging from low-level explntions sed on locl contrst nd multi-scle sptil filtering (Blkeslee & McCourt, 1999; Cornsweet, 197) to those sed on the role of orders or luminnce junctions etween or cross strips (Adelson, 1993; Adelson, 2; Anderson, 1997), high-level explntions where the explntion is sed on how the visul system dels with illumintion (Gilchrist et l., 1999; Logvinenko & Ross, 25) nd multi-level explntions (Kingdom, 23). There re prticulrly striking cses where simple predictions from the SBC effect (lck surroundings induce lighter trgets, etc.) seem to e completely reversed. One exmple of these is the Necker cue presented y Agostini nd Glmonte (22) whose dshed sides re perceived lighter even when they re completely surrounded y white ckground nd vice vers. Other exmples re the dungeon illusion (Bressn, 21) nd the Checkerord contrst illusion (De Vlois & De Vlois, 1988), where grey fetures surrounded y white look lighter nd grey fetures surrounded y white look drker. These were interpreted in terms of higher level grouping fctors where, for exmple ech set of dshes in the Necker cue is nchored y the cue (Gilchrist, 26). In the following section, we review severl ttempts to model rightness induction within computtionl frmework Modelling ttempts Some of the most successful computtionl models of rightness perception were developed using multi-scle pproches to low-level vision. They postulte tht edges nd lines re the driving fetures of erly vision nd set of opertors (receptive fields) re in chrge of detecting these (du Buff, 1994; Fiorentini, Bumgrtner, Mgnussen, Schiller, & Thoms, 199; Morrone & Burr, 1988; Tolhurst, 1972). These models my differ in the wy these opertors interct with ech other. For exmple, oth the models of Tolhurst (1972) nd Morrone nd Burr (1988) employ pirs of orthogonl opertors ut the former pplies mutul inhiition etween them while the ltter pools their responses. The model of Fiorentini et l. (199) employs single filter type t different sptil scles while the model proposed y du Buff (1994) uses opertors

3 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx 3 tht resemle pirs of simple cells centred t the sme loction ut in qudrture. A second type of model is sed on the frmework originlly proposed y Mrr (1982). An exmple of these is MIRAGE (Wtt & Morgn, 1985) which filters the stimuli t vrious sptil scles nd genertes list of primitives nd uses set of rules to detect lines nd edges. A more sophisticted version ws proposed y Kingdom nd Moulden (1992) nd clled MIDAAS, which includes gin control mechnism (light dpttion), sptil scle filtering, thresholding nd symolic descriptions t ech sptil scle efore pplying set of rules nd comining the outputs cross scles. A third type of model propose tht the min tsk of the visul system is not to extrct slient fetures of scenes (s do the other two types of models) ut to uild perceptul representtions tht keep the geometric structure of scenes (Pesso et l., 1995). This model uses contrst-driven nd luminnce-driven representtion. The first representtion is then filtered to produce oundries. The filtering overshoots nd undershoots trpped y these oundries re filled-in efore the contrst nd luminnce signls re recomined to provide the model s output, which is ment to resemle the sptil distriution of the percept. These models cn ccount for severl rightness induction effects such s the Mch nds nd the Chevreul illusion with vrying degrees of ccurcy (for review see Pesso et l., 1995 & Gilchrist, 26). A unified rightness model sed on low-level isotropic filters (difference of Gussins or DOG) sensitive to contrst t multiple sptil scles ws proposed y Blkeslee nd McCourt (1997) to explin the GI effect which, they rgue, cnnot e explined y fill-in type of model. The min difference etween this model nd the previous ones (Kingdom & Moulden, 1992; Moulden & Kingdom, 1991) ws the presence of more s.f. filters (sensitive to lower sptil frequencies) nd weighting scheme djusted to mtch the psychophysicl dt. This simple model is cple of ccounting for other rightness effects such s SBC nd the Hermnn grid illusion. A more sophisticted version, which includes non-linerly pooled nisotropic filters (oriented difference of Gussins or ODOG) nd normlistion to equlise the glol response t ech orienttion, ws developed to ccount for vriety of rightness effects tht require oriented filters such s the White effect (Blkeslee & McCourt, 1999; Blkeslee & McCourt, 21; Blkeslee & McCourt, 24; Blkeslee, Psiek, & McCourt, 25). The ltest extension of ODOG ws mde y Roinson, Hmmon, nd de S (27), who constrined normlistion to mke it more neurlly plusile nd expnded the rnge of illusions predicted y the model. Another multiresolution perceptul model is the one developed y D Zmur nd Singer (1998) nd D Zmur nd Singer (1999). Here the visul spce is decomposed into s.f. nd orienttion xis, nd sudivided into severl regions ccording to their sptil properties. The uthors use four (octve-wide) s.f. chnnels nd six orienttions of 3 deg width. The contrst of the surround is introduced in this model s Gussin lurring of the full-wve rectified frequency chnnel (clled sptil pooling of contrst). The Brightness Induction Wvelet Model (BIWM) we present here shres some similrities with oth the ODOG nd the D Zmur nd Singer (1998, 1999) models. Although multiresolution decomposition of the stimulus is performed, the output of the s.f. chnnels is processed differently (see elow) nd the contrst sensitivity function nd stimulus distnce re introduced explicitly. In the ODOG model the interction of the centrl stimulus nd its surround is performed through normlistion of the totl visul spce (in the cse of the D Zmur model this explicit comprison is not performed). In our model, we introduced precise dependency on the contrst energy of the surround compred to the centrl stimulus. Another crucil feture of our model is tht for ech of the illusion simultions descried elow ll prmeters were held constnt. 2. The Brightness Induction Wvelet Model (BIWM) In this work, we propose new low-level rightness induction model (the BIWM) tht comines three importnt stimulus fetures, nmely sptil frequency, sptil orienttion nd surround contrst to explin rightness ssimiltion/contrst phenomen. This is done through multiresolution wvelet decomposition which seprtes the chromtic input imge into different sptil frequency nd orienttion components (reminiscent of prvocellulr s.f. chnnels nd corticl orienttion-selective receptive fields). The recovery of the perceptul rightness imge is done y weighting the wvelet coefficients using modified version of the contrst sensitivity function (CSF). This modified CSF tkes into ccount the (sptil) surround informtion, so tht the vlue of the contrst sensitivity increses when surround contrst decreses nd vice vers. Oservtion distnce is lso tken into ccount to generlise the model. The choice of the wvelet trnsform s the min frmework for this work ws motivted y the fct tht wvelets shre severl mthemticl properties tht fit nicely with those of the erly visul system (e.g. two-dimensionl receptive field profiles re well descried y two-dimensionl Gor functions (Jones & Plmer, 1987)). Although there is considerle vriility in the receptive field shpes cross neurons (Tolhurst & Thompson, 1982; De Vlois, Alrecht, & Thorell, 1982) nd no single sis set cn cpture this vriility, wvelets provide sis functions tht re well loclised (oth in spce nd frequency) mong other mthemticl properties (such s self-similrity, ese of mthemticl representtion, etc.) which mke them populr mong modellers (Field, 1999; Hrvey & Don, 199; Olshusen & Field, 1997; Vn Rullen & Thorpe, 21; Zetzsche & Nuding, 25). We im to produce the simplest mthemticl formultion (nd to include the lest possile numer of

4 4 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx free prmeters) which models these three properties in mnner comptile with current physiologicl nd psychophysicl reserch with unified set of prmeters. Despite shring mny of its chrcteristics nd philosophy with other multiresolution models of the sme type (minly with the ODOG nd D Zmur s models), BIWM hs some importnt differences from those, such s: It is sed on wvelet decomposition which llows full reconstruction of the originl imge (i.e. n invertile trnsform) with sis tht hve profile similr to Gor function. It reltes rightness induction to the CSF y weighting ech sptil scle (see next section). It ccounts for the oservtion distnce, which is not explicitly defined in the other models (it predicts different rightness induction effects depending on the oserver s physicl position, which is the true ehviour of the humn visul system (HVS)). This issue ws recognised s prolemtic y other uthors (e.g. FLODOG model, Roinson et l., 27). It explicitly introduces the stimulus-surround contrst energy. ODOG fmily models use normlistion, either glol or locl, nd the D Zmur s model performs locl gin control in order to modify the different scles. Its prmeters re fixed nd ccount (simultneously) for ll the results descried elow Contrst sensitivity function The detection threshold for sinusoidl grtings depends on the grting s.f. nd this reltionship is descried y the contrst sensitivity function CðmÞ, where m represents the sptil frequency, which is nd pss for chromtic stimuli (Mullen, 1985; Simpson & McFdden, 25). Experiments with squre-wve periodic ptterns hve shown tht rightness ssimiltion effects increse when the s.f. of the trget feture is higher thn certin induction threshold. For rightness induction, this trnsition point m thr ws estimted to e ner 4 cpd (Smith, Jin, & Pokorny, 21; Wlker, 1978) nd for chromtic contrst induction etween 4 nd 6.7 cpd (Fch & Shrpe, 1986; Mullen, 1985). Given n oservtion distnce d the psychophysicl CSF CðmÞ function cn e defined in the scle spce s function C d ðsþ, where s is the sptil scle, nd pproximted y piecewise function defined y two Gussins. We cn lso define prticulr scle s thr ssocited to m thr, the sptil frequency (see Appendix A) where the CSF peks. A choice of m thr ¼ 4 cpd ws mde tking into ccount the chnges tht occur to the retinl CSF with men field luminnce (its pek vries etween 1 cpd t low luminnce levels nd 8 cpd for intense photopic ckgrounds) (Vnnes & Boumn, 1967). This llows us to define the function C d ð_sþ, eing _s ¼ s s thr. The sptil decomposition of the visul stimuli into the one octve-ndwidth independent chnnels tht form the sis of the CSF is modelled y multiresolution wvelet trnsform, s descried in the next section Multiresolution wvelet nlysis A common pproch to model the responses of visul corticl res involves the use of Gor functions (Dugmnn, 198), which re good descriptions of the erly visul system s receptive fields profiles ut hve the disdvntge of eing non-invertile (i.e. the originl imge cnnot e fully recovered). Our wvelet sis functions (i.e. the mother wvelet) re not strictly Gor functions, ut hve similr profile (smooth, symmetric, nd highly concentrted in oth spce nd frequency). They re sed on n lgorithm hlfwy etween the Mllt decomposition (Mllt, 1989; Mllt, 1998) nd the à trous lgorithm (Holschneider, Kronlnd-Mrtinet, Morlet, & Tchmitchin, 1989; Otzu & Vnrell, 26). We decided ginst using full Mllt decomposition (such s the Duechies wvelets, which re orthogonl nd compct) minly ecuse its sis functions re not smooth or symmetric like the Gor function. On the other side, the à trous lgorithm is very flexile multiresolution wvelet (equivlent to i-orthogonl wvelet decomposition) tht llows us to define smooth nlysis filters tht my led to smooth nd symmetric wvelets. Furthermore, it does not require us to define the synthesis filter, ecuse the synthesis of the originl imge is performed y simple ddition. It hs lso the dvntge of eing n undecimted lgorithm which, in contrst to decimted lgorithms, llows us to otin trnsltion invrint decomposition. For the present work, we used decimted version of the à trous lgorithm in order to reduce the computtionl complexity of the model ( common pproch in computer vision pplictions, e.g. JPEG2 imge compression) nd to simplify future implementtions. Despite these mthemticl differences, the min concept nd philosophy of the two lgorithms is the sme. BIWM decomposes its input in series of new imges x o s (or wvelet plnes) tht contin fetures of the originl imge t different sptil frequencies (indexed y s), nd sptil orienttions (indexed y o). Our lgorithm lso decomposes the imge in 3 orienttions with 45 deg orienttion ndwidths (i.e. verticl, horizontl nd digonl orienttions). In Fig. 1 we show grphicl scheme of this decomposition. The terms x v 1 ; xh 1 nd xd 1 represent the highest frequency components of the input imge for the verticl, horizontl nd digonl orienttions, respectively. The terms x v 2 ; xh 2 nd x d 2 represent similr orienttion wvelet plnes of s.f. one octve lower. The c 2 imge is residul plne, which is smoothed version of the originl imge nd cn e similrly decomposed, s shown in Fig. 1(). After eing decomposed, the originl imge I is

5 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx 5 Fig. 1. Multiresolution decomposition. Pnel () the White effect imge from Fig. 2() is decomposed into severl wvelet plnes x which contin fetures of certin s.f. nd orienttion. Pnel () shows the representtions of these plnes (only 3 multiresolution levels re shown). represented s summ of wvelet plnes of different s.f. nd orienttion s follows: I ¼ Xn s¼1 x v s þ xh s þ xd s þ cn ; ð1þ where n is the numer of wvelet plnes. The term c n is the residul plne, which is low resolution version of the originl imge. This expression cn e written more compctly s I ¼ Xn s¼1 X o¼v;h;d x o s þ c n; eing the index o the severl orienttions verticl, horizontl nd digonl, i.e. o ¼ v; h; d. The s.f. chnnels (or wvelet plnes) of our model hve ndwidth nd lyout similr to the visul system chnnels tht determine the shpe of CðmÞ Assumptions As mentioned efore, there is mple evidence tht the perception of centrl stimulus cn e modified y the sptil content of the surroundings (Chu, Sperling, & Solomon, 1989; D Zmur & Singer, 1998, 1999; Heeger, 1992; De Vlois et l., 1982; Werner, 23; Yu, Klein, & Levi, 21; Yu, Klein, & Levi, 22). In this section we descrie how the centre surround interction of the three min stimulus properties: sptil frequency, sptil orienttion nd contrst ws modelled. ð2þ similr, rightness contrst of the centrl stimulus is reduced (rightness ssimiltion) nd when these frequencies re different the centrl stimulus contrst is enhnced (rightness contrst). Therefore, rightness ssimiltion is only performed when oth centrl nd surround stimuli hve similr sptil frequencies within frequency rnge of out n octve (Blkemore & Cmpell, 1969; D Zmur & Singer, 1998, 1999; Grhm & Nchmis, 1971; De Vlois et l., 1982; Werner, 23; Wilson, McFrlne, & Phillips, 1983; Yu et l., 21, 22). Pnel () in Fig. 2 shows this property. In this figure, the grey ptches hve the sme horizontl s.f. s the surrounding lck nd white stripes. The left ptch is perceived drker ecuse of the induced rightness ssimiltion with the contiguous drk verticl stripes, nd similrly for the right grey ptch mong contiguous white stripes which is perceived righter. Pnel () shows how douling the s.f. of the grey ptches (i.e. one octve difference with the ckground), wekens the effect. Considering this, we modify the CSF ccording to the first of the three ssumptions of our model: Stimulus-surround reltive sptil frequency The sptil frequency content of the surroundings is one of the min contriutors to the perceived rightness chnges in centrl stimulus. As shown y grting perception studies (Chu et l., 1989; D Zmur & Singer, 1998, 1999; Werner, 23; Yu et l., 21, 22), when the sptil frequencies of oth centrl nd surround stimulus re Fig. 2. () Exmple of the influence of surround sptil frequency. Grey ptches hve different rightness ecuse of their different locl sptil informtion content. Both grey ptches nd verticl stripes hve the sme width (i.e. horizontl size or sptil scle) nd this produces strong induction effect. () When the widths of the grey ptches re different to tht of the lck nd white stripes, the rightness induction effect is lrgely reduced.

6 6 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx Assumption 1. Brightness ssimiltion is only performed when oth centrl nd surround stimuli hve similr sptil frequencies within frequency rnge of out one octve. The multiresolution wvelet frmework llows us to decompose the visul stimulus into one octve-ndwidth sptil frequency components nd estimte the influence of every sptil feture on fetures of the sme sptil scle. The wvelet scles s here re relted to the s.f. chnnels tht constitute the C d ð_sþ (oserver-trget distnce is included in this CSF, see definition ove) Stimulus-surround reltive sptil orienttion Another importnt contriution to rightness ssimiltion in grtings comes from the reltive orienttion of centrl nd surround stimulus. Severl studies (Cnnon & Fullenkmp, 1991; Solomon, Sperling, & Chu, 1993; Yu et l., 21, 22; Yu, Klein, & Levi, 23) show tht rightness ssimiltion of centrl stimulus is strongest when this nd the surround stimulus hve identicl orienttions. On the contrry, when the reltive sptil orienttions re orthogonl, rightness ssimiltion of the centrl stimulus is wekest (rightness contrst is strongest). This effect cn e oserved in Fig. 3. Following this, we define our second ssumption: Assumption 2. Brightness ssimiltion is strongest when the centrl stimulus nd the surround stimulus hve identicl orienttions. Furthermore, when the reltive sptil orienttions re orthogonl, rightness ssimiltion of the centrl stimulus is wekest (rightness contrst is strongest). This ssumption ws implemented y weighting nd pooling the different orienttion components of the wvelet trnsform, e.g. x h s ; xv s ; xd s. in rightness ssimiltion effects (Cnnon & Fullenkmp, 1991; Chu et l., 1989; Ejim & Tkhshi, 1985; Ellemerg, Wilkinson, Wilson, & Arsenult, 1998; Klein, Stromeyer, & Gnz, 1974; McKy, 1973; Nchmis & Snsury, 1974; Yu et l., 21, 22, 23). Brightness ssimiltion in centrl test stimulus increses s its surround contrst increses (nd vice vers), efore reching sturtion stte. This effect cn e oserved in Fig. 4 where the two verticl grey lines re plced in ckgrounds with different contrst. The left grey line is lwys in contct with drk stripes, nd right grey line is lwys in contct with white stripes. When the surrounding luminnce is uniform (i.e. surround contrst is null), rightness contrst is induced in the lines with the left line eing perceived righter nd the right line drker (simultneous contrst). When the contrst of the surrounding verticl stripes increses (downwrds direction in Fig. 4) our perception of the grey stripes chnges, clerly reversing their previously perceived difference when the contrst of the surrounding rs is mximum, i.e. when they re lck nd white. This wy, we define the third ssumption of our method s: Stimulus-surround reltive contrst energy Surround contrst is the third contriution to rightness induction considered y our model. It hs een shown tht the contrst of the surround stimuli plys n importnt role Fig. 3. The verticl grey stripe () is perceived drker thn the horizontl grey stripe () ecuse of rightness ssimiltion with its surrounding verticl lck nd white stripes (i.e. clssicl White effect). If the grey ptch is orthogonl (i.e. t 9 deg) to the lck nd white stripes, rightness ssimiltion does not occur. Fig. 4. In this series of imges, the surround contrst of the two verticl grey lines is incresed downwrds in four steps. The left grey line is in contct with lck stripes, nd the right grey line is in contct with white stripes. As the line extends downwrds (nd surround contrst increses) rightness induction increses, i.e. the left grey line ecomes drker nd the right grey line ecomes righter.

7 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx 7 Assumption 3. When the rightness contrst of the surround fetures increses, rightness ssimiltion increses (i.e. rightness contrst decreses) nd vice vers. To trnslte this ssumption to the model we will need to dd the contriution of the surround contrst. The next section shows how we do this Recovery of the perceived imge All the ssumptions mentioned ove re implemented in our model y modifying the wvelet coefficients in Eq. (2), where the gol ws reconstructing the perceived or induced imge from the decomposed originl imge. This weighting function ttempts to emulte some perceptul properties. As first pproximtion, we ssume it hs shpe similr to the CSF, such s: I percep ¼ Xn s¼1 X o¼v;h;d C o d ð_sþxo s þ cn : ð3þ eing I percep the recovered perceptul imge, nd indexes s nd o represent the multiple sptil frequency nd orienttion plnes. As seen efore, the reltive s.f. nd orienttion etween centrl nd surround stimulus re importnt contriutors to rightness induction. In the multiresolution frmework, imge components re grouped y oth their s.f. nd orienttion, giving us representtion in which similr fetures re grouped into the sme dt set (i.e. wvelet plne). In this wy, Assumption 1 nd Assumption 2 re nturlly implemented within this frmework. On the other hnd, Assumption 3 introduces the concept of surround contrst. Since the coefficients t sptil scle s nd orienttion o of the wvelet decomposition represent the vrition of these imge fetures t certin scle nd orienttion round men vlue, the mesured energy of these coefficients is relted to the contrst energy of the corresponding feture. Therefore, we cn esily estimte the reltive contrst of centrl feture compred to the contrst of its surround fetures y defining r ¼ r 2 cen =r2 sur, eing r2 cen nd r2 sur the stndrd devition of the wvelet coefficients on two concentric nnuli tht represent centre surround interction round ech point (x, y). The regions determined y these nnuli (referred s U nd W, respectively) were modelled s squres of 5 5 nd points wide, enclosing N U nd N W points inside, respectively. Region U ws chosen to e 5 5 points wide in order to include 2 complete Nyquist periods ðt s Þ, when mesuring the vrition of the centrl region d s ðx; yþ, therefore, N U ¼ 25. Its surrounding region W is points wide, tht is out three times lrger thn the inner region, n pproximte rtio suggested y Spitzer nd Semo (22) nd Shpley nd Enroth-Cugell (1984) nd psychophysiclly mesured y Yu et l. (21). Although the surround region W is centred in the sme point, it does not overlp with the inner region U (it includes only N W ¼ 144 points). A study y Nchmis nd Snsury (1974) on how contrst msking vried with msk contrst suggested the presence of contrst non-linerities in visul s.f. chnnels. These contrst non-linerities were modelled with function similr to the Nk nd Rushton (1966) function (which ws lso successful in reproducing the responses of corticl V1 neurons (Alrecht & Hmilton, 1982; Sclr, Munsell, & Lennie, 199; Tolhurst & Heeger, 1997)). Since we re ttempting to model the influence of surrounding imge fetures on the perception of centrl stimulus (in wys tht re relted to grting contrst msking) we defined similr non-linerity which provides n ccelertion t su-threshold contrst levels nd compression t supr-threshold: z ctr ¼ r2 1 þ r ; ð4þ 2 where z ctr is non-liner nd fulfils 6 z ctr ðx; y; s; oþ 6 1. The previous expression cn e seen s non-lineristion of the r vrile. As seen in the previous section, locl contrst of centrl test feture decreses s the contrst of its surround fetures increses nd vice vers (see Fig. 4). Since r is n estimtion of the centrl feture contrst reltive to its surround contrst, z ctr in Eq. (4) it cn e interpreted s nonliner estimtion of the degree of rightness contrst induced y the surround contrst into centrl feture. To introduce the effect of surround contrst fetures into the C d ð_sþ we use the vrile z ctr ðx; y; s; oþ where s represents the s.f. nd o is the orienttion involved. A new CSF C cn e written s follows: C ð_s; z ctr Þ¼z ctr C d ð_sþþc min ð_sþ: ð5þ In this expression, C ð_s; z ctr Þ reches its minimum when z ctr ¼ (i.e. minimum rightness contrst or mximum rightness ssimiltion). To void C ð_s; z ctr Þ ecoming null for some sptil frequencies s (minly for low sptil frequencies see Fig. 5) we hve introduced the term C min ð_sþ in Eq. (5), defining 8 n o >< exp C d ð_sþ ¼ n >: exp s2 2r 2 1 s2 2r 2 2 ; _s _s s thr 6 ; o ; _s _s s thr > : where prmeters r 1 nd r 2 re the stndrd devition of the piecewise Gussin function for s 6 s thr nd s thr < s, respectively. To reproduce the pproximte profile of the psychophysicl CðsÞ functions otined from the literture (Mullen, 1985), we mde r 2 ¼ 2r 1 nd r 1 ¼ 2. We defined C min ð_sþ s 8 n o < 1 s2 exp ; _s _s s 2 2r C min ð_sþ ¼ 2 thr 6 : 1 : ð7þ : 1 ; _s _s s 2 thr > In this wy, C ð_s; z ctr Þ tends to C min ð_sþ when z ctr tends to. This voids high degree of ssimiltion eing performed t low s.f. (i.e. lrge scle fetures), which would mke ð6þ

8 8 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx α( ) We otin I percep ðx; yþ ¼ Xn s¼1 X o¼v;h;d þ c n ðx; yþ C ð_s; z ctr ðx; y; s; oþþ x o s ðx; yþ ð9þ them invisile. We show this function C d ð_sþ in Fig. 5. In the opposite sitution, C ð_s; z ctr Þ is mximum when z ctr ¼ 1. Spitzer nd Semo (22) estimte tht the mximum enhncement fctor, i.e. mximum perceived rightness contrst, is round 1.5. It is the pek vlue of C ð_s; z ctr Þ (see Fig. 5). Another importnt property of the C ð_s; z ctr Þ is tht it reproduces the dip function for grting dpttion nd msking effects. A wvelet sis is usully represented in the s.f. plne y Heinsenerg ox with certin ndwidth in oth spce nd frequency. The function tht defines the spred s influence is defined y verging the Wigner Wille distriution (Mllt, 1998, Chpter 4), which cn itself e pproximted y Gussin function. In order to show wht is the effect on the continuous C ðþ function when weighting down n individul wvelet coefficient from prticulr discrete scle, we multiplied Gussin function with one octve frequency ndwidth y this weight (see dshed line in Fig. 5). To construct this function we defined certin oservtion distnce d nd otined the corresponding s thr vlue (see Appendix A). Following this, we found the prticulr wvelet plne s tht fulfils ð_s ¼ s thr s ¼ 1Þ. The dshed line in Fig. 5 shows how the C ðþ is modified when we force z ctr < 1 for certin feture elonging to this prticulr wvelet plne. The resulting plot is very similr to the dip function otined y Grhm nd Nchmis (1971) nd Nchmis nd Snsury (1974) for grting dpttion nd msking effects, using mthemticl expression qulittively equivlent the ours (Eq. (4)). Eq. (3) shows the generl expression to recover perceptul imge I percep represented y set of wvelet plnes x o s. Replcing the set of weights y our own weighted CSF C ðs; Þ C ð_s; z ctr ðþþ: scle s induction threshold s =s-s thr (6 cpd) low frequency high frequency Fig. 5. Continuous function: contrst sensitivity function. Dshed function: profile of CSF C ð_s; z ctr ðþþ with z ctr ðx; y; s; oþ ¼:75. Tht is, when pplying z ctr ðþ ¼ :75 just on prticulr s wvelet plne which fulfils _s s s thr ¼ 1, see text for detils. Dshed-dotted function: profile of C min ð_sþ. Dotted line: vlues ove this vlue implies rightness contrst, nd vlues elow it implies rightness ssimiltion. ð8þ which defines the perceptul imge recovered from the wvelet components of the originl imge. 3. Model predictions This section shows the model s predictions (oth quntittively nd qulittively) for the rightness induction effects mentioned in Section 1 (e.g. the simultneous rightness contrst (SBC), the White effect (W), the grting induction (GI), the Todorović effect (T), the Mch nds (MB), the Chevreul effect (C), the Adelson Logvinenko tile, the dungeon illusion nd the checkerord illusion). To e le to compre BIWM predictions to psychophysicl dt, we djusted the input imge nd the oservers distnce prmeter to e consistent with the physicl dimensions (size, visul ngle, oservtion distnce, etc.) reported y the experimenters for their ctul stimuli. For the SBC, GI, W nd T effects, we compred our model to pulished psychophysicl results (Blkeslee & McCourt, 1999) supplied y McCourt. For the MB nd the C effect we otined dt from Lu nd Sperling (1996) (Tle 2) SBC nd White effect Pnels () nd (c) in Fig. 6 show two versions of the SBC effect. The grey rectngle is seen drker when it is in front of right ckground, nd righter when it is in front of drk ckground. For this exmple, we used the sme stimulus geometry nd oservtion distnce s Blkeslee nd McCourt (1999), nd otined vlue of s thr ¼ 2:65 for our model. The continuous line in Fig. 6 (pnels () nd (d)) shows the luminnce profile of the centrl row from pnels () nd (c) which contins the grey ptch surrounded y the light/ drk uniform ckground. The dshed lines in the plots show the perceptul profile predicted y our model. We see tht the BIWM predicts the increse/decrese of the perceived rightness of this grey ptch over its originl vlue. The opertion of the BIWM cn e summrised s follows. Consider the grey ptch t the left of pnel (), which is reltively well represented in the wvelet plne tht est corresponds to its s.f. nd orienttion. Since the grey ptch is not surrounded y similr fetures (or it my e sid tht is surrounded y similr fetures with zero contrst), rightness contrst is induced t this prticulr sptil scle. Given tht the grey squre is drker thn its locl surround, it ecomes even drker (perceptully). The sme resoning cn e pplied to the other grey ptch upon the drker ckground, which ecomes perceptully lighter (nd the ottom pnels in Fig. 6). The model lso predicts locl mxim nd minim running prllel to

9 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx c d Fig. 6. Pnels () nd (c) show n exmple of the simultneous contrst effect nd pnels () nd (d) show our model s results. The solid lines in the plots on the right show the profile of the centrl row tken from the corresponding left pnel. The dshed lines show our model s predicted rightness profile (drker left grey ptch nd righter right grey ptch). the verticl light/drk edge t the centre of figures () nd (c), i.e. round column 5. In this imge, this crisping effect is not perceived y the oserver. This edge nd the two lck nd white plteus re defined y severl sptil frequency components. The BIWM considers some of these frequency components (minly the highest s.f.), s isolted fetures of given sptil frequency, i.e. rightness contrst is induced. It implies tht different weighting fctors hve een pplied to the different sptil frequency components tht define oth the edge nd the plteus, thus the finl rightness profile predicted y the BIWM in this edge is not psychophysiclly perceived y the oserver. In some cses, this crisping effect cn e perceived y the oserver. For exmple, on the edge etween the wide white rectngle nd the grey ckground, we perceive drker zone where the grey ckground is closer to the white ptch, nd similrly we perceive righter zone where the grey ckground is closer to the drk rectngle. Another extensively studied effect (shown in Fig. 7 pnels () nd (c), where the left grey rectngle is perceptully righter thn the right one in pnel () nd drker in pnel (c)) is the White effect. This effect is generlly considered prticulr cse of SBC (Moulden & Kingdom, 1991; Zidi, 1989) nd cn e explined using sptilly-oriented filters (Blkeslee & McCourt, 1999; Blkeslee et l., 25). It hs een suggested (Blkeslee et l., 25) tht the White effect is not n ssimiltion effect ecuse the direction of rightness chnge is kept, even when the height of the test ptch is reduced so tht it hs more order contct with the r on which it is situted. In our formultion, the White effect cn e modelled s sptilly-oriented rightness ssimiltion. The continuous lines in Fig. 7 (pnels () nd (d)) show the luminnce profile of row from pnels () nd (c), respectively, contining the grey ptches. The dshed line in the sme figure shows the perceptul profile predicted y the BIWM. We see tht our method correctly predicts tht left grey ptches re perceived drker nd right ptches righter. As comprison, we hve dded in Fig. 7 the psychophysicl rightness vlues otined y Blkeslee nd McCourt (1999) for the sme imge. We cn see tht the rightness predicted y the present method fits the psychophysicl dt well.

10 1 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx Psychophysicl rightness c d Fig. 7. Pnels () nd (c) show n exmple of the White effect, where the grey rs re equl ut perceived with different rightness ecuse of their different surrounds. Pnels () nd (d) show the profile of row from () nd (c), respectively, contining the grey ptches. The dshed lines re the model s predictions, showing perceptul rightness increse of the left grey ptch nd decrese on the right ptch. Similrly, perception of the verticl stripes is modified (i.e. rightness increse of white verticl stripes nd rightness decrese of lck verticl stripes). The sme effects cn e oserved in pnels (c) nd (d). Our model reproduces the White effect in similr wy. Consider for exmple, the ptch on the right side of pnel () (i.e. the one in front of white verticl stripes nd in lterl contct with drk verticl stripes). There is wvelet plne where this ptch is well represented given its prticulr s.f. nd orienttion. This is the sme wvelet plne where the ckground grting is lso est represented, since it shres the sme horizontl s.f. Since here the surround contrst is greter thn the locl contrst, rightness ssimiltion is induced on the grey ptch (it ecomes perceptully drker). On wvelet plnes corresponding to different orienttions (e.g. verticl) the opposite interction my occur, since t these orienttions the ckground is not est represented in the sme wvelet plne s the grey ptch. As result, horizontl fetures induce rightness ssimiltion nd verticl fetures induce rightness contrst. The totl perceived rightness is comintion of these orienttion-dependent interctions. Fig. 7 (pnels () nd (d)) lso shows tht the model s prediction for right verticl stripes is righter tht the originl vlue nd similrly, the prediction for drk verticl stripes is drker. The reson for this my e the rightness contrst induced y the surrounding (low s.f.) plin grey ckground on the verticl stripes. Fig. 8 illustrtes this effect, where the originl imges (grey ckground) re represented on the left while version of the sme imges on lck ckground is on the right. The presence of lck ckground induces rightness contrst effect on the verticl stripes, mking the perceived light rs lighter nd the perceived drk rs drker Grting induction Grting induction (GI) (McCourt, 1982) produces perceived rightness vrition ( grting) on n sptilly extended test field, see Fig. 9(). The centrl thin horizontl test ptch hs constnt luminnce, ut its rightness is perceived s n horizontl sinusoidl in counterphse with the upper nd lower sinusoidl extended ptches. As shown y Blkeslee nd McCourt (1997) (who modelled it with their

11 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx 11 Fig. 8. Influence of drk ckground on the White effect. Imges () nd (d) re the sme s () nd (c) ut surrounded y drk ckground. The perceived rightness of verticl drk stripes in drk ckground surroundings is different thn in the cse of the grey ckground. ODOG model), this effect my e interpreted s prticulr cse of rightness contrst. Fig. 9() shows the profile of the centrl row of oth the uniform rightness of the centrl thin horizontl ptch, the rightness profile predicted y our method nd row of the extended sinusoidl luminnce grting. As we cn see, the perceived rightness of the centrl grey stripe is predicted s sinusoidl rightness profile in counterphse with the extended ptches. Pnels (c) nd (d) show similr exmple for lower s.f. sinusoidl ptch. In the GI exmple, the sitution is similr to the simultneous contrst: since the horizontl grey ptch is orthogonl to the grting, it is not well represented in the sme verticlly-oriented wvelets plnes s the grting nd therefore contrst effect is induced. The overll result is sinusoidl rightness grting in counterphse with the sinusoidl luminnce grting Todorović effect In Fig. 1, pnel () we show version of the Todorović effect (Todorović, 1997). This imge is the sme s in Fig. 6 pnel (c), except for the superimposed lck nd white squres which mke the grey ptches tke the form of cross (ordered y equl mounts of lck nd white). The test ptch on the lck ckground ppers righter thn the test ptch on the white ckground despite the fct tht oth ptches hve the sme mount of lck nd white order contct. In Fig. 1, pnel () we show the rightness predicted y our model. It correctly predicts tht the grey ptch on the white ckground is perceived drker thn the grey ptch on the lck ckground. In this exmple, the grey ptches do not shre fetures with the rest of the figure within the sme orienttion nd sptil-scle wvelet plnes, nd therefore rightness contrst is induced. All other squres do shre some similrities nd therefore hve tendency to e ssimilted Mch nds In the Mch nds effect (Mch, 1865), see Fig. 11(), right nd drk nds re perceived ner the righter nd drker order, respectively, of rmp edge etween two uniform regions of different luminnce. Pnel () in Fig. 11 shows two plteus of different luminnce with wedge etween them. A righter verticl cusp is perceived where the wedge meets the righter plteu nd drker verticl cusp is perceived where the wedge meets the drk plteu. Our model correctly predicts this ehviour, s shown in pnel (). The sme pnel lso shows the luminnce profile for the centrl row of oth the originl imge nd the perceptul rightness predicted y our method. The Mch nds re reproduced t the region etween the centrl wedge nd the lterl plteus. In this cse the s.f. fetures defined y the edges of the wedge nd the two plteus will e prominent in prtic-

12 12 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx Sinusoidl luminnce ptch c d Fig. 9. Pnel () in the GI effect, the thin horizontl stripe with constnt luminnce etween the horizontl sinusoidl grtings is perceived s sinusoidl rightness stripe in counterphse with the grting. Pnel () profile of row of pnel () showing the constnt luminnce of the horizontl stripe (continuous line) nd the rightness predicted y our method (dshed function) in counterphse with row of the horizontl sinusoidl luminnce grting (dotted function). A similr effect is shown in pnels (c) nd (d) Fig. 1. Pnel () in the Todorović effect (Todorović, 1997) the left grey ptch is perceived drker tht right one, even when they hve the sme mount of order contct with lck nd white surfces. Pnel () predicted rightness from our model. ulr group of wvelet plnes of the optiml sptil scle nd orienttion. There will e no other such prominent feture in the sme wvelet plnes nd this will determine rightness contrst induction, producing the cusps.

13 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx Fig. 11. Pnel () exmple of the Mch nds effect. Pnel () originl dt luminnce (solid lines) nd perceptul rightness predicted y our model (dshed line) Chevreul effect In the Chevreul effect (see Fig. 12), series of stripes with stircse profile is perceived s swtooth, tht is, ech stripe is perceived with rightness incresing regulrly from one stripe to the next. This effect hs een modelled with vrious ccurcy levels (Keil, 26; Morrone & Burr, 1988; Morrone et l., 1994; McArthur & Moulden, 1999). In Fig. 12, pnel () we show the originl stircse profile (continuous function) nd the rightness predicted y our method (dshed function), which follows n pproximtely swtooth profile. The luminnce step etween two ptches is outlined y severl s.f. components (minly high s.f. components) tht will feture highly in group of wvelet plnes of the optiml sptil scle nd horizontl orienttion. Since t these prticulrly high sptil frequencies they re not surrounded y similr components (the size of the steps mke interctions etween edges wek), rightness contrst is induced in the edges, leding to the finl swtooth profile Adelson Logvinenko tile In Fig. 13, pnel () we show version of the Adelson Logvinenko tile pttern (Logvinenko, 1999). This imge consists of 2D representtion of severl 3D cues modulted y horizontl sinusoidl grting, where the upper (or lower, depending on how the oserver solves the cue s miguity) sides of the cues hve equl luminnce ut re perceived with different rightness. The grey level vlue of these top surfces is 134. Our method predicts grey level vlue of 134 for the pprently light surfces nd 99 for the pprently drk ones. The presence of the verticl (low s.f.) modulting sinusoidl grting mens tht there will e prticulr verticlly-oriented wvelet plne where this feture will e represented est with very little influence of the rest of the imge. This will induce strong rightness contrst effect in the rows tht re coincident with the peks nd vlleys of this sinusoidl (precisely where the tops of the cues re locted), therefore producing the finl perceived effect Fig. 12. Pnel () exmple of the Chevreul effect. The originl imge hs luminnce stircse profile, ut it is perceived with swtooth profile. Pnel () shows the originl dt vlues nd the perceptul rightness predicted y our model Dungeon illusion There is suset of illusions where the direction of contrst does not fit the one predicted y trditionl contrst theories (Gilchrist, 26). One of these is the dungeon illu-

14 14 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx sion (Fig. 14, Bressn, 21) where the perceived difference etween the grey squres on the left nd the right of the picture is the opposite of wht one would predict from nlysis of individul squres nd their immedite surroundings (Gilchrist, 26). Pnel () in Fig. 14 shows n exmple of the dungeon illusion nd pnel () shows the rightness predicted y our model for the centrl row, where ll grey rectngles on the left side re represented with drker shde of grey thn those on the right side of the imge. This effect cn e qulittively explined y the fct tht ll grey rectngles re surrounded y other rectngles of the sme size nd t distnce similr to its size, oth verticlly, horizontlly nd digonlly. Therefore, they will e strongly represented in the sme wvelet plnes, leding to rightness ssimiltion effect. This in turn will produce drker rectngles in the left side of the imge (where the rectngles rightness will tend towrds tht of the rs) nd slightly righter rectngles on the right side. Fig. 13. Pnel () exmple of the Adelson Logvinenko tile. The prts of the cues t the crests nd vlleys of the modulting sinusoidl hve ll the sme luminnce ut re perceived differently. The model we present qulittively predicts these differences (see text) Checkerord Another exmple of complete reversl of contrst is the checkerord illusion (shown in pnel () of Fig. 15). Here, the grey squre in contct with white squres is perceived righter thn the grey squre in contct with the lck squres (n effect similr to the dungeon illusion). A simplified explntion cn e given in terms of the fetures tht surround ech of the squres, since the grey squres re horizontlly nd verticlly surrounded y elements of equl size nd high contrst, they will gin e represented in the sme sptil scles nd orienttions wvelet plnes which will induce rightness ssimiltion on them. If the squre is surrounded y lck squres (left), its rightness will tend to go in the direction of the locl surroundings (drker). The other grey squre will e ssimilted towrds the other end (it will look righter). 4. Discussion: Comprison with psychophysics Fig. 14. Pnel () shows n exmple of the dungeon effect. Individul grey squres on the left side re completely surrounded y lck pixels nd should e seen lighter thn individul grey squres on the right side, which re in turn surrounded y white pixels, in prctice the opposite occurs. Pnel () shows our model s predictions for the centrl row of the figure in pnel (), demonstrting the power of multiresolution wvelet pproch to provide qulittive explntion of the effect. To mke more quntittive ssessment of our model s predictions, we tested our model ginst psychophysicl mesures from the literture of the reltive rightness increse/decrese produced y rightness induction. We simulted the physicl conditions (imge size nd oserver s distnce) in our model nd produced set of predictions tht were compred to the mesurements. All other prmeters were kept the sme for ll conditions. Figs. 16 nd 17 show the psychophysiclly-mesured vlues pulished y Blkeslee nd McCourt (1999) nd our predicted vlues for some of the visul effects descried ove (e.g. simultneous rightness contrst, grting induction, White effect nd Todorović effect). In Fig. 16, experimentl vlues (nd their ssocited 95% confidence limit error rs) re represented y rs. Our predicted vlues re represented y squres (empty squres for test ptches on drk ckground, nd solid squres for test ptches on right ckground). The ordinte xis shows the difference etween the mtching luminnce nd the men luminnce expressed s proportion of the men luminnce, consistent with Blkeslee nd

15 X. Otzu et l. / Vision Reserch xxx (28) xxx xxx 15 Men mtching luminnce (proportion men) SBC3 SBC1 GI3 GI1 W4 W2 T Stimulus Fig. 16. Representtion of the vlues predicted y our model (empty squres for ptches on drk ckground nd filled squres for ptches on right ckground) nd the psychophysicl vlues otined y the BB oserver in Blkeslee nd McCourt (1999) (rs) for simultneous rightness contrst (SBC), grting induction (GI), White effect (W) nd Todorović effect (T). Mtching luminnce (proportion men) y =.898 * x -.7 r =.868 Fig. 15. Pnel () shows n exmple of the checkerord contrst effect. The imge shows two grey squres (of the sme luminnce) emedded in checkerord, tht re perceived differently y the oserver. The right squre is perceived righter thn the left one. Pnel () shows the rel nd perceptul rightness profiles of these two squres s predicted y our model Model predictions Fig. 17. Representtion of the vlues predicted y our model (ciss xis) nd the psychophysicl vlues otined y Blkeslee nd McCourt (1999) (ordinte xis) for ll the visul effects considered in Fig. 16. McCourt (1999). The. vlue represents the luminnce of the test ptches. We cn see tht our method pproximtely predicts the direction nd mgnitude of the rightness induction. The gretest devition from the psychophysicl vlues is for the GI3 (our model underestimtes the effect) nd W2 (our model overestimtes the effect) results. It is possile to otin etter fits y dding more degrees of freedom (e.g. modifying the CSF C to djust for these two sets of results) to our model, ut for the ske of consistency nd simplicity, we prefer to keep the lowest numer of degrees of freedom (nd the simplest mthemticl expression) in ll cses. In Fig. 17 we show plot of the vlues predicted y our model (sciss) versus the psychophysicl vlues (ordinte) for ll considered visul effects. Ech point in the plot represents n oserver (either MM or BB) from Blkeslee nd McCourt (1999). We lso show the digonl line (dotted line) where ll points would lie should our model s predictions e 1% ccurte. The points show n pproximtely liner ehviour. The solid line represents the est fitting line (liner regression) with slope of round.9, correltion coefficient r ¼ :87, nd the sum of squres of the residuls eing.77. We lso used psychophysicl dt y Lu nd Sperling (1996) in order to nlyse the predictions of our model for the Mch nds nd the Chevreul effect. As with the previous exmples, we simulted in our model the physicl conditions of Lu nd Sperling (1996). In the cse of the Mch nds effect, these uthors report rightness increment/decrement (in the right nd drk plteu, respectively) which is round 8% of the luminnce difference etween the right nd the drk plteu. Our model lso predicts rightness increment round 13%. For the Chevreul effect, these uthors report n increment/decrement of the rightness round 54% of the luminnce difference etween consecutive steps. Our model predicts n